MMSE Journal Vol.6 2016

Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

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Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

Sankt Lorenzen 36, 8715, Sankt Lorenzen, Austria

September 2016

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Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

Mechanics, Materials Sciences & Engineering Journal, Austria, Sankt Lorenzen, 2016

Mechanics, Materials Science & Engineering Journal (MMSE Journal) is journal that deals in peerreviewed, open access publishing, focusing on wide range of subject areas, including economics, business, social sciences, engineering etc.

MMSE Journal is dedicated to knowledge-based products and services for the academic, scientific, professional, research and student communities worldwide.

Open Access model of the publications promotes research by allowing unrestricted availability of high quality articles.

All authors bear the personal responsibility for the material they published in the Journal.

The Journal Policy declares the acceptance of the scientific papers worldwide, if they passed the peer-review procedure.

Editor-in-Chief Mr. Peter Zisser Dr. Zheng Li, University of Bridgeport, USA Prof. Kravets Victor, Ukraine Ph.D., Shuming Chen, College of Automotive Engineering, China Dr. Yang Yu, University of Technology Sydney, Australia Prof. Amelia Carolina Sparavigna, Politecnico di Torino, Italy

ISSN 2412-5954

Design and layout: Mechanics, Materials

e-ISSN 2414-6935

Science & Engineering Journal, www.mmse.xyz Technical support: hotmail@mmse.xyz ©2016, Magnolithe GmbH © Copyright, by the authors

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Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

CONTENT I. Materials Science MMSE Journal Vol. 6 ..................................................................................... 6 Effect of Temperature and Strain Rate on Dynamic Re-Crystallization of 0.05C-1.52Cu1.51Mn Steel. Pawan Kumar, Peter Hodgson .................................................................................... 7 Nanostructure Formation in Anodic Films Prepared on a β Alloy Ti39Nb PVD Layer. Zdenek Tolde, Vladimír Starý, Petr Kozák ..................................................................................................... 15 Diagnostics of Argon Injected Hydrogen Peroxide Added High Frequency Underwater Capillary Discharge. Muhammad Waqar Ahmed, Sooseok Choi, Jong-Keun Yang, Rai Suresh, Heon Ju Lee ....................................................................................................................................... 27 Optimizing the Parameters in Heat Treatment for Achieving High Hardness and Efficient Bending of Thin BS 2014 Aluminium Alloy Sheets. Abirami Priyadarshini B. ........................... 39 The Effects of Ukam (Cochlospermum Planchonii) Plant Fiber Variation on the Properties of Polyester Matrix Fiber Reinforced Composite. Ihom A.P., Dennis O. Onah .......................... 46 Effect of Alternating Bending and Texture on Anisotropic Damage and Mechanical Properties of Stainless Steel Sheets. V.V. Usov, N.M. Shkatulyak, E.A. Dragomeretskaya, E.S. Savchuk, D.V. Bargan, G.V. Daskalytsa ................................................................................... 56 II. MECHANICAL ENGINEERING & PHYSICS MMSE JOURNAL VOL. 6 .......................................... 64 The Influence of Cutting Speed on Concordant and Discordant Tangential Milling of MDF. Priscila Roel de Deus, Manoel Cleber de Sampaio Alves, Luciano Rossi Bilesky ............................ 65 Substantiating of Rational Law of Hydrostatic Drive Control Parameters While Accelerating of Wheeled Tractors with Hydrostatic and Mechanical Transmission. Taran I.O., Kozhushko A.P ................................................................................................................................... 70 Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function. Pawan Kumar, Vaneshwar Kumar Sahu, P.K.Ray, B.B.Verma ....................................... 77 Fundamental Solutions for Micropolar Fluids with Two-Temperature. M. Zakaria ............ 86 Calibration of COD Gauge and Determination of Crack Profile for Prediction of Through the Thickness Fatigue Crack Growth in Pipes Using Exponential Function. Pawan Kumar, Hemendra Patel, P.K.Ray, B.B. Verma ............................................................................................. 99 Numerical Solution of Nonlinear Fredholm Integro-Differential Equations using LeibnitzHaar Wavelet Collocation Method. C. Shiralashetti, R. A. Mundewadi ...................................... 108 An Equivalent Beam Model for the Dynamic Analysis to a Feeding Crane of a Tall Chimney.

Application in a Coal Power Plant. Viorel-Mihai Nani, Ioan Cires............................................. 120 Determination of Bond Capacity in Reinforced Concrete Beam and Its Influence on the Flexural Strength. Mohammad Rashidi, Hana Takhtfiroozeh ....................................................... 135 Prediction of Rubber Element Useful Life under the Long-Term Cyclic Loads. Dyrda V.I., Loginova A.A., Shevchenko V.G. ..................................................................................................... 145 Calculation of Strength and Stiffness of Sports Equipment for Games in a Radial Basketball.

V. P. Ovchinnikov, A. A. Nesmeyanov, A. N. Chuiko ....................................................................... 151 Development of Force Monitoring Transducers Using Novel Micro-Electromechanical Sensor (MEMS). Dimitar Chakarov, Vladimir Stavrov, Detelina Ignatova, Assen Shulev, Mihail Tsveov, Rumen Krastev, Ivo Vuchkov .................................................................................. 158 Analytical Simulation of Dynamical Process in One-Dimension Task. Kravets V.V., Kravets T.V., Fedoriachenko S.A., Loginova A.A. ........................................................................... 169 MMSE Journal. Open Access www.mmse.xyz

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VI. ENVIRONMENTAL SAFETY MMSE JOURNAL VOL. 6 .............................................................. 177 The Impact of Vehicular Emissions on Air Quality in Uyo, Nigeria. Aondona Paul Ihom, Ogbonnaya Ekwe Agwu, John Akpan John ..................................................................................... 178 Utilization of Point Clouds Characteristics in Interpretation and Evaluation Geophysical Resistivity Surveying of Unstable Running Block. Marcel Brejcha, Petr Zbíral, Hana Staňková, Pavel Černota .................................................................................................................................. 185 Atmosphere Re-Entry Simulation Using Direct Simulation Monte Carlo (DSMC) Method.

Francesco Pellicani ......................................................................................................................... 195 VII. INFORMATION TECHNOLOGIES VOL. 6 ................................................................................... 204 Comparison of Modeling and Simulation Results Management Microclimate of the Greenhouse by Fuzzy Logic Between a Wetland and Arid Region. Didi Faouzi), N. Bibi-Triki, B. Draoui, A. Abène ........................................................................................................................ 205 IX. ECONOMICS & MANAGEMENT MMSE JOURNAL VOL. 6........................................................ 218 The Role of Education in Formation of Knowledge Economy. Tetiana Chumachenko, Olena Hladun ................................................................................................................................... 219 Selection of the Reconstruction Options for Industrial Power Supply System under Uncertainty Conditions on the Basis of the Game Theory Criteria. Alina Iuldasheva, Aleksei Malafeev .............................................................................................................................. 223

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I . M a t e r i a l s S c i e n c e M M S E J o u r n a l V o l . 6

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Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

Effect of Temperature and Strain Rate on Dynamic Re-Crystallization of 0.05C1.52Cu-1.51Mn Steel Pawan Kumar1, a, Peter Hodgson1, b 1 – Institute For Frontier Materials, Deakin University, Australia a – pkumar@deakin.edu.au b – peter.hodgson1@deakin.edu.au DOI 10.13140/RG.2.1.4905.2403

Keywords: dynamic re-crystallization, strain rate, temperature

ABSTRACT. Dynamic re-crystallization (DRX) is one of the most efficient methods to achieve ultra-fine ferrite grain in the steel. The DRX associated with the formation of new grains in hot working condition. The factors influencing the grain size achievable through thermo-mechanical controlled processing are known to be work hardening and softening by dynamic process of recovery. The point at which the combine effect of strain hardening and recovery are unable to accommodate more immobile dislocation is the starting point of DRX process. In present investigation, critical stress for initiation of DRX is calculated for 0.05C-1.52Cu-1.51Mn steel and the influence of strain rate and temperature is studied.

It was observed that at lower strain rate, critical stress for initiation of Dynamic re-crystallization (DRX) is increases initially and then it become saturated at higher strain rate. It is also absorbed that higher temperature and lower strain rates are the favourable condition for typical DRX process. It is also hinted that Cu precipitation take place process adopted in the experiments.

Introduction. Dynamic re-crystallization (DRX) is one of the most efficient method to achieve ultrafine ferrite grain in the steel [1-2]. The DRX associated with the formation of new grains (in hot working condition); the size of grain is expressed as: đ?œŽ = đ??´đ??ˇđ?‘› đ??ş where A – is a constant; G – is the shear modulus; n – is the grain size exponent, which is about 0.7 for hot working conditions [3-5].

The factors influencing the grain size achievable through thermo-mechanical controlled processing are known to be work hardening and softening by dynamic process of recovery [6]. The three mechanisms with strain hardening, dynamic recovery and dynamic re-crystallization are different in their softening mechanisms.

When the combine effect of strain hardening and recovery are unable to accommodate more immobile dislocation is the starting point of DRX process. Low stacking-fault energy materials generally exhibit discontinuous DRX. The mechanism corresponding to DDRX is bulging (local migration).

Bulging of grain boundaries generate nuclei which further grows and consumes at deformed matrix; leading to increase in the dislocation density. The morphology governing by DDRX shows nearly constant average grains size, which is due to the further deforming of large grains due to further straining and taken up by new DRX nuclei. This process considered as a Discontinuous process [7, 8-9].

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High stacking-fault energy materials generally exhibit Continuous DRX [10-13]. In this phenomenon the formation of three-dimensional arrays of deformation low-angle boundaries (LABs) takes place, which is further transformed into high-angle grain boundaries (HABs). The high orientation gradient and the strain incompatibility between joint grains evolves the strain induced LABs. Upon further straining their mis-orientation increases leading to transformation into HABs; this leads to the development of recrystallized grains. The CDRX phenomena generally exhibits an equi-axed morphology throughout the structure.

The Cu is use to provide precipitation hardening in steel. Setuo Takaki et. al has studied the effect of pre-strain with Cu addition on 0.007C-0.01Mn-1.5Cu steel aged at 300C at 20 mins [14]. There is no Cu clusters/precipitates observed in non-prestrained steel; although existence of Cu clusters of size around 0.7nm are reported in prestrained steel, it has shown any change in distribution upon ageing as 500 oC for 20 mins. It is observed that Cu clusters tend to distribute coarsely in non-pre-strained steel. It is also observed that at peak age condition; clusters of copper tend to grow homogeneously in pre-strained samples. However it found that in non-pre-strained samples; a coarsening behavior is observed. The mechanism of grain refinement in steel by Cu precipitation is not known till now. It is proposed by some workers that precipitate –dislocation interaction tends to create deformation bands during straining and this leads to fine re-crystallized grains [14]. Setuo Takaki et. al. also reported strengthening of heavily deformed and re-crystallized ferrite due to precipitates of copper [14].

In the present investigation the effect of temperature and strain rate is studied for the flow behaviour of material under investigation. The critical stress and strain is also calculated for the initiation of dynamic re-crystallization process. Also the influence of temperature and strain rate on the critical stress and strain for DRX is investigated.

Materials and Methods.

The material under investigation is 0.05C-1.52Cu-1.51Mn steel. Thermo-mechanical simulator (Gleeble) was used for hot compression test in plain strain condition. The specimens were austenitized at 1100oC for 5 min and cooled at the rate of 5oC/Sec; it is then subjected to hot compression as shown in Fig. 1. Single hot compression tests were conducted at temperature 800-1000oC with strain rates of 0.01, 0.1, 1 s-1.

Fig 1. Thermo-Mechanical process used in experiments.

Result and Discussion. From Fig. 2, It is observed that DRX taken place at strain rate of 0.01/Sec at different temperature up to 800 0C. Effect of temperature and strain rate on DRX of experimental steel can also be observed from fig. 3, fig. 4 and fig. 5. When the deformation temperature is comparatively low , DRX seemingly take place only at a slower strain rate of 0.01/Sec; for higher MMSE Journal. Open Access www.mmse.xyz

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strain rates however offend indication of dynamic recovery is noticed is noticed for higher strain rate of 0.1/sec. Increasing the strain rate at low deformation temperature as restricted DRX is observed from Fig. 3. Upon increasing deformation temperature to 950 0C the dynamic re-crystallization is recorded at low strain rates till 0.1/Sec; whereas at higher strain rates of 1/Sec occurrence of dynamic recovery is indicated as shown in Fig. 4.

As expected higher deformation temperature like 1000 0C envisages the occurrence of DRX at all strain rates which 0.01/Sec, 0.1/Sec and 1/Sec. It therefore follows from the above diagram that dynamic re-crystallization of the experimental steel is favored at higher temperature and lower strain rate. The combination of deformation temperature and strain rate is essentially an important aspect in deciding dynamic re-crystallization is set in or not. It is known that DRX is thermally activated from therefore it is accentuated by higher deformation temperature and higher availability of time at deformation temperature. It is obvious that the slower strain rates provides longer time for DRX phenomena to take place and hence above observations are made in present investigation.

The Ć&#x;-ÎŁ Analysis to Calculate Critical Stress for Initiation of DRX: From true stress/ true strain Curve; plot of work hardening rate Vs true stress (Ć&#x;-Ďƒ) is given in Fig. 6 as: The inflection point is detected by fitting 3rd degree polynomial to Ć&#x;-Ďƒ curve Ć&#x; = đ??´đ?œŽ 3 + đ??ľđ?œŽ 2 + đ??śđ?œŽ + đ??ˇ

(1)

At critical stress for initiation of DRX the second derivative becomes zero; so đ?‘‘2 Ć&#x; đ?‘‘2 đ?œŽ

= 6đ??´đ?œŽ + 2đ??ľ

Becomes zero, therefore, Ďƒ (critical) = –B/3A

(2)

(3)

Following the same argument the critical stress for DRX as well as critical strain for the same has been calculated for all cases where DRX could be observed. It appears from Fig. 7 that the critical stress decreases with increase in deformation temperature. Fig. 8 exhibits that critical strain for occurrence of DRX at a constant strain rate of 0.01/Sec decreases with deformation temperature tending to assume some constant value at higher deformation temperature. Fig. 9 shows that effect of strain rate on critical stress for occurrence of DRX at fixed highest deformation temperature 10000C; rise in the magnitude of critical stress for DRX with increasing strain rate is logically consistent with the fact that higher strain rate provides less time for DRX to take place at any specific deformation temperature.

In fig. 10 transmission electron micrographs of steel deformed at strain rates of 0.01/Sec at 900 0C shows that precipitation of Cu has taken place concurrently with DRX or just after DRX and during austenite to ferrite transformation. In the first case the precipitates would have sited at the grain boundaries while in the second case the precipitate impend transformation growth of DRX grains although conclusive evidence has not been derived in the present investigation. The either of the above two events could lead to achievement of fine grained ferrite from austenite this is why SEM image by Fig. 10 shows that ferrite grain size of 2-3 Âľm that the precipitates of Cu forms in specimens deformed at 900 0C. MMSE Journal. Open Access www.mmse.xyz

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180 160 140

stress(Mpa)

120 100 80 (950C-0.01/sec)

60

(1000C-0.01/sec)

40

(850C-0.01/sec) 20

(800C-0.01/sec)

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

strain

Fig. 2. Flow curve at temperature 800 oC-1000 oC and strain rate of 0.01/sec.

850C-strain rate 0.01/sec 850C-strain rate 0.1/sec

300

850C-strain rate 1/sec

stress(Mpa)

250 200 150 100 50 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

strain

Fig. 3. Flow currve at temperature 850 oC and strain rate of 0.01, 0.1 and 1/sec.

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0.8

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250

stress(Mpa)

200

150

100 950C-strain rate 0.01/sec 50

950C-strain rate 0.1/sec 950C-strain rate 1/sec

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

strain

Fig. 4. Flow curve at temperature 950 oC and strain rate of 0.01, 0.1 and 1/sec.

180 160 140

stress(Mpa)

120 100 80 60

(1000C- strain rate 0.01/sec)

40

1000c-strain rate 0.1/sec

20 1000C-strain rate 1/sec

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

strain

Fig. 5. Flow curve at temperature 1000 oC and strain rate of 0.01, 0.1 and 1/sec.

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0.8

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300

work hardening rate

250 200 150 100 50 0 100

110

120

130

140

150

stress (Mpa)

Fig. 6. Work hardening rate Vs True stress.

critical strain for DRX

0.35 0.3 0.25 0.2 0.15 0.1 800

850

900

950

1000

1050

1100

Temperature in C

Fig. 7. Critical strain for DRX Vs Temperature.

140

critical stress for DRX

130 120 110 100 90 80 70 60 50 40 800

900 Temperature in C

Fig. 8. critical stress for DRX Vs temperature.

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1000

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160

critical stress for DRX

140 120

100 80 60 40 20 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

strain rate

Fig. 9. critical stress for DRX Vs strain rate at 1000 0C.

Fig. 10. TEM micrograph at temperature 900 oC and strain rate of 0.01/sec.

Summary.

1. High deformation temperature and low strain rate is the favorable condition for dynamic recrystallization for the material under investigation which is 0.05C-1.52Cu-1.51Mn steel.

2. The critical stress for dynamic re-crystallization decreases with increase in deformation temperature. The critical strain for occurrence of DRX at a constant strain rate of 0.01/Sec decreases with deformation temperature tending to assume some constant value at higher deformation temperature.

3. Precipitation of Cu has taken place concurrently with DRX or just after DRX and during austenite to ferrite transformation. The ferrite grain size of 2-3 Âľm is formed in the process adopted in the experimentation.

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References [1] H. Beladi, P. Cizek, and P. D. Hodgson, “On the characteristics of substructure development through dynamic recrystallization,” Acta Materialia, Vol. 58, pp. 3531-3541, 2010, doi: 10.1016/j.actamat.2010.02.026 [2] R. D. Doherty, D. A. Hughes, F. J. Humphrey, and J. J. Jonas, et al., “Current issues in recrystallization: A review,” Materials Science and Engineering A, Vol. 238, pp. 219-274, 1997, doi: 10.1016/S0921-5093(97)00424-3 [3] Sakai T, Jonas JJ., Overview no. 35 dynamic recrystallization: mechanical and microstructural considerations, Acta Metallurgica, 1984 Vol. 32, pp. 189-209 [4] Derby B. Acta Metallurgica, Grain Refinement in a Copper Alloy by Shaped Charge Explosion, 1991, Vol. 39, p. 955.

[5] Sakai T. J Mater Process Technology, 1995, Vol. 53, p.349 [6] Kentaro IharaYasuhiro Miura. Dynamic recrystallization in Al-Mg-Sc alloys. MaterialsSCience & Engineering A, 2003, doi:10.1016/j.msea.2004.05.082 [7] Gourdet S, Montheillet F., An experimental study of the recrystallization mechanism during hot deformation of aluminium, Materials Science and Engineering A 283(1):274-288, doi: 10.1016/S0921-5093(00)00733-4 [8] Sakai T, Jonas JJ. In: Buschow KH, Cahn RW, Flemings MC, Ilschner B, Kramer EJ, Mahajan S, editors. Encyclopedia of materials: science and technology, vol. 7. Oxford: Elsevier; 2001. p. 7079.

[9] Solberg JK, McQueen HJ, Ryum N, Nes E. Philos Mag. A 1989; 60:447.

[10] Hales, S.J., McNelley, doi:10.1007/BF02661097

T.R.

&

McQueen,

H.J.

MTA

(1991)

22:

1037.

[11] Tsuji N, Matsubara Y, Saito Y., Dynamic recrystallization of ferrite in interstitial free steel, Scripta Materialia,Volume 37, Issue 4, 1997, pp. doi:10.1016/S1359-6462(97)00123-1 [12] McNelley TR, McMahon ME. Metall. Mater. Trans. A, 1997; 28: 1879.

[13] Setuo Takaki, Masaaki Fujioka, Shuji Aihara, Yasunobu Nagataki, Takako Yamashita,Naoyuki Sano, Yoshitaka Adachi, Masahiro Nomura and Hiroshi Yaguchi, Effect of Copper on Tensile Properties and Grain-Refinement of Steel and its Relation to Precipitation Behavior, Materials Transactions, Vol. 45, No. 7 (2004) pp. 2239- 2244, doi: 10.2320/matertrans.45.223 Cite the paper Kumar, P., & Hodgson, P. (2016). Effect of Temperature and Strain Rate on Dynamic ReCrystallization of 0.05C-1.52Cu-1.51Mn Steel. Mechanics, Materials Science & Engineering, Vol 6. doi:10.13140/RG.2.1.4905.2403

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Nanostructure Formation in Anodic Films Prepared on a β Alloy Ti39Nb PVD Layer Zdenek Tolde1, a, Vladimír Starý1, Petr Kozák1 1 – Department of Materials Engineering, Faculty of Mechanical Engineering, CTU in Prague, Karlovo nám. 13, CZ-121 35 Prague 2, Czech Republic a – zdenek.tolde@fs.cvut.cz DOI 10.13140/RG.2.1.2756.8883

Keywords: TiNb, oxide layer, Ti alloys, nanostructured surface, anodic oxidation.

ABSTRACT. Ti alloys are widely used for construction of bone implants. Some of them can be prepared without any toxic elements containing only Nb, Zr and Ta. At suitable composition they have the beta (BCC) structure with low modulus of elasticity and high corrosion resistance. The oxidation of their surface can increase the biocompatibility and enable the preparation of nanostructured surface morphology.

The β-alloy Ti39wt.%Nb alloy was melted eight times by electric discharge, annealed at 850°C for 30 minutes and quenched to water. The substrates of the TiNb layers were prepared from bulk Ti39Nb and commercial cpTi and Ti6Al4V. They were cut using a SiC cutting wheel, ground with abrasive papers and then polished with a suspension of colloidal SiC. The TiNb layers were prepared by cathodic sputtering in a Hauser Flexicoat 850 unit. The thickness of the TiNb layer was measured by Calotest. Surface roughness was measured by a Hommel T1000 Basic roughness tester. The sample surface was observed by a JSM7600F scanning electron microscope. Samples were anodically oxidized in (NH4)2SO4 + 0,5wt%NH4F electrolyte at DC voltages 10, 20 and 30 V using a stabilized voltage source.

The morphology of the nanostructured surface of a PVD layer depends particularly on the oxidation voltage and time, but also on the type of substrate. The surface morphology containing nanotubes appeared only on TiNb layer with a TiAlV substrate prepared at certain oxidation voltage and time.

The morphology of oxidized layers is heavily influenced by substrate material even though the surface roughness of PVD layer and substrate is identical for all oxidation processes.

TiNb alloy have very suitable properties for bioapplications and the study of surface properties contribute to the practical use of this material.

Introduction. Titanium alloys have very suitable properties for bioapplications including high specific strength, high corrosion resistance and due to these properties also excellent biocompatibility.

Until now the classical biomedical material – stainless steel (e.g. AISI 316L, (E ~ 210 GPa)) and pure Ti and α and α+β Ti alloys (E ~ 110-120 GPa) are usually used for the production of implants [1; 2].

Recently Ti alloys, which have a lower modulus of elasticity, are intensively studied. Since for bone implants is very useful to obtain maximum similarity of the moduli of elasticity of the material f the implant and the bone [3] and, simultaneously, the high corrosion resistance, the aim of these studies is the fabrication of a material with these properties. The moduli of β-Ti alloys, especially TiNb alloys (β-TiNb), can be about 60 GPa. Layers of these β-alloys with prospective properties can be prepared applying an appropriate method, particularly PVD (Physical Vapour Deposition) [4].

A relatively thin film of oxide(s) of the basic material is present on the surface of titanium and its alloys. In a TiNb alloy these are usually titanium oxide TiO2 and a niobium oxide, usually Nb2O5 [4; 5]. These oxides form a layer, which decreases potential corrosion in a corrosive environment. By anodic oxidation a basic thicker oxide layer is created. The properties of this layer (crystalline MMSE Journal. Open Access www.mmse.xyz

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structure, thickness, porosity and/or nanoporosity) depend on the conditions of oxidation and can be selected with respect to the required properties of the final application. One of the advantageous properties of the surface is its porosity, since a better adhesion of cells was observed on the surface with a porous morphology (structure). The structure with suitable pores can also be used as a reservoir of appropriate healing drugs. All these properties can improve the process of healing after implantation [5].

In our work we studied β-Ti39Nb, i.e., a Ti alloy with 39 wt.% Nb (further only TiNb). Both elements in the alloy are nontoxic and highly biocompatible [4; 6; 7]. At present research is concentrated on a material which could replace the until now widely used Ti alloy Ti6Al4V, which can hypothetically cause damage to the tissue due to the content of potentially toxic elements Al and V [3].

Generally, titanium alloys containing 10 - 15% of β stabilizers are in a metastable state and they are denoted as β-metastable. According to the phase diagram these alloys are formed by the unstable phase β´ and the phase ω or by a mixture of the unstable phase β´ and the stable phase β [1]. This depends on the concentration of the β stabilizer. The phase ω is created by the decomposition of the unstable phase β´. The mechanical properties of the alloys are influenced by the appearance of unstable and martensitic phases which depends on the concentration of β stabilizing elements and on the cooling rate during heat treatment. Only high concentrations of β-stabilizers (30% or more) create a stable form of the β phase in the alloy at room temperature. However, it is responsible for an increase of the density and weight of the alloy [8].

At room temperature the strength of the alloy increases with the content of the β-phase. The mechanical properties of β-alloys, e.g., strength and fatigue resistance, can be improved by heat treatment.

Fig. 1. Phase diagram of TiNb alloys [9].

Oxidation is frequently used to improve and/or optimize the surface properties of cell adhesion. Also, the surface of the alloy samples containing Nb related sites improves cell adhesion and growth.

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Oxygen creates a thin film of oxides, which at room temperature can be several nanometers thick. By continuing the process the thickness of oxide layer increases up to micrometer range. There are two basic methods of the oxidation of metals: an electrochemical (anodic) one in an appropriate electrolyte, and a thermal one, at high temperature approximately in the range of hundreds of degrees of Celsius. The chemical reaction in anodic oxidation can be described by the chemical equation (1) (M - metal ions, O - oxygen ions) [10]:

đ?‘?

đ?‘Žđ?‘€ + 2 đ?‘‚2 → đ?‘€đ?‘Ž đ?‘‚đ?‘?

(1)

The anodic growth of an oxide layer on Ti (i.e., in electrolyte 1 M (NH 4)2SO4 + 0.5 wt.% NH4F) is shown in Figs. 2 and 3. In the anodic oxidation of Ti it is necessary to add fluoride ions to obtain a nanostructured surface (i.e. a surface with a morphology with objects of nanometer size).

Simultaneously with the growth of the oxide layer, Ti dissolves in the basic material of the anode according to equation (2) [11].

đ?‘‡đ?‘– → đ?‘‡đ?‘– 4+ + 4đ?‘’ −

(2)

The cathodic reaction is described by equation (3), where H2O is decomposed into hydrogen and hydroxyl anions [11] 4đ??ť2 đ?‘‚ + 4đ?‘’ − → 2đ??ť2 + 4đ?‘‚đ??ť −

(3)

Firstly the interaction of Ti4+, OH- and O2- takes place on the surface in contact with the electrolyte and later also on the interface of the oxide layer and the metal and an oxide layer is created according to chemical equation (4); also titanium hydroxide can be created according to (6) [12]: đ?‘‡đ?‘– 4+ + 2đ?‘‚2− → đ?‘‡đ?‘–đ?‘‚2 đ?‘‡đ?‘– + đ?‘‚2 → đ?‘‡đ?‘–(đ?‘‚) → đ?‘‡đ?‘–6 đ?‘‚ → đ?‘‡đ?‘–3 đ?‘‚ → đ?‘‡đ?‘–2 đ?‘‚ → đ?‘‡đ?‘–đ?‘‚ → đ?‘‡đ?‘–2 đ?‘‚3 → đ?‘‡đ?‘–3 đ?‘‚5 → đ?‘‡đ?‘–đ?‘‚2 đ?‘‡đ?‘– 4+ + 4ĐžĐ?− → đ?‘‡đ?‘–(đ?‘‚đ??ť)4

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(4) (5) (6)

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Fig. 2. Formation of an anodic layer of titanium oxide.

Titanium hydroxide is converted into an oxide according to (7) [12]: đ?‘‡đ?‘–(đ?‘‚đ??ť)4 → đ?‘‡đ?‘–đ?‘‚2 + 2đ??ť2 đ?‘‚

(7)

The whole process of anodic oxidation and creation of the oxide layer is described by equation (8) [11]. đ?‘‡đ?‘– + 2đ??ť2 đ?‘‚ → đ?‘‡đ?‘–đ?‘‚2 + 4đ??ť + + 4đ?‘’ −

(8)

Fig. 3 Schematic view of titanium anodization (a) and dissolution of titanium inside pores.

In the case of the formation of the oxide layer, electrons and ions pass through the film, i.e. the electric current through the oxide necessary for the growth of the film is strongly limited by the thickness of the film. This electric current is influenced by a decrease of the electric field on the film and approximately follows an exponential law.

Niobium oxidation and the growth of the Nb oxide in layer are given by equations (9) – (11) [13]:

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đ?‘ đ?‘? + đ??ť2 đ?‘‚ − 2đ?‘’ − → đ?‘ đ?‘?đ?‘‚ + 2đ??ť +

(9)

đ?‘ đ?‘?đ?‘‚ + đ??ť2 đ?‘‚ − 2đ?‘’ − → đ?‘ đ?‘?đ?‘‚2 + 2đ??ť +

(10)

2đ?‘ đ?‘?đ?‘‚2 + đ??ť2 đ?‘‚ − 2đ?‘’ − → đ?‘ đ?‘?2 đ?‘‚5 + 2đ??ť +

(11)

The growth rate of the oxide layer is given by the Faraday law and the rate of the dissolution of the layer. The Faraday law defines the mass of the TiO2 and Nb2O5 oxides in the layer assuming zero production of hydrogen in the process. At simultaneous growth and dissolution the thickness of the layer practically [13] does not depend on the time and the final thickness depends on the oxidation potential U đ?‘‘ =đ?‘˜âˆ™đ?‘ˆ

(12)

where k – is the constant of the growth of the layer [14] In the presence of fluoride ions the behaviour of the dependence of the current on the time of oxidation depends on their content. The dependence is shown in Fig. 4 [14]. In an electrolyte without fluoride ions we can observe an exponentially decreased current density up to the final equilibrium state with a certain minimum value of the current density. In an electrolyte with fluoride ions the current density after a certain minimum value begins to increase again. This is caused by the interaction of fluoride ions with the formed oxide layer. This increase also signalizes the creation of nanostructures on the surface of the specimen [14; 15].

Fig. 4. Behaviour of current density in an electrolyte without fluoride ions (dashed line) and with an addition of fluoride ions (solid line curve).

If the voltage increases, the structure of the oxide changes from an amorphous into a crystalline one.

During this process the conductivity changes from ionic to electronic one which retards the growth of the film. The film growth is finished by electrical breakdown of the film [16].

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The thermal oxidation of a β-TiNb alloy sample creates on the surface two distinct crystalline oxide phases: T-Nb2O5 and TiO2 (usually rutile) [3]. No evidence was found of the presence of the Ti–Nb mixed oxide phase even though the single-crystalline Ti1−xNbxO2 with x as large as 0.3 and with even larger values of x for the nanocrystalline oxide have been reported in literature [16].

The purpose of our work is to compare the surface structure/morphology of TiNb films prepared on TiNb, Ti and TiAlV substrates using PACVD (cathodic sputtering).

Experiments. Preparation and characterization of samples. β-Ti39Nb alloy samples were prepared by arc-melting 61 wt.% Ti (ingot, 99.55%, Frankstahl, Austria) with 39 wt.% Nb (ingot, 99.85%, TIC, Brussels, Belgium). The melting proceeded eight times at 800–1000A/ 23V with subsequent solution annealing at 850 °C for 30 minutes and water quenching to achieve the defined homogeneity.

The substrates for the TiNb layer were ground and polished coupons of cpTi ISO 5832-2 and Ti alloy Ti6Al4V ISO 5832-3 (the basic material was supplied by Beznoska Ltd., Kladno, CR). Using a SiC cutting wheel the as-prepared ingot was sliced into coupons (diameter either 10.5 mm or 14 mm and thickness ~1.5 mm). The surfaces of the coupons were ground sequentially with abrasive papers (240, 600, 800, 1000 and 4000 grit) and then polished with a suspension of colloidal SiC (0.05 μm, Colloidal Sillicat, Leco, CR) into a mirror-like sheen, using a Leco machine.

The TiNb layer was prepared by cathodic sputtering (PVD) in a Hauser Flexicoat 850 unit (Hauser, Netherlands). The time of deposition was 2.5 hrs, the temperature of the substrate 250°C, rotation of the substrate 2 rpm, and the working pressure 2.10-3 mbar (2 Pa). The thickness of the TiNb layer was measured by Calotest (CSM, Switzerland).

Surface roughness was measured by a Hommel T1000 Basic roughness tester (Jenaoptic, Germany).

For a general evaluation the surface morphology all samples were observed by a JSM7600F scanning electron microscope (JEOL Ltd, Japan) at several magnifications (usually between 1000 and 50 000x), using SEI detectors and LEI detectors. The SEI detector shows a general overview of the surface, while the LEI detector shows the irregularities with higher sensitivity.

The samples were oxidized by anodic oxidation in 1M (NH4)2SO4 + 0, 5 wt% NH4 F, the resultant pH was 4.7. The potentiostatic process was carried out at constant voltages (DC) 10, 20 and 30 V using a stabilized voltage source SZ 20 110/400 – 19 I2 KZ C230 (NES Nová Dubnica, SR). From the beginning of oxidation, the potential was increased to the final value approximately at a rate of 100 mV/s. The time dependence of the oxidizing current was measured and recorded by a UT 804 (TIPA Ltd, CR) digital multimeter and the current density was calculated for all samples.

Results and discussion. The parameters of substrates is in table 1.

Table 1. List of substrates.

Substrate material

Phases substrate

in Composition

TiNb Ti TiAlV

Ti – beta Ti + 39wt.%Nb Ti –alpha Ti Ti – Ti + 6wt% Al, alpha+beta 4wt.%V,

Modulus of elasticity Er[MPa] 0.058+/-0, 006 95+/-2.5 0.047+/-0.005 135+/-2.1 143+/-1.3 0.035+/-0.003

Substrate roughness Ra [μm]

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Substrate hardness (GPa) 3.1+/-0.09 2.3+/-0.1 4.0+/-0.11

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The PVD coating was 1.45 μm thick.

After depositing the TiNb layer on a Ti no grain boundaries were observed on surface of TiNb/Ti while on the layer of TiNb on a TiAlV substrate both alpha and beta phases and the grain boundaries of the substrate could be observed on electron microscope images.

Next we compared the electron microscope images of the structure and surface morphology of the oxide film on a bulk Ti39Nb alloy and of an oxidic film on the TiNb layer which was sputtered by means of the PVD process on different Ti substrates (TiNb/Ti) and Ti6Al4V (TiNb/TiAlV).

Table 2. Values of several parameters of TiNb layers on different substrates.

Substrate

Modulus of elasticity Film hardness Substrate Er [MPa] [GPa] size [μm]

grain Film grain sizeFilm roughness by [μm] Ra [μm]

TiNb/TiNb

100+/-2.5

3.8+/- 0.14

600

600

0.093+/-0.047

TiNb/Ti

116+/-3.37

4.1+/-0.22

10

6–7

0.13+/-0.008

TiNb/TiAlV

115+/-3.15

4.1+/-0.19

3

3–5

0.10+/-0.005

The anodic voltage and time of oxidation was monitored. Different details of the surface morphology are apparent on images at various magnifications.

At an anodic voltage of 10 V individual pores with a 10 nm size are clearly apparent (Fig. 5a) in the oxide film on a TiNb/Ti layer. This is due to the presence of fluoride ions in the electrolyte. The oxide film on TiNb/TiAlV has a similar structure with pores of comparable size (Fig. 5b).

Fig. 5. SEM images of oxide films prepared at anodic voltage 10 V, magn. 50 000x, time 1 hr, a) TiNb/Ti, b) TiNb/TiAlV.

During anodic oxidation at a constant 20V voltage a porous layer of titanium and niobium oxides [3, 13] which has a similar surface structure as in the previous experiment at a 10V voltage (Fig. 5, 6a) is apparent on the coating deposited on the Ti substrate. A more clearly apparent porous nanostructure can be seen on the Ti6Al4V substrate (Fig. 5, 6b).

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Fig. 6. SEM images of oxide films prepared at an anodic voltage of 20 V, magn. 50 000x, time 1hr, a) TiNb/Ti, b) TiNb/TiAlV.

In the TiNb/Ti sample at an anodic voltage of 30 V (Fig. 7a) the arrangement and topography of the oxide layer is different from previous results. The pores are accumulated into characteristic corrugations which cover the entire surface of the sample. Also the microscope images of the TiNb/TiAlV samples (Fig. 7b) clearly show the grain boundaries of alpha and beta phases of the substrate which remained in the layer after deposition.

Fig. 7. SEM images of oxide films prepared at anodic voltage 30 V, magn. 50 000x, time 1 hr, a) TiNb/Ti, b) TiNb/TiAlV.

Since maximum pore density and their best regularity were observed at an anodic voltage of 20 V, the time of oxidation was increased to 2 hours in the next set of experiments. On TiNb/Ti pores were present but their density was relatively low. We observed localized growth of nanotubes (Fig. 8a).

On TiNb/TiAlV the oxide layer is formed by a nanoporous selforganized system of oxide nanotubes (Fig. 8b, c). The diameter of the nanotubes is within the range 50 – 100 nm and they are distributed uniformly over the sample surface.

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Fig. 8. SEM images of oxide films prepared at anodic voltage 20 V, time 2 hrs, a) TiNb/TiAlV, magn.

30 000x, b) TiNb/TiAlV, magn.100 000x.

We also can compare the electron microscope images of an anodic layer of TiNb/TiNb, oxidized at an anodic voltage of 20 V for 1 hour, taken at different magnifications. The results are in Figs. 9a-f.

It can be seen that on the TiNb layer on bulk TiNb a porous anodic film was not formed and the surface of sample remained practically unchanged. This could be explained by the mechanical treatment of the surface layer during grinding and polishing of the surface of the sample before oxidation, which limited the growth of the porous oxide film.

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Fig. 9. SEM images of oxide films prepared at anodic voltage 20 V, time 1hr, a) bulk TiNb, magn.10 000x, b) bulk TiNb, magn.100 000x, c) TiNb/Ti, magn. 10 000x, d) TiNb/Ti, magn.

100 000x, e) TiNb/TiAlV, magn. 10 000x, f) TiNb/TiAlV, magn. 100 000x.

Besides the surface structure also the dependence was studied of the oxidation current on the anodic voltage and on the time. The dependences of the current density on time during the oxidation of the TiNb layer (both on Ti and TiAlV) at anodic voltages 10, 20 and 30 V in the studied electrolyte are in Figs. 10a, b. The rate of the voltage increase was 100 mV.s-1. We found that the critical passivation current density ikp on TiNb/Ti and TiNb/TiAlV is 11.9 – 12.2 mA.cm-2 and 10.2 – 10.4 mA.cm-2, respectively. It can be observed that the magnitude of the anodic voltage has no influence on the critical passivation current density ikp at the given parameters of the process.

The magnitude of the anodic voltage obviously affects the rate of the decrease of the current density; at higher voltages the decrease is slower. Also at higher anodic voltages the value of the density of the passivation current increases (Fig. 10a, b). From the diagram of the dependence of the current density on the time, the value of the critical passivation current density ikp lies within the time intervals 65 – 75 s and 75 – 85 s for TiNb/Ti and TiNb/TiAlV respectively. Using these values and the rate of growth of the anodic voltage we can calculate the passivation voltages 6.5 ÷ 7.5 V and 7.5 ÷ 8.5 V for TiNb/Ti and TiNb/TiAlV respectively.

a

b

Fig. 10. The dependence of current density on time during oxidation of a) TiNb/Ti, b) TiNb/TiAlV. Summary. On anodic oxide films prepared on layers of TiNb on various substrates we found a porous oxide layer. After 1 hour of growth at anodic voltages of 10, 20 and 30 V the layers were porous without apparent nanotubes with a various degree of porosity and different directions of the growth MMSE Journal. Open Access www.mmse.xyz

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of the coating. In TiNb/Ti the porous structure is homogeneous over the entire surface of the substrate.

In the oxide layer on TiNb/TiAlV the phase boundaries in the substrate are visible.

After 2 hours of growth at an anodic voltage of 20 V, the oxide film on TiNb/Ti has a relatively low density of pores. In spite of this in TiNb/TiAlV a nanostructured surface morphology with a regular set of nanotubes with diameters in the range 50 – 100 nm was observed. The boundaries of phase grains disappeared.

No nanostructural features were found on TiNb/TiNb samples.

Finally, we can state that the sputtered TiNb layer is not influenced by the mechanical treatment of the substrate and by the potential impurities due to this treatment. Moreover a nanostructured oxide layer (without or with nanotubes) can grow at suitable conditions of growth on a deposited layer of TiNb. Acknowledgment This study was supported by the Grant Agency of the Czech Republic (grant no. 15-01558S) and by the Ministry of Education, Youth and Sport of the Czech Republic, Program NPU1, project No. LO1207. We are grateful to Mr Ivan Šiman for the English language review.

References [1] Lutjering G. A J. Williams Titanium. 2nd ed. New York: Springer, 2007. ISBN 978-354-0713975.

[2] Ozaki Tomomichi, Hiroaki Matsumoto, Sadao Watanabe a Shuji Hanada. Beta Ti Alloys with Low Young's Modulus. MATERIALS TRANSACTIONS. 2004, 45(8), 2776-2779. DOI: 10.2320 [3] Oshida, Yoshiki. Bioscience and bioengineering of titanium materials. 2nd ed. Waltham, MA: Elsevier, 2013. Elsevier insights. ISBN 04-446-2625-5.

[4] Jirka Ivan, Marta Vandrovcová, Otakar Frank, Zdeněk Tolde, Jan PLŠEK, Thomas Luxbacher, Lucie Bačáková a Vladimír Starý. On the role of Nb-related sites of an oxidized β-TiNb alloy surface in its interaction with osteoblast-like MG-63 cells. 2013, 33(3), 1636-1645. DOI: 10.1016/j.msec.2012.12.073. ISSN 09284931.

[5] Liu X., P. Chu, C. Ding. Surface modification of titanium, titanium alloys, and related materials for biomedical applications. Materials Science and Engineering: R: Reports. 2004, 47(3-4), 49-121.

DOI: 10.1016/j.mser.2004.11.001. ISSN 0927796x [6] Mani A, L Vaidhyanathan, Y Hariharan, M Janawadkar, T Radhakrishnan. Structural and superconducting properties of Nb-Ti alloy thin films. Bulletin of Materials Science. 1997, 20(4), 503507. DOI: 10.1007/BF02744763. ISSN 0250-4707.

[7] Vlcak P, F Cerny, Z Weiss, S Danis, J Sepitka, Z Tolde, V Jech. The Effect of Nitrogen Ion Implantation on the Surface Properties of Ti6Al4V Alloy Coated by a Carbon Nanolayer. Journal of Nanomaterials. 2013, 2013, 1-8. DOI: 10.1155/2013/475758. ISSN 1687-4110 [8] Bönisch Matthias, Mariana Calin, Thomas Waitz, Ajit Panigrahi, Michael Zehetbauer, Annett Gebert, Werner Skrotzki a Jürgen Eckert. Thermal stability and phase transformations of martensitic Ti–Nb alloys. Science and Technology of Advanced Materials. 2016, 14(5), 055004-. DOI: 10.1088/1468-6996/14/5/055004. ISSN 1468-6996.

[9] Murray Joanne L. The Nb−Ti (Niobium-Titanium) system. Bulletin of Alloy Phase Diagrams.

1981, 2(1), 55-61. DOI: 10.1007/BF02873704. ISSN 0197-0216.

[10] Izman S., M Rafiq, M Anwar, E.M. Nazim, R. Rosliza, A. Shah, M.A. Hass. Surface Modification Techniques for Biomedical Grade of Titanium Alloys: Oxidation, Carburization and MMSE Journal. Open Access www.mmse.xyz

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Ion Implantation Processes. Titanium Alloys - Towards Achieving Enhanced Properties for Diversified Applications. InTech, 2012. DOI: 10.5772/36318. ISBN 978-953-51-0354-7.

[11] Macák, J. Growth of Anodic Self-Organized Titanium Dioxide Nanotube Layers. Der Technischen Fakultät der Universität Erlangen-Nürnberg zur Erlangung des Grades, 2008.

[12] Liu Guohua, Kaiying WANG, Nils HOIVIK, Henrik JAKOBSEN. Progress on free-standing and flow-through TiO2 nanotube membranes. Solar Energy Materials and Solar Cells. 2012, 98, 2438. DOI: 10.1016/j.solmat.2011.11.004. ISSN 09270248.

[13] Krasicka-Cydzik, E. Anodic Layer Formation on Titanium and Its Alloys for Biomedical Applications. Titanium Alloys - Towards Achieving Enhanced Properties for Diversified Applications. InTech, 2012. DOI: 10.5772/34395. ISBN 978-953-51-0354-7.

[14] Macak J.M., H. Tsuchiya, A. Ghicov, K. Yasuda, R. Hahn, S. Bauer, P. Schmuki. TiO2 nanotubes: Self-organized electrochemical formation, properties and applications. Current Opinion in Solid State and Materials Science. 2007, 11(1-2), 3-18. DOI: 10.1016/j.cossms.2007.08.004. ISSN 13590286.

[15] Minagar S, Ch. Berndt, J Wang, E Ivanova, C Wen. A review of the application of anodization for the fabrication of nanotubes on metal implant surfaces. Acta Biomaterialia. 2012, 8(8), 28752888. DOI: 10.1016/j.actbio.2012.04.005. ISSN 17427061.

[16] Zorn, G., A. Lesman a I. Gotman Oxide formation on low modulus Ti45Nb alloy by anodic versus thermal oxidation. Surface and Coatings Technology. 2006, 201(3-4), 612-618. DOI: 10.1016/j.surfcoat.2005.12.009. ISSN 02578972.

Cite the paper Tolde Z., Starý V. & Kozák, P. (2016). Nanostructure Formation in Anodic Films Prepared on a β Alloy Ti39Nb PVD Layer. Mechanics, Materials Science & Engineering Vol.6, 6. doi:10.13140/RG.2.1.2756.8883

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Diagnostics of Argon Injected Hydrogen Peroxide Added High Frequency Underwater Capillary Discharge Muhammad Waqar Ahmed1, Sooseok Choi1, Jong-Keun Yang1, Rai Suresh1, Heon Ju Lee1 1 – Department of Nuclear, Energy and Chemical Engineering, Jeju National University, 690-756, Republic of Korea DOI 10.13140/RG.2.2.16147.27684

Keywords: high frequency plasma discharge, hydrogen peroxide, emission spectroscopy, OH radicals.

ABSTRACT. The effects of hydrogen peroxide addition and Argon injection on electrical and spectral characteristics of underwater capillary discharge were investigated. The flowing water discharge was created in a quartz tube (Φ = 4mm outer; Φ = 2mm inner; thickness 1mm) by applying high frequency (25 kHz) alternating current voltage (0-15kV) across the tungsten electrodes (Φ=0.5mm), in pin-pin electrode configuration, separated by a gap distance of 10 mm. The results of no hydrogen peroxide addition and no Argon gas injection were compared with addition of hydrogen peroxide and Argon injection for different values. The emission spectrum was taken to present the increase in concentration of •OH radicals with and without hydrogen peroxide addition under different argon injection rates. The results demonstrated that addition of hydrogen peroxide do not remarkably affected the conductivity of water, but its addition increased the yield rate of •OH radicals generated by plasma discharge. The addition of Argon generated bubbles and gas channels reduced the high power consumption required for inducing flowing water long gap discharge. The results showed large concentration of •OH radicals due to hydrogen peroxide addition, less required input power for generating flowing water discharge by using high frequency input voltage and due to Argon injection.

1. Introduction. The generation of reactive species like •OH radicals, ozone, reactive hydrogen and oxygen through electrical discharge in water is of large interest and has been widely investigated by many researchers [1-3]. Through various diagnostics phenomena different kinds of reactive species were detected [4-5]. Among them ozone and •OH radicals are of larger interest due to high redox potential (2.07V and 2.80V respectively) and their high sterilization rate [6]. These reactive species have wide range of environmental, biological, medical, Nano-technology and industrial applications [7-12]. •OH radicals are widely used for controlling environmental pollution including drinking water and waste water treatment [13]. High redox potential reactive species are useful in blood treatment and E.Coli degradation when generated in water and other liquids through electrical discharge [1415]. Synthesis of Nano-particles and polymers surface modifications is another useful application of underwater plasma discharge and these reactive species can act as antibacterial agents [16-17]. The hydroxyl radicals can be generated by various mechanisms [18] else than electrical discharge in water, but researchers proved that the electrical discharge method is most effective method where high intensity of •OH radicals can be obtained [19-20]. Therefore in this research electrical discharge in water was used to induce highly reactive oxidant species especially •OH radicals. When plasma generated then highly intensive shock waves, high temperature, strong electric field generation and electron impact dissociation can cause water molecule to split into •OH, other reactive species, ionization and excitation process. Under various chemical reactions the splitting and recombination of generated radicals and ionic species takes place to form •OH, ozone and H2O2. Following are some common chemical reactions that occur in aqua system while inducing some highly reactive oxidant species [21]: H2O → H+ •OH H+H→H2 MMSE Journal. Open Access www.mmse.xyz

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OH +•OH →H2O2

•

OH +•OH → H2O +O O +•OH → O2 +H e-+ H2O2 →•OH +H

H2O2 + H2O →2•OH + H2+ O2 In the table 1 the possible chemical reactions list, that takes place when discharge occurs in liquid is shown [22].

Table 1. The possible chemical reactions list that takes place when discharge occurs in liquid (Dors et al. 2005; Chen et al., 2002;2009; Grymonpre et al., 2001; Mok et al., 2008).

Reactants → Products

Reactants → Products

2H2O → H2O2+H2 H2O → H++eaq+·OH eaq+ ·OH→H2+OH. eaq +·OH→OH. eaq+ HO2→HO2eaq+ O2.→HO2- + OH. eaq+ H2O2→ OH. + OHeaq+ HO2- →O- + OHeaq+ O2 → O2eaq+ H+ → H. eaq + H2O → OH- + H. 2eaq → H2O+2 OH2 H. →H2 H.+·OH → H2O H. + HO2 → H2O2 H. + O2- → HO2H. + H2O2 → H2O +·OH H. + O2 → HO2.

OH- + H. → eaq + H2O · OH+ ·OH → H2O2 · OH +O- → HO2· OH + HO2. →O2 + H2O · OH+ O2- → O2 + OH· OH+ H2O2 → HO2. + H2O · OH+ HO2- → HO2. + OH-

O- + HO2- → O2- + OHO- + H2 → H. + OHO- + H2O → ·OH + OH2HO2. → O2 + H2O2 HO2. + H2O2 → ·OH + O2 + H2O O2- + HO2. → HO2- + O2 HO2. → H+ + O22O2- → H2O2 + O2 + 2OHO2- + H2O2 → ·OH + O2 + OHO2- + HO2- → O- + O2 + OHH+ + O2- → HO2.

H2O2 → 2·OH H2O2 + OH- → HO2- + H2O HO2- + H2O → H2O2 + OHHO2- + H+ → H2O2 H+ + OH- → H2O H2O → H+ + OHO- + O2 → O3O- + O3 → 2O2H2O2 + O3- → O2- + O2 + H2O HO2- + O3 → O2- + O2 + OHO3- → O2 + OH2 + O3- → H. + O2 + OHO- + H+ → OH .

HO2 + OH- → O2- + H2O H2O → H.+·OH

·

OH+ H2 → H. + H2O · OH + OH- → O- + H2O 2O- → OH- + HO2O2- + O- → O2 + 2 OHO- + H2O2 → O2- + H2O The addition of hydrogen peroxide in water can enhance the reaction rates for generating •OH radicals and other reactive species. In this research the standard value of hydrogen peroxide (0.35ml/L) [23] MMSE Journal. Open Access www.mmse.xyz

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was added at different amounts starting from (0-0.35) ml/L. This addition enhanced the yield rate of • OH radicals. It is important to measure the intensity of reactive species especially •OH radicals.

Several methods exist for the measurement of •OH radicals among them the most convenient method is optical emission spectroscopy (OES) [24]. Beside that other complicated methods like spin-trap electron-spin resonance (ESR) [25], indirect measurement of •OH radicals using chemical probe [26], laser induce fluorescence (LIF) [27] and •OH radicals dissolved in liquid were observed indirectly using fluorescent properties of hydroxyl-terepethalic acid (HTA) formed in the reaction of Terepethalic acid (TA) [28]. Among all of them OES is simple and convenient method that was used in this experiment. The properties of •OH radicals and other reactive species observed by several researchers by different mechanisms. Table 2 represents the properties of reactive species generated by electrical discharge in water [29].

This research work is useful to present the effect of H2O2 addition in water along with plasma discharge to enhance the yield of •OH radicals. Also the electrical characteristics of H2O2 added water discharge were presented.

Fig. 1. (Color online) Schematic view of experiment set-up.

Fig. 2. (Color online) Visual view of the capillary discharge.

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Table 2. Properties of selected species involved in AOP are through electrical discharge (Buxton et al., 1998; Lide, 2006; Petri et al., 2011).

SPECIES

OH

STANDARD ELECTROCHEMICAL POTENTIAL (v) +2.59

Hydrogen Peroxide

H2O2

+1.77

pH (where present) pH < 11.9 pH<11.6

Superoxide anion

O2-

-0.33

pH<4.8

Strong oxidant Week reductant Week reductant

Per-hydroxyl radical

HO2.

+1.49

pH<4.8

Strong oxidant

Hydro-peroxide anion

HO2-

+0.88

pH>11.6

Week oxidant Week reductant

Hydroxyl Radical

Singlet oxygen

FORMULA ·

1

Role

Strong oxidant

O2

Ozone gas

O3

+2.07

Strong oxidant

Atmospheric oxygen (normal triplet form) Solvated electrons

O2

+1.23

Week oxidant

e(aq)-

-2.77

pH > 7.85

Strong reductant

2. Materials and Methods. Fig. 1 represents the experimental set-up used while Fig. 2 shows the visual view of the discharge. The inter-electrode gap where plasma generated was kept 10mm, a liquid flow meter and controller (Dwyer-RM series) was used to control the flow rate of water (0.1L/min). Hydrogen peroxide (H2O2) was added to the water reservoir that was to be treated at standard rates starting from 0ml/L to 0.35ml/L. A conductivity meter (OAKTON-CON6) was used for observing conductivity of water during experiment specially after adding hydrogen peroxide.

Mass flow controller (LINE TECH M3030V) along with display unit was used to control and provide Argon gas. A Neon transformer (15 kV, 25 kHz) was used to provide required input power for generating discharge at 10 mm inter-electrode gap in tab water.

A Tektronix digital oscilloscope (DPO 2024) with high voltage and current probes and having data storage facility was used for recording Volt-Ampere characteristics. An Avantes Avaspec-NIR256 miniature fiber-optic spectrometer was used to record the emission spectrum of hydrogen emitted lines.

A mixture of water and hydrogen peroxide was taken in one liter water tank (H 2O2 was added for different amounts), and water was allowed to flow through the quartz tube. The two terminals of electrodes were connected at the output of the Neon transformer. The discharge was created inside capillary between two electrodes carrying flowing water and after discharge occurrence the electrical and spectral data was recorded. The electrical data taken by oscilloscope was evaluated by Matlab codes to find volt-ampere characteristic curves, electrical power of discharge pulses, frequency of discharge pulses and time difference between the occurrences of discharge pulses under different experimental conditions. Argon gas was injected at 0-500sccm injection rates through injection syringe and bubbles were created to reduce required power for generating discharge in flowing water long gap discharge. The emission spectrum was recorded to find the intensity of •OH radicals and other reactive species. The Gaussian distribution was applied on •OH emission spectrum peaks to determine the intensity of •OH radicals. The results were tabulated and presented graphically as well.

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Under different experimental conditions, electrical and spectral data was taken simultaneously, compared and presented.

3. Results and Discussion. Electrical results Fig. 3 represents typical Volt-Ampere characteristics of non-gas injected discharge for different amounts of hydrogen peroxide addition. The addition of hydrogen peroxide does not influence remarkably water conductivity; therefore the required breakdown voltage was almost same at without argon injection. Fig. 4 represents typical Volt-Ampere characteristics of argon injected discharge for various amounts of hydrogen peroxide addition.

Fig. 3. (Color online)Typical Volt-Ampere characteristics of non-gas injected discharge for different amounts of hydrogen peroxide addition (a) 0ml/l H2O2 (b) 0.05 ml/L H2O2 (c) 0.20 ml/L H2O2 (d) 0.35 ml/L H2O2.

Fig. 4. (Color online)Typical Volt-Ampere characteristics of 500 sccm Ar gas injected discharge for different amounts of hydrogen peroxide addition (a) 0ml/l H2O2 (b) 0.05 ml/L H2O2 (c) 0.20ml/L H2O2 (d) 0.35 ml/L H2O2.

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The addition of argon gas generated bubbles and gas channels drastically reduced the required breakdown voltage due to small dielectric constant of gas compared to pure water medium having high dielectric strength. When no gas was injected, after applying electrical fields across the electrodes, the jouleâ&#x20AC;&#x2122;s heating cause evaporation and micro bubbles generation that assist the discharge occurrence process. Moreover electron impact dissociation was another cause of electrical breakdown in water medium. In volt-ampere curves the sharp peaks represents the stage when evaporation, micro bubbles density and electron density due to electron impact dissociation was at maximum, after discharge occurrence the voltage drops and discharge current raises. Sharpe peaks represents quick breakdown process. After argon injection, bubbles and gas channels were generated, that participated mainly in creating low voltage breakdown. Discharge occurred within that bubbles and channels and inside water or liquid-gas interface. The generation of bubbles and gas channels, occurrence of discharge in theses gas channels and bubbles and in liquid-water interface was a quick and random process, so underwater discharge was of pulsating nature. The addition of argon gas generated bubbles and gas channels that drastically reduced the required breakdown voltage due to small dielectric constant of gas compared to pure water medium having high dielectric strength. Fig. 5 represents the reduction in breakdown voltage. Due to high dielectric constant of pure water medium, the required breakdown voltage was larger compared to the gas injected discharge, where gas channels and gas bubbles created low voltage breakdown. With increase in gas injection rate, breakdown voltage reduced enormously.

Fig. 6 represents the variation in electrical power of the discharge pulses, under different experimental conditions.

The addition of hydrogen peroxide had no remarkable effect on the breakdown voltage therefore, the electrical power of discharge pulses depends upon the medium of discharge i.e. pure water medium or argon injected medium. In case of argon injection due to rise in bubbles size and number density, and gas channels, the discharge strength increased and dimensionally more expanded discharge was obtained. This increased the strength of electrical power of discharge pulses. The electrical power of pulses becomes higher with increase in gas injection rates.

Fig. 5. (Color online). Variation in breakdown voltage Fig.6. (Color online). Variation in for different Ar injection rates and various hydrogen electrical power of discharge pulses for peroxide addition. different Ar injection rates and various hydrogen peroxide additions.

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Fig. 7 shows the variation of discharge pulse frequency. Due to argon injection high frequency of discharge pulses were obtained compared to the pure water discharge.

The increase in gas injection rate can cause high frequency discharge. The time difference between the occurrences of discharge pulses under different experimental conditions is shown in Fig. 8. After gas injection quick discharge pulses were obtained compared to the non-gas discharge.

The results presented that without gas injection high break down voltage was needed, while after argon injection that is non-reactive the chemical characteristics of discharge were not altered, but the physical characteristics varied. At higher gas injection rates the breakdown voltage reduced, electrical power of discharge pulses raised, frequency was increased while time difference between occurrences of discharge pulses reduced. The addition of hydrogen peroxide had no remarkable influence on the electrical characteristics of the discharge.

Fig. 7. (Color online). Variation in frequency of Fig. 8. (Color online). Average time difference discharge pulses for different Ar injection rates and between the occurrence of discharge pulses for various hydrogen peroxide additions. different Ar injection rates and various hydrogen peroxide additions.

Spectral results.

Fig. 9 (a-d) represents the emission spectrum results of the discharge. The emission spectrum was obtained after discharge occurrence by setting spectrometer wavelength range 250-1000 nm. The peaks of •OH radicals at 309 nm, Hα at 656 nm and reactive oxygen at 777 nm and 844 nm were observed more dominant among required reactive species peaks. The intensity of •OH radicals and other reactive species was too high when H2O2 was added. Fig. 9 (a) represents the emission spectrum without hydrogen peroxide addition. Without H2O2 addition only the splitting of water molecule by electrical field, ultra violet (UV) radiations and electron impact dissociation caused the generation of these reactive species. Fig. 9 (b) represents the emission spectrum at 0.05ml/L hydrogen peroxide addition. While Fig. 9 (c) for 0.20 ml/L hydrogen peroxide addition and Fig. 9 (d) for MMSE Journal. Open Access www.mmse.xyz

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0.35ml/L hydrogen peroxide addition. Comparison outcome that the •OH radicals, reactive hydrogen and reactive oxygen were quite high for the case of hydrogen peroxide addition.

Fig. 9. (Color online). Emission Spectrum of •OH radicals and other reactive species for different Ar injection rates and various hydrogen peroxide addition (a) 0ml/l H2O2 (b) 0.05 ml/L H2O2 (c) 0.20 ml/L H2O2 (d) 0.35 ml/L H2O2.

The results also demonstrated that with increase in argon gas injection since the strength of the discharge, electrical power of discharge pulses and frequency of discharge pulses was high, therefore MMSE Journal. Open Access www.mmse.xyz

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intensity and concentration of reactive species increased as well. The addition of hydrogen peroxide along with argon injection generated more reactive species. Fig. 10 represents the concentration of OH radicals by applying Gaussian distribution function on â&#x20AC;˘OH radicalâ&#x20AC;&#x2122;s emission peaks at 309nm [30]: â&#x2C6;&#x17E;

â&#x2C6;Ťâ&#x2C6;&#x2019;â&#x2C6;&#x17E; đ??šđ?&#x2018;&#x2013; (đ?&#x2018;Ľ) = đ??´đ?&#x2018;&#x2013; đ?&#x153;&#x17D;đ?&#x2018;&#x2013; â&#x2C6;&#x161;2đ?&#x153;&#x2039;

(1)

Increase in argon injection and hydrogen peroxide addition resulted in high concentration of OH radicalâ&#x20AC;&#x2122;s.

Fig. 10. (Color online). Variation in concentration of â&#x20AC;˘OH radicals for different Ar injection rates and various hydrogen peroxide addition.

At larger argon injection rates, since power of discharge pulses and frequency of discharge pulses was observed increasing therefore, higher dissociation rate of water molecules was obtained, that resulted in higher concentration of â&#x20AC;˘OH radicals.

Summary. Following conclusions have been made from the results: 1. The addition of hydrogen peroxide along with argon injection generated stronger plasma and high intensity of reactive species especially â&#x20AC;˘OH radicals.

2. Addition of hydrogen peroxide effected chemical properties and have no remarkable effect on electrical characteristics, especially conductivity of water.

3. Argon gas injection generated bubbles and gas channels that reduced the required breakdown voltage for long gap flowing water discharge.

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4. The frequency and electrical power of discharge pulses increased while time difference between the occurrence of discharge pulses and breakdown voltage was reduced at higher argon injection rates.

Acknowledgements This study was supported by Plasma Diagnostics Using Fast Thomson Scattering through the National Research Foundation of Korea (NRF) funded by Ministry of Education, Science and Technology (2014M1A7A1A03045383) and Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education, Science and Technology (20100020077).

References [1] Mayank Sahni and Bruce R. Locke , “Quantification of Hydroxyl Radicals Produced in Aqueous Phase Pulsed Electrical Discharge Reactors”, I & EC research, 2006. DOI: 10.1021/ie0601504 [2] Petr Lukes, Martin Clupek, Vaclav Babicky, Vaclav Janda and Pavel Sunka,” Generation of ozone by pulsed corona discharge over water surface in hybrid gas–liquid electrical discharge reactor”, J. Phys. D: Appl. Phys., 2005. DOI:10.108/0022-3727/38/3/010 [3] Robert J. Wandell, Bruce R. Locke,” Hydrogen Peroxide Generation in Low Power Pulsed Water Spray Plasma Reactors”, Ind. Eng. Chem. Res., 2014. DOI: 10.1021/ie402766t.

[4] Mitsuru Tahara, Masaaki Okubo,” Detection of Free Radicals Produced by a Pulsed Streamer Corona Discharge in Solution Using Electron Spin Resonance”, Proc. 2012 Joint Electrostatics Conference. DOI: 10.1016/j.elstat.2014.03.002.

[5] Vít Jirasek, Petr Lukes, Halyna Kozak, Anna Artemenko, Martin Clupek, Jan Cermak, Bohuslav Rezek, Alexander Kromka ,” Filamentation of diamond nanoparticles treated in underwater corona discharge”, 2015.DOI: 10.1039/C5RA23292A. [6] United States Environmental Protection Agency (USEPA) Report No. EPA-815-R-99-014(1999).

[7] S. Li, I. V. Timoshkin, M. Maclean, S. J. MacGregor, M. P. Wilson, M. J. Given, T. Wang and J. G. Anderson,” Fluorescence Detection of Hydroxyl Radicals in Water Produced by Atmospheric Pulsed Discharges”, IEEE Transactions on Dielectrics and electrical insulation, 2015.

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[8] Yong Yang, Young I. Cho, Alexander Friedman, Plasma Discharge in Liquid: Water Treatment and Applications”, by CRC Press, January 24, 2012.

[9] Xiujuan J. Dai, Cormac S. Corr, Sri B. Ponraj, Mohammad Maniruzzaman, Arun T. Ambujakshan, Zhiqiang Chen, Ladge Kviz, Robert Lovett, Gayathri D. Rajmohan, David R. de Celis, Marion L. Wright, Peter R. Lamb, Yakov E. Krasik, David B. Graves, William G. Graham, Riccardo d’Agostino, Xungai Wang,” Efficient and Selectable Production of Reactive Species Using a Nanosecond Pulsed Discharge in Gas Bubbles in Liquid”, Plasma Process. Polym. 2016.

DOI: 10.1002/ppap.201500156 [10] Genki Saito and Tomohiro Akiyama,” Nanomaterial Synthesis Using Plasma Generation in Liquid”, Journal of Nanomaterials, 2015. DOI:10.1155/2015/123696.

[11] W F L M Hoeben, E M van Veldhuizen, W R Rutgers and G M W Kroesen,” Gas phase corona discharges for oxidation of phenol in an aqueous solution”, Phys. D: Appl. Phys. Nov. 1999. DOI: 10.1088/0022-3727/32/24/103.

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[12] Joanna Pawłat, Dr. Thesis, “Electrical discharges in humid environments Generators, effects, application”, Lublin University of Technology, Poland.

[13] Masayuki Sato,” Environmental and biotechnological applications of high-voltage pulsed discharges in water”, Plasma Sources Sci. Technol. Oct. 2007.DOI; 10.1088/0963-0252/17/2/024021.

[14] M. Sato,” Degradation of organic contaminants in water by plasma”, International Journal of Plasma Environmental Science and Technology Vol. 03, No.1, MARCH 2009.

[15] P. Baroch, T. Takeda, M. Oda, N. Saito and O. Takai1,” Degradation of Bacteria Using Pulse Plasma Discharge in Liquid Medium”, 27th international power modulator symposium, 2006. DOI; 10.1109/MODSYM.2006.365289.

[16] Paul Y. Kim, Yoon-Sun Kim, Il Gyo Koo, Jae Chul Jung, Gon Jun Kim, Myeong Yeol Choi, Zengqi Yu, George J. Collins*,”Bacterial Inactivation of Wound Infection in a Human Skin Model

by Liquid-Phase Discharge Plasma”, PloS One, 2011. DOI:10.1371/journal.pone.0024104 [17] Vasil I. Parvulescu, Plasma Chemistry and catalysis in Gases and Liquids, Wiley-VCH, Germany, 2012. DOI: 10.1002/9783527649525.index.

[18] Magnus Ingelman-Sundbergz and Inger Johansso,” Mechanisms of Hydroxyl Radical Formation and Ethanol Oxidation by Ethanol-inducible and Other Forms of Rabbit Liver Microsomal Cytochromes”, 1984.DOI: 10.1124/mol.110.067652 [19] J. Kornev, N. Yavorovsky, S. Preis, M. Khaskelberg, U. Isaev & B.-N. Chen,” Generation of Active Oxidant Species by Pulsed Dielectric Barrier Discharge in Water-Air Mixtures”, Ozone: Science and Engineering, 2007. DOI: 10.1080/01919510600704957.

[20] Pankaj Attri , Yong Hee Kim , Dae Hoon Park , Ji Hoon Park , Young J. Hong , Han Sup Uhm , Kyoung-Nam Kim , Alexander Friedman & Eun Ha Choi,” Generation mechanism of hydroxyl radical species and its lifetime Prediction during the plasma-initiated ultraviolet (UV) photolysis”, Scientific Reports, 2015. DOI: 10.1038/srep09332.

[21] Svetlana Gasanova, Dr. Thesis, Aqueous-phase electrical discharges: generation, investigation and application for Organics removal from water, 2013.

[22] Shigeo Daito, Fumiyoshi Tochikubo and Tsuneo Watanabe,” Improvement of NOx Removal Efficiency Assisted by Aqueous-Phase Reaction in Corona Discharge”, Jpn. J. Appl. Phys. 2000.

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[23] Dip. Ing. Dr. Otto Zajic,” Disinfection of Drinking Water with Hydrogen Peroxide / Silver”, Fourth International Water Technology Conference IWTC 99, Alexandria, Egypt, 1999.

[24] Dong Nam Shin, Chul Woung Park, and Jae Won Hahn,” Detection of OH (A2S+) and O (1D) Emission Spectrum Generated in a Pulsed Corona Plasma”, Bull. Korean Chem. Soc., 2000.

[25] Weiwei He, Yitong Liu, Wayne G. Wamer, Jun-Jie Yin, “Electron spin resonance spectroscopy for the study of nanomaterial-mediated generation of reactive oxygen species”, journal of food and drug analysis, 2014. DOI:10.1016/j.jfda.2014.01.004.

[26] Michael S. Elovitz1 and Urs von Gunten,” Hydroxyl Radical/Ozone Ratios During Ozonation Processe”, 1998. DOI: 10.1080/01919519908547239.

[27] Ryo Ono, and Tetsuji Oda,” Measurement of Hydroxyl Radicals in an Atmospheric Pressure Discharge Plasma by Using Laser-Induced Fluorescence”, IEEE Transactions on Industry Applications, 2002. DOI:10.1109/28.821800.

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[28] Seiji Kanazawa, Hirokazu Kawano, Satoshi Watanabe, Takashi Furuki, Shuichi Akamine, Ryuta Ichiki, Toshikazu Ohkubo, Marek Kocik and Jerzy Mizeraczyk,” Observation of OH radicals produced by pulsed discharges on the Surface of a liquid”, Plasma Sources Sci. Technol. Vol. 20, No. 03, Jan. 2011. DOI:10.1088/0963-0252/20/3/034010.

[29] Petr Lukes, Martin Clupek, Vaclav Babicky, Vaclav Janda and Pavel Sunka,” Generation of ozone by pulsed corona discharge over water surface in hybrid gas–liquid electrical discharge reactor”, J. Phys. D: Appl. Phys. 2004.DOI: 10.1088/0022-3727/38/3/010.

[30] Muhammad Waqar Ahmed, Jong-Keun Yang, Young-Sun Mok and Heon Ju Lee∗,”Underwater Capillary Discharge with Air and Oxygen Addition”, Journal of the Korean Physical Society, 2014.

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Cite the paper Ahmed, M. W., Choi, S., Yang, J., Suresh, R., & Lee, H. J. (2016). Diagnostics of Argon Injected Hydrogen Peroxide Added High Frequency Underwater Capillary Discharge. Mechanics, Materials Science & Engineering Vol.6, 6. doi:10.13140/RG.2.2.16147.27684

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Optimizing the Parameters in Heat Treatment for Achieving High Hardness and Efficient Bending of Thin BS 2014 Aluminium Alloy Sheets Abirami Priyadarshini B. 1 – GTN Engineering India Ltd, Tamil Nadu, India DOI 10.13140/RG.2.2.10632.42242

Keywords: hardness measurement, aluminium alloys, bending, aging.

ABSTRACT. The present work targets in setting a standard heat treatment procedure for obtaining high hardness values of the order of 80 HRB in BS 2014 aluminium alloy sheets of 2mm thick commonly used in aerospace industries. A hardness range of 60HRB to 72HRB is possible in low thickness sheets as stated in the standard BS EN 485-2:2013.

Experiments were performed to achieve higher hardness values by controlling the heat treatment temperatures thereby understanding the ageing mechanism of the Al-Cu alloy to a wider extent. The validated process sequence in turn resulted in complications where bending of the sheets resulted in cracking. Further investigation was performed and it was found that the BS 2014 alloy has to be bent within two hours of solution annealing in order to have an efficient bending. The results showed that the natural ageing is so rapid in this alloy, which strengthens the material so quickly by the formation of CuAl2 precipitates, thereby, demanding the bending procedure to be performed before the growth of precipitates becomes dominant.

Introduction. The research and innovation at the aircraft industry focuses on reducing the weight of the aircraft for improving the efficiency, safety and performance. It also demands a positive step in environmental and economic factors thereby resulting in a favorable combination of high corrosion resistance, fatigue resistance, formability and strength coupled with low density[1]. Aluminium is one of the most important materials facing these challenges where it finds a wide variety of applications in the aerospace industry depending on their complexity and performance requirements.

With copper as the main alloying element, the 2xxx series of aluminium alloys are of significant interest possessing high strength to density ratio and thereby being used for structural applications in variety of fields such as the aviation and military sectors[2], [3].

Aluminium, being a sheet material, demands a predominant level of bending and forming. Among the 2xxx series, the BS 2024 is the most popular alloy used in the manufacturing of aircraft skins, cowls and structures[4]. Currently, the BS 2014 aluminium alloy is gathering attention due to its ability to achieve higher hardness and therefore it is used mainly for the interface beam assembly in aircraft structures and casings. These applications require a balance to be struck between the higher degree of hardness produced with the ability to bend and form the alloy. Work was performed in identifying the precipitates that are responsible for hardening the 2014 Al alloy where the precipitation of θ‫ ׳‬and θ‫ ״‬due to the presence of copper was of significant importance[5]. A good combination of mechanical properties can be achieved by controlling the precipitation mechanism where the elements such as Magnesium and Silicon are also responsible in improving the hardness of this alloy[6].

In the present work, the BS 2014 alloy was targeted to produce an increased hardness by optimizing the heat treatment factors thereby having an efficient control over the ageing mechanism. BS EN 4852:2013 states a maximum hardness of 72 HRB that could be achieved in a 2014 alloy[7]. However, this alloy has been studied widely for its ageing process where the precipitates are solely responsible in hardening the alloy resulting from the copper addition[2], [5]. Sadeler et al. studied the effect of T4 (solution treated and naturally aged) and T6 (solution treated and artificially aged) tempers where they concluded that the T6 temper has positive effects on the mechanical properties of this alloy[8].

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Various failures were reported from the aircraft industries where there were issues in the bending of this alloy even if the required hardness was achieved. This defect from the application side of industries demanded a more appropriate methodology in order to successfully bend these alloys considering the mechanism of hardening. Hence, work was also done in performing a successful 90o bend without the formation of cracks where an efficient heat treatment cycle was investigated and the process was optimized.

Experimental work. The BS 2014 aluminium alloy used in this investigation was procured at the T6 temper condition owing to its better properties compared to the T4 temper[8] and was subjected to a chemical analysis treatment which had the composition as shown in table 1. The alloy plates with the dimensions of 2mm x 65mm x 300mm were considered for the experiments. Also, the standard BS EN 485-2:2013 states a hardness value of 72 HBW for a thickness of 1.5-3 mm and hence a thickness of 2 mm was considered for performing a comparison in the achieved hardness.

Table 1. Chemical analysis of 2014 aluminium alloy.

Elements Specified values[9] (%) Observed values (%) Cu

3.8-5.0

4.051

Si

0.5-1.2

0.941

Fe

0.70 max

0.151

Mn

0.3-1.2

0.714

Mg

0.2-0.8

0.546

Cr

0.3 max

0.004

Zn

0.2 max

0.034

Ti

0.3 max

0.023

Al

Remainder

93.458

Initially seven sample plates were considered and prepared for undergoing the heat treatment trials.

The surface was cleaned to remove any foreign bodies, oxides and impurities if present. The 2014 aluminium sheets that satisfied the standard composition were cut into the required dimensions using laser-cutting process. The trials that were performed had the following sequence as depicted in table 2. It should be noted that the various trials performed had different process parameters where every trial sequence followed the standard heat treatment procedure for Aluminium alloys.

Various standards such as the AMS-H-6088 B, MIL-S-10699B, ASTM-B597-1992 and IS: 88601978 were considered in selecting the appropriate temperatures of heat treatment. These standards provided the code of practice for the heat treatment of aluminium alloys and the required conditions that are maintained throughout the process. These factors essentially include the salt composition, heat treatment baths and the procedure of heat treatment.

Furnace annealing was done at certain trials at a temperature of 410oC for two hours. The sample was then furnace cooled with a maximum cooling rate of 28oC per hour until the specimen reached 260oC which was then followed by air cooling. This is an important pre-step to solution annealing for the effective dissolution of precipitates in order to avoid cracking.

Solution annealing was done at 510oC for 35-40 minutes with water as the quenching medium. The main purpose of solution annealing was to achieve proper homogenization of the alloy to facilitate an efficient aging mechanism.

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Table 2. Sequence of trials for the heat treatment process.

Trial number

Process sequence

Trial 1

Laser cutting  Bending

Trial 2

Laser cutting  Solution annealing  Aging  Bending

Trial 3

Laser cutting  Solution annealing  Bending within 12 hours of solution annealing  Aging

Trial 4

Laser cutting  Solution annealing  Bending within 6 hours of solution annealing  Aging

Trial 5

Laser cutting  Furnace annealing  Solution annealing  Bending within 6 hours of solution annealing  Aging

Trial 6

Laser cutting  Furnace annealing  Solution annealing  Bending within 4 hours of solution annealing  Aging

Trial 7

Laser cutting  Furnace annealing  Solution annealing  Bending within 2 hours of solution annealing  Aging

The bending of the alloy sheets was performed using the Yawei bending machine with a capacity of 220 tons for various trials as mentioned in table 2 depending on the time after solution treatment. The sample under trial 7 (see table 3) that passed the bending test was approved and considered for studying the aging mechanism to achieve the required hardness.

As the heat treatment procedure for bending is now validated, eight other samples of 2014-T6 were subjected to the process of laser cutting, furnace annealing, solution annealing, straightening and bending within 2 hours of solution annealing according to trial 7. These samples were successfully bent and were subjected to the aging process with a temperature of 175oC. The soak time varied from 2 hours to 18 hours to see the variation in hardness produced depending on the temperature changes (see table 5). The hardness was measured using a standard Rockwell hardness tester at B scale.

Results and discussion Bending factors and parameters. The trials performed using different sequences of heat treatment yielded the following results after bending.

Table 3. Bending results of the trials performed Trial number

Process validation

Trial 1 -Trial 6

Failed due to the formation of cracks

Trial 7

A successful 90o bend performed without the formation of crack or irregularities.

The sample under the trial 1 methodology failed as expected, as there were no surface modifications performed. The bending of this sample resulted in the obvious formation of crack thereby leading to breakage. The sample under trial 2 had a complete heat treatment cycle following the theoretical reasoning where the solution annealing, quenching and ageing resulted in a significant formation of the precipitates. This sample also failed due to the formation of cracks, which is a result of the precipitation of CuAl2 along with various other insoluble compounds. The material has a tendency to MMSE Journal. Open Access www.mmse.xyz

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naturally age at a rapid rate thereby resulting in the growth of precipitates leading to a significant improvement in hardness. However, this results in increasing the brittleness of the material leading to the formation of cracks.

A different methodology was followed where bending of the material was performed within a certain specific time after solution annealing to assess the rate of growth of the precipitates that affects bending. In order to achieve this, a time limit has to be deduced for performing the bending before the precipitates start to age. Trails 3 to 7 were bent within a specified time after solution annealing with the time available for bending reduced from 12 hours to 2 hours where a successful bend was performed.

Fig. 1. Failure of the sheets.

Time is a critical parameter where the bending was totally dependent on the rate of growth of precipitates which in turn increases the hardness and brittleness of the thin sheet. Thus, for an efficient bending of the Aluminium 2014 alloy, the thin sheet has to be bent within two hours of solution annealing so that the material can be formed before the rate of growth of precipitates reaches the critical limit. The failure of the sheets is as depicted in Fig. 1 when undergoing the process from trial 1 to trial 6.

Parameters for high hardness. It was also noted that the aerospace industries require a certain minimum hardness that has to satisfy the component working conditions. After performing a successful bend, the target was laid on achieving a high hardness value for the bent sheet and this was satisfied by setting the proper ageing time, restricting the over ageing process. In order to establish this target, the hardness values were recorded as shown in table 5.

Table 4. Hardness obtained.

TRAIL NOS

TEMP

SOAK TIME

OBTAINED HARDNESS

1 2 3 4 5 6 7 8

175°C 175°C 175°C 175°C 175°C 175°C 175°C 175°C

2.0 Hrs 4.0 Hrs 6.0 Hrs 8.0 Hrs 10.0 Hrs 12.0 Hrs 14.0 Hrs 18.0 Hrs

48 HRB 49 HRB 59 HRB 68 HRB 78 HRB 67 HRB 65 HRB 63 HRB

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It was seen that the hardness value at 175oC for an ageing time of 10 hours yielded high hardness value of 78 HRB for the heat treatment parameters that was considered throughout this experiment.

Beyond this time, over ageing occurred where the hardness values dropped down. The sample surfaces were polished with the use of Silicon carbide paper and diamond paste and then etched by Kellerâ&#x20AC;&#x2122;s reagent for ten seconds. A microstructural analysis was performed using an optical microscope after every ageing cycle so that the growth of the precipitates can be efficiently related to its hardness as shown in Fig. 3. It can be seen that the growth of precipitates is so rapid where they reach a maximum hardness at the time of ten hours. The precipitates might act as a stress raiser where the crack propagation starts to initiate. The microstructures confirmed the rapid growth of precipitates, which supports the experimental hardness values that are obtained.

Precipitate formation. The evolution of hardness at the performed trials is directly proportional to the Cu-Al precipitates that are formed. Various research in the past confirms the precipitates to be CuAl2 phase where during the process of quenching, Cu is contained as a super saturated solid solution in the Aluminium rich phase at room temperature[8]. During the aging phase, the combination of copper and aluminium results in the formation of fine crystals of CuAl2 in the solution.

The increase in hardness values are a result of the formation of these crystals owing to the solubility of copper in aluminium. From the microstructures obtained in Fig. 3, it is evident that the CuAl2 phase is present by the difference in contrast that is produced. The main elements of the microstructure are characterized as dark, insoluble precipitates composed of complex compounds such as Fe, Mn, Al, Si and also the presence of particles of CuAl2 phase which are the white areas in a matrix of solid solution[8]. It can also be seen that the condition of reduced hardness obtained after the aging time of 11 hours produced a state of over aging as shown in Fig. 2. Hence, for achieving the maximum hardness, the region showing a maximum peak was utilized thereby fixing the aging time to 10 hours at a temperature of 170oC. These parameters yielded hardness values that were higher than the hardness mentioned in the standard[7]. It has to be noted that aging was performed after the samples were bent so that the required shape of the component can be progressed to the desired level of hardness without failure.

80

Hardness (HRB)

75 70 65 60 55 50

45 40 0

2

4

6

8

10

12

14

16

Aging time (Hours)

Fig. 2. Hardness vs Aging time.

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a

b

c

d

Fig. 3. Optical microscope images showing the growth of precipitates at (a) 2h (b) 8h (c) 10h (d) 14h taken at 200x.

Summary. The various samples of 2mm thin sheets of Aluminium 2014 alloy were optimized for heat treatment and bending parameters, where the following conclusions were drawn.

(1) The bending of aluminium 2014 alloy has to be performed within two hours of solution annealing, as the degree of natural aging in this alloy is significantly high. A successful bend can be performed within that time before the rapid growth of precipitates significantly increases the hardness resulting in cracking of the sheets during bending.

(2) The aging of the specimen after bending yields high hardness values than the standard hardness mentioned in BS EN 485-2:2013 when the process of aging is carried out for 10 hours at a temperature of 175oC. The furnace annealing proved to have a positive impact by being a successful pre-process to solution annealing. Proper control of temperature and environment resulted in a hardness of 78 HRB which is more than the value that was previously achieved.

The improved parameters for achieving high hardness value with successful bending will be highly desired by the aerospace industries where thin sheets of aluminium 2014 alloy plays a significant role. The scope of the future work lies in improving the aging conditions to perform a successful bend and achieve higher hardness values for thinner sheets of the order of 1mm that will significantly improve the efficiency of weight reduction in an aircraft. Work is also demanded in areas of fracture mechanics where the mode and mechanics of fracture in this alloy can be analyzed to a greater extent.

Acknowledgement. I take this opportunity to express my gratitude to Mr.K.B. Babu, CEO, GTN Engineering India Ltd for permitting me to undertake a project at his reputed industry. His constant support and guidance is highly appreciated. I also thank Mr. K. Vijayabaskar, Chief Operations officer, GTN Engineering India Ltd for his extended support throughout the course of this project.

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In addition, I gratefully acknowledge the assistance and contribution of my guide, Mr. Gowtham, Operations manager, GTN Engineering India Ltd, for his cordial support, valuable information and guidance, which helped me in completing this task through various stages.

References [1] A. Davidkov, R. H. Petrov, P. De Smet, B. Schepers, and L. A. I. Kestens, “Microstructure controlled bending response in AA6016 Al alloys”, Material Science Enineering. A, vol. 528, no. 22– 23, pp. 7068–7076, 2011, doi: 10.1016/j.msea.2011.05.055 [2] S. Wenner, J. Friis, C. D. Marioara, and R. Holmestad, “Precipitation in a mixed Al - Cu - Mg / Al - Zn - Mg alloy system,” Journal of Alloys and Compounds, vol. 684, pp. 195–200, 2016, doi: 10.1016/j.jallcom.2016.05.132 [3] I J Polmear, Light alloys: Metallurgy of the light metals. 1995.

[4] “Experimental Aircraft Info.” [Online]. Available: info/articles/aircraft-aluminum.php. [Accessed: 10-Aug-2016].

http://www.experimentalaircraft.

[5] M. Zeren, “Effect of copper and silicon content on mechanical properties in Al – Cu – Si – Mg alloys,” vol. 169, pp. 292–298, 2005.

[6] R. Ciach, S. Yu, and J. Kr, “Effect of ageing on the evolution of precipitates in AlSiCuMg alloys,” vol. 2344236, pp. 165–168, 1997.

[7] BS EN 485-2:2013, “Aluminium and aluminium alloys — Sheet , strip and plate —,” Part 2: Mechanical properties.

[8] R. Sadeler and M. Öcal, “Influence of Relative Slip on Fretting Fatigue Behaviour of 2014 Aluminium Alloy with the Age-Hardened Conditions T4 and T6,” vol. 18, no. 2, pp. 273–277, 2014.

[9] ASTM B209, “Standard Specification for Aluminum and Aluminum-Alloy Sheet and Plate ,” 2014.

Cite the paper Abirami Priyadarshini B. (2016). Optimizing the Parameters in Heat Treatment for Achieving High Hardness and Efficient Bending of Thin BS 2014 Aluminium Alloy Sheets. Mechanics, Materials Science & Engineering Vol.6, doi:10.13140/RG.2.2.10632.42242

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The Effects of Ukam (Cochlospermum Planchonii) Plant Fiber Variation on the Properties of Polyester Matrix Fiber Reinforced Composite Ihom, A.P.1, a, Dennis .O. Onah1 1 – Department of Mechanical Engineering, University of Uyo, Uyo, PMB 1017 Uyo-Nigeria a – ihomaondona@gmail.com, draondonaphilip@gmail.com DOI 10.13140/RG.2.2.35903.92320

Keywords: Cochlospermum planchonii fiber, polyester, composite; matrix, reinforcement, properties.

ABSTRACT. The work titled ‘The Effects of Ukam (Cochlospermum Planchonii) Plant Fiber Variation on the Properties of Polyester Matrix Fiber Reinforced Composite has been undertaken. The work delved into the production of the composites, this was accomplished by the separate activities of fiber production, mould production and matrix preparation. The fiber was from cochlospermum planchonii (Ukam) plant and the matrix was from polyester. Samples of the produced composites were used in preparing standard test specimens, which were subjected to various tests in order to characterize the composite. In all the properties tested, it was observed that Ukam fiber content had a major role in determining the properties of the composite. The variation of the fiber weight fraction affected all the tested properties of the composite. The results showed that to produce a polyester composite with optimized properties using Ukam fiber, which is biodegradable, the fiber content should be 40%.

Introduction. There are very many situations in engineering where no single material will be suitable to meet a particular design requirement. However, two materials in combination may possess a feasible solution to the materials selection problem. The principle of composite materials is not new.

The use of straw in the manufacture of dried mud bricks, and the use of hair and other fibers date back to ancient civilizations [1]. A typical composite material is a system of materials comprising of two or more materials mixed and bonded together. For example, concrete is made up of cement, sand, stones and water. If the composition occurs on a microscopic scale (molecular level), the new material is called an alloy for metals or a polymer for plastics [15]. Types of composites are fiber reinforced composites, metal matrix composites, polymer matrix composites, and ceramic matrix composites [16].

Generally, a composite material is composed of reinforcements. These reinforcements are generally classified into two; synthetic and natural. Synthetic reinforcements include glass, carbon and aramid fibers. Mass production of glass strands was discovered in 1932 when Games Slayter, a researcher at Owens-Illinois accidentally directed a jet of compressed air at a stream of molten glass and produced fibers [16].

Nowadays, natural fibers are an interesting option for the most widely applied fibers in the composite technology. Examples of Natural fibers are jute, hemp, flax, kenaf, coconut, Ukam, sisal, and banana, pineapple fibers from the leaf; cotton and kapok from seed; coir and coconut from the fruit; oil palm and bamboo fibers. The components of natural fibers are cellulose, hemicellulose, lignin, pectin, waxes and water soluble substances. The cellulose, hemicellulose and lignin are the basic components of natural fibers, governing the physical properties of the fibers. In order to fully utilize the natural fibers, understanding their physical and mechanical properties is vital. A unique characteristic of natural fibers reinforced plastic is dependent on the variations in the characteristics and amount of these components, as well as difference in its cellular structure. Therefore, to use natural fibers to its best advantages and most effectively in automotive and industrial application, physical and mechanical properties of natural fibers composite must be considered [2, 3, 4, 5].

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Fibers are made to exist in many forms such as; chopped strand mat, chopped strands, woven roving, surface tissues and continuous strand mat. The matrix holds the reinforcements to form the desired shape of a composite. In the case of polymer based composites, matrix materials are resins. A suitable resin for combining “fiber glass" with a plastic to produce a composite material was first developed in 1936 by du Pont [6]. The matrix binds the fibres together, protect them against damages and transmit load from fibre to fiber. Examples of Matrix materials are Polyester, Epoxy, Vinyl ester, etc.

Polyester is a term often defined as “long-chain polymers chemically composed of at least 85% by weight of an ester, a dihydric alcohol and a terephthalic acid. It is a category of polymers that contain the ester functional group in their main chain. Polyester also refers to the various polymers in which the backbones are formed by the “esterification condensation of polyfunctional alcohols and acids” [1, 16, 7, 8 9].

The fabrication and properties of composites are strongly influenced by the proportions and properties of the matrix and the reinforcement. The impact strength of fiber reinforced composite increases as the fibre volume fraction increases [23]. The strength also improves with increase in fibre volume fractions, fibre treatment, fibre length, fibre orientation and the addition of additives [10, 11, 13].

Rasheed et al [20] found that the tensile strength of the composite increases with the fiber volume fraction up to 40% and after which it decreases slightly. Experimental analysis of coir-fiber reinforced polymer composite materials have shown that the mechanical properties of the composite are dependent on the content or the volume fraction of fibers [14]. Based on the experiments, it was found that the tensile strength and the young’s modulus decreased with the increasing fiber volume after a particular value of fiber content. It was also seen that the failure strain increases with the increase in the fiber content. The fiber length is another parameter affecting the mechanical properties of the composite. The fiber length also has an impact on the tensile property, flexural property and impact strength of the composite. Homogeneity is an important characteristic that determines the extent to which a representative volume of the material may differ in physical and mechanical properties from the average properties of the material. The amount of reinforcement that can be incorporated in a given matrix is limited by a number of factors. For example with particulate reinforced metals the reinforcement content is usually kept to less than 40 vol. % (0.4 volume fraction) because of processing difficulties and increasing brittleness at higher contents. On the other hand, the processing methods for fiber reinforced polymers are capable of producing composites with a high proportion of fibers, and the upper limit of about 70 vol. % (0.7 volume fraction) is set by the need to avoid fiberfiber contact which results in fiber damage [12, 16, 17, 18, 19, 21, 22].

Cochlospermum Planchonii known locally as Ukam plants grow in savannah and forest savannah mosaic in West Africa. The plant is a perennial plant with a woody subterranean rootstock, from which, in the rainy season, annual leafy shoots growing around 2 metres tall are produced. The height of the plant depends on the particular habitat and the age. The people of the area where these plants are found use their fibers as sponge and also to reinforce clay with which they produced intricate earthen pots and silos [2, 3, 4, 5]. The objective of this research work is to investigate the effect of Ukam fiber variation on the properties of the composites produced using the fibers.

Materials and Method.

Materials.

The materials used for this work were: polyester resin, Ukam fiber (Cochlospermum Planchonii fibers), sodium hydroxide, acetic acid, releasing agent, methyl ethyl ketone peroxide, calcium carbonate, cobalt naphthenate and water.

Equipment.

The equipment used for the study were as follows: rule, digital weighing balace, Moulds, Tensile Strength Tester, Scanning Electron Microscope, Universal Testing Machine, Flexural Testing Machine, Compression Testing Machine, Rockwell – B scale, and Impact Testing Machine MMSE Journal. Open Access www.mmse.xyz

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Method.

The work commenced with the production of the composite using polyester as the matrix and cochlospermum planchonii as the fibers. Cut stems of the plants were soaked inside flowing water for thirty days. This enhanced the decay and removal of the thin back of the plant leaving behind, white fibrous stems (see Plate I). The fibers were removed from the fibrous stems with hands (see Plate II). The density, tensile strength, SEM analysis, and water absorption characteristics of the produced fibers were all determined. The produced fibers were then used in the development of polyester composite using various weight fractions of the fiber which were randomly oriented in the matrix (see table 1). The produced composites were allowed to cure for 24 hours before the commencement of their processing into standard test specimens which were used for characterization of the produced composites. Plates III-VIII show some equipment, the developed composites, and some specimens which were used for the characterization of the produced composites.

Plate I: Cut Stems of Cochlospermum Planchonii Fibers

Plate II: Treated and Dried Cochlospermum Planchonii Fibers

Plate III: Scanning Electron Microscope

Plate IV: Developed Samples of Cochlospermum Planchonii Reinforced Polyester Composites

Plate V: Test Specimens of Cochlospermum Plate VI: Universal Strength Testing Machine Planconii Reinforced Composite for tensile test (Testometric)

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Plate VII: Test Specimen of Cochlospermum Plate VIII: Flexural Test Machine Planconii Fiber Reinforced Polyester Composite for Flexural Test

Results and Discussion.

Results.

The results of the work are as presented in Fig.s 1-10 and Plates IX-X

Fig. 1. Ultimate tensile strength variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 2. Extension variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

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Fig. 3. Flexural strength variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 4. Deflection variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 5. Compressive strength variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

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Fig. 6. Maximum load variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 7. Hardness in HRB of the composites with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 8. Toughness variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

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Fig. 9. Density variation with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Fig. 10. Water absorption capacity with % reinforcement of cochlospermum planchonii (Ukam) fiber in polyester composites.

Plate IX: Scanning electron micrograph of 30% fiber content, the clusters in the plate show how the fibers were arranged.

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Plate X: Scanning electron micrograph of 40% fiber content. The arrangement of the fibers in the polyester show higher volume of fiber than in 30% fiber content above.

Discussion.

Figs. 1-6 show the variation of ultimate tensile strength, extension, flexural strength, deflection, compressive strength, and maximum load properties with % reinforcement of cochlospermum planchonii (Ukam) fibers in polyester ccomposites. All the properties of the developed composites show a steady increase as the Ukam fiber content was increased. This is depicted by the curve which rises steadily, peaking at 40% Ukam fiber content and then falling gradually after 40% Ukam fiber content. All the six properties of the composite are optimized at 40% reinforcement in the polyester composite.

Figs. 7-10 show the variation of hardness, toughness, density, and water absorption capacity properties with % reinforcement of cochlospermum planchonii (Ukam) fibers in polyester composite.

The plot of the hardness against % reinforcement in polyester show the hardness of the composite decreasing as the fiber reinforcement was increased. The hardness property has an inverse relationship with % fiber reinforcement and this is depicted by the curve which falls gradually from left to right. Fig. 8 shows that the toughness of the composites has a direct proportion relationship with % Ukam fiber content. As the fiber content was increased, the toughness property kept increasing up to 40% fiber content, not much significant increase was noticed after 40% fiber content. The same trend is seen in fig. 9. The only difference is that significant reduction in the density property after 40% fiber content can be sighted after 52% Ukam fiber content down. Fig. 10 shows that water absorption capacity property has a direct proportionality relationship with % fiber content. As the fiber content is increasing, so is the water absorption capacity increasing. This is depicted by the continuous rising of the curve from left to right.

Too much water in the composite has degrading effects on the composite which includes swelling and weakening of the strength property. This may call for the selection of an optimum fiber content which will optimize other properties and minimize the amount of Ukam fiber in the composite and from the above results 40% fiber content is the best. The result of the work has shown that the variation of Ukam fiber content in polyester has a major influence on the tested properties of the composite. This is in agreement with previous work by several authors [20, 23, 10, 11, 13].

Matthews and Rawlings [12] argued that the fabrication and properties of composites are strongly influenced by the proportions and properties of the matrix and the reinforcement. Other properties which may significantly affect the properties of a composite are the shape, size, orientation, and distribution of the reinforcement and various features of the matrix such as grain size for polycrystalline matrices. These, together with volume fraction, constitute what is called the microstructure of the composite. It should be noted that even for properties which are microstructure dependent, and which do not obey the law of mixtures, the volume fraction still plays a major role in MMSE Journal. Open Access www.mmse.xyz

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determining properties. The volume fraction is generally regarded as the single most important parameter influencing the composite properties. Also, it is an easily controllable manufacturing variable by which the properties of a composite may be altered to set the application [20, 23, 10, 11].

Plates xi-xii show the Scanning Electron Microscope (SEM) micrograph of the produced polyester composite with 30% and 40% Ukam fiber content. Looking at the two plates it can be seen that the plate with 40% Ukam fiber has more fiber content than the one with 30% fiber content and the distribution is more uniform in the plate with 40% fiber content. Homogeneity in the distribution of the fibers in composite promotes uniform properties. According to these researchers [12, 16, 17, 18, 19, 21, 22], Homogeneity is an important characteristic that determines the extent to which a representative volume of the material may differ in physical and mechanical properties from the average properties of the material. Non- uniformity of the system should be avoided as much as possible because it reduces those properties that are governed by the weakest part of the composite.

The plate also shows the orientation of the fibers. The orientation of the reinforcement within the matrix affects the isotropy of the system [12, 16]. The microstructure as earlier mentioned contributes to the overall properties of the composite.

Summary. The work titled ‘The Effects of Ukam (Cochlospermum Planchonii) Plant Fiber Variation on the Properties of Polyester Matrix Fiber Reinforced Composite‘ has been undertaken and the following conclusions drawn from the work: 1. A set of polyester composites were produced by varying the reinforcement with cochlospermum planchonii (Ukam) fiber which is a natural fiber and biodegradable. This makes it environmentally friendly.

2. The work has succeeded in proving that Ukam fiber content in polyester plays a major role in determining the properties of the developed polyester composite reinforced with Ukam fibers.

3. The work has established that using Ukam fiber to produce polyester composite, the amount of Ukam fiber to use in order to optimize the properties of the produced composite is 40% fiber, i.e. a volume fraction of 60 vol.% matrix (polyester) – 40 vol.% reinforcement (Ukam fiber) References [1] John, 1972. Introduction to Engineering materials. London: Max Pub., pp 234-250.

[2] Balarami, R. (2013). Mechanical performance of green coconut fiber/HDPE Composites. Int. Journal of Engineering Research and Applications 3: 1262-1270.

[3] Bascom, W.D. (1987). Fiber sizing. Engineered Materials Handbook – Volume 1: Composites.

Metals Park, OH: American Society of Metals, pp 34 – 35.

[4] Benjamin, C. and Tobias (1990) Fabrication and Performance of Natural Fiber-Reinforced Composite Material. 35th International SAMPLE Symposium and Exhibition, Anaheim, pp. 970978.

[5] Burkill, H.M., (1985). The useful plants of West Tropical Africa. United Kingdom: Families A– D. Royal Botanic Gardens, Kew Richmond, 2nd Edition, Vol. 1, pp. 960.

[6] Ferguson, A.: In Onah, D.O. (2016) Development and Characterisation of Cochlospermum Planchonii, Fiber Reinforced Polyester Composite, M.Eng Degree Dissertation submitted to the Department of Mechanical and Aerospace Engineering, University of Uyo, Uyo- Nigeria [7] Giuseppe, C., Alberta, G. Latteri, and Gianluca, C. (2011). Composites Based on Natural Fiber Fabrics, Woven Fabric Engineering, Polona Donik Dubrovski (Ed.), ISBN:978-953-307-194-7, pp.

317 – 342 [8] Hull, D and Clyne, T.W. (1996) An Introduction to Composite Materials (2nd edition)Cambridge: University Press, p 65.

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[9] Irawan, A.P., Soemardi, T.P., Widjajalaksmi, K and Reksoprodjo, A.H.S. (2011) Tensile and flexural strength of ramie fiber reinforced epoxy composites for socket prosthesis application.

International Journal of Mechanical and Materials Engineering, Vol. 6 (1): 46-50.

[10] Ku, H., Wang, H., Pattarachaiyakoop, N. and Trada, M. (2009) A review on the tensile properties of natural fiber reinforced polymer composites. Journal of Reinforced Plastics and Composites 28:1169-1189.

[11] Kumar, D. (2014) Mechanical characterization of treated bamboo natural fiber composite.

International Journal of Advanced Mechanical Engineering. Vol. 4( 5):551-556.

[12] Matthews, F.L. and Rawlings, R. D. (2005) Composite Materials: Engineering and Science, 5th Edition London: WoodHead Publishing Limited, pp 1- 300.

[13] Munikenche, T. G., Naidu, A.C.B., Rajput, C. (1999). Some mechanical properties of untreated jute fabric-reinforced polyester composites. Science Direct Journals 30:227 -284.

[14] Naveen, P. N. E. and Yasaswi, M. (2013) Experimental analysis of coir-fiber reinforced polymer composite materials. International Journal Of Mechanical Engineering & Robotics Research, 2( 1): 10-18.

[15] Nick, I. and Mark, J. (2004). Low environ-mental impact polymers. Int. Automotive Research Center, University of Warwick, Vol. 2, No. 30, pp. 99-108.

[16] Onah, D.O. (2016) Development and Characterisation of Cochlospermum Planchonii Fiber Reinforced Polyester Composite, M.Eng Degree Dissertation submitted to the Department of Mechanical and Aerospace Engineering, University of Uyo, Uyo- Nigeria.

[17] Onuegbu, T.U., Umoh, E.T. and Okoroh, N.C. (2013) Tensile behaviour and hardness of coconut fiber-ortho unsaturated polyester composites. Global Journal of Science Frontier Research Chemistry, Vol.13( 1): 1.

[18] Olusegun, D. S., Agbo, S. and Adekanye, T.A. (2012) Assessing mechanical properties of natural fiber reinforced composites for engineering application. Journal of Minerals and Materials Characterization and Engineering 11: 780-784.

[19] Osman, E., Vakhguelt, A., Sbarski, I. and Mutasher, S. (2012) Water absorption behavior and its effect on the mechanical properties of kenaf natural fiber unsaturated polyester composites. 18th International Conference on Composite Materials International Conference on Composite Materials.

2pp.

[20] Rasheed, H. M. M.A., Islam, M. A. and Rizvi, F. B. (2006) Effects of process parameters on tensile strength of jute fiber reinforced thermoplastic composites. Journal of Naval Architecture and Marine Engineering ( 3) 1: 105 â&#x20AC;&#x201C; 117.

[21] Senthiil, P.V. and Sirshti, A. (2014) Studies on Material and Mechanical Properties of Natural Fiber Reinforced Composites. The International Journal of Engineering and Science. Volume 3, pp 18-27.

[22] Tuttle, M. (2004) Introduction. In: Structural analysis of Polymeric Composite Materials University of Washington, USA, ISBN 0-8247-4717-8, pp. 1-40.

[23] Ugoamadi, C.C. (2011) factors that improve the impact responses of Ukam plant fiber reinforced composite. Nigerian Journal of Technology, Vol. 30( 3) : 111-117.

Cite the paper Ihom A.P. & Dennis O. Onah. (2016). The Effects of Ukam (Cochlospermum Planchonii) Plant Fiber Variation on the Properties of Polyester Matrix Fiber Reinforced Composite. Mechanics, Materials Science & EngineeringVol.6, doi: 10.13140/RG.2.2.35903.923202 MMSE Journal. Open Access www.mmse.xyz

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Effect of Alternating Bending and Texture on Anisotropic Damage and Mechanical Properties of Stainless Steel Sheets V.V. Usov1, N.M. Shkatulyak1, E.A. Dragomeretskaya1, E.S. Savchuk1, D.V. Bargan1, G.V. Daskalytsa1 1 – South Ukrainian National Pedagogical University named after K.D. Ushinsky, Odessa, Ukraine DOI 10.13140/RG.2.2.35491.04640 Keywords: alternating bending, texture, Young’s modulus, anisotropy, damage, stainless steel.

ABSTRACT. Effect of alternating bending and the crystallographic texture on the anisotropy of damage and mechanical properties of stainless steel sheets X5CrNi18-10 at subsequent uniaxial tensile tests were studied. The symmetric tensor of damage of the second order D was used for the analysis of anisotropy damage of sheet material. The only one non-zero component of this tensor D at uniaxial tensile was determined by the defect of the Young's modulus from the mechanical test data. The value of D was found on relation D  1  E E0 . Here E0 and E are Young’s modules of the undamaged and tested material, respectively. It was established the anisotropy of the damage and mechanical properties of steel sheets at uniaxial tensile tests of initial sheet as well of sheets after alternating bending. This anisotropy is caused by the texture that is formed in sheets of investigated steel as was showed by correlation analysis.

Introduction. Stainless steels are widely used in various fields of engineering: architecture, construction, transport engineering, medicine, food industry, energy [1]. An important role is played stainless steel in the petroleum refining [2]. This steel is practically irreplaceable in high-temperature processes, when the raw material is heated to 600°C [2]. In this regard, the operational forecasting of durability of materials refining industry remains an important problem in relation to the requirements of increasing the depth and quality of oil processing. In the operating conditions under the influence of alternating prolonged loading of equipment inevitably arise damage or irregularities of its working capacity even in the absence of defects in workmanship and compliance in the operation of regulatory requirements. Over last years has been proposed the calculation model of resource estimation of coilpipes furnace of pyrolysis with considering forming quasi-multilayer shell that is formed due the diffusion of carbon in surface layers of steel pipes 20CrNi23-18 at the furnace operation [3]. Articles [4-6] are focused on corrosion and protection from it. The above review shows that proposed methods of predicting damage of structural materials and residual life of process equipment usually are based on monitoring of mechanical properties, metal thickness, morphology and distribution of structural components and structural defects in the steel. It is known that final properties of steel and products depends on many factors such as the chemical composition and its distribution in thickness, metal structure (average size of grains and sub-grains deviousness their borders) [7], crystallographic texture [8], operating temperature, duration of thermal action etc. The emergence during the operation of equipment large number of different defects indicates that is implemented several mechanisms of damage accumulation in metals. In the same time certain characteristics such as crystallographic texture, damage, which could be used for monitoring of the structural condition of the steel rarely taken into account. The impact of above characteristics on corrosion [9] of structural materials requires a more detailed study in terms of degradation and forecasting of metal state. Not investigated also effect of alternating bending (AB) on the anisotropy of damage accumulation in the sheet metal under uniaxial tension. The alternating bending is usually applied before using of roll metal for the straightening of sheets, reducing residual stresses and imparting to the metal of optimal flat characteristics. During the straightening of the metal in him arise and accumulate uncontrollable micro defects, such as micro cracks, micro pores that are found already at tensile on 3-10% [10]. The MMSE Journal. Open Access www.mmse.xyz

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occurrence and accumulation of micro defects indirectly reflected in changing of material properties, in particular on Young’s modulus defect, which can be used to measure the accumulation of damage in the metal [11].

In this paper, we investigate the effect of alternating bending and the crystallographic texture on the anisotropy of damage and mechanical properties of stainless steel sheets X5CrNi18-10 in the process of subsequent uniaxial tensile tests.

Experimental Procedure. As initial material for investigation were used sheets of stainless steel X5CrNi18-10 of 1 mm thickness in delivery conditions after recrystallization annealing. Sheets measuring 100 × 100 mm have been subjected to the alternating bending (AB) by means of roller diameter of 50 mm in the rolling direction (RD). The speed of movement of the metal during bending was about 150 mm / s, which corresponds to a strain rate of ~ 10-2 s-1. From initial sheets and sheets after bending on 0,5; 1; 3 and 5 cycles were cut three batches of samples for mechanical test in the RD, in the diagonal direction (DD is direction, that is deflected from the RD on 45º), and transverse direction (TD), and also samples for study of texture. Testing machine Zwick Z250 / SN5A with power sensor on 20 kN at room temperature was used for mechanical tests on a tensile of samples cut in the RD, DD, and TD. Samples for mechanical testing have had total length of 90 mm, the width of working part was of 12.5 mm. Values of mechanical properties were found by averaging the test results of at least three specimens in each direction.

The X-ray method [12] with the construction of inverse pole Fig.s (IPF) was used for investigation of the crystallographic texture. On the diffractometer DRON-3m in the filtered Mo Ka radiation was performed the theta-2-theta scanning of sample without texture, as well as of samples after corresponding cycles of the AB. The scanning carried out from two opposed surfaces of sheets, as well and in the RD. These data were used for the construction of IPF ND and IPF RD, respectively.

Samples were chemically polished to a depth of 0.1 mm for removing distorted surface layer before texture investigation. Sample without texture was prepared from the fine powder of studied steel after recrystallization. Composite samples in the form of glued each other strips wide of 3 mm cut perpendicular to RD were prepared for the texture investigation in the RD. The microscope Axioplan 2 of the firm KARL ZEISS was used for examine of the metallographic structure from end surfaces of samples cut in the RD and TD. A symmetric damage D tensor of the second order [13, 14] was used for the analysis of the anisotropy of sheet material damage. Only one nonzero component of the tensor D exists for the case of uniaxial stress. This nonzero component D is determined by the formula [13, 14]: D  1  E E0 .

(1)

where E 0 and E – are elastic modules of intact material and the current modulus determined at uniaxial tensile tests, respectively.

Results and discussion. On fig. 1 are shown mechanical properties and damage after different cycle’s number of the AB. On fig. 2 are presented corresponding IPF’s. It is seen that IPF of the initial sample (Fig. 1, a, b) are typical for the rolling texture of FCC metals. Texture undergoes marked changes after various stages of the AB (Fig. 2, c-l). Fig. 3 shows appropriate microstructure. The presence of twins is seen in the initial sample (Fig. 3, a, b). The tendency to increase amounts of twins is traced with increasing number of the AB cycles (Fig. 3, c-l). Therefore, one should expect the development of twins orientations during the alternating bending, since the role of twinning is amplified at deformation of materials with low stacking fault energy [15], to which belongs the investigated here steel.

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Anisotropy of mechanical properties and damage D take place (fig. 1). Anisotropy coefficient k that was determined by relation (1) decreases with increasing number cycles of the AB.

k  Fmax  Fmin  / Fmin   100% .

(2)

where F – is the appropriate property.

The minimum value of k is observed after 5 cycles of the AB (Fig. 1). The character of the tensile strength anisotropy does not change with increasing number cycles of AB. In all cases, the ultimate strength in the RD has a higher value than in the TD, and in the DD has intermediate value. There is probably manifested an effect of the mechanical texture, namely, the preferential elongation of the grains in the RD. Coefficient anisotropy of the tensile strength initially increases with increasing number of cycles, taking the value of 5.0 % in the initial sheet; 2.9 % after 0.5 cycle; 6.8% after one cycle, and then decreases to 4.1% after 5 cycles.

Yield strength 0.2 in RD exceeds its value in the TD in the initial sample. Coefficient of anisotropy k has made 3.6%. Anisotropy character changed after 0.5 cycle of AB. Yield strength in RD is smaller than in TD, and in a diagonal direction has intermediate value. Anisotropy ratio had decreased. Its value was 2.9%. A similar pattern of anisotropy persists after the one AB cycle. At the same time the anisotropy coefficient grew to 6.4%. Anisotropy character of yield stress is similar to him in the initial sample, and the anisotropy ratio decreased to 1.6% after 5 AB cycles.

Absolute values of yield strength and tensile strength of the studied steel also are increased with increasing number of AB cycles, and reach a maximum after 1 cycle of AB. Absolute values of the strength properties of the investigated steel are decreased with further increase in the number of AB cycles. Elongation shows an opposite tendency with respect to the tensile strength (Fig.1).

Fig. 1. Dependence of tensile strength, proof strength, uniform elongation and damage D on the number of AB cycles.

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Analysis of initial sample IPF (Fig. 1, a, b) showed that its texture consists of two limited axial components. The first component with the axis <110> parallel to the ND extends from {011} <100> up to {011} <112>. The second component may be characterized by an axis of <110> inclined toward ND on approximately 60Âş. It extends from ~ {112} <111> through {135} <211> up to {011} <112>.

Fig. 2. Experimental IPF of the studied steel; (a, b) are the initial state, respectively IPF (ND) and IPF (RD); (c - l) are IPF (ND) after the alternating bending: (c, d); (e, f); (g, h); (k, l) are after 0.5; 1; 3; and 5 cycles of AB, respectively.

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The development of these two limited axial components is in the agreement with the Taylor prediction model on the base of the normal octahedral sliding [16]. In addition, there are the twinned orientations {113} <211> that were formed probably during the annealing [16].

Fig. 3. Microstructure of steel sheets: (a, b) are corresponded to the initial state; on (c - l) are shown states after the AB: (c, d), (e, f), (g, h), and (k, l) are shown microstructure after 0.5, 1, 3, and 5 cycles of AB, respectively. a, c e g k are filmed in the cross section perpendicular to the RD; b d f h l are filmed in a cross section perpendicular to the TD. MMSE Journal. Open Access www.mmse.xyz

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Texture is undergoing significant changes after different stages of the AB. The following deformation model of samples was basis of the texture changes interpretation. Grains of metal in layers on the convex sheet side are exposed to the action of tensile stresses at bending of the sample in one direction (0.25 cycles). Meanwhile, on the concave side occur compressive stresses. In the strip are initiated shear deformations as a result of the action of opposite sign stresses. Direction of acting stresses is reversed when the strip bends in the opposite direction. Thus, in the metal strip arise alternating shear deformations, which lead to the formation of the shear texture components during the AB. In FCC metals are formed following components of shear texture: A - {111} <hkl>; B - {hkl} <110>; C - {001} <110>. Orientation {hkl} <uvw>, listed here indicate that the plane {hkl} coincide with the shear plane, and the direction <uvw> coincide with the shear direction [16].

Component B of shear texture formed after 0.5 cycle of the AB in the sample at one side of the sample [16]. The formation of shear bands in the rotated twin-matrix regions changes the orientations {332} <113> and {111} <110> near to {011} <100> and {011} <112> positions respectively in metals and alloys with the low stacking fault energy (SFE) [17]. On the corresponding IPF (Fig. 2, c) pole density <110> increased to 2.38 while in the initial sample it was 1.81 (Fig. 2, a). On the opposite side of the sample also take place twins orientations (Fig. 2, c).

Component C of shear texture is formed in the sample after 1 cycle of the AB (Fig. 2, e, f). At the same time on the opposite side of the same sample (Fig. 6, f) orientations of the initial sample are observed (Fig. 2, a).

Texture that is similar to the texture of the initial sample (Fig. 1, a) was formed at one side of the sample after three cycles of the AB (Fig. 2, g). Sufficiently intense component C of shear texture (Fig. 2, h) is present on the opposite side of this same sample.

Texture on the one sample side after five cycles of the AB is characterized by orientations of C shear texture and by orientations of twins {113} <211> (Fig. 2, k). The area of increased pole density on the corresponding IPF is greatly expanded in comparison with the other samples probably due to the twinning [18]. The texture of same sample on the opposite side is characterized by orientations of twins (Fig. l, 2). In general, scattering of the texture had increased when considering of both surfaces of the sheet after 5 cycles of the AB (Fig. 2, k, l), as compared with the initial state of the sheet in Fig. 1, a.

The above described anisotropy of the proof strength corresponds to the texture formed in samples.

In the IPF (RD) of the initial sample (Fig. 1, b) there is a high pole density of <111>. This means that there is a significant volume fraction of crystals axis <111> of which coincides with the RD. In this case <112> and <110> crystals axis are oriented along the TD. The crystals that have axes <111> oriented along the applied stresses are characterized by of high flow stresses in comparison with other crystal orientations [18].

Number of crystals with axes <111> oriented along the RD is decreases, and in the TD is increases with increasing of AB cycles number from the 0.5 to 1 inclusive, due to the increasing of shear texture.

Consequently, the proof strength in TD is becoming greater than in the RD. Increasing of the AB cycles number up to 5 leads not only to the development of shear texture components but also to the strengthening of the twinning, as it’s mentioned above. This promotes again to the formation of texture components similar to initial orientations but with more significant scattering. Initial character of proof stress anisotropy is restored, but its absolute value is decreased.

Fig. 2 shows that orientations of <110> have the highest values of the pole density on IPF ND. Significant correlations of averaged through the direction of sheets at uniaxial tensile tests of values av av of ultimate strength  av m , proof stress  0.2 , relative uniform elongation l / l uni , and damages D av. with values of pole density of <110>, averaged on both sides of sheets P110av. take place. The

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corresponding regression equations and approximations reliability coefficients R 2 are represented by relations 2 2  av m  292.6P110av.   884.6 P110av.  12.9; R  0.94

(3)

 0av.2  463.2P110av. 2  1362.0P110av.  667.8; R 2  0.81

(4)

av l / l uni  32.5P110av. 2  99.6P110av.  131.1; R 2  0.66

(5)

Dav.  0.43P110av. 2  1.43P110av.   0.73; R 2  0.61

(6)

Summary. Effect of alternating bending and the crystallographic texture on the anisotropy of damage and mechanical properties of stainless steel sheets X5CrNi18-10 in the process of subsequent uniaxial tensile tests was studied.

Texture of stainless steel X5CrNi18-10 of 1 mm thickness in delivery conditions after recrystallization annealing includes two limited axial components and twinning orientations {113} <211>. The first component with the axis <110> parallel to the ND extends from {011} <100> up to {011} <112>. The second component may be characterized by an axis of <110> inclined toward ND on approximately 600. It extends from ~ {112} <111> through {135} <211> up to {011} <112>.

Various combinations of the original texture of rolling, components of shear texture {001} <110> and twinned orientations are formed in sheets during the alternating bending.

The twinning role is enhanced at the increasing of number alternating bending cycles that is confirmed by metallographic data.

Anisotropy of damage and mechanical properties take place in initial sheet as and in sheets after alternating bending. Anisotropy decreases with increasing of number alternating bending cycles. The minimal anisotropy was observed after 5 cycles of the alternating bending.

Anisotropy is caused mainly by texture formed in steel sheets. Significant quadratic correlations take place between values of ultimate strength, proof stress, relative uniform elongation and damage, averaged through the direction of sheets at uniaxial tensile tests with values of <110> pole density averaged on both sides of sheets.

References [1] “Most common uses of stainless steel” [Online]. https://www.metalsupermarkets.com/most-common-uses-of-stainless-steel/

Aviable:

[2] “Role of Stainless Steel in Petroleum Refining (9021)”. [Online]. Aviable: https://www.nickelinstitute.org/~/media/Files/TechnicalLiterature/RoleofStainlessSteelinPetroleum Refining_9021_.ashx [3] A. Chirkova, N. Makhutov, M. Gadenin, M. Kuzeev, V. Farkhutdinov, “A computational and experimental method for estimating degradation of mechanical characteristics of steels under the conditions of high-temperature pyrolysis”. Inorg Mater, vol. 46, pp. 1688–1691, 2010, DOI: 10.1134/S002016851015015X [4] V. Mertinger, M. Benke, Sz. Szabo, O. Banhidi, B. Bollo, A. Kovacs, “Examination of a failure detected in the convection zone of a cracking furnace”. Engineering Failure Analysis, vol. 18, pp.

1675–1682, 2011, DOI: 10.1016/j.engfailanal.2011.02.003

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[5] G. Hughes, Materials of crude oil Refining: corrosion Problems and prevention [Online]. Aviable: http://www.dunand.northwestern.edu/courses/Case%20study/Gareth%20Hughes%20%20Materials%20of%20Crude%20Oil%20Refining.pdf [6] I. Kucora, L. Radovanović, “Pyrolysis furnace tube damaging and inspection”. Acta tehnica corviniensis – Bulletin of Engineering, vol. 7, (3), pp. 19–24, 2014.

[7] V. Usov, E. Gopkalo, N. Shkatulyak, A. Gopkalo, T. Cherneva, “Texture, Microstructure, and Fractal Features of the Low Cycle Fatigue Failure of the Metal in Pipeline Welded Joints”. Russian Metallurgy (Metally), (9), pp. 759–770, 2015, DOI: 10.1134/S0036029515090128 [8] V. Usov, N. Shkatulyak, “Fractal nature of the brittle fracture surfaces of metal”, Materials Science, vol. 41, (1), pp. 62-66, 2005, DOI: 10.1007/s11003-005-0132-8 [9] N. Shkatulyak1, O. Tkachuk, “A role played by the crystallographic texture in the process of corrosion of hot-rolled rods made of carbon steel”. Materials Science, vol. 48, (2), pp. 153-161, 2012, DOI: 10.1007/s11003-012-9485-y.

[10] G. Gerstein, A.A. Bruchanov, D.V. Dyachok, F. Nürnberger, “The effect of texture in modelling deformation processes of bcc steel sheets”. Materials Letters, pp. 356-359, 2016, DOI: 10.1016/j.matlet.2015.11.007.

[11] S. Murakami, Continuum damage mechanics: A continuum mechanics approach to the analysis of damage and fracture, Springer Sciences + Business Media, Dordrecht, Heidelberg, London, New York, 2012. DOI 10.1007/978-97-007-2666-6_1 [12] V. Randle, O. Engler, Introduction to Texture Analysis: Macrotexture, Microtexture and Orientation Mapping, CRC PRESS, Boca Raton, London, New York, Washington, D.C., 2000.

[13] K. Rashid, Abu Al-Rub, G. Voyiadjis, “On the coupling of anisotropic damage and plasticity models for ductile materials”, International Journal of Solids and Structures, pp. 2611-2643, 2003, DOI: 10.1016/S0020-7683(03)00109-4 [14] M. Bobyr, O. Khalimon, O. Bondarets, “Phenomenological damage models of anisotropic structural materials”, Journal of mechanical engineering NTUU “Kyiv Polytechnic Institute”, pp. 513, 2013.

[15] E. Tadmor, N. Bernstein, “A First-Principles Measure for the Twinnability of FCC Metals”, J. Mech. Phys. Solids, vol. 52, pp. 2507-2519, 2004, DOI: 10.1016/j.jmps.2004.05.002 [16] N. Shkatulyak, “Effect of Stacking Fault Energy on the Mechanism of Texture Formation during Alternating Bending of FCC Metals and Alloys”. International Journal of Nonferrous Metallurgy, vol.

2, (2), pp. 35-40, 2013, DOI:10.4236/ijnm.2013.22005.

[17] Y. Vishnyakov, A. Babareko, S. Vladimirov and I. Egiz, Teoriya Obrazovaniya Tekstur v Metallakh i Splavakh (Theory of Textures Formation in Metals and Alloys), 1979, Nauka, Moscow, (russian).

[18] Springer Handbook of Metrology and Testing, Editors: H. Czichos, T. Saito, L. Smith, Springer Science & Business Media, 2011, DOI 10.1007/978-3-642-16641-9 Cite the paper V.V. Usov, N.M. Shkatulyak, E.A. Dragomeretskaya, E.S. Savchuk, D.V. Bargan & G.V. Daskalytsa. (2016). Effect of Alternating Bending and Texture on Anisotropic Damage and Mechanical Properties of Stainless Steel Sheets. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.35491.04640

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I I. M ec hanic al Engi neeri ng & Physic s M M S E J o u r n a l V o l . 6

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The Influence of Cutting Speed on Concordant and Discordant Tangential Milling of MDF Priscila Roel de Deus 1, Manoel Cleber de Sampaio Alves 2, Luciano Rossi Bilesky 1 1 – Professor in forestry course, Faculty of Technology of Capão Bonito, SP, Brazil 2 – Professor in Mechanical Engineering Course, UNESP, Guaratinguetá Campus, Brazil DOI 10.13140/RG.2.1.2114.3286

Keywords: roughening, panels, wood, MDF.

ABSTRACT. The tangential milling is consistent when the direction of forward movement is equivalent to the movement of the cutter. But the dissenting milling is when the sense of forward movement is contrary to movement of the cutter.

The way the material is removed differentiates and may cause different results in apereza the surface. For Medium Density Fiberboard - MDF material and which is composed of pressed lignocellulosic fibers with resin and presence of heat, concordant and discordant response to milling with diferetntes surfaces presents results. The objective of this study was to analyze the milling results in consistent direction and discordant through the MDF surface analysis with the average roughness parameter (Ra), given in units of micrometer (μm) The MDF panels were milled tangentially on concordant and discordant direction with six repetitions in each direction. The tests were carried out with four cutting speed in the forward speed of 2 m/min and 1 mm machining depths. The results of surface roughness in the cutting speeds in concordant direction are larger by 50% than in the discordant direction.

Introduction. MDF (Medium Density Fiberboard) is an industrial product manufactured from lignocellulosic fibers and resin through the joint action of heat and pressure. It is a material used in the furniture industry, since it presents homogeneity, dimensional stability and mechanical strength next to medium density solid wood. It also receives various types of coating, maintaining the quality, besides reacting positively to machining processes.

The growth in demand for industrial wooden products and their derivatives is clear, due to this fact, the research of technological innovations is necessary. With this technology, the industry is able to offer state-of-art products while increasing the competitiveness in the market. Machining stands out among these innovations, once it evolves notably and there are machines that provide the automation of processes within wood sector, producing higher quality machined workpieces.

The MDF machining in Computer Numerical Control (CNC) centers represents technology that combines materials, machines and tools, which results in more accurate and with quality finishes workpieces.

The cutting parameters are numerical quantities related to the movement of the tool and workpiece during milling, such activities must be suited to each material both tool and workpiece. From these parameters, it is possible to make use of the milling process as a form of productivity and quality improvement.

Understanding the machining forces is primordial for the determination of the cutting conditions, machine and tool lifespan and the workpiece quality [1]. Also cites the importance of machining because it determines the quality of the workpiece and tool wear [2].

The use of a suitable machining technique for the transformation of wood can minimize or even correct problems due to its variability [3].

The literature for the most appropriate cutting parameters in order to optimize processes, reduce costs, and increase utilization of the workpiece, tool and machine. The experiment studied the influence of MMSE Journal. Open Access www.mmse.xyz

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machining depth (63 mm) and cutting speed (396 m/min) in cutting force and tangential milling with carbide. The results showed that the depth of cut and cutting speed significantly influence the cutting strength, tool lifespan, and particularly the costs [4].

In a studied the influence of the cutting speed and feed rate parameters through MDF surface roughness. The surface roughness decreases as the cutting speed increases. The MDF milling shows the advantage of using a high cutting speed [5].

Highlighting the importance of high quality surface in machining, as it can influence the cost of the final product in the industry, particularly for high durability materials. Thus, milling operations are almost indispensable, consuming much of the processes time and affecting significantly on the quality of the surface finish and final product costs. The surface roughness is influenced by the cutting speed, cutting depth, tool and workpiece conditions [6]. The same investigation, the surface roughness decreases as the number of revolutions increases, along with the reduction of the machining depth [6].

The high surface quality can be achieved by the proper selection of cutting parameters [7]. The influence of the cutting forces and their required intensity for wood milling reported [8]. The cutting parameters that influence the surface quality must be understood as every specific need, mainly related to wood [9].

In a study the values of MDF artificial finishing through the peripheral milling process with the feed rate of 2,90; 4,10; 5,80; 8,20; 10,90; 15,15; 21,80 and 30,30 m/min and the section thicknesses of 1, 3 and 5 mm in concordant and discordant direction. The roughness values are lower in consistent direction [10].

In [11] conducted measuring the feed force in chipboard panels, which is commonly used in the furniture industry. The surface characteristics are strongly influenced by machining parameters. Feed rate values near to 2 m/min and cutting speed near to 800 m/min have proved more suitable for finishing, benefiting more effectively the furniture industry.

The concordant milling occurs when the direction of the forward movement is equivalent to the mill movement. The discordant milling consists in the direction of forward movement being contrary to the mill movement [11].

In [9] studied the concordant and discordant milling in CNC machining center with solid carbide tool.

The result of the average roughness - Ra shows statistically significant differences for the two cutting directions. It is concluded that discordant direction provided the lowest roughness but with high power consumption.

The objective of this study is to investigate the influence of cutting speed on concordant and discordant tangential milling of MDF through the average and overall roughness.

Material and Method. Commercial medium density fiberboard – MDF by Duratex was used, with average basic density of 736.22 kg/m3, average moisture of 8.33%, 15 mm thickness and coated plate.

Tangential milling tests were performed on a machining center with Computerized Numerical Control (CNC) TECH Z1 model by SCM brand. The mill used was a solid carbide cutter top finishing type with three-helix cutting teeth, HWM- Premium - Upcut Spiral Bit model.

The MDF specimens, dimensions of 300x65x15 mm, were milled tangentially on concordant and discordant direction with six repetitions in each one. The tests were carried out with four cutting speed in the feed rate of 2 m/min and 1 mm machining depth.

For the measurement of workpieces average roughness represented by the average roughness parameter (Ra), a rugosimeter Taylor Hobson 25sultronic model was used including measuring probe with diamond cone-spherical tip, and 2 μm nose radius. The rugosimeter parameters are 2,5 mm cutoff, 12,5 mm measuring length, robust Gaussian filter and Range (resolution) of 300 μm. The measures emphasized the average roughness parameter (Ra), given in units of micrometer (μm).

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Results and discussion. In the concordant direction, the average roughness values (Ra) of 1.0 mm in depth were analyzed by Tukey test and result in a coefficient of variation of 16,38%. The feed rate does not differ significantly with FVa = 1,47; p-value >5%, as well as the relation between cutting speed and feed rate with FVcxVa = 1,93; p-value >5%. However, among the cutting speeds occur statistical differences (FVc = 13,34; p-value >5%).

It is shown the results of the average roughness (Ra) according to the machining in concordant direction with 1 mm machining depth. In Fig. 1 is illustrated the results regarding the average roughness (Ra).

Average Roughness (Ra) 30

Roughness Ra (µm)

25

20

4000 rpm 8000 rpm

15

12000 rpm 16000 rpm

10 a

a

a

a

b ab a

a

b

ab a

a

5 2 m/min

4 m/min

6 m/min

Feed rate

Fig. 1. Values of average roughness Ra for 1 mm machining depth regarding to cutting speed and feed rate in concordant tangential milling.

It is noted that the cutting speed 804 m/min (16000 rpm) is related to lower average surface roughness (Ra), ranging between 12 µm and 22 µm in concordant direction. The lowest roughness values were 14,93 µm for cutting speed of 804 m/min and feed rate of 4 m/min. During Pinus elliottii milling, observed average surface roughness (Ra) within a range of 1,2 µm to 2,8 µm [9]. These values are inferior to those found for panels such as MDF. During a peripheral cylindrical milling with vertical milling machine (router) and cutting speed of 312 m/min, achieved results close to 25 and 30 µm.

These results highlight the satisfactory outcome for MDF surface finishing in CNC milling [10].

During MDF tangential milling and feed rate of 4 m/min, observed the average surface roughness between 14 and 18 µm with cutting speed of 213 m/min [13].

In discordant direction, the average roughness (Ra) values with 1.0 mm depth and tangential milling were analyzed through Tukey test and result in a coefficient of variation of 16.46%. The feed rate has significant differences with FVa = 6.94; p-value >5%. The relationship between cutting speed and feed rate with FVcxVa = 0,72; p-value >5% and cutting speed with FVc = 0,59; p-value >5%, do not present significant statistical differences. For total roughness (Rt) and 1,0 mm depth, it is observed a coefficient of variation of 21.10%. There were no statistical differences between feed rates (FVa = 1.23; p-value >5%) and in the relationship of cutting speed and feed rate with FVcxVa = 0,5; p-value >5%. In the cutting speed, there were no significant statistical differences (FVc = 3,62; p-value >5%).

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In Fig. 2, it is observed that in all feed rates do not occur statistically significant differences and the lowest roughness values occur in the feed rate of 2 m/min. It is observed a tendency of lower roughness values occur in the cutting speed 12000 rpm and 8000 rpm (603 and 201 m/min).

Roughness Ra (µm)

Average Roughness (Ra) 23 21 19 17 15 13 11 9 7 5

4000 rpm 8000 rpm 12000 rpm

a a a a

a a a a

b ab a ab

2 m/min

4 m/min

6 m/min

16000 rpm

Feed Rate

Fig. 2. Values of average roughness Ra for 1 mm machining depth regarding to cutting speed and feed rate in discordant tangential milling.

Fig. 2 shows the feed rate of 2 m/min with the lowest surface roughness values. Only with feed rate of 6 m/min that is observed difference in cutting speed of 201 m/min. The lowest roughness values were 13 µm in Ra for cutting speed of 804 m/min and feed rate of 2 m/min. During peripheral milling of Eucalyptus sp in discordant direction for feed rate of 3 m/min and cutting speed of 292 m/min, found Ra values of approximately 2 µm for E. Dunni, 1.6 µm for E. Uroplhylla and 1.5 for E. Grandis [14].During discordant milling and feed speed of 5 m/min with cutting speed of 504 m/min showed an average roughness of 4 µm; and with cutting speed of 654 m/min presented average roughness of approximately 5 µm [15]. These studies are related to wood and present lower values in relation to the MDF. During discordant tangential CNC milling in MDF with feed rate of 8 m/min, was found that using cutting speed of 703 m/min, the average roughness was 17.63 µm. With the cutting speed of 603 m/min, average roughness was 15,67 µm [16]. During tangential milling in MDF, noted that with feed rate of 3 m/min and cutting speed of 527 m/min, the average roughness was approximately 10 µm; and with cutting speed was 904 m/min, the average roughness was approximately 11 µm in discordant direction [5].

Summary. In conclusion, the parameter that most influenced the surface quality was the cutting speed. All tests demonstrated that the cutting speeds of 603 and 804 m/min, i.e. the higher cutting speeds used in the experiments, correspond to lower values of roughness.

The dissenting tangential milling corresponds to the lower roughness values, showing up to 50% lower results than the concordant tangential milling.

References [1] Rigatti, A. M. Y. Avaliação da força de usinagem e energia específica de corte no fresamento com alta velocidade de corte. 2010. 87f Dissertation (Master in Mechanical Engineering) Faculty of Engineering, State University Paulista, Ilha Solteira, 2010.

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[2] Cardoso, F. G. Análise de forças de fresamento de roscas API. 2012, 96p. Dissertation (Master in Mechanical Engineering and Materials Technology). Federal Center of Technological Education Celso Suckow da Fonseca CEFET / RJ. Rio de Janeiro – RJ. 2012.

[3] Zamarian, E. H. C.; Alburquerque, C. E.; Matos, J. L. M. Usinagem Da Madeira De Bracatinga Para Uso Na Indústria Moveleira. Floresta, Curitiba, Pr, V. 42, N. 3, P. 631 - 638, Jul./Set. 2012.

[4] Lima, D.O. Araújo, A.C. Silveira, J.L.L. Influência da profundidade de corte e do avanço na força do corte no fresamento de faceamento. CONEM – Congresso nacional de Engenharia Mecânica. São Luís – Maranhão. 2013.

[5] Davim, J. P. Clemente, V. C. Silva, S. Surface Roughness Aspects In Milling MDF (Medium Density Fiberboard). Int J Adv Manuf Technol. 40:49–55. 2009.

[6] Chen, Chih-Chern; Liu, Nun-Ming; Chiang, Ko-Ta; Chen, Hua-Lun. 2012. Experimental Investigation Of Tool Vibration And Surface Roughness In The Precision End-Milling Process Using The Singular Spectrum Analysis. Int J Adv Manuf Technol. 63:797–815. 2012.

[7] Kiswanto, G. Zariatina, D.L. Ko, T.J. The effect of spindle speed, feed-rate and machining time to thesurface roughness and burr formation of Aluminum Alloy 1100 in micro-milling operation.

Journal of Manufacturing Processes. JMP-243; 16p. 2014.

[8] Eyma, F. Méausoone, P.J. Martin, P. Strains and cutting forces involved in the solid wood rotating cutting process. Journal of Materials Processing Technology 148: 220–225. 2004.

[9] Pinheiro, C. Efeitos do teor de umidade da madeira no fresamento de Pinus elliottii. 2014. 122f.

Dissertation (Master in Mechanical Engineering) Faculty of Engineering Guaratinguetá Campus, UNESP - Univ. Estadual Paulista, Guaratinguetá, 2014.

[10] Castro, E. M.; Gonçalves, M. T. T. Estudo do acabamento superficial em chapas MDF usinadas em processo de fresamento. MADEIRA: arquitetura e engenharia, ano 3, n.8, 2002.

[11] Valarmathi, T. N.; Palanikumar, K.; Latha, B. Measurement and analysis of thrust force in drilling of particle board (PB) composite panels. Measurement 46: p 1220–1230. 2013.

[12] Diniz, A. E.; Marcondes, F. C.; Coppini, N. L. Tecnologia da Usinagem dos Materiais. 8. ed.

São Paulo: Art Liber, 2013. 272 p.

[13] Aguilera, A.; Muñoz, H. Rugosidad superficial y potencia de corte en el cepillado de acacia melanoxylon y sequoia sempervirens. Maderas. Ciencia Y Tecnología, Concepción, v. 13, n. 1, p.1928, 2011.

[14] Lopes, C.S.D. Nolasco, A.M. Tomazello, M.F., Dias, C.T.S. Avaliação da rugosidade superficial da madeira de Eucalyptus sp submetida ao fresamento periférico. Cerne. v. 20 n. 3. p. 471-476. 2014.

[15] Camilo, R. S. Fresamento de Eucalyptus grandis. Work Completion of course (Graduation) Industrial Engineering wood, Universidade Estadual Paulista, Campus Experimental Itapeva, Itapeva, 2013. 105 p.

[16] Barros, V. R. Fresamento de madeiras de média densidade- MDF. 2013. 76 f. Work Completion of course (Graduation) - Industrial Engineering wood, Universidade Estadual Paulista, Campus Experimental Itapeva, Itapeva, 2013.

Cite the paper Priscila Roel de Deus, Manoel Cleber de Sampaio Alves & Luciano Rossi Bilesky (2016). The Influence of Cutting Speed on Concordant and Discordant Tangential Milling of MDF. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.1.2114.3286

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Substantiating of Rational Law of Hydrostatic Drive Control Parameters While Accelerating of Wheeled Tractors with Hydrostatic and Mechanical Transmission Taran I.O.1, a, Kozhushko A.P.2 1 – Department of Transport Management, National Mining University, Dnipropetrovsk, Ukraine 2 – Department of Automobiles and Tractor Industry, National Technical University "Kharkiv Polytechnical Institute", Kharkiv, Ukraine a – taran_70@mail.ru DOI 10.13140/RG.2.1.3590.9362

Keywords: wheeled tractor, acceleration, hydrostatic and mechanical transmission, law of variation, power distribution.

ABSTRACT. The paper explains a process to determine rational laws of change in parameters to control hydraulic units of hydrostatic drive while accelerating wheeled tractors with hydrostatic and mechanical transmissions operating according “input differential” and “output differential”schemes.

The paper substantiates application of rational laws of change in parameters to control hydromachines by determining power distribution within hydraulic and mechanical branches of hydrostatic and mechanical transmissions produced according to “input differential” and “output differential” schemes. Decrease in a zone of power circulation within hydrostatic and mechanical transmissions with output differential has been determined while applying rational laws of changes in control parameters.

Introduction. Constant progress of technologies in the world tractor building makes home tractor manufacturers implement innovative technical solutions to improve technical and economic performance. It results in the necessity to upgrade or modify tractors.

Trying to widen the range of power stream control from a power unit to engines, world tractor manufacturers continue designing stepped mechanical transmissions with the great number of transmission mechanisms; however, they neglect application of less number of shafts, gears, and other mechanical components. Nevertheless, it should be noted that year by year the number of tractors equipped with stepless transmissions, in particular, with hydrostatic and mechanical transmissions (HSMT). That can be explained by advantages of HSMT to compare with stepped transmissions from the viewpoint of smooth motion, increase in ergonomic properties while performing technological operations, automated control etc.

Statement of the problem. As it is known, according to their design HSMTs are divided into “input differential”, “output differential” schemes and those with varied structure. “Input differential” and “output differential” schemes are the most popular as it depends on simple design and less number of mechanical components within transmission.

Consideration of wheeled tractor being integral part of machine and tractor system should involve paying attention to acceleration process while performing technological operation called “plowing” as it means increase in propulsion forces which factors into following significant changes in technical and economic performance: increase in fuel consumption, efficiency decrease as well as effectiveness of machine and tractor system, increase in working pressure difference within hydrostatic drive MMSE Journal. Open Access www.mmse.xyz

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(HSD), angle velocities and power parameters of HSMT components. Basing upon above it is expedient to determine rational law of change in parameters to control hydromachines while accelerating and substantiate it at the expense of power stream distribution determination within HSMT operating according to “input differential” and “output differential” schemes.

Analysis of the research and publications. Analysis of [1-4] papers has helped develop mathematical model of wheeled tractor accelerating process while performing technological operation called “plowing”. Among other things, paper [1] is applied to determine dynamics of internal combustion engine operation. Motion equation demonstrating changes in crankshaft acceleration has been given to do that. The paper also uses mathematical model of transmission taking into consideration changes in parameters of HSD hydromachines control, volumetric capability of hydromachines, and loss point in hydromachines. It makes it possible to explain an accelerating process of transmissions components in terms of various scheme designs. To explain interaction between wheels and ground in a function of design parameters of tires and physical and mechanical properties of support surface, mathematical model of single traction wheel dynamics while accelerating shown in [1-4] publications is used. They explain interaction of wheels and support surface.

Model of ХТЗ-170/240 line was chosen as base wheeled tractor meant for internal combustion engines with 125 ... 176 kW (170 ... 240 h.p.) motor capacity. [3 – 4] publications have helped select HSMT to be analyzed (Fig. 1) on the criterion of peak efficiency and minimum required motor capacity of internal combustion engine.

а)

b) Fig. 1 Structural patterns of two-flow HSMT: а – “input differential”; b – “output differential”.

According to [5 – 9] papers, distribution of power passing within mechanical branch and hydraulic branch of two-flow HSMT is determined using following equations: MMSE Journal. Open Access www.mmse.xyz

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– for HSMT operating with “input differential”:

N gid

iingid 

Nk

iinmeh 



M 2 a  2 ; M 5c  5

N meh M 4 a  4  ; Nk M 5c  5

(1) (2)

– for HSMT operating with “output differential”

iexgid  iexmeh 

N gid Nk



M 3a  3 ; M1a  1

N meh M 4 a  4  .

Nd M1a  1

(3) (4)

Definition of rational law to control parameters of hydromachines while accelerating.

Calculations are made within MATLAB system with the help of Simulink subsystem to simulate dynamic processes where generalized mathematical model for wheeled tractor with HSMT accelerating while performing “plowing” operation has been developed.

To form rational laws of changes in parameters to control HSD hydromachines introduce generalized criterion (KΣ) (with the help of partial criteria it characterizes both efficiency and effectivity of MTS while performing technological operation called “plowing”, and should be maximum)

n

m

i 1

j 1

K    i  Ki    j  Pj ,

(5)

where i ,  j are weight coefficients;

K i – is partial criteria; Pj – is penalty function which decreases value of generalized criterion when varying parameter

is beyond admissible values.

MTS efficiency is estimated on fuel consumption value (K1(e1, e2)); to estimate MTS efficiency while performing technological operation called “plowing”, MTS efficiency indices (K2(e1,e2)) and temporal value while MTS accelerating(K3(e1,e2)) are used:

K1 (e1 , e2 )  1 

QP* (e1 , e2 )   (e , e ) t * (e1 , e2 ) ; K 2 (e1 , e2 )  MTA 1 2 ; K3 (e1 , e2 )  1  , QP max max MTA (e1 , e2 ) tmax

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(6)

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where QP* (e1 , e2 ) – is current value of fuel consumption;

QP max – is maximum of fuel consumption;  MTA (e1 , e2 ) – is current value of MTS efficiency;

max MTA (e1 , e2 ) – is maximum of MTS efficiency; t * (e1 , e2 ) – is current value of MTS accelerating period;

tmax – is maximum of MTS accelerating period being determined while applying linear law of change in parameters to control HSD hydromachines.

Changes in e1 (t ), e2 (t ) parameters either increase or decrease factors working upon a process of MTS accelerating; namely: effective pressure difference within HSD ( P ), angle velocity on a shaft of hydraulic pump ( e1* ), hydraulic motor ( e 2* ) and satellite in planetary gear ( S ).

Bound violation factors into the fact that HSMT is out of service; inaccurate results are obtained.

Accordingly, penalty functions ( Pj ) are introduced to show excess of maximum values while optimizing.

Selection of values of weight coefficients for partial criteria involves the fact that sum of weight coefficients i should be equal to 1. Selection of weight coefficients  j for penalty functions takes into account that HSMT is out of order when penalty function is beyond the range of change. Thus, identification of rational law of changes in parameters to control HSD hydromachines is to apply one of the methods of optimization theory, i.e. direct search method. The search consists of consequent stages of research around basic point. If it is successful, the next step if the search according to certain sample.

While accelerating MTS in the process of “plowing” operation, generalized criterion characterizing technical and economic performance in the function of parameters controlling HAD hydromachines is

 Q* (e , e )   t * (e1 , e2 )   * (e , e ) K  (e1 ,e2 )  Z1  1  P 1 2   Z 2  MTA 1 2  Z3  1   QP max  МТА max tmax     Z P  PP ( P )  ZS  PS ( S )  Ze1*  PN ( e1* )  ZM  Pe 2* ( e 2* ).

(8)

Optimization process has formed rational laws of change in parameters to control HSD hydromachines for soil preparation, i.e. crop remains on light, medium-textured, and heavy loams as it is shown in Fig. 2.

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а)

b) Fig. 2 Changes in parameters to control HSD hydromachines (dependences of controlling parameters e1 , e2 of HSD hydromachines on t time): а – for “input differential”HSMT; b – for “input differential”HSMT; А is a zone of hydraulic pump control; B is a zone of hydromotor control; 1 are straight functional dependences of change in parameters to control HSD hydromachines; 2 is rational law of change in parameters to control HSD hydromachines for soil preparation in the context of light loams; 3 is rational law of change in parameters to control HSD hydromachines for soil preparation in the context of medium-textured loams; 4 is rational law of change in parameters to control HSD hydromachines for soil preparation in the context of heavy loams.

A process of (1 – 4) equations calculation has identified power distribution for wheeled tractor with HSMT as a part of machine and tractor system while performing “plowing” operation as it is shown in Fig. 3.

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a)

b) Fig. 3 Power distribution in HSMT: а – for “input differential”HSMT; b – for “output differential”HSMT; А is circulation zone while applying rational law; В is circulation zone while applying straight functional dependence.

Analysis of Fig. 3 demonstrates that in the initial stage of accelerating when excitation forces are maximal, acceleration process of wheeled tractor with HSMT needs iingid and iingid values be peak ones; that is, all forces initiated at accelerating stage and reacting against acceleration where smoothed out by HSD. Moreover, from Fig. 3, b decrease in a zone where power circulation is observed takes place. It is the result of application of rational law of change in parameters to control HSD hydromachines.

Summary. It has been determined that application of rational parameters laws of change in parameters to control HSD hydromachines while accelerating MTS with HSMT according to “output differential” and “input differential” is the utmost: fuel consumption is decreased by 36.2%; MTS efficiency is increased by 29.7%; accelerating period for MTS is decreased by 79.7%; and working pressure difference is decreased by 48.7% to compare with the use of linear functional dependence of change in parameters to control HSD hydromachines.

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Interrelation between power distribution within hydraulic and mechanical branches of two-flow HSMT and functional dependences of change in parameters to control HSD hydromachines has been identified.

Decrease in a zone within which power circulation is observed in “input differential” HSMT has been proved owing to application of rational law of change in parameters to control HSD hydromachines.

References [1] Kozhushko, A.P. (2016) Improving technical and economic indicators of wheeled tractors with continuously variable transmission by a rational change of regulation hydromachines parameters during acceleration: the thesis of dissertation for obtaining a scientific degree of Candidate of Science (Technology) on the specialty 05.22.02 – automobiles and tractors / Kozhushko Andriy Pavlovich. – Kharkiv. – 24 pp.

[2] Kozhushko, A.P. (2015) Determination of technical and economic parameters wheel tractor Fendt 936 Vario in the performance of technological operations "plowing" //Privolzhskiy nauchnyiy vestnik. – # 10 (50). – Pp. 20 – 25.

[3] Kozhushko, A.P. (2015) Features of work wheeled tractors with hydrostatic mechanical transmissions // Sil’s’kogospodars’ki mashyny. – #31. – Pp. 70 – 82.

[4] Kozhushko, A.P. (2015) Simulation results of work wheel tractor with hydrostatic mechnical transmissions working on schemes for "planetary gear input" // Advances in Mechanical Engineering and Transport. – №1 (3). – Pp. 93 – 102.

[5] Samorodov, V.B., Taran, I.A. (2012) Analysis of the distribution power flow considering the efficiency of hydraulic continuously variable two-flow hydrovolumetric-mechanical transmission with differential output // The bulletin of the National Technical University "KhPI". – # 64. – Pp. 3 – 8.

[6] Taran, I.O. (2012) Laws of power transmission on branches of double-split hydrostatic mechanical transmissions // Naukoviy visnyk NGU. – Dnipropetrovsk: SHEI «NMU». – #2. – Pp. 69 – 75.

[7] Taran, I.O. (2013) System of integral stochastic criteria for transmissions of transport vehicles// Naukoviy visnyk Khersons’koi derzhavnoi mors’koi akademii. – Kherson: Kherson state maritime academy. – # 2 (9). – Pp. 277 – 283.

[8] Transmission of mine locomotive: Monograph. – Dnipropetrovsk: published by SHEI «NMU». – 256 pp.

[9] Taran, I.O. (2013) Automated analysis of the distribution of power flow transmission locomotive // Ugol’ Ukraine. – #12. – Pp. 34 – 38 [10] Sunghyun Ahn, Suchul Kim, Jingyu Choi, Jinwoong Lee, Hyunsoo Kim, Development of an integrated engine-hydro-mechanical transmission control algorithm for a tractor, Advances in Mechanical Engineering 7(7), July 2015, doi: 10.1177/1687814015593870 Cite the paper Taran I.O. & Kozhushko A.P. (2016) Substantiating of Rational Law of Hydrostatic Drive Control Parameters While Accelerating of Wheeled Tractors with Hydrostatic and Mechanical Transmission. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.1.3590.93620

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Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function Pawan Kumar 1,a, Vaneshwar Kumar Sahu2, P.K.Ray2, B.B.Verma2 1 – Institute For Frontier Materials, Deakin University, Australia 2 – National Institute of Technology, Rourkela, India a – pkumar@deakin.edu.au DOI 10.13140/RG.2.2.16973.03043

Keywords: fatigue crack propagation, part-through cracked pipes, gamma model.

ABSTRACT. In the present investigation a gamma model has been formulated to estimate the fatigue crack growth in part-through cracked pipe specimens. The main feature of the model is that the gamma function is correlated with various physical variables like crack driving parameters and materials properties in non-dimensional form so that the proposed model can be used for different loading conditions. The validation of model has been done with experimental data in order to compare its accuracy in predicting fatigue crack growth.

Introduction. The fatigue crack growth Nomenclature behaviour of surface crack in a pipe in a – semi circumferential crack length; radial direction is one of the most serious problems associated with piping systems ai – initial crack length (mm); as it is responsible for detection of leak aj final crack length (mm);; before break. There are different piping integrity systems used in aircrafts, A, B, C, D – curve fitting constants offshore oil drilling, and coolant pipes in da/dN – crack growth rate; high pressure nuclear reactors which encounter fluctuating loading condition. K – stress intensity factor (MPa); Due to this kind of loading condition a new KC –fracture toughness (MPa m); surface crack can generate or an existing crack can propagate. This leads to damage ΔK – stress intensity factor range (MPa m); in structure and integrity. Many M – specific growth rate; researchers have studied fatigue crack growth problems in pipes. Different mij – specific growth rate in interval( j-i); techniques like numerical analysis, finite N – number of cycles; element method, boundary integral have P been used to address fatigue crack growth N j – predicted fatigue life using exponential model; in pipes. Jhonson et al. [1] used boundary R – load ratio; integral method for tension loading and providing a numerical solution to the Ri – Internal radius of specimen; problem. Daond and Cartwright [2] t – Thickness of specimen; applied strain energy release rate method w – width of specimen in uniform tension as well as pure bending condition in pipes. Delate et al. [3] reported numerical results for exterior cracks of semi-elliptical shape using line spring model.

In present research a modified gamma function is used to model fatigue crack growth in part- through cracked pipe in radial direction.

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Experimental Procedure. In the present investigation TP316L grade of stainless pipe was used. The chemical composition and mechanical properties of the material is presented in Tables 1 and 2 respectively. A part-through notch of angle 45o was machined by wire EDM and shown in Fig.1.

Fatigue crack growth test was conducted using a servo-hydraulic dynamic testing machine Instron 8800 on part-through cracked pipe specimens. The tests were conducted in air and at room temperature under constant amplitude 4-point bend loading condition. The schematic loading diagram is shown in Fig. 2. Seven specimens were tested in order to formulate the model and the 8th specimen was tested for validation of the model. Fig. 3 shows the fractured surface of the break opened specimen after fatigue test.

Table 1. Chemical Composition of TP316L stainless steel.

Element

C

Mn

Si

Weight (%)

0.03

2.00

0.75

Cr

Ni

16-18 10-14

P

S

Mo

N

Fe

0.045

0.030

2-3

0.10

balance

Table 2. Mechanical Properties ofTP316L stainless steel.

Modulus of elasticity (E), GPa

Poisson’s ratio (ν)

220

0.3

Yield strength (σys), MPa 366

Fig. 1. Cross sectional view of specimen (All dimensions are in mm).

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Ultimate tensile strength (σut), MPa 611

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Fig. 2. Four-point bend schematic loading diagram (All dimensions are in mm).

Fig. 3. Fracture surface of specimen.

Formulation of Model. Fatigue crack propagation, a natural physical process of material damage, is characterised by rate of increase of crack length (a) with number of cycles (N). It requires a discrete set of crack length vs. number of cycle data generated experimentally. The experimental a-N data is shown in Fig. 5.

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Crack length, a (mm)

5.8 4.8 3.8

2.8 1.8 0.8 0.00E+00

2.00E+04

4.00E+04

6.00E+04

No. of cycles, N

Fig. 5. Experimental a-N curve.

Gamma function is defined as the generalization of the factorial function to non-integral values, introduced by the Swiss mathematician Leonhard Euler in the 18th century. For a positive whole number n, the factorial (written as n!) is defined by n! = 1 Ă&#x2014; 2 Ă&#x2014; 3 Ă&#x2014; â&#x20AC;Ś Ă&#x2014; (n â&#x2C6;&#x2019; 1) Ă&#x2014;n. To extend the factorial to any real number n > 0 (whether or not n is a whole number), the gamma function is defined as [5-7]: đ?&#x2018;

Î&#x201C;(z) = â&#x2C6;Ť0 đ?&#x2018;Ą đ?&#x2018;§â&#x2C6;&#x2019;1 đ?&#x2018;&#x2019; â&#x2C6;&#x2019;đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą, đ?&#x2018;&#x2026;đ?&#x2018;&#x2019;(đ?&#x2018;§) > 0

(1)

In our present investigation a modified gamma model has been proposed to predict crack growth in part-through cracked pipe. Here term t is replaced by number of cycles N. The term z is chosen in such a way that it becomes non-dimensional and represents the parameters that affect crack growth.

The integral is chosen so that it is non-dimensional and represents crack growth at the end of a fixed number of loading cycles. Generally fatigue crack growth depends on the initial crack length, material properties and specimen geometry, loading conditions etc. The non-dimensional parameter is chosen in such a way so that it includes all these variables and properties. The expression for predicting the final crack length at the end of N cycle is given by: đ?&#x2018;&#x161;đ?&#x2018;&#x17D;1 đ?&#x2018;¤

đ?&#x2018;

= â&#x2C6;Ť0 đ?&#x2018; (

đ?&#x2018;&#x161;đ?&#x2018;&#x17D;0 đ?&#x2018;¤

â&#x2C6;&#x2019;1) â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2019;

đ?&#x2018;&#x2018;đ?&#x2018; ,

(2)

where z â&#x20AC;&#x201C; has been replaced by [mĂ&#x2014;(a/w)]; w â&#x20AC;&#x201C; is the specimen thickness; m â&#x20AC;&#x201C; is defined as a non-dimensional parameter whose value remains approximately constant for a given cycle interval.

The value of m includes all the properties which affect crack growth.

Fatigue crack growth behaviour depends upon initial crack length and load history. Therefore, while using gamma model each previous crack length is taken as initial crack length for the present step and non-dimensional parameter m is calculated for each step in incremental manner. The experimental a-N data have been used to determine the parameter m for each step using MATLAB programming. At first, initial value of m is assumed for a given interval of cycle. The iteration is continued till the value of m for which LHS and RHS of the equation (2) becomes nearly equal (within +0.01) is taken as the value of m for given cyclic interval.

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The RHS of equation (2) was solved with the help of MATLAB programming.

The expression of m is given by: đ?&#x2018;&#x161; = đ??´đ?&#x2018;&#x2122; 3 + đ??ľđ?&#x2018;&#x2122; 2 + đ??śđ?&#x2018;&#x2122; + đ??ˇ ,

(3)

where A, B, C, and D are curve fitting constants. The non-dimensional number m is correlated with another parameter l which takes into account two crack driving forces Î&#x201D;K and Kmax as well as material parameters Kc, E and Ď&#x192;YS. đ??ž

đ?&#x2018;&#x2122; = [(đ??ž ) (

đ??žđ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ đ??žđ??ś

đ??ś

đ?&#x153;&#x17D;đ?&#x2018;&#x152;đ?&#x2018;&#x2020;

)(

đ??¸

1/4

)]

.

(4)

The stress intensity factor K is calculated by equation (5) [8]: đ?&#x2018;&#x17D; 2đ?&#x2018;? đ?&#x2018;&#x2026;đ?&#x2018;Ą

đ??ž = â&#x2C6;&#x161;đ?&#x153;&#x2039;đ?&#x2018;&#x17D; (â&#x2C6;&#x2018;3đ?&#x2018;&#x2013;=1 đ?&#x153;&#x17D;đ?&#x2018;Ą đ?&#x2018;&#x201C;đ?&#x2018;Ą ( đ?&#x2018;Ą

đ?&#x2018;&#x17D;

đ?&#x2018;Ą

đ?&#x2018;&#x17D; 2đ?&#x2018;? đ?&#x2018;&#x2026;đ?&#x2018;Ą

) + đ?&#x153;&#x17D;đ?&#x2018;?đ?&#x2018;&#x201D; đ?&#x2018;&#x201C;đ?&#x2018;?đ?&#x2018;&#x201D; ( đ?&#x2018;Ą

đ?&#x2018;&#x17D;

đ?&#x2018;Ą

)) ,

(5)

where đ?&#x153;&#x17D;đ?&#x2018;?đ?&#x2018;&#x201D; â&#x20AC;&#x201C; is the bending stress; đ?&#x153;&#x17D;đ?&#x2018;Ą â&#x20AC;&#x201C; is the axis-symmetrical stress which is zero in present case.

The predicted crack length was calculated as: đ?&#x2018;¤

đ?&#x2018;

đ?&#x2018;&#x17D;1 = đ?&#x2018;&#x161; Ă&#x2014; â&#x2C6;Ť0 đ?&#x2018; (

đ?&#x2018;&#x161;đ?&#x2018;&#x17D;0 đ?&#x2018;¤

â&#x2C6;&#x2019;1) â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2019;

đ?&#x2018;&#x2018;đ?&#x2018; ,

(6)

where m has been given by equation (3) after putting average value of curve fitting constants (for seven specimens) and validate it by using eq. (6) for 8th specimen.

Discussion. The predicted a-N curve obtained from proposed gamma model was compared with experimental test data (Fig. 6). The experimental a-N data of specimen no. 1, 2, 3, 4, 5, 6, and 7 were used for formulation of model, and its validation has been checked by 8th specimen. The experimental test data was compared with predicted a-N curve obtained from proposed gamma model (Fig. 6). The average values of curve fitting constants (for seven different specimens) for gamma model have been given in Table 3. The predicted and experimental da/dN-Î&#x201D;K curves for the 8th specimen are presented (Fig. 7) for comparison. It can be seen that results from predicted gamma model are in good agreement with experimental data.

Table 3 Values of coefficients for gamma model.

A 23.33 E+07

B

C

35.12E+06

-17.65E+05

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D 29.63E+03

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6

crack length, a (mm)

5.5 5

a ( experimental) a ( predicted)

4.5 4 3.5 3 5.00E+04

7.00E+04

9.00E+04

1.10E+05

No. of cycle, N

Fig. 6. a-N curve (gamma model).

5.5E-05

da/dN (mm/cycle)

5E-05

da/dN experimental da/dN predicted

4.5E-05 4E-05 3.5E-05 3E-05 17

18

19

20

21

Î&#x201D;K (Mpa*m^1/2)

Fig. 7. da/dN-Î&#x201D;K curve (gamma model).

The performance of gamma model is evaluated by comparing the predicted results with experimental data for part-through cracked pipes under constant amplitude loading condition. There are three criteria that have been used for comparison of predicted results and experimental data, which are: 1. Percent deviation of predicted result from the experimental data as:

% đ??ˇđ?&#x2018;&#x2019;đ?&#x2018;Łđ?&#x2018;&#x2013;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; =

đ?&#x2018;&#x192;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;Ąđ?&#x2018;&#x2019;đ?&#x2018;&#x2018; đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;Ą â&#x2C6;&#x2019; đ??¸đ?&#x2018;Ľđ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2122; đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;Ą Ă&#x2014; 100 đ??¸đ?&#x2018;Ľđ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2122; đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;Ą

2. Prediction ratio which is defined as ratio between experimental data to predicted result as:

đ?&#x2018;&#x192;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153; đ?&#x2018;&#x192;đ?&#x2018;&#x; =

đ??¸đ?&#x2018;Ľđ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x2122; đ?&#x2018;&#x2018;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x17D; đ?&#x2018;&#x192;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;?đ?&#x2018;Ąđ?&#x2018;&#x2019;đ?&#x2018;&#x2018; đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;Ą

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3. Error bands i.e. the scatter of the predicted life on either side of experimental line within certain error limits.

The percentage deviations and the prediction ratio of exponential model are presented in Table 4 and Table 5.

Table 4. Model Performances (for crack length).

Test specimen

% Dev

Prediction ratio

TP316L stainless steel

1.74

0.99

Table 5. Model performances (for number of cycle).

Test specimen TP316L stainless steel

% Dev model) 1.02

Prediction ratio 1.01

The error band scattered is plotted in order to evaluate performance of gamma model as shown in Fig. (8-9). It has been observed that the error band scatter of gamma model lies in the range of +0.03% to -0.03% for experimental number of cycles and +0.03 to -0.03% of experimental crack length.

1.20E+05

Prpredicted N to a given crack length

1.10E+05 1.00E+05

N (Predicted)

9.00E+04 8.00E+04 7.00E+04 6.00E+04 60000

80000

100000

Experimental N to a given crack length

Fig. 8. Error band scatter for number of cycle (gamma model).

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120000

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6

Crack length, a (mm) Predicted

5.5 Predicted 5

4.5

4

3.5

3 3

3.5

4

4.5

5

5.5

6

crack length, a (mm) Experimental

Fig. 9. Error band scatter for crack length (gamma model).

Summary 1. The fatigue crack propagation in part-through cracked pipes can be determined effectively using gamma model of the form

đ?&#x2018;&#x161;đ?&#x2018;&#x17D;1 đ?&#x2018;¤

đ?&#x2018;

= â&#x2C6;Ť0 đ?&#x2018; (

đ?&#x2018;&#x161;đ?&#x2018;&#x17D;0 đ?&#x2018;¤

â&#x2C6;&#x2019;1) â&#x2C6;&#x2019;đ?&#x2018;

đ?&#x2018;&#x2019;

đ?&#x2018;&#x2018;đ?&#x2018;

2. Specific growth rate ((m) is expressed as đ?&#x2018;&#x161; = đ??´đ?&#x2018;&#x2122; 3 + đ??ľđ?&#x2018;&#x2122; 2 + đ??śđ?&#x2018;&#x2122; + đ??ˇ where l is correlated with various crack driving forces and material properties is an important parameter for gamma model.

3. Using Gamma model it is possible to predict the crack extension corresponding to a given number of cycles or to predict the number of cycles required for a given crack extension.

References [1] Johnson, RN. "Fracture of a cracked solid circular cylinder (Investigation of cracks in radial plane of solid right circular cylinder)[Ph. D. Thesis]." (1972).

[2] Daoud, O. E. K., and D. J. Cartwright. "Strain energy release rates for a straight-fronted edge crack in a circular bar subject to bending." Engineering Fracture Mechanics,19.4 (1984): 701-707.

[3] Delale, F., and F. Erdogan. "Application of the line-spring model to a cylindrical shell containing a circumferential or axial part-through crack." Journal of Applied Mechanics 49.1 (1982): 97-102.

[4] Andrews, George E., Richard Askey, and Ranjan Roy. "Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications." (1999).

[5] Rice, J. A. (1995). Mathematical Statistics and Data Analysis (Second Edition). p. 52â&#x20AC;&#x201C;53 [6] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of mathematical functions: with formulas, graphs, and mathematical tables. No. 55. Courier Corporation, 1964.

[7] J. R. Mohanty, B. B. Verma, and P. K. Ray, â&#x20AC;&#x153;Prediction of fatigue crack growth and residual life using an exponential model: Part I (constant amplitude loading),â&#x20AC;? International. Journal of Fatigue., Elsevier, vol. 31, pp. 418-424, 2009, doi: 10.1016/j.ijfatigue.2008.07.015

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[8] Al Laham, S., and Structural Integrity Branch. Stress intensity factor and limit load handbook.

British Energy Generation Limited, 1998.

Cite the paper Pawan Kumar, Vaneshwar Kumar Sahu, P.K.Ray & B.B.Verma (2016). Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.16973.03043

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Fundamental Solutions for Micropolar Fluids with Two-Temperature M. Zakaria1, 2, a 1 – Mathematics department, Faculty of Science, Al-Baha University, Al-Baha, P.O. Box 1988, Kingdom of Saudi Arabia 2 – Mathematics department, Faculty of Education, El-Shatby Alexandria University, Alexandria, 21526, Egypt a – zakariandm@yahoo.com DOI 10.13140/RG.2.2.28685.95201

Keywords: micropolar, two-temperature, low-Reynolds-number ABSTRACT. New fundamental solutions for steady two temperature micropolar fluids are derived in explicit form for two- and three dimensional steady unbounded Stokes and Oseen flows due to a point force, a point couple, and time harmonic sources including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a lowReynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow.

1. Introduction. In the theory of micropolar fluids [1], rigid particles contained in a small volume element can rotate about the centroid of the volume element. An independent microrotation vector describes the rotation. Micropolar fluids can support body couples and exhibit microrotational effects. The theory of micropolar fluids has shown promise for predicting fluid behaviour at microscale. Papautsky et al. [2] found that a numerical model for water flow in microchannels based on the theory of micropolar fluids gave better predictions of experimental results than those obtained using the Navier– Stokes equation. Micropolar fluids can model anisotropic fluids, liquid crystals with rigid molecules, magnetic fluids, clouds with dust, muddy fluids, and some biological fluids [3]. In view of their potential application in microscale fluid mechanics and non-Newtonian fluid mechanics, it is worth exploring new fundamental solutions.

NOMENCLATURE  – del operator;

V – velocity vector of the fluid; P – pressure; μ – viscosity; μr – coupling viscosity coefficient or vortex viscosity; F – constant vector; xi – coordinates system, i=1,2…n; U∞ – free stream; Ir – micro-inertia; cr,cm – two angular viscosities; ω – microrotation vector; q – heat flux vector;

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τo – relaxation time;

Srinivas and Kothandapani [4] studied the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Chen [5] performed a magnetohydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption. Many authors have extended the above model to the flow of micropolar fluid past an infinite horizontal moving plate, in the presence of a magnetic under different physical situations (see for instance [6–10]).

k – thermal conductivity; CE – specific heat at constant pressure; ρ – density of the fluid; T – dynamical temperature of the fluid; To – reference temperature chosen; Q – strength of the applied heat source per unit mass; θ – the conductive temperature;

The physical mechanisms of heat, mass, and a – a > 0 two-temperature parameter momentum transport in small-scale units may differ significantly from those in micro-scale equipment [11, 12]. Fundamental and applied investigations of micro-scale phenomena in fluid mechanics are motivated by developments in the areas of biological molecular machinery, atherogenesis, microcirculation, and microfluidics. At scales larger than a micron, the fluid can be treated as a continuum, and the flow is governed by the Navier–Stokes equation. The continuum model assumes that the properties of the material vary continuously throughout the flow domain. In Newtonian continuum mechanics, the fluid is modeled as a dense aggregate of particles, possessing mass, and translational velocity. However, the field equation, such as the Navier–Stokes equation, does not account for the rotational effects of the fluid micro-constituents.

In the theory of micropolar fluids [13], rigid particles contained in a small volume element can rotate about the centroid of the volume element. The rotation is described by an independent micro-rotation vector. Micropolar fluids can support body couples and exhibit microrotational effects. The theory of micropolar fluids has shown promise for predicting fluid behaviour at microscale. Papautsky et al.

[11] found that a numerical model for water 2. The mathematical model. Consider a point force in an unbounded, quiescent, incompressible micropolar fluid. Without loss of generality, the point force is placed at the origin, and the free-stream velocity U∞ is taken to be (U∞, 0, 0). Based on the Oseen approximation, the governing equations reduce to [11] The equations of motion have the form

  V  o,

U 

(1)

V   p     r     V  2 r      F ( x),  x1

(2)

  2 r   V  2   cr       cm     ,  x1

(3)

 I rU 

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The equation of energy is given by:

  q   Q ( x)   ToU 

 ,  x1

(4)

The entropy η may be written in terms of temperature as follows

 

CE To

T

 To ,

(5)

The generalized Fourier’s law including the current density effect is given by [12]-[14]

q   k ,

(6)

a 2    T .

(7)

By eliminating η between (4) and (5) and using (6) we get the equation of heat conduction for the linear theory as follows

k  2   C EU 

T   Q ( x) .  x1

(8)

The pressure, p, translational velocity, V, and microrotation velocity, ω, are required to decay as |x|→∞ and temperature T → T∞ in an unbounded flow, as follows p  0,

| V |  0,

|  |  0,

| T |  T

as | x |  .

3. The formulation of the problem in the Fourier transform domain The Fourier transform f   of a function f x  in the n-dimensional complex Fourier is defined as:

f ( )   f ( x) 2 



n 2  n

f ( x)e i  x d x,

where ξ – is the transformed variable of x, ξ, x = ξ1x1 + · · · + ξnxn i – is the imaginary unit, i=  1 . The divergence of (2) yields MMSE Journal. Open Access www.mmse.xyz

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 2 p    F ( x),

(9)

which states that p is harmonic everywhere except at the pole. To solve (9) for p, we take the Fourier transform, finding n    2 ( 2  ) , p    ( i )  F   2   

Taking the inverse Fourier transform, we find

  1   Fx n  2,      F ln r   2   2  2 r p ,       F 1    F  x n  3   4 r  4 r 3  

(10)

Stokes and Oseen flows due to a point force have the same pressure field, regardless of whether the fluid is Newtonian or micropolar. From (3), we have

 V  2 

cr I       r U     .

2r 2r

(11)

Taking the curl of (2) gives

  r       V  2r         F ( x)    U  V .

(12)

Substituting (11) in (12), we derive a partial differential vector equation containing only one unknown, ω,

 4  2     a0 2  a1  2  a  a3 2 2   x1  x1  x1 

    a4   F ( x)  

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(13)

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where a4 

ao 

4  r , cr    r 

I 1  a1  U   r  ,  cr    r 

a2 

 2U 2 I r , cr   r 

a3 

4 r U  , cr    r 

2 r . cr    r 

To solve partial differential equations of higher order, such as (13), we may factorize the higher-order partial differential operator into products of lower order [4]. This method was used by Olmstead and Majumdar [5]. Formally, it is proposed that

     L    2  A1  B1    2  A2  B2   x1  x1    where L – is a fourth-order partial differential operator; A1, A2, B1 and B2 – are constants.

While the method of factorization is attractive, a certain relationship between the parameters must exist for L to admit the desired factorization. To factorize the differential operator in (16), the following must be true: L   4  a0 2  a1  2

 2   a2 2  a3  x1  x1 x

 2 2    4  B1  B2  2   A1  A2    A1 A2 2   A1B2  A2 B1   B1B2 .  x1  x1  x1

(14)

Consequently, it is required that

B1  B2   ao , A1  A2   a1, A1 A2  a2 , A1B2  A2 B1  a3 , B1B2  0.

We have five equations and only four unknowns. To expedite the solution, the value of B1 is taken to be zero since B1B2 = 0. Therefore,

B2   ao ,

Then, from A1B2 + B1A2 = a3,

A1  

a3  U  . ao 

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Consequently,

A2 

a2  U  I r  .

A1 cr   r 

However, A1 + A2 = −a1, or

I  U   U  I r 1    U   r  ,  cr   r   cr   r 

which gives

Ir 

cr



.

(15)

Hence, the partial differential operator in (14) can be factorized as follows:

 4  a0 2  a1  2  2  U       x1 

 2   a2 2  a3   x1  x1 x

  2 U  I r        ao    r   x1  

This allows (13) to be rewritten as

 2     2no  x1 

 2       2mo  ao    a4   F ( x)  x1  

(16)

where 2no = ρU∞/μ and 2mo = ρ U∞/(μ + μr ). The above factorization is valid under the physical constraint of the parameters given by (14). To solve (15) for ω, it is convenient to take the Fourier transform.



2





n

 2i1no  2  2 imo1  ao   ia4 2  2   F ,

which gives MMSE Journal. Open Access www.mmse.xyz

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

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





n

ia4 2  2   F 2



 2i1n  2  2 im1  ao

.

(17)

The inverse Fourier transform gives n      2  2  1     a4   F , 2 2    2i1n   2 im1  ao      







which is the micropolar Oseenlet of ω

     x  t  e nt K o ns   e mt K o  o s  dt   e o 1 2U  2  x1      F  cr  e n (t  s )  e mt  1 s    e o  x1  t  dt , 4sU    x1

where  o  4 2 U cr ,  o  m2  ao2 , s  t 2 

n



j 2

 n  2, ,  n  3 

(18)



x 2j , and K o ( )   e  cosh d , is the o

modified Bessel function of the second kind. Because ω is expressed as the curl of the product of a scalar function and the constant vector F, the divergence of ω is zero. Hence, the assumption made earlier about the divergence of ω being zero is satisfied.

As U∞ → 0, Eq. (17) produces the micropolar Stokeslet.

   4   F n r  K o ao r     ao r         F 1 e     8  r    

n  2, (19)

n  3.

Equation (19) gives the microrotation velocity in the presence of a point force. The curl of (3) gives the curl of the curl of V as

    V  2   

cr  I rU      ,   2 r 2 r  x1

which can be written as MMSE Journal. Open Access www.mmse.xyz

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 c  I rU      2 V  2  r  2      , 2  2   x r r 1  using vector identities (1) and the assumption that the divergence of ω is zero. Taking the Fourier transform, we find

  c I U  2 V  2  r  2  i r  1     , 2 r 2 r  

Substituting (14) and (16) in the above equation leads to   2  2 2 2 2  n      i 2 no1   i 2mo1  ao  V  2  2 a4 i   i    F cr    2 2 2    2 r   i 2mo1  ao











   ,   

The inverse Fourier transform of this expression yields u in the form n      2  2   2 2 2      i 2 no1   i 2mo1  ao2 V  a4 i   i    2 F1  n   cr 2  2   2 2 2   2 r   i 2mo1  ao











     .    

We find the translational velocity u is given by the micropolar Oseenlet of V









   1  eo  x1 t  e not K o ( no s )  e o  x1  t e mot K o ( o s )  n s        F  dt  , n  2 , 2 U       x1  V     1  eo  x1  t  e no ( t  s )  eo  x1  t e mot  o s  1        F dt  , n  3.    4 s U      x1  

(20)

In the limit μr → 0, the translational velocity and microrotation velocity fields decouple. Then, mo → no, ωo → no, and the expression (19) of V simplifies to

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   e no t K o (no s)  n s  dt  , n  2,      F  2 U    x1  V  n ( t  s )  e o 1       F dt  , n  3.     x1 4 s U   

(21)

We see that the Newtonian Oseenlet is recovered. In the limit U∞ → 0, the micropolar Oseenlet of V in (19) becomes the micropolar Stokeslet of V,

    r 2n r    cr       n r  K a r     F    , n  2, o o 8  2   8     V     r  cr  1  e  ao r        F     2    , n  3.

8   r 16          

(22)

Finally, By eliminating  between (7) and (8) for T, we obtain it is convenient to take the Fourier transform.

 2  k 1     2  a  x1   x1 aCEU 





 Q  T   1  a  2  ( x), a CEU  

(23)

It is convenient to take the Fourier transform.



n





Q(2 ) 2 a 2  1 T  .  2 k 1 ia C EU        ia CEU  a 

The inverse Fourier transform gives

T 

n  2

2 

ia C EU 

1  ai  i 

      1  1   Q  ,    k 1 2             ia  C U a     E     

which is the dynamical temperature of the fluid T

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(24)

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   e o t K o ( o s )  2 dt  , n  2,  a  1 Q    x1 2 aCEU   T   t   e o  s o   2 a   1 Q dt    , n  3,   x1 4 s aC EU   









(25)

a 2 o2  1 . a2

where  o  k 2aCEU  , and  o 

Now, by eliminating T between (7) and (8) we get the equation of heat conduction for the linear theory as follows

 2  k 1     2  a  x1   x1 aCEU 

 Q     ( x), a CEU  

(26)

Taking the Fourier transform, we find.

 

Q(2 )



n 2

 k 1 ia C EU    2    ia CEU  a 

  

.

(27)

The inverse Fourier transform gives

 

2 



n 2

ia C EU 

      1  1    , Q   k 1      2         ia C EU  a      

which is the conductive temperature of the fluid  ,

  e o t K o ( o s )  dt  , n  2, Q   x1 2 aC EU      t     e o  s o Q dt    , n  3,  4  s a C U x 1 E   

(28)

Finally, we find the translational velocity u is given by the micropolar Oseenlet of V The solution of  for a micropolar fluid is much more complicated than that for a Newtonian fluid.

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4. Drag on a translating solid sphere in a micropolar viscous flow Consider the flow produced by a solid sphere of radius R translating with velocity U∞ in an ambient micropolar fluid of infinite expanse. The flow due to the sphere may be obtained in terms of a point force and a potential dipole, both placed at the center of the sphere, as in the case of Stokes flow [15, 16]. Hence, the velocity is given by

  cr   r V       F   2    8  16  

        B    4 r  O( Re ),   

 1  e  r   r 

(25)

where B – is the vectorial strength of the potential dipole;

Re 

 U 2R – is the Reynolds number, assumed to be small. 

Requiring the boundary condition V = U∞ at r = R to be satisfied on the surface of the sphere, yields two algebraic equations for the coefficients F and B,

  F  B  U  1  O( Re ) , 8 R    r  4 R 2  3 F  B  O 1  O( Re ) .

3 8 R    r  4 R 5 whose solution is

 F  6 R   r U  1  O( Re ) ,

 B   R3   r U  1  O( Re ) 

The drag comes exclusively from the point force. The dimensionless drag coefficient is

CD 

2F U 2

R

2



24   r 1 O( Re ), Re 

Putting μr = 0, we recover the result for the classical viscous flow

CD 

2F U 2

R

2



24 .

Re

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Summary. New fundamental solutions for micropolar fluids with two temperature have been derived in explicit form. The problem of two- and three-dimensional, steady, unbounded Stokes and Oseen flows of a micropolar fluid due to a point force and a point couple has been considered. The new fundamental solutions for Stokes and Oseen flows are the two-dimensional micropolar Stokeslet, given by (17), (20), (24) and (27) , the three-dimensional micropolar Oseenlet, given by (18) and (19).

These fundamental solutions are possible due to the existence of microrotation velocity fields in micropolar fluids. The fundamental solutions can generate further fundamental solutions by successive differentiation with respect to the singular point [15,16].

These fundamental solutions for micropolar fluids can be used as the basic building blocks to construct new solutions of microscale flow problems by employing the boundary-integral method or the singularity method. It was demonstrated that these fundamental solutions can be used to calculate the drag coefficients for a translating solid sphere and circular cylinder, respectively, in a micropolar fluid at low Reynolds numbers. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid by the factor (μ + μr)/μ.

References [1] I. Papautsky, J. Brazzle, T. Ameel, A. B. Frazier, Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors Actuators A Physical (1999) 73(1-2):101–108, DOI 10.1016/S09244247(98)00261-1 [2] J. J. Shu , Microscale heat transfer in a free jet against a plane surface. Superlattices Microstruct (2004) 35: 645–656, DOI 10.1016/j.spmi.2003.12.005 [3] A. C. Eringen, Microcontinuum field theories II: fluent Media. Springer-Verlag, New York, Inc (2001).

[4] S. Srinivas, M. Kothandapani: The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, App. Math. Comp. 213 (2009) 197–208, DOI 10.1016/j.amc.2009.02.054 [5] Chien-Hsin: Magneto-hydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption, Int. J. Non-Linear Mech. 44 (2009) 596 – 603, DOI 10.1016/j.ijnonlinmec.2009.02.004 [6] S. Srinivas, M. Kothandapani: The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Appl. Math. Comp.213 (2009) 197–208, DOI 10.1016/j.amc.2009.02.054 [7] R. C. Chaudhary, Abhay Kumar Jha: Effects of chemical reactions on MHD micropolar fluid flow past a vertical plate in slip-flow regime, Appl. Math. Mech. -Engl. Ed., 2008, 29(9):1179–1194, DOI 10.1007/s10483-008-0907-x [8] Youn J. Kim: Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium, Transport in Porous Media 56: 17–37, 2004, DOI 10.1023/B:TIPM.0000018420.72016.9d [9] A. Ishak , R. Nazar , I. Pop: MHD boundary-layer flow of a micropolar fluid past a wedge with variable wall temperature, Acta Mech 196, 75–86 (2008), 10.1007/s00707-007-0499-8 [10] A. Ishak, R. Nazar, I. Pop: Mixed convection stagnation point flow of a micropolar fluid towards a stretching sheet, Meccanica (2008) 43: 411–418, DOI 10.1007/s11012-007-9103-5 [11] Papautsky I, Brazzle J, Ameel T, Frazier AB: Laminar fluid behavior in microchannels using micropolar fluid theory. Sensors Actuators A Physical 73(1–2):101–108 (1999) MMSE Journal. Open Access www.mmse.xyz

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[12] Shu J-J: Microscale heat transfer in a free jet against a plane surface. Superlattices Microstruct 35(3–6):645–656 (2004) [13] Eringen AC: Microcontinuum field theories II: fluent Media. Springer- Verlag, New York, Inc (2001) [14] Zwillinger D: Handbook of differential equations. Academic Press (1998) [15] Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow.

Cambridge University Press [16] Kohr M, Pop I (2004) Viscous incompressible flow for low Reynolds numbers. WIT Press Cite the paper M. Zakaria (2016). Fundamental Solutions for Micropolar Fluids with TwoTemperature. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.28685.95201

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Calibration of COD Gauge and Determination of Crack Profile for Prediction of Through the Thickness Fatigue Crack Growth in Pipes Using Exponential Function Pawan Kumar1, a, Hemendra Patel2, P.K.Ray2, B.B.Verma2 1 – Institute for Frontier Materials Deakin University, Australia 2 – National Institute of Technology, Rourkela a – pkumar@deakin.edu.au DOI 10.13140/RG.2.2.23243.18724

Keywords: fatigue crack propagation, Crack opening displacement calibration, crack profile, exponential function

ABSTRACT. In present investigation the calibration of COD gauge and study of crack profile for part-through cracked pipes subjected to four-point bending has been done. The results show that crack profile is semi-elliptical in nature for lower crack depth and is flattened with the increase in crack depth. The linear relationship is obtained between crack depth and COD gauge. The COD calibration curve is used to study fatigue crack propagation by exponential function.

The material of the pipes under investigation was TP316L grade stainless steel. The specimens were subjected to fourpoint bend fatigue load in air and at room temperature. The predicted results were compared with experimental crack growth data. It has been observed that the results obtained using exponential function is in good agreement with experimental data.

Introduction. In industries pipe installations are used to transport pressurized fluids. Therefore, it is possible that these pipes experience stresses developed by the pressurized fluid. They may also experience seismic vibration as well as fluctuating bending stresses. It is possible that these stresses may promote extension of an existing crack or initiate a new fatigue crack from a highly stressed region [1-6]. In several industries the pipe installations carry hazardous fluids. Therefore, monitoring of these crack propagation in pipes is important in terms of safety and stability [7-8]. The study of fatigue crack growth requires determination of crack length/depth and number of cycles. The number of cycles can be obtained from data acquisition system, integrated with the fatigue testing machine.

For crack growth measurement various techniques are available like potential drop method, compliance method etc. To use compliance method the relation between COD gauge deflection and crack depth must be known. For some standard specimens like compact tension (CT), single edge notched (SEN) and others, the relationship between dimensionless compliance and normalized crack length are known. The relation between dimensionless compliance and normalized crack length are available for standard specimen given by ASTM standard data book; and this relationship can be used for the standard specimen and geometry for which they were developed. But in case of pipes there is no such a relation is available in the software supplied with the machine. In present investigation the calibration of cod gauge for straight notched pipe subjected to four point bending has been done.

Different techniques like finite element method, numerical analysis, boundary integral have been used to address fatigue crack growth in pipes. Athanassiadis et al. [9] reported numerical solution to a near circular crack front problem. Nezu et al. [10] applied finite element method with experimental results and studied circular shape of crack front. There are also finite element simulation software like CASCA and FRANC2D used to study fatigue crack growth in pipes [11]. There are methods in which it is proposed to convert 3D problems into 2D problems like spring model used by Rice and Levi [12] and conformal transformation methods by Wall Brink et al. [13]. These models are able to analyze partial circumferential crack as well as complex circumferential cracks in pipes. Mohanty et al. [14MMSE Journal. Open Access www.mmse.xyz

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17] developed an exponential model for prediction of fatigue crack growth in SENT specimen under constant amplitude loading and variable amplitude loading. Pawan Kumar et al. recently developed a fatigue crack propagation model for pipes using gamma function [18]. In present work exponential function is used instead of gamma function to estimate the fatigue crack propagation life in pipes.

Experimental procedure The material under investigation was TP316L stainless steel. The specimens were part-through cracked pipes having notch angle 45o. The notches were prepared by wire-EDM process. The fatigue crack growth tests were conducted in servo-hydraulic dynamic testing machine (Instron 8800) under load control mode. A four-point bend fixture as shown in Fig. 1 was fabricated for conducting fatigue crack growth tests. The Instron da/dN software is not calibrated for COD output and crack extension for pipe geometry. Therefore before conducting the test, COD gauge was calibrated using multiple specimens. The notch-geometry of the material is presented in Table 1.

Test Condition Pipe test have been carried out at room temperature and air environment under load control mode using sinusoidal waveform loading. The constant amplitude method with stress ratios of 0.1 with frequency 4 Hz has been followed. The load range applied during the fatigue crack initiation and growth test was of the order of 40.5 KN, which is below the yield strength of the piping material which corresponds given notch dimensions. This is to ensure that the crack growth is under gross elastic conditions.

Fig. 1. Four-point bend fixture and specimen.

Table 1. Specimen and notch dimension of pipe.

Specimen parameters

Outer radius (R0)

Inner radius (Ri)

Inner radius (Ri)

Crack depth

Crack length (L)

Length of the specimen

Angle (2θ)

Dimension (mm)

30

21

9

2.28

23

505

45o

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Calibration of cod gauge Pipes with straight notches were used for calibrating COD gauge. Multiple specimens were used for this purpose. The COD calibration curve (Î&#x201D;COD vs. measured crack length along the pipe thickness) is shown in Fig. 2. The crack length was measured with the help of a travelling microscope.

Crack length,a (mm)

6 5

4 3 2 1 0 0

10 20 Del. COD (mm)

30

Fig. 2. Calibration of COD gauge.

Determination of crack profile With the help of optical travelling microscope crack profile are measured. The crack profiles of all the fractured pipes are shown in fig 3. From the crack profile it is clear that the crack propagates along thickness direction first, than the crack propagates in circumferential direction. It is also clear that the crack front profile is semi-elliptical in nature for lower crack depth, but the crack front shape is flattened as the crack depth increases. After initial crack propagation, there is crack growth in the circumferential direction as well. This reduces the SIF at the main crack front. This is the probable cause of flattening of the main crack front after the crack has grown some distance in the throughthickness direction. The SEM image of fractured surface is shown in fig. 4 and no beach mark is observed.

Fig. 3. Crack profile.

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Fig. 4. SEM image of fractured surface shows no beach mark.

Formulation and validation of model Fatigue crack propagation, is characterised by rate of increase of crack length (a) with number of cycles (N). It requires a discrete set of crack length vs. number of cycle data generated experimentally.

Fig. 5 shows experimental a-N data.

Crack length, a (mm)

5.8 4.8 3.8 2.8 1.8 0.8 0

20000 40000 No. of cycles, N

60000

Fig. 5. Experimental a-N curve.

Finally the validation has been done with experimental data in order to compare its accuracy in predicting fatigue life in part-through cracked pipes.

Formulation of model This model is based on the exponential growth of fatigue crack with number of loading cycles. The modified exponential equation is given as [15-16] đ?&#x2018;&#x17D;đ?&#x2018;&#x2014; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2013; đ?&#x2018;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; (đ?&#x2018; đ?&#x2018;&#x2014; â&#x2C6;&#x2019; đ?&#x2018; đ?&#x2018;&#x2013; )

(1)

And, đ?&#x2018;&#x17D;đ?&#x2018;&#x2014; đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;

đ?&#x2018;&#x2122;đ?&#x2018;&#x203A;( )

đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; = (đ?&#x2018;

đ?&#x2018;&#x2014; â&#x2C6;&#x2019; đ?&#x2018; đ?&#x2018;&#x2013; )

Here, Nj and Ni represent number of cycles in jth step and ith step respectively; MMSE Journal. Open Access www.mmse.xyz

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(2)

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aj and ai â&#x20AC;&#x201C; are the crack lengths in jth step; ith step respectively; mij â&#x20AC;&#x201C; is specific growth rate in the interval (i-j).

The specific growth rate m is calculated for each step from experimental result of fatigue test (a-N data) using equation (2).

The exponent mij (known as specific growth rate) of the proposed exponential model has been correlated with various physical variables like crack driving parameters, crack resisting parameter, and material properties in non-dimensional forms. The specific growth rate is correlated with a parameter l, which takes into account two crack driving forces Î&#x201D;K and Kmax as well as material parameters KC, E, Ď&#x192;ys and is represented by equation: đ??ž

đ?&#x2018;&#x2122; = [(đ??ž ) (

đ??žđ?&#x2018;&#x161;đ?&#x2018;&#x17D;đ?&#x2018;Ľ

C

đ??žC

đ?&#x153;&#x17D;ys

)(

đ??¸

1/4

)]

(3)

The different m and l values are fitted by a polynomial equation.

The predicted m values are calculated for seven specimens by a polynomial fit as follows: đ?&#x2018;&#x161; = đ??´đ?&#x2018;&#x2122; 3 + đ??ľđ?&#x2018;&#x2122; 2 + đ??śđ?&#x2018;&#x2122; + đ??ˇ

(4)

where A, B, C, and D are curve fitting constants whose average value for seven specimens have been presented in the Table 3.

The stress intensity factor K has been calculated by equation [19]: đ?&#x2018;&#x17D; 2đ?&#x2018;? đ?&#x2018;&#x2026;đ?&#x2018;Ą

đ??ž = â&#x2C6;&#x161;đ?&#x153;&#x2039;đ?&#x2018;&#x17D; (â&#x2C6;&#x2018;3đ?&#x2018;&#x2013;=1 đ?&#x153;&#x17D;đ?&#x2018;Ą đ?&#x2018;&#x201C;đ?&#x2018;Ą ( đ?&#x2018;Ą

đ?&#x2018;&#x17D;

)+ đ?&#x2018;Ą

đ?&#x2018;&#x2026;đ?&#x2018;Ą đ?&#x153;&#x17D;đ?&#x2018;?đ?&#x2018;&#x201D; đ?&#x2018;&#x201C;đ?&#x2018;?đ?&#x2018;&#x201D; (đ?&#x2018;&#x17D;đ?&#x2018;Ą 2đ?&#x2018;? )) đ?&#x2018;&#x17D; đ?&#x2018;Ą

(5)

Here đ?&#x153;&#x17D;đ?&#x2018;?đ?&#x2018;&#x201D; is bending stress; đ?&#x153;&#x17D;đ?&#x2018;Ą is axis-symmetrical stress which is zero in present case.

Validation of model The predicted number of cycles or fatigue life is given by:

đ?&#x2018; đ?&#x2018;&#x2014;đ?&#x2018;&#x192; =

đ?&#x2018;&#x17D;đ?&#x2018;&#x2014; đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;

đ?&#x2018;&#x2122;đ?&#x2018;&#x203A;( ) đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014;

+ đ?&#x2018; đ?&#x2018;&#x2013;

(6)

The predicted values of specific growth rate (mij) of the tested specimen have been calculated by putting the average values of the curve fitting constants (for specimen no. 1, 2, 3, 4, 5, 6, 7) in equation (4). For validation of proposed exponential model; fatigue life is calculated (for specimen no. 8) by using the equation (6) .

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Discussion. An attempt has been made to develop a fatigue crack propagation model for part-through cracked pipe using exponential function. The specific growth rate (m) which relates crack growth to material properties is an important parameter for exponential model. The experimental a-N data of seven specimens were used for formulation of model, and its validation was checked for 8th specimen.

Table 2 shows the average value of curve fitting constants. These constants have been used to predict fatigue life of a through wall cracked pipe specimen. A comparative study of a-N curve is made for proposed exponential model and experimental data (Fig. 6).The da/dN – ΔK curves are also compared (Fig. 7). It is found that the predicted results are conservative in nature.

Table 2. Value of coefficients for exponential model.

Material

A

B

C

D

TP316L

-359.484

+52.708

-2.578

0.0421

9

crack length, a (mm)

8

a ( experimental)

a ( predicted)

7 6 5 4 3 50000

70000

90000

110000

130000

No. of cycles, N

Fig. 6. a-N curves (experimental and predicted).

0.0001

da/dN (mm/cycle)

9E-05 8E-05

da/dN ( experimental)

7E-05

da/dN (predicted)

6E-05 5E-05 4E-05 3E-05 2E-05 17

18

19 ΔK (MPa*m^1/2)

20

Fig. 7. da/dN-ΔK curves (experimental and predicted).

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Mechanics, Materials Science & Engineering, September 2016 â&#x20AC;&#x201C; ISSN 2412-5954

Three different criteria have been followed to evaluate performance of exponential model by comparing the predicted results with experimental data for part-through cracked pipes under constant amplitude loading condition. These criteria are Percent deviation, Prediction ratio and Error bands In order to predict fatigue crack propagation in part-through cracked specimen, seven specimens were used; the validity of proposed exponential model is checked for specimen no. 8. The percentage deviations and the prediction ratio of exponential model are presented in Table 3 and Table 4.

Table 3. Model Performances (for crack length).

Test specimen

% Dev

Prediction ratio

TP316L stainless steel

5.80

0.94

Table 4. Model performances (for number of cycle).

Test specimen TP316L stainless steel

% Dev model) 3.66

Prediction ratio 1.038

Performance of exponential model is evaluated by error band scatter, which is shown in Figs. 8 & 9.

The error band lie within +0.0% to -0.09% of experimental number of cycles and + 0.0% to+0.06% of experimental crack length for exponential model.

Fig. 8. Error band scatter for number of cycle (exponential model).

Fig. 9. Error band scatter for crack length (exponential model).

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Summary. The crack front profile is nearly semi elliptical in nature for lower crack depth, but the crack front shape is flattened as the crack depth increases. The calibration curve of COD gauge is found to be straight line, which shows linear relationship between COD gauge and crack depth of pipe specimens. Fatigue crack propagation in part-through cracked pipes can be determined by using exponential function of the form đ?&#x2018;&#x17D;đ?&#x2018;&#x2014; = đ?&#x2018;&#x17D;đ?&#x2018;&#x2013; đ?&#x2018;&#x2019; đ?&#x2018;&#x161;đ?&#x2018;&#x2013;đ?&#x2018;&#x2014; (đ?&#x2018; đ?&#x2018;&#x2014; â&#x2C6;&#x2019; đ?&#x2018; đ?&#x2018;&#x2013; ). Exponential function can be effectively used to predict the fatigue life of part-through cracked pipe.

References [1] Beden S M, Abdullah S, Ariffin A K, Review of fatigue crack propagation models for metallic components, European Journal of Scientific Research 28.3 (2009) 364-397.

[2] Wang G S, Blom A F, A strip model for fatigue crack growth predictions under general load conditions, Engineering Fracture Mechanics 40.3 (1991) 507-533, DOI: 10.1016/00137944(91)90148-T [3] Rice J R, Mechanics of crack tip deformation and extension by fatigue, fatigue crack propagation, ASTM, ASTM STP (1966) 415.

[4] Paris P C, Erdogan F, A critical analysis of crack propagation laws, Journal of Fluids Engineering 85.4 (1963) 528-533, DOI: 10.1115/1.3656901 [5] Paris P C, Gomez M P, Anderson W E, A Rational Analytical Theory of Fatigue, The Trend in Engineering, U. of Washington, Seattle, Wa 13.1 (1961).

[6] Walker E K, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7076T6 aluminum. In: Effect of environment and complex load history on fatigue life, ASTM STP 462.

Philadelphia: American Society for Testing and Materials, (1970) 1â&#x20AC;&#x201C;14.

[7] Shibata K, Results of reliability test program on light water reactor piping, Nuclear engineering and design, 153.1 (1994) 71-86.

[8] Yeon-Sik Y, Ando K, Circumferential fatigue crack growth and crack opening behavior in pipe subjected to bending moment, SMIRT-15, Seoul, Korea 15.5 (1999) 343-350.

[9] Athanassiadis A, Boissenot J M, Brevet P, Francois D, Raharinaivo A, Linear elastic fracture mechanics computations of cracked cylindrical tensioned bodies, International Journal of Fracture 17.6 (1981) 553-566.

[10] Kikuo N, Machida S, Nakamura H, SIF of surface cracks and fatigue crack propagation behaviour in a cylindrical bar, Japan Congress on Materials Research, 25 th, Tokyo, Japan (1982).

[11] Sharan A, Prediction of fatigue crack propagation in circumferentially cracked pipe specimen using casca and Franc2D, Diss. National Institute of Technology Rourkela, (2012).

[12] Rice J R, and Nouri L, The part-through surface crack in an elastic plate, Journal of Applied Mechanics 39.1 (1972): 185-194.

[13] Wallbrink C D, Peng D and Jones R, Assessment of partly circumferential cracks in pipes, International Journal of Fracture (2005): 167-181 [14] Mohanty J R, Verma B B, and Ray P K, Prediction of fatigue crack growth and residual life using an exponential model: Part II (mode-I overload induced retardation), International Journal of Fatigue 31.3 (2009): 425-432, DOI: 10.1016/j.ijfatigue.2008.07.018 [15] Mohanty J R, Verma B B, and Ray P K, Evaluation of overload-induced fatigue crack growth retardation parameters using an exponential model, Engineering Fracture Mechanics 75.13 (2008): 3941-3951.

[16] Mohanty J R, Verma B B, and Ray P K, Prediction of fatigue life with interspersed mode-I and mixed-mode (I and II) overloads by an exponential model: extensions and improvements, Engineering Fracture Mechanics 76.3 (2009): 454-468.

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[17] Mohanty J R, Verma B B, and Ray P K, Determination of fatigue crack growth rate from experimental data: a new approach, International Journal of Microstructure and Materials Properties 5.1 (2010): 79-87.

[18] Pawan Kumar, Vaneshwar Kumar Sahu, P.K.Ray and B.B.Verma , Modelling of Fatigue Crack Propagation in Part-Through Cracked Pipes Using Gamma Function, Mechanics, Materials Science & Engineering, Vol. 6 (2016), DOI: 10.13140/RG.2.2.16973.03043 Al Laham S, Structural Integrity Branch. Stress intensity factor and limit load handbook. British Energy Generation Limited, (1998).

Cite the paper Pawan Kumar, Hemendra Patel, P.K.Ray & B.B. Verma (2016). Calibration of COD Gauge and Determination of Crack Profile for Prediction of Through the Thickness Fatigue Crack Growth in Pipes Using Exponential Function. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.23243.18724

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Numerical Solution of Nonlinear Fredholm Integro-Differential Equations using Leibnitz-Haar Wavelet Collocation Method S. C. Shiralashetti1, a, R. A. Mundewadi1 1 – P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, India a – shiralashettisc@gmail.com DOI 10.13140/RG.2.2.31444.19848

Keywords: Leibnitz-Haar wavelet collocation method, operational matrix, nonlinear Fredholm integro-differential equations.

ABSTRACT. In this work, we present a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equation of the second kind. Haar wavelet and its Operational matrix are utilized to convert the differential equations into a system of algebraic equations, solving these equations using MATLAB to compute the required Haar coefficients. The numerical results obtained by the present method have been compared with those obtained by [3, 4] of the illustrative examples, which shows the efficiency of the method.

1. Introduction. In recent years, there has been a growing interest in the integro-differential equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations. IDEs play an important role in many branches of linear and nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics. The mentioned integro-differential equations are usually difficult to solve analytically, so a numerical method is required. There are several numerical methods for approximating the solution of nonlinear Fredholm integro-differential equations are known and many different basic functions have been used [1-4].

Wavelets theory is a relatively new and an emerging tool in applied mathematical research area. It has been applied in a wide range of engineering disciplines; particularly, signal analysis for waveform representation and segmentations, time-frequency analysis and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators.

Moreover, wavelets establish a connection with fast numerical algorithms [5, 6]. Since from 1991 the various types of wavelet method have been applied for the numerical solution of different kinds of integral equations, a detailed survey on these papers can be found in [7]. The solutions are often quite complicated and the advantages of the wavelet method get lost. Therefore, any kind of simplification is welcome. One possibility for it is to make use of the Haar wavelets, which are mathematically the simplest wavelets. Haar wavelet methods are applied for different type of problems in [8-14]. Sirajul-Islam et al. [15], V. Mishra et al. [13] and Lepik [16] and Tamme [17] have applied the Haar wavelet method for solving nonlinear Fredholm integro-differential equations. In the present work, a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equations of the second kind is proposed.

The article is organized as follows: In Section 2, the basic formulation of Haar wavelets and its operational matrix is given. Section 3 is devoted to the method of solution. In section 4, we report our numerical results and demonstrated the accuracy of the proposed scheme. Conclusion is discussed in section 5.

2. Properties of Haar Wavelets.

2.1. Haar wavelets MMSE Journal. Open Access www.mmse.xyz

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Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

The scaling function h1 ( x) for the family of the Haar wavelet is defined as

1 h1 ( x)   0

for x   0, 1 otherwise

(1)

The Haar Wavelet family for x [0,1) is defined as, for x  [ ,  ), 1  hi ( x)  1 for x  [  ,  ), 0 elsewhere, 

where  

k , m



k  0.5 , m



(2)

k 1 , m

where m  2l , l  0,1,..., J , J is the level of resolution;

k  0,1,..., m 1 is the translation parameter.

Maximum level of resolution is J . The index i in (2) is calculated using i  m  k  1. In case of minimal values m  1, k  0 then i  2 . The maximal value of i is N  2

J 1

.

j  0.5 , j  1, 2,..., N , Haar coefficient matrix N H  i, j   hi ( x j ) which has the dimension N  N . For instance, J  3  N  16 , then we have

Let us define the collocation points x j 

1  1 1  0 1  0 0  0 H 16,16    1 0  0 0  0 0  0 0 

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1 1

1 1

1 1

1 -1

1 -1

1 -1

1 -1

-1 0

-1 0

-1 0

-1 0

-1 0

-1 0

-1 0

0 1

0 -1

0 -1

0 0

0 0

0 0

0 0

1 0

1 0

1 0

1 0

-1 0

-1 0

-1 0

0 0

0 0

0 0

1 0

1 0

-1 0

-1 0

0 1

0 1

0 -1

0 -1

0 0

0 0

0 0

0 -1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

1 0

-1 0

0 0

1 0

-1 0

0 1

0 -1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

-1 0

0 1

0 -1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

-1 0

0 1

0 -1

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1  -1  0  -1  0   0  0   -1  0   0   0  0   0  0   0  -1  

Any function f(x) which is square integrable in the interval (0, 1) can be expressed as an infinite sum of Haar wavelets as

MMSE Journal. Open Access www.mmse.xyz

109

Mechanics, Materials Science & Engineering, September 2016 â&#x20AC;&#x201C; ISSN 2412-5954 ď&#x201A;Ľ

f ( x) ď&#x20AC;˝ ď&#x192;Ľ ai hi ( x)

(3)

i ď&#x20AC;˝1

The above series terminates at finite terms if f(x) is piecewise constant or can be approximated as piecewise constant during each subinterval. Given a function f(x) â&#x2C6;&#x2C6; L2(R) a multi-resolution analysis (MRA) of L2(R) produces a sequence of subspaces V j ,V j ď&#x20AC;Ť1 ,... such that the projections of f(x) onto these spaces give finer and finer approximations of the function f(x) as j ď&#x201A;Ž ď&#x201A;Ľ.

2.2. Operational Matrix of Haar Wavelet The operational matrix P which is an N square matrix is defined by x

P1,i ( x) ď&#x20AC;˝ ď&#x192;˛ hi (t ) dt

(4)

0

often, we need the integrals x x

x

x

1 Pr ,i ( x) ď&#x20AC;˝ ď&#x192;˛ ď&#x192;˛ ... ď&#x192;˛ hi (t ) dt ď&#x20AC;˝ ( x ď&#x20AC;­ t ) r ď&#x20AC;­1 hi (t ) dt ď&#x192;˛ (r ď&#x20AC;­ 1)! A AA A r

(5)

r ď&#x20AC;­times

r ď&#x20AC;˝ 1, 2,..., n and i ď&#x20AC;˝ 1, 2,..., N.

For r ď&#x20AC;˝ 1, corresponds to the function P1,i ( x) , with the help of (2) these integrals can be calculated analytically; we get đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ź đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;ź, đ?&#x203A;˝) đ?&#x2018;&#x192;1,đ?&#x2018;&#x2013; (đ?&#x2018;Ľ)={ đ?&#x203A;ž â&#x2C6;&#x2019; đ?&#x2018;Ľ đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;˝, đ?&#x203A;ž) 0 đ?&#x2018;&#x201A;đ?&#x2018;Ąâ&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;¤đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x2019; 1

đ?&#x2018;&#x192;2,đ?&#x2018;&#x2013; (đ?&#x2018;Ľ)=

1

2

4đ?&#x2018;&#x161;2

(6)

(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ź)2 đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;ź, đ?&#x203A;˝) 1

â&#x2C6;&#x2019; 2 (đ?&#x203A;ž â&#x2C6;&#x2019; đ?&#x2018;Ľ)2 đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;˝, đ?&#x203A;ž) 1

4đ?&#x2018;&#x161;2

{

(7)

đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;ž, 1)

0 đ?&#x2018;&#x201A;đ?&#x2018;Ąâ&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;¤đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x2019;

In general, the operational matrix of integration of r th order is given as 1

đ?&#x2018;&#x192;đ?&#x2018;&#x;,đ?&#x2018;&#x2013; (đ?&#x2018;Ľ)= {

1 1 đ?&#x2018;&#x;!

đ?&#x2018;&#x;!

đ?&#x2018;&#x;!

(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ź)đ?&#x2018;&#x; đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;ź, đ?&#x203A;˝)

{(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ź)đ?&#x2018;&#x; â&#x2C6;&#x2019; 2(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;˝)đ?&#x2018;&#x; } đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;˝, đ?&#x203A;ž) đ?&#x2018;&#x;

đ?&#x2018;&#x;

đ?&#x2018;&#x;

{(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ź) â&#x2C6;&#x2019; 2(đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;˝) + (đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x203A;ž) } đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;Ľ â&#x2C6;&#x2C6; [đ?&#x203A;ž, 1) 0 đ?&#x2018;&#x201A;đ?&#x2018;Ąâ&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;¤đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x2019; MMSE Journal. Open Access www.mmse.xyz

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(8)

Mechanics, Materials Science & Engineering, September 2016 â&#x20AC;&#x201C; ISSN 2412-5954

For instance, đ??˝=3 â&#x2021;&#x2019; N = 16, then we have ď&#x192;Ś ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ 1 ď&#x192;§ P1,i (16,16) ď&#x20AC;˝ 32 ď&#x192;§ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;¨ ď&#x192;Ś ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ 1 ď&#x192;§ P2,i (16,16) ď&#x20AC;˝ 2048 ď&#x192;§ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;§ ď&#x192;¨

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

1 1

3 3

5 5

7 7

9 7

11 5

13 3

15 1

15 0

13 0

11 0

9 0

7 0

5 0

3 0

0 1

0 3

0 3

0 1

0 0

0 0

0 0

0 0

1 0

3 0

5 0

7 0

7 0

5 0

3 0

0 0

0 0

0 0

0 0

1 0

3 0

3 0

1 0

0 1

0 3

0 3

0 1

0 0

0 0

0 0

0 1

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

3 0

3 0

0 0

0 0

1 0

1 0

0 1

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

1 0

0 1

0 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

1 0

0 1

0 1

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

9

25

49

1 1

9 9

25 25

49 49

0 1

0 9

0 23

0 31

0 0

0 0

0 0

0 0

0 1

0 7

0 8

0 8

0 0

0 0

1 0

7 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

31 ď&#x192;ś ď&#x192;ˇ 1 ď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 1 ď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 0 ď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 1 ď&#x192;ˇ 0 ď&#x192;ˇď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 0 ď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 0 ď&#x192;ˇ 0 ď&#x192;ˇ ď&#x192;ˇ 0 ď&#x192;ˇ and 1 ď&#x192;ˇď&#x192;¸

81 121 169 225 289 361 441 529 625 729 841 961 ď&#x192;ś ď&#x192;ˇ 81 121 169 225 287 343 391 431 463 487 503 511 ď&#x192;ˇ 79 103 119 127 128 128 128 128 128 128 128 128 ď&#x192;ˇ ď&#x192;ˇ 0 0 0 0 1 9 25 49 79 103 119 127 ď&#x192;ˇ 32 32 32 32 32 32 32 32 32 32 32 32 ď&#x192;ˇ ď&#x192;ˇ 1 9 23 31 32 32 32 32 32 32 32 32 ď&#x192;ˇ 0 0 0 0 1 9 23 31 32 32 32 32 ď&#x192;ˇ ď&#x192;ˇ 0 0 0 0 0 0 0 0 1 9 23 31 ď&#x192;ˇ 8 8 8 8 8 8 8 8 8 8 8 8 ď&#x192;ˇď&#x192;ˇ 8 8 8 8 8 8 8 8 8 8 8 8 ď&#x192;ˇ ď&#x192;ˇ 1 7 8 8 8 8 8 8 8 8 8 8 ď&#x192;ˇ 0 0 1 7 8 8 8 8 8 8 8 8 ď&#x192;ˇ ď&#x192;ˇ 0 0 0 0 1 7 8 8 8 8 8 8 ď&#x192;ˇ 0 0 0 0 0 0 1 7 8 8 8 8ď&#x192;ˇ ď&#x192;ˇ 0 0 0 0 0 0 0 0 1 7 8 8ď&#x192;ˇ 0 0 0 0 0 0 0 0 0 0 1 7 ď&#x192;ˇď&#x192;¸

3. Method of Solution In this section, we present a Leibnitz-Haar wavelet collocation method (LHWCM) for solving nonlinear Fredholm integro-differential equations of the second kind, 1

u '( x) ď&#x20AC;˝ f ( x) ď&#x20AC;Ť ď&#x192;˛ K ( x, t , u (t ))dt ,

(9)

0

where K(x, t, u(t)) is a nonlinear function defined on [0, 1]Ă&#x2014;[0, 1] are the known function K(x, t, u(t)) is called the kernel of the integral equation and f(x) is also a known function while the unknown function u(x) represents the solution of the integral equation.

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Leibnitz rule: The conversion of the integral equation into an equivalent differential equation. The conversion is achieved by using the well-known Leibnitz rule [17] for differentiation of integrals. h( x)



Let, F ( x) 

K ( x, t ) u (t ) dt

(10)

g ( x)

Then differentiation of the integral in (10) exists and is given by

dF dh( x) dg ( x) K ( x, t ) F '( x)   K ( x, h( x)) (u (h( x)))  K ( x, g ( x)) (u ( g ( x)))   u (t ) dt dx dx dx x g ( x) h( x)

(11)

For Fredholm, If g(x) =0 and h(x) =1 where 0 & 1 are fixed constants, then the Leibnitz rule (11) reduces to

dF K ( x, t ) F '( x)   u (t ) dt . dx 0 x 1

(12)

A numerical computation procedure is as follows: Step 1: Differentiating (9) thrice w.r.t x, using Leibnitz rule (11) we get, u ''( x)  f '( x)  F '( x)

(13)

u '''( x)  f ''( x)  F ''( x)

(14)

uiv ( x)  f '''( x)  F '''( x)

(15)

subject to initial conditions, u(0)   , u '(0)   , u "(0)   , u '''(0)   (16) Step 2: Applying Haar wavelet collocation method, Let us assume that, N

u iv ( x)   ai hi ( x)

(17)

i 1

Step 3: By integrating (17) four times and using (16), we get, N

u '''( x)     ai p1,i ( x) i 1

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u ''( x)     x   ai p2,i ( x)

(19)

i 1

u '( x)     x  

x2 N   ai p3,i ( x) 2 i 1

(20)

u ( x)     x  

x2 x3 N    ai p4,i ( x) 2 6 i 1

(21)

Step 4: Substituting (17) – (21) in the differential equation (15), which reduces to the nonlinear system of N equations with N unknowns and then Newton’s method can used to find the Haar coefficients ai , i  1, 2,..., N . Substituting Haar coefficients in (21) to obtain the required solution of equation (9).

4. Illustrative Examples In this section we consider the some of the examples to demonstrate the capability of the method in section 3 and error function is presented to verify the accuracy and efficiency of the following numerical results:

Error function  Emax  ue ( xi )  ua ( xi )

max



n

 u ( x )  u i 1

e

i

a

( xi ) 

2

where ue and ua are the exact and approximate solution respectively.

Example 4.1. Let us first consider the Nonlinear Fredholm Integro-differential equation [3], 1

u '( x) 

5 1 2  x   ( x 2  t )(u (t ))2 dt , 0  x  1, 4 3 0

(22)

with initial conditions u(0)  0, u '(0)  1, u ''(0)  0, u '''(0)  0. Which has the exact solution u( x)  x .

We applied the Haar wavelet technique presented in section 3 and solved (22) as follows.

Successively differentiating (22) w.r.to x and using Leibnitz rule (11) which reduces to the differential equation.

Let us first consider the given

f ( x) 

5 1 2  x 4 3

differentiating w.r.to x

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2 f '( x)   x , 3

again differentiating twice w.r.to x

f ''( x)  

2 3,

f '''( x)  0 .

(23)

Next, consider 1

F ( x)   ( x 2  t )(u (t )) 2 dt 0

differentiating w.r.to x using Leibnitz rule (11), 1

F '( x)   2 x (u (t )) 2 dt 0

again differentiating twice w.r.to x using Leibnitz rule (11), 1

F ''( x)   2(u (t )) 2 dt 0

F '''( x)  0 .

(24)

Substituting (23) and (24) in (15), we get the differential equation

u iv ( x)  0

(25)

assume that, N

u iv ( x)   ai hi ( x) i 1

integrating (26) twice, we get

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u '''( x) ď&#x20AC;­ u '''(0) ď&#x20AC;˝ ď&#x192;Ľ ai p1,i ( x)

(27)

i ď&#x20AC;˝1

N

u '''( x) ď&#x20AC;˝ ď&#x192;Ľ ai p1,i ( x)

(28)

i ď&#x20AC;˝1

N

u ''( x) ď&#x20AC;­ u ''(0) ď&#x20AC;˝ ď&#x192;Ľ ai p2,i ( x)

(29)

i ď&#x20AC;˝1

N

u ''( x) ď&#x20AC;˝ ď&#x192;Ľ ai p2,i ( x)

(30)

i ď&#x20AC;˝1

again integrating (30) twice, we get N

u '( x) ď&#x20AC;­ u '(0) ď&#x20AC;˝ ď&#x192;Ľ ai p3,i ( x)

(31)

i ď&#x20AC;˝1

N

u '( x) ď&#x20AC;˝ ď&#x192;Ľ ai p3,i ( x) ď&#x20AC;Ť 1

(32)

i ď&#x20AC;˝1

N

u ( x) ď&#x20AC;­ u (0) ď&#x20AC;˝ ď&#x192;Ľ ai p4,i ( x) ď&#x20AC;Ť x

(33)

i ď&#x20AC;˝1

N

u ( x) ď&#x20AC;˝ ď&#x192;Ľ ai p4,i ( x) ď&#x20AC;Ť x

(34)

i ď&#x20AC;˝1

Substituting (26), (28), (30), (32) and (34) in the differential equation (25), we get the system of N equations with N unknowns N

ď&#x192;Ľ a h ( x) ď&#x20AC;˝ 0 i ď&#x20AC;˝1

i i

(35)

solving (35) using Newtonâ&#x20AC;&#x2122;s Method to obtain Haar wavelet coefficients đ?&#x2018;&#x17D;đ?&#x2018;&#x2013; â&#x20AC;&#x2122;s for any N values the coefficients are zero. Substituting these coefficients in (34) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 1. This justifies the efficiency of the LHWCM. Example 4.2. Next, consider the Nonlinear Fredholm Integro-differential equation [3], 1

u '( x) ď&#x20AC;˝ 1 ď&#x20AC;­

1 x x ď&#x20AC;­ ď&#x20AC;Ť ď&#x192;˛ xt exp(ď&#x20AC;­(u (t ))2 )dt , 0 ď&#x201A;Ł x ď&#x201A;Ł 1, 2 2e 0

(36)

with initial conditions u(0) ď&#x20AC;˝ 0, u '(0) ď&#x20AC;˝ 1, u ''(0) ď&#x20AC;˝ 0. Which has the exact solution u( x) ď&#x20AC;˝ x .

Differentiating (36) twice w.r.t x and Using Leibnitz rule (11), its equivalent differential equation, MMSE Journal. Open Access www.mmse.xyz

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1 1 u ''( x) ď&#x20AC;˝ ď&#x20AC;­ ď&#x20AC;­ ď&#x20AC;Ť ď&#x192;˛ t exp(ď&#x20AC;­(u (t )) 2 )dt 2 2e 0 u '''( x) ď&#x20AC;˝ 0

(37)

Assume that, N

u '''( x) ď&#x20AC;˝ ď&#x192;Ľ ai hi ( x)

(38)

i ď&#x20AC;˝1

Integrating (38) thrice, we get

N

u ''( x) ď&#x20AC;˝ ď&#x192;Ľ ai p1,i ( x)

(39)

i ď&#x20AC;˝1

N

u '( x) ď&#x20AC;˝ ď&#x192;Ľ ai p2,i ( x) ď&#x20AC;Ť 1

(40)

i ď&#x20AC;˝1

N

u ( x) ď&#x20AC;˝ ď&#x192;Ľ ai p3,i ( x) ď&#x20AC;Ť x

(41)

i ď&#x20AC;˝1

Substituting (38) - (41) in the differential equation (37), we get the system of N equations with N unknowns N

ď&#x192;Ľ a h ( x) ď&#x20AC;˝ 0 i ď&#x20AC;˝1

(42)

i i

solving (42) using Newtonâ&#x20AC;&#x2122;s Method to obtain Haar wavelet coefficients đ?&#x2018;&#x17D;đ?&#x2018;&#x2013; â&#x20AC;&#x2122;s for any N values the coefficients are zero. Substituting these coefficients in (41) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 2. This justifies the efficiency of the LHWCM. Example 4.3. Now, consider the Nonlinear Fredholm Integro-differential equation [4], 1

1 u '( x) ď&#x20AC;˝ 2 x ď&#x20AC;Ť (ď&#x20AC;­ď ° ď&#x20AC;Ť log(4)) ď&#x20AC;Ť ď&#x192;˛ t arctan(u (t )) dt , 8 0

0 ď&#x201A;Ł x ď&#x201A;Ł1

(43)

with initial conditions u(0) ď&#x20AC;˝ 0, u '(0) ď&#x20AC;˝ 0. Which has the exact solution u( x) ď&#x20AC;˝ x 2 .

Differentiating (43) w.r.t x and Using Leibnitz rule (11), its equivalent differential equation, u ''( x) ď&#x20AC;˝ 2

assume that, MMSE Journal. Open Access www.mmse.xyz

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u ''( x) ď&#x20AC;˝ ď&#x192;Ľ ai hi ( x)

(45)

i ď&#x20AC;˝1

Integrating (45) twice, we get N

u '( x) ď&#x20AC;˝ ď&#x192;Ľ ai p1,i ( x)

(46)

i ď&#x20AC;˝1

N

u ( x) ď&#x20AC;˝ ď&#x192;Ľ ai p2,i ( x)

(47)

i ď&#x20AC;˝1

Substituting (45) - (47) in the differential equation (44), we get the system of N equations with N unknowns N

ď&#x192;Ľ a h ( x) ď&#x20AC;­ 2 ď&#x20AC;˝ 0 i ď&#x20AC;˝1

(48)

i i

Solving (48) using Newtonâ&#x20AC;&#x2122;s Method to obtain Haar wavelet coefficients đ?&#x2018;&#x17D;đ?&#x2018;&#x2013; â&#x20AC;&#x2122;s for any N values the coefficients are zero. Substituting these coefficients in (47) and obtained the required LHWCM solutions, which gives the exact solutions is presented in fig 3. This justifies the efficiency of the LHWCM. 1 LHWCM EXACT

0.9 0.8

Numerical Solution

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

Fig. 1. Comparison of LHWCM with exact solution for N = 64 of example 1.

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1

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0.9 0.8

Numerical Solution

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Fig. 2. Comparison of LHWCM with exact solution for N = 64 of example 2.

1 LHWCM EXACT

0.9 0.8

Numerical Solution

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Fig. 3. Comparison of LHWCM with exact solution for N = 64 of example 3.

Summary. The aim of this work is to apply the Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equations of the second kind. Nonlinear integro-differential equations are usually difficult to solve analytically. In many cases, it is required to obtain the approximate solutions, for this purpose the presented method is proposed, which gives the exact ones.

Our present method avoids the tedious work, it minimizes the computational calculus and supplies quantitatively reliable results. Using Leibnitz rule, converts integral equations into differential equations with initial conditions. The Haar wavelet function and its operational matrix were employed to solve the resultant differential equations. The results obtained by the proposed method have been compared with the method [3, 4]. The illustrative examples have been included to justify the efficiency and which confirms plausibility of new technique.

Acknowledgement The authors thank for the financial support of UGCâ&#x20AC;&#x2122;s UPE Fellowship vide sanction letter D. O. No. F. 14-2/2008(NS/PE), dated-19/06/2012 and F. No. 14-2/2012(NS/PE), dated 22/01/2013.

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References [1] Y. Ordokhani, S. Davaei, Application of the Bernstein Polynomials for Solving the Nonlinear Fredholm Integro-Differential Equations, J. Appl. Math. Bio. 1(2) (2011) 13-31.

[2] O. A. Arqub, M. Al-Smadi, N. Shawagfeh , Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method, Appl. Math. Comp. 219 (2013) 8938–8948.

[3] A. Shahsavaran, A. Shahsavaran, Application of lagrange interpolation for Nonlinear integro differential equations, Appl. Math. Sci. 6 (2012) 887 – 892.

[4] F. Mirzaee, E. Hadadiyan, A collocation method to the solution of Nonlinear fredholmhammerstein integral and integro-differential equations, J. Hyper. 2 (1) (2013) 72-86.

[5] C. K. Chui, Wavelets: A Mathematical Tool for Signal Analysis, SIAM. Philadelphia. PA. 1997.

[6] G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Commun. Pure Appl. Math. 44 (1991) 141–183.

[7] Ü. Lepik, E. Tamme, Application of the Haar wavelets for solution of linear integral Equations, Ant. Turk–Dynam. Sys. Appl. Proce. (2005) 395–407.

[8] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, Numerical solution of stiff systems from nonlinear dynamics using single term haar wavelet series, Inter. Jour. Nonlin. Dynam. 51 (2008) 595605.

[9] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J. Comp. Appl. Math. 219 (2008) 90-101.

[10] N. M. Bujurke, S. C. Shiralashetti, C. S. Salimath, An Application of Single Term Haar Wavelet Series in the Solution of non-linear oscillator Equations, J. Comp. Appl. Math. 227 (2010) 234-244.

[11] N. M. Bujurke, C. S. Salimath, R. B. Kudenatti, S. C. Shiralashetti, A Fast Wavelet- Multigrid method to solve elliptic partial differential equations, Appl. Math. Comp. 185 (2007) 667-680.

[12] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinear integral equation, Appl. Math. Comp. 167(2) (2005) 1119–1129.

[13] V. Mishra, H. Kaur, R. C. Mittal, Haar wavelet algorithm for solving certain differential, integral and integro-differential equations, Int. J. of Appl. Math. Mech. 8 (6) (2012) 69-82.

[14] S. Islam, I. Aziz, B. Sarler, The numerical solution of second order boundary value problems by collocation method with the Haar wavelets, Math. comp. Model. 52 (2010) 1577-1590.

[15] S. Islam, I. Aziz, M. Fayyaz, A new approach for numerical solution of integro-differential equations via Haar wavelets, Int. J. Comp. Math, 90 (9) (2013) 1971-1989.

[16] Ü. Lepik, Haar wavelet method for nonlinear integro-differential equations, Appl. Math. Comp 176 (2006) 324-333.

[17] A. M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, 2011.

Cite the paper C. Shiralashetti & R. A. Mundewadi (2016). Numerical Solution of Nonlinear Fredholm IntegroDifferential Equations using Leibnitz-Haar Wavelet Collocation Method. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.31444.19848

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An Equivalent Beam Model for the Dynamic Analysis to a Feeding Crane of a Tall Chimney. Application in a Coal Power Plant Viorel-Mihai Nani1, 2, a, Ioan Cires3, b 1 – University Politehnica Timisoara, Research Institute for Renewable Energy, G. Muzicescu Street, no 138, Timisoara 300774, Romania 2 – University ″Ioan Slavici″ Timisoara, Faculty of Engineering, Paunescu Podeanu Street, no 144, Timisoara, 300568 Romania 3 – SC PRO ATLAS ING SRL Timisoara, Liniștei Street, no 52, First Room, Timisoara, Romania 300288 a – viorel.nani@upt.ro, viorelnani@yahoo.com b – proatlasing@yahoo.com DOI 10.13140/RG.2.2.33544.62720 Keywords: feed crane, latticed boom, equivalent beam, balancing motion, strain gauge, conservation of energy.

ABSTRACT. The paper presents a dynamic analysis for a special crane, which serves a coal power plant. The steel cables for the lifting mechanisms of crane are long and flexible. For this reason, when is feeding the tall chimney, its can appear dangerous dynamic effects due to the suspended load. This load can perform oscillations or vibration movements. As a result, the suspended load position is sometimes difficult to control. Through experimental researches, using a special fitting with strain gauges and accelerometers assembled along the crane’s arm as a beam, we have obtained relevant information. Using the initial design data, we were able to develop an optimal nonlinear dynamic model. This one was the experimental support for other simulations in extremely dangerous situations, like: the accidental fall of the suspended load from the crane hook or a mechanical strong shock due to the collision between the suspended load and the tall chimney wall or the power plant wall, under the strong wind conditions, for example.

Introduction. A crane is a type of machine, equipped in the main with a hoist rope, wire ropes or chains and pulleys, which can be used both to lift the lower materials but also to move them horizontally. It is mainly used for lifting heavy loads and transporting them to other places. Generally, by their construction, the cranes are strong mechanical structures, which must allow loads handling, regardless of the external environmental conditions. A crane model is shown in Fig. 1. Graziano F. and Michel G. [1] studied more applications for the cranes under loads moving, Maczynski A. and Wojciech S. [2] have stabilized the load's position to offshore cranes and Zhou Y. and Chen S. [3] have investigated the cables breakage, but very important is that the cranes should provide the elimination of injury risk, even if this one results from abnormal predictable situations. The cranes and their lifting accessories are subject to random dynamic loads, whose emergence, size and direction of movement are very difficult to control. According to Cioara T.Gh. et al [4] and Gabbal R.D. and Simin E. [5], speed, intensity and the wind direction or the malfunction of lifting loads systems (as vibratory shock load), represent causes, which may endanger the strength crane structure.

The study of percussive systems movements with a linear or nonlinear elastic characteristic was the subject of the dynamic models with more freedom degrees, which were developed by Awrejcewicz [6] and Kwon D.K. et al [7].

Getter D.J. et al [8] showed that the wind maybe causes certain dynamic and complex loads relative MMSE Journal. Open Access www.mmse.xyz

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to some structures located in free air conditions, as: buildings, industrial installations, communications antennas and many others. These structures must to be designed so that to withstand at wind loads corresponding to the maximum intensity of these areas, as a strong gale (mean speed 22-25 m/s) or a storm (30-35 m/s), e.g. For the design, control and verification of the cranes working in such conditions, the usual methods for the vibrations study are no longer applicable.

Being equivalence with a mechanical structure having several degrees of freedom, every crane can be modeled by associating with a function of unilateral connections which was formulated by Paraskevopoulos E. and Natsiavas S. [9]. The function of unilateral connections is the analytical form for representation the degrees of freedom and allows the study all possible cases of movements for the mechanical structures requested to variable loads. Mainly, the function of unilateral connections allows the analytical transposition of strikes (percussions), so that to be applied the general methods of analytical mechanics, like Lagrangeâ&#x20AC;&#x2122; equations or the conservation of energy, e.g. In this way, studying the dynamic behavior of crane in the strong wind conditions or under vibrational shock loads of the load lifting system, Harris C.M. and Piersol A.G. [10] and Silva C.W. [11] have imposed the need for determination of own frequencies to eliminate the resonance frequencies and to ensure the stability in operation.

Fig. 1. A feeding crane model used to serving a coal power plant.

Modelling of the testing plant.

The modelling and investigation of the feed crane was conducted in normal operating conditions. An overview of the crane that feeds the tall chimney of a coal power plant is shown in Fig. 2.

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Fig. 2. Overview of a crane, serving the tall chimney of a coal power plant.

Into detailed shape, a bottom view of the loading crane is illustrated in Fig. 3.

Fig. 3. Bottom view of crane, loaded with a load P. For a correct application of the unilateral connections function, we have attached to the mathematical model a coordinate system conventional established, 0XYZ. From constructively point of view, the crane is composed of a pivoting latticed boom 1 which is located at approx. 111.5 m above the ground, on the coal power plant platform serving a tall chimney (see Fig. 2 and 3). The pivoting latticed boom 1 can be considered as a special structure because the suspended load 5 is lifted from ground to the supply platform situated about 120 m, without to be guided. For local manipulations of low loads, is used another supplementary pivoting lattice boom 2. The lifting of load 5, it is achieved with the help of two pulley tackle 3 (upper sheave) and 4 (lower sheave).

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The load space is limited by the power plant wall and body of the tall chimney. In a possible oscillation movement of suspended load during lifting, it may come in contact with the walls of neighboring constructions. For this reason, Lukasz D. [12] showed that the position of the suspended load should be kept permanent under control.

In view to get information on modal components tensioned in the working time of crane, we have used two sensors - see Fig. 4 -. A strain gauge T1 it was applied on one of the four longerons of the pivoting latticed boom 1, in the middle distance L/2. This one is placed in a Wheatstone bridge circuit (the strain gauge itself is one of four resistances), and it measures elastic deformations of the pivoting jib in a critical section Su. We used the DY1x type as strain gauge, with the 2 parallel measuring grids for measurements on bending beams.

Fig. 4. Schematic diagram of the crane, serving the tall chimney of a coal power plant.

To the same distance on longeron we placed an accelerometer Ac having its sensitivity axis to normal on longitudinal axis of the lattice boom. Like accelerometer we used the ADXL103 model. ADXL103 is high precision, low power, dual axis, with signal conditioned of voltage outputs, in a monolithic structure. The measuring range started from the nil frequency, so that the accelerometer was able to measure the dynamic component of static or quasi-static inclination. Both signals, from strain gauge and the accelerometer, they were amplified, modulated, converted and then wireless transmitted to a receiver, which was connected to a computer.

The constructive modelling.

According to the equations of the rotating motion of a composite beam, Georgiadis F. et al [13] demonstrated that for a discrete approximation of dynamic model, the pivoting latticed boom 1 can be considered as an equivalent elastic bending beam. This one - see Fig. 5 - it hinged under the angle Îą0 of the pivoting position due to the pulley tackle 3. From the dynamic study of a special crane, Cioara T.Gh. et al [4] showed that pulley tackle have an equivalent stiffness kt.

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Fig. 5. The scheme of the equivalent beam model of the feeding crane.

Lifting the load 5 is performed by another pulley tackle 4, having an equivalent stiffness kp. According to observed experimental data by Buckham B. et al [14] and Crellin E. et al [15], the instantaneous position of any section Su, along the equivalent beam at the pivoting angle α + α0, can be expressed in a reference system XOZ, by coordinates:

xu  u cos (   0 )  f (u ) q(t ) sin (   0 ) zu  u sin (   0 )  f (u ) q(t ) cos (   0 )

(1)

where u – coordinate of the section Su in undeformed status; f(u) – the arrow elastic deformation of the equivalent beam requested to bending according to first natural mode of the vibration motion, q(t); α0 – the initial geometric position of the equivalent beam; α – the angular variation of the equivalent beam position under dynamic loads The shape of the first vibration’ structure mode, where the stiffness of the tall chimney and the coal power plant wall are bigger that the crane elements, it can be approximated by a static distribution.

In case where the load P has a uniform distribution along the beam of length L = 42.7 m, the elastic arrow can be calculated after Bruno D. et al [16] with:

f st 

R  (1  2  2   3 ) 24 ,

with the notations MMSE Journal. Open Access www.mmse.xyz

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ď ¸ď&#x20AC;˝

u ; L

Rď&#x20AC;˝

p L4 cos ď Ą 0 EI

(3)

where E â&#x20AC;&#x201C; a constant (Young's modulus); I â&#x20AC;&#x201C; the inertial moment into a cross section of equivalent beam For the equivalent beam section, inertial moment is: â&#x201E;&#x17D; 2

đ??ź = 4 (2) đ??´;

đ?&#x153;&#x2039;

đ??´ = 4 (đ??ˇđ?&#x2018;&#x2019;2 â&#x2C6;&#x2019; đ??ˇđ?&#x2018;&#x2013;2 )

(4)

The mathematical modelling.

Based on the fundamentals of structural dynamic established from Craig R. and Kurdila A. [17], the mathematical modelling of crane was made by analyzing the energy balance in critical sections.

Therefore: Kinetic energy Using the interface stresses and fracture energies by Bruno D. et al [16], kinetic energy of the equivalent beam has the integral form:

Ek ď&#x20AC;˝

1L 2 2 ď&#x192;˛ ď ˛ (x ď&#x20AC;Śu ď&#x20AC;Ť yď&#x20AC;Ś u ) du 20

(5)

where Ď = p/g (g = 9.81 m/s2) is mass per unit length of the equivalent beam.

Taking into account a modal distribution along the length L of equivalent beam and the decomposition of governing the equations of nonlinear system formulated by Awrejcewicz J. et al [18], the kinetic energy expression (5) becomes:

Ek ď&#x20AC;˝

1 1 J Îąď&#x20AC;Ś 2 ď&#x20AC;Ť S q qď&#x20AC;Ś 2 ď&#x20AC;Ť SÎąq Îąď&#x20AC;Ś qď&#x20AC;Ś 2 2

(6)

Where L

L

L

0

0

0

J ď&#x20AC;˝ ď&#x192;˛ u 2 Ď du ; S q ď&#x20AC;˝ ď&#x192;˛ Ď (f(u)2 ) du ; S Îąq ď&#x20AC;˝ ď&#x192;˛ Ď f(u) u du

(7)

and these values can be calculated from the design data.

The second kinetic energy of dynamic system which was studied by Cires I. and Nani V.M. [19], is a load considered as a point mass in the gravity center Gs, having the coordinates:

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xs ď&#x20AC;˝ L cos ď Ą ď&#x20AC;­ ( w0 ď&#x20AC;Ť w) sin ď § z s ď&#x20AC;˝ L sin ď Ą ď&#x20AC;­ ( w0 ď&#x20AC;Ť w) cos ď §

(8)

so that the kinetic energy of load becomes: 1

đ??¸đ?&#x2018;&#x2DC; = 2 đ?&#x2018;&#x161;đ?&#x2018; (đ?&#x2018;Ľđ?&#x2018; 2 + đ?&#x2018;§Ě&#x2021;đ?&#x2018; 2 )

(9)

which, by substitution (8) into (9), leads to final form:

1 1 1 Ek ď&#x20AC;˝ ms ( Lď Ąď&#x20AC;Ś ) 2 ď&#x20AC;Ť ms ( wď&#x20AC;Ś ) 2 ď&#x20AC;Ť ( w0ď §ď&#x20AC;Ś ) 2 ď&#x20AC;Ť ms L ď Ąď&#x20AC;Ś wď&#x20AC;Ś cos (ď Ą 0 ď&#x20AC;­ ď § ) ď&#x20AC;­ 2 2 2 ď&#x20AC;­ ms l ď Ąď&#x20AC;Ś w0 ď §ď&#x20AC;Ś sin (ď Ą 0 ď&#x20AC;Ť ď § ) ď&#x20AC;Ť ms w0 wď&#x20AC;Ś ď §ď&#x20AC;Ś sin 2ď § )

(10)

Strain energy The strain energy of the dynamic system under the equivalent beam form it consists of three components: a) The binding energy of the equivalent beam 1 under its own weight Using the computational dynamics of an elastic string pendulum attached to a rigid body, which was formulated by Taeyoung L. et al [20], the binding energy of the equivalent beam under its own weight, has the form: đ??¸đ?&#x2018; đ?&#x2018;? =

1 2

đ?&#x2018;&#x2DC;đ?&#x2018;? đ?&#x2018;&#x17E; 2

(11)

Where đ??ż đ?&#x153;&#x2022;2 (đ?&#x2018;&#x201C;(đ?&#x2018;˘))

đ?&#x2018;&#x2DC;đ?&#x2018;? = đ??¸đ??ź â&#x2C6;Ť0 [

đ?&#x153;&#x2022;đ?&#x2018;˘2

2

] đ?&#x2018;&#x2018;đ?&#x2018;˘

(12)

is the equivalent stiffness of the equivalent beam 1. b) The strain energy of beam due to stretching of the hoist ropes in pulley tackles 3 According to Koh C. et al [21] who studied the low-tension cables dynamics, the kt elastic stiffness of the hoist ropes in the pulley tackle 3 is:

đ?&#x2018;&#x2DC;đ?&#x2018;Ą = đ?&#x2018;&#x203A;đ?&#x2018;?

đ??¸đ?&#x2018;?đ?&#x2018;Ą đ??źđ?&#x2018;?đ?&#x2018;Ą đ??´đ??ľ

where Ept and Ipt are Youngâ&#x20AC;&#x2122;s modulus and the inertial moment of a single strand; nc â&#x20AC;&#x201C; number of stranded wire MMSE Journal. Open Access www.mmse.xyz

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The strain energy due to stretching of the hoist ropes in pulley tackle 3 is after Buckham B. et al [14] and Koh C. et al [21]:

đ??¸đ?&#x2018; đ?&#x2018;?đ?&#x2018;Ą 3 =

1 2

đ?&#x2018;&#x2DC;đ?&#x2018;Ą [(đ??´đ??ľ)đ?&#x2018;Ą â&#x2C6;&#x2019; (đ??´đ??ľ)0 ]2

(14)

where: (AB)t - instantaneous distance at time t between two points, A and B; (AB)0 - is same distance having the system unloaded corresponding to angular position Îą0 of the equivalent beam From the geometric conditions:

( AB) 0 ď&#x20AC;˝ ( L cos ď Ą 0 ) 2 ď&#x20AC;Ť ( L sin ď Ą 0 ď&#x20AC;­ H ) 2

(15)

the instantaneous position can be considered as a small variation of the Îą parameter (Îą < 50, sinÎą â&#x2030;&#x2C6; Îą, cosÎą â&#x2030;&#x2C6; 1), which is added to Îą0, resulting: ( AB)t ď&#x20AC;­ ( AB) 0 ď&#x20AC;˝ Rď Ą ď Ą

(16)

where:

Rď Ą ď&#x20AC;˝

ď&#x20AC;­ L H cos ď Ą 0 ( L cos ď Ą 0 ) 2 ď&#x20AC;Ť ( L sin ď Ą 0 ď&#x20AC;­ H ) 2

(17)

Finally, from (14), we obtained đ??¸đ?&#x2018; đ?&#x2018;?đ?&#x2018;Ą 3 =

1

đ?&#x2018;&#x2DC;đ?&#x2018;Ą đ?&#x2018;&#x2026;đ?&#x203A;ź2 đ?&#x203A;ź 2

2

(18)

c) The strain energy of beam due to stretching of the hoist ropes in pulley tackle 4 This energy is resulting from the load lifting motion variation. The elastic kp stiffness of the hoist ropes in the pulley tackle 4 is according to Buckham B. et al [14] and Koh C. et al [21]:

k p ď&#x20AC;˝ nc

E pt I pt w0

(19)

where: Ept and Ipt are Youngâ&#x20AC;&#x2122;s modulus and the inertial moment of a single strand; nc â&#x20AC;&#x201C; number of stranded wire; w0 - initial position of load 5 Thus, like section b), the strain energy due to stretching o the hoist ropes in the pulley tackle 4 is MMSE Journal. Open Access www.mmse.xyz

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đ??¸đ?&#x2018; đ?&#x2018;?đ?&#x2018;Ą 5 =

1 2

đ?&#x2018;&#x2DC;đ?&#x2018;? đ?&#x2018;¤ 2

(20)

Potential energy Because we studied the movements in a vertical plane, for the energetic balancing of mechanical ensemble, we must take into account from the potential energy. After the applications of the beams under moving loads described in Graziano F. and Michel G. [1], the potential energy for the beam deformation, it has the form:

E p b ď&#x20AC;˝ g mb

L sin (ď Ą ď&#x20AC;Ť ď Ą 0 ) ď&#x20AC;Ť ď ˛ g q cos (ď Ą ď&#x20AC;Ť ď Ą 0 ) F ; 2

F ď&#x20AC;˝

L

f (u ) du

(21)

đ??¸đ?&#x2018;?đ?&#x2018;&#x2122; = đ?&#x2018;&#x161;đ?&#x2018; đ?&#x2018;&#x201D; đ?&#x2018;§đ?&#x2018; = đ?&#x2018;&#x161;đ?&#x2018; đ?&#x2018;&#x201D; [đ??ż đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x203A;ź + đ?&#x203A;ź0 ) â&#x2C6;&#x2019; (đ?&#x2018;¤ + đ?&#x2018;¤0 )(1 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x203A;ž)]

(22)

ď&#x192;˛

0

and for any given load, we can calculate:

The calculation formulas of all energies developed above were used to build a system of differential equations governing the feeding crane movements, which are defined by the column vector developed

by Georgiadis F. et al [13]: {đ?&#x203A;š(đ?&#x2018;Ą)} = {đ?&#x203A;ź(đ?&#x2018;Ą)

đ?&#x2018;&#x17E;(đ?&#x2018;Ą)

đ?&#x2018;¤(đ?&#x2018;Ą)

đ?&#x153;&#x2020;(đ?&#x2018;Ą)}đ?&#x2018;&#x2021;

(23)

and using Lagrange's relationship: đ?&#x2018;&#x2018; đ?&#x153;&#x2022;đ??¸ ( đ?&#x2018;? ) đ?&#x2018;&#x2018;đ?&#x2018;Ą đ?&#x153;&#x2022;{đ?&#x203A;šĚ&#x2021;}đ?&#x2018;&#x2021;

â&#x2C6;&#x2019;

đ?&#x153;&#x2022;đ??¸đ?&#x2018;&#x2DC; đ?&#x153;&#x2022;{đ?&#x203A;š}đ?&#x2018;&#x2021;

+

đ?&#x153;&#x2022;đ??¸đ?&#x2018; đ?&#x153;&#x2022;{đ?&#x203A;š}đ?&#x2018;&#x2021;

+

đ?&#x153;&#x2022;đ??¸đ?&#x2018;? đ?&#x153;&#x2022;{đ?&#x203A;š}đ?&#x2018;&#x2021;

= {0}

(24)

we obtained the general equation of free vibrations undamped for the entire mechanical structure:

[đ?&#x2018;&#x20AC;]{đ?&#x203A;šĚ&#x2C6;} + [đ??ž]{đ?&#x203A;š} = {0}

(25)

In assumption that the tall chimney stiffness and the coal power plant wall stiffness are too bigger compared to the crane elements, then [M] and [K] are:

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ď&#x192;Š J ď&#x20AC;Ť ms L2 S ď Ąq ď&#x192;Ş S ď Ąq 0 ď&#x192;Ş [M ] ď&#x20AC;˝ ď&#x192;Ş 0 0 ď&#x192;Ş ď&#x192;Şď&#x192;Ť ď&#x20AC;­ ms L w0 sin ď Ą 0 0

ď&#x192;Š 2 ď&#x192;Şkt Rď Ą ď&#x20AC;­ gmb ď&#x192;Ş [K ] ď&#x20AC;˝ ď&#x192;Ş ď&#x192;Ş ď&#x192;Ş ď&#x192;Şď&#x192;Ť

ď&#x20AC;­ ms L w0 sin ď Ą 0 ď&#x192;š ď&#x192;ş 0 0 ď&#x192;ş ď&#x192;ş ms 0 ď&#x192;ş ď&#x192;şď&#x192;ť 0 ms

(26a)

ď&#x192;š 0ď&#x192;ş ď&#x192;ş 0 ď&#x192;ş kb 0 ď&#x192;ş ď&#x192;ş 0 ms gw0 ď&#x192;şď&#x192;ť

(26b)

0

L sin ď Ą 0 ď&#x20AC;Ť gms L cos ď Ą 0 0 0 2 0 kb 0 0

0

0

0

where [M] â&#x20AC;&#x201C; is equivalent mass; [K] â&#x20AC;&#x201C; is equivalent stiffness of the crane; {đ?&#x203A;š} â&#x20AC;&#x201C; is a random conventional variable and represents the function of unilateral connections Simulation and experimental results.

Experimental simulation was performed under actual conditions operating of the feeding crane. It has been placed into a dangerous area. When the tall chimney of the coal power plant is feeding can appear more rocking situations of load during the load lifting. The load can touch such the power plant walls or the tall chimney construction. In this way, the danger of an accident is imminent.

At various heights, we intentionally simulated few oscillations of the lifting load, in a plane parallel to the walls of adjacent buildings. The amplitude of oscillation it was high enough to enter into the dangerous area operating.

Experimentally, it was raised a load of 6000 kg. The load was subjected to forced oscillations in 5 different areas on the lifting height: 5, 30, 60, 90 and 110 m above the ground. In each area, the two sensors (strain gauge and the accelerometer) have provided on-line information regarding to the vibrational amplitude variation and stresses induced into the pivoting latticed boom 1 of the feeding crane, namely in Su critical section. The relative position of the crane boom compared to the plane XOZ it was determined using a MTS360-232 clinometer.

Results were considered relevant to a loading height of 5 m above the ground. To this height, the oscillation motion was sufficient to determine energies, frequency and vibration amplitude which led to structural changes in the critical section.

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Fig. 6. The time variation of stresses and the angle position α=230-720.

In Fig. 6 are presented the diagrams of the two signals. In the first phase, it is observed a slow form of tilt to the angular position α = 230 to 700 during of 600 s, without that the load ms to be hanged. In these conditions, close to a static functioning, the stresses T into hoist ropes of the pulley tackle 4 from the pivoting latticed boom 1, are:

T  g

L L cos  0 2 sin ( 0   )

(27)

The two components of the tension T are: the normal tension Tn, which is perpendicular to the beam axis, and also the axial tension N: Tn  T sin ( 0   ) ;

N  T cos ( 0   )

(28)

The bending moment in the middle section of the beam, at L/2 is:

M i  Tn

L L2   g cos  0 2 4

(29)

and the axial force in the same section has the form:

L N m  N   g sin  0 2 MMSE Journal. Open Access www.mmse.xyz

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Strain ε, at the location where sensors were positioned on the longeron, can be calculated using formula developed by Crocker [22]:

 

1  M i h Nm     E  2 I 4 A 

which is in accordance with experimental researches, the maximum compression strain being attained at -80 μstrain (μPa), relative to position α0 of the beam.

The dynamic effect was more intense at the test end, when the rocking movement of the beam it was stopped to angular position α0≈ 300 (detail A of Fig. 6). At that time, we recorded a transient dynamic strain by magnitude 40 μstrain (graph a), Fig. 7). This one contain a modal component frequency at f = 0.12378 Hz (graph b), Fig. 7), which was confirm in Nastac S. and Leopa A [23].

Fig. 7. Time history a), and its spectrum b), for strain during a transient break in a particular position of the pivoting latticed boom.

In detail B of Fig. 6 and in the next chart of Fig. 8 are presented the diagrams of time history for dynamic strain and its spectrum, which were recorded during lifting the load having the mass of ms = 6 000 kg.

The value of transient dynamic strain is approximately 60 μstrain (graph a), Fig. 8). From the time history diagrams and the spectrum’s strain, results two peaks, one which corresponds to frequency f= 0.04503 Hz (graph b), Fig. 8), and the other having same frequency of 0.12378 Hz as in spectrum b) (graph b), Fig. 7). Measured value 0.04503 Hz it corresponds to the loads of a pendulum hinged by a long cable w0 = 110-120 m, resulting from

f 

1 2

g  0.045 Hz w0

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Fig. 8. Time history a), and its spectrum b) of a strain during lifting the load.

The second measured value of frequency 0.12378 Hz it corresponds to a rocking denoted α of the vibration motion; this frequency component belong to both situations, when the rocking movement of the beam is stopped at angular position α0 ≈ 300 during lifting the load, having a mass ms = 6 000 kg.

Discussions.

The simulation experiment confirmed the mathematical model presented at point 3 (see relationships 24 and 25), based on the function of unilateral connections and conservation of energy applied on the equivalent beam. Results obtained have provided sufficient reliable data for adopting some constructive solutions to limit and avoid the mechanical stresses close to critical values.

In extremely dangerous situations, automatically crane it stops and it is locks lifting the load, eliminating the human operator intervention. If the dynamic structure of the crane not it stabilizes, then all automatically the load is lowered urgently on the ground.

For this reason, in such industrial applications, is sometimes necessary the adoption of a special construction for the feeding crane safety.

Summary. From the experimental research made in normal operating conditions of the feeding crane, the following conclusions can be drawn:  Dynamic effect is more intensive at the end of operation, when the rocking motion of the beam is stopped at angular position α0 ≈ 300 and during lifting the load having a mass ms = 6 000 kg.  During the transient dynamic period, can appear a variable strain having the magnitude 40- 80 μstrain, which is sometimes dangerous for the mechanical structure of crane.  From graph time, results two peaks of low frequency: - the frequency f=0.04503 Hz correspond to the oscillating load, where the load is hinged by a long cable; in this case, the crane operator should avoid the starting/stopping with shocks; - the frequency f=0.12378 Hz correspond to the pendulum motion, so the crane operator should avoid the crane's working in strong winds.

An actual and important objective for designers is to design a new pivoting jib crane as a lattice beam in a lighter structure. From the experimental simulation data, we found that there is the possibility to adopt a flexible constructive structure for the same nominal load lifted.

This paper presents only a portion of experimental undertaken researches, which can continues in the future. The results obtained are useful for designer to optimizing the crane conception in a light structure, flexible and resistant to variable and random loads.

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This objective refers to improving the performances of crane, in particular to the operational reliability at high speeds by operating.

Acknowledgments: The authors acknowledge the contribution of Gh. Silas and T.Gh. Cioara for their help in system theoretical development.

Funding: The study was funded from private funds according to Research Contract no 120/20122013 entitled „Optimal design of a pivoting beam”, between University Politehnica Timisoara and ROMINEX SA Timisoara.

References [1] Graziano F., Michel G., Applications of influence lines for the ultimate capacity of beams under moving loads, Engineering Structures, 103, 125-133, (2015), DOI: 10.1016/j.engstruct.2015.09.003 [2] Maczynski A., Wojciech S., Stabilization of load’s position in offshore cranes, Journal of Offshore Mechanics and Arctic Engineering 134(2), 1– 10, (2012), DOI: 10.1115/1.4004956 [3] Zhou Y., Chen S., Numerical investigation of cable breakage events on long-span cable-stayed bridges under stochastic traffic and wind, Engineering Structures, 105, 299-315, (2015), DOI: 10.1016/j.engstruct.2015.07.009 [4] Cioara T. Gh., Cires I., Nicolae I., Cristea D., Tirlea A., Timar L., Dynamic study of a special crane serving a power plant tall chimney, IMAC – XXVI:A Conference and Exposition on Structural Dynamics, Rosen Shingle Creek Resort and Golf Club Orlando, Florida, USA, February 4-7, (2008) [5] Gabbal R.D., Simiu E., Aerodynamic damping in the along-wind response of tall buildings, Journal of Structural Engineering, 136 (1): 117-9, (2010), DOI: 10.1061/(ASCE)07339445(2010)136:1(117) [6] Awrejcewicz J., Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems, Springer Science Business Media B.V., (2008) [7] Kwon D.K., Kareem A., Stansel R., Bruce R.E., Wind load factors for dynamically sensitive structures with uncertainties, Engineering Structures, 103, 53-62, (2015), DOI: 10.1016/j.engstruct.2015.08.031 [8] Getter D.J., Davidson M.T., Consolazio G.R., Patev R.C., Determination of hurricane-induced barge impact loads on floodwalls using dynamic finite element analysis, Engineering Structures, 104, 95-106, (2015), DOI: 10.1016/j.engstruct.2015.09.021 [9] Paraskevopoulos E., Natsiavas S., Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems, International Journal o Non-Linear Mechanics, 77, 208-222, (2015), DOI: 10.1016/j.ijnonlinmec.2015.07.007 [10] Harris C.M., Piersol A.G., Shock and Vibration Handbook, McGrower Hill Book Co, (2002) [11] Silva C.W., Vibration and Shock Handbook, Taylor&Francis Group, LLC, (2005) [12] Lukasz D., Application of dynamic optimization to the trajectory of a cable-suspended load, Nonlinear Dynamic, Publish online (Jan 14, 2016), DOI: 10.1007/s11071-015-2593-0 [13] Georgiadis F., Latovski J., Warminski J., Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle, Meccanica, 49, 1833-1858, (2014), DOI: 10.1007/s11012-014-9926-9 [14] Buckham B., Driscoll F., Nahon M., Development of a finite element cable model for use in lowtension dynamics simulation, Journal of Applied Mechanics, 71, 476–485, (2004), DOI: 10.1115/1.1755691

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[15] Crellin E., Janssens F., Poelaert D., Steiner W., Troger H., On balance and variational formulations of the equations of motion of a body deploying along a cable, Journal of Applied Mechanics, 64, 369–374, (1997), DOI: 10.1115/1.2787316 [16] Bruno D., Greco F., Lo Feudo S., Blasi P.N., Multi-layer modeling of edge debonding in strengthened beams using interface stresses and fracture energies, Engineering Structures, 109, 2642, (2016), DOI: 10.1016/j.engstruct.2015.11.013 [17] Craig R., Kurdila A., Fundamentals of structural dynamics, John Wiley&Sons Inc., (2006) [18] Awrejcewicz J., Starosta R., Sypniewska-Kaminska, G., Decomposition of governing equations in the analysis of resonant response of a nonlinear and non-ideal vibrating system, Nonlinear Dynamics, 82, 299–309, (2015), DOI: 10.1007/s11071-015-2158-2 [19] Cires I., Nani V.M., Stability control of a huge excavator for surface excavation, Applied Mathematical Modelling, 40, 388–397, (2016), DOI: 10.1016/j.apm.2015.04.056 [20] Taeyoung L., Melvin L., McClamroch N.H., Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism, Nonlinear Dynamics, 64, 97–115, (2011), DOI: 10.1007/s11071-010-9849-5 [21] Koh C., Zhang Y., Quek S., Low-tension cable dynamics. Numerical and experimental studies, Journal of Engineering Mechanics, 125(3), 347–354, (1999), DOI: 10.1061/(ASCE)07339399(1999)125:3(347) [22] Crocker M., Handbook of Noise and Vibration Control, John Wiley&Sons Inc., (2007) [23] Nastac S., Leopa A., Structural Optimization of Vibration Isolation Devices for High Performances, International Journal of Systems Applications, Engineering&Development, Issue 2, Volume 2, 66-74, (2008) Cite the paper Viorel-Mihai Nani & Ioan Cires (2016). An Equivalent Beam Model for the Dynamic Analysis to a Feeding Crane of a Tall Chimney. Application in a Coal Power Plant. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.33544.62720

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Determination of Bond Capacity in Reinforced Concrete Beam and Its Influence on the Flexural Strength Mohammad Rashidi1, Hana Takhtfiroozeh2 1 â&#x20AC;&#x201C; Department of Civil Engineering, Sharif University of Technology, Tehran, Iran 2 â&#x20AC;&#x201C; Department of Civil Engineering, Building and Housing Research Centre, Tehran, Iran DOI 10.13140/RG.2.2.18300.95361

Keywords: flexural strength, bond capacity, tensile bars, reinforced concrete beam, compressive strength

ABSTRACT. This paper presents results of an experimental investigation of actual performance of the reinforced concrete beam in bond under flexure, when reinforced with tension steel is going to consider. In this experiment four specimens of beam and a bar in the middle of the width of the beam has been used and 2.5 cm of concrete cover has been considered from the center of the bar. In addition, transverse bars have been used to reassure lack of shear yield at the two ends of the beam. Flexural bar has been put in the middle of the beam symmetrically and the length of the flexural bar in each of the samples shall be: 15, 20, 30 and 40 cm. Three cylindrical samples were made in order to determine f'c and were examined at 28 days and the compressive strength of concrete used in this study was about 35 MPa. The beam samples were examined after 28 days via two-point loading system. Based on the results, increasing the length of bar causes increase of flexural strength. The presence of longitudinal rebar resulted in the ultimate momentum to be more than the crack momentum of the cross-section in parts which have broken at the point of longitudinal bar cut.

Introduction. Concrete is of a lot of use in constructions due to availability, appropriate compressive strength and ease of implementation; although, its weakness in traction has resulted in not being able to use this material solely in construction. In order to eliminate the weak traction of concrete, usually bar is used in the tensile area of the concrete. The goal of this experiment is to determine the bond strength between steel reinforcing bars and concrete. The main parameters that influence this bond strength are well documented in the technical literature. Important among these parameters include development/splice length, diameter of the reinforcing bar, and concrete compressive strength [1, 2, 3, 4].

The type of cracking leading to failure has been investigated using deformed bars in tension by injecting ink around the bars [5, 6]. The bond strength of rebars in concrete decreases as the embedment length increases, and decreases with increasing the bar diameter [7]. The previous investigations proved that the bond strength of rebars in concrete is influenced by the development length rather than the bar diameter [8]. The ultimate bond strength seems to be a function of c f 'c when other parameters are constant, since the bond strength is related to the tensile strength of concrete. Studies on understanding the nature of bond, modes of failure and factors influencing the failure, bar spacing and beam width, end anchorage, flexural bond and anchorage bond with high strength ribbed bars have been reported [9]. The slip of deformed bars is due to (i) splitting of concrete by wedge action, and (ii) crushing of concrete in front of the ribs [9].

Nilson [10] used slope of steel strain curve to evaluate the bond stress at a given load in reinforcing bar, and a new test method was adopted to study the local slip, secondary cracking and strain distribution in concrete [11]. A bond stress-slip model has been proposed to predict the load end slip and anchorage length of bars extended from adjoining beams in to exterior columns under large nonlinear actions [12]. Effect of bar diameter, confinement and strength of concrete on the bond behaviour of bar hooks in exterior beam-column joints has been reported [13]. The bond strength decreases as the bar diameter increases. The post-peak bond-slip response was not influenced by the bar diameter [14], while confinement has direct influence on the local bond stress [15]. A new bond MMSE Journal. Open Access www.mmse.xyz

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stress-slip response has been simulated recently by Abrishami and Mitchel [16]. However, consistent bond stress-slip response was obtained on short embedment length [17]. A mathematical model for bond stress-slip response of a reinforcing bar due to cyclic load has been reported [19]. Other models to predict the tensile strength of concrete from the pullout load has been reported [20]. Confinement by ordinary steel reinforcement has improved the bond strength with significant ductility [21]. Several studies on bond in normal strength concrete (NSC) have been reported [22]. In high strength concrete (HSC), increasing the development length does not seem to increase the bond strength of deformed bars when the concrete cover is relatively small. A minimum confinement reinforcement needs to be provided over the splice length in RC members when HSC is used [23]. An expression has been proposed to estimate the extra confinement reinforcement [24]. Also more general information on the local bond can be seen in CEB-FIP Report [25].

This paper studies bond capacity in reinforced concrete beam and its influence on the flexural strength. Firstly it introduces materials and test methods. Then it presents the comparison of the results of the experiment with the existing theories.

Materials and methods. In this experiment four specimens of beam and a bar in the middle of the width of the beam has been used and 2.5 cm of concrete cover has been considered from the center of the bar. Also, transverse bars have been used to reassure lack of shear yield at the two ends of the beam. Flexural bar has been put in the middle of the beam symmetrically and the length of the flexural bar in each of the samples shall be: 15, 20, 30 and 40 cm. Three cylindrical specimens were made in order to determine f 'c and were examined at 28 days and the compressive strength of concrete used in this study was about 35 MPa. The beam samples were examined after 28 days via two-point loading system.

The considered mix for the samples has been shown in table 1 below. According to the instructions, coarse aggregates have been sieved via a 2-cm sieve. Also, the samples considered in construction are three cylindrical samples in 30×15 cm dimensions and four beams samples in 60×10×10 cm dimensions.

Due to the fact that the goal of this experiment is to determine the capacity of sliding bar from within the beam; therefore, bars with different lengths in each bar have been applied. Longitudinal bars are of 8mm and transverse bars are of 6mm. The longitudinal bars’ cover for all samples is 2.5 cm and for observing this space, spacer has been used. The existing spacers in the laboratory were of more height; therefore, in order to convert this height to 2.5 cm, we cut them. All the beams have the same shear bar and their design was conducted as over design. Shear bars were placed 5 cm from the bar up to 20 cm with the distance of 5 cm between according to Fig.s 1 to 4.

Table 1. The considered mix for the samples.

Part

Weight Ratio (kg/m3)

Cement

500

Sand

800

Gravel

800

Water

220

Total

2320

With regard to the fact, that the goal of this experiment was determination of the bond strength between steel reinforcing bars and concrete, 4 produced beam samples during the length of the main bar are different and the flexural bar shall be placed symmetrically from the middle of the beam in a manner, that the length of the flexural bar in each of the samples is: 15, 20, 30, 40 cm.

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After reinforcement of samples according to Fig.s 1 to 4, the stages of concreting and curing of concrete shall be conducted and then the samples shall be examined after 28 days of curing.

Dimensions of cylindrical samples and beam samples are also shown in table 2 and 3 respectively.

Fig.1. Samples No. 1, longitudinal bar of 40 cm.

Fig.2. Samples No. 2, longitudinal bar of 30 cm.

Fig.3. Samples No. 3, longitudinal bar of 20 cm.

Fig.4. Samples No. 4, longitudinal bar of 15 cm.

Table 2. Dimensions of Cylindrical Samples.

Sample No. 1 2 3

The Average Diameter (Cm) The Average Height (Cm) 15.2 30.4 15.2 30.3 15.2 30.6

Table 3. Dimensions of the Beam Samples.

Sample No. 1 2 3 4

Length (Cm) 60.20 60.30 60.20 60.25

Width (Cm) 9.95 10.25 10.1 9.95

Height (Cm) 10.10 10.15 10.02 10.25

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It should be noted that the compressive strength test of the samples shall be conducted after cappingthe goal of which is to create a flat surface on the sample.

All the beams which were experimented on were 60 centimetres long they were placed on a 55centimetre- wide support and were loaded and tested. Two concentrated symmetrical loads which were 25 centimetres away from each other were used for loading purposes. The weight of the rods which are placed on the beam was 37.8 kg. The used bars in this experiment are of type A2 and the current strength of 300 mpa. The loading model of the beams can be seen in Fig. 5.

Fig. 5. The loading model of the beam.

With regard to the suggested relationship in the regulations, the amount of modulus of elasticity of concrete is: E  500 f ' c

(1)

Table 4. The Result Modulus of Elasticity of Concrete for the Samples Sample No.

Compressive force KN

Stress Mpa

Ec

1

706.0

39.95

31603

2

730.0

41.30

32132

3

511.6

28.15

26530

Average

649.2

34.46

29351

Discussion of test results. In the flexural load of the reinforced beam, at the beginning of loading due to the lowness in the amount of tensile and compressive stress in concrete, the part operates in an elastic and linear manner. The linear behavior continues until when the stress in the last warp of the tensile area of the cross-section reaches the concrete tensile strength of σr. This stage of the behavior of the bar is known as the elastic stage.

In order to calculate the crack momentum of the concrete cross-sections (meaning the least of flexural momentum which causes the fraction of the cross-section), an approximate but simple method, which is based on the distribution of linear stress and applying module of rupture of concrete, is used. Until the time when the most tensile stress in one flexural cross- section does now exceed the tensile strength of the concrete, the cross- section will remain in the elastic mode. In this mode, the crossMMSE Journal. Open Access www.mmse.xyz

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section will remain uncracked and distribution of stress is linear. Therefore, in order to determine the stress, it is possible to use the classic relations of material strength.

Another matter which is discussed in relation to the elastic stage is the calculation of crack momentum. A simple and practical method which is applied for this calculation, is the use of the tensile equation and limiting the tensile stress of concrete to the module of rupture, σr. Therefore, the order of the relation of determining crack momentum of a tensile cross- section is according to equation 2:

M cr 

fr  I y'

(2)

In which σr is the moment of inertia of the converter cross- section in relation to the neutral axis and y’ is the distance of the furthest tensile warp from the neutral axis.

It is worth mentioning that in the regulations, for the ease of calculation, a simpler relation is suggested instead of the above relation:

M cr 

r  Ig yt

(3)

In which Ig is the moment of inertia of the whole cross- section in relation to the central axis of the cross- section without taking steel into account and yt is the distance of the furthest tensile warp from the central axis of this cross- section. According to the regulations, the module of rupture for concretes with normal weight is kg/ cm2:  r  2 f 'c

(4)

The results of the experiment for different samples of beam are as follows: As can be seen in Fig. 6, the crack has begun precisely from under the load in Sample number 1 in a flexural manner and by increasing the load, the crack progressed and ended at the point of applying the load and the beam failed completely.

Fig. 6. Failure of beam No. 1 MMSE Journal. Open Access www.mmse.xyz

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For sample number 2 according to Fig. 7, the flexural crack was established at the point of cutting the longitudinal bar and developed through increasing the load and caused the beam to break.

Fig. 7. Failure of beam No. 2 Based on Fig.s 8 and 9 for sample number 3 and 4 respectively, the flexural crack was established at the point of cutting the longitudinal bar and developed through increasing the load and caused the beam to break.

Fig. 8. Failure of beam No. 3

Fig. 9. Failure of beam No. 4.

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With regard to the existing methods in the discussion of traction of the parts of the reinforced concrete, in order to calculate crack momentum of the cross- section and the ultimate momentum of the crosssection, the following equations can be applied. While the required equation for calculation the ultimate momentum of the cross- section, with regard to the ultimate compressive force which was tolerated by the cross- section is mentioned below Crack momentum:

đ?&#x2018;&#x20AC;đ?&#x2018;?đ?&#x2018;&#x; =

0.63â&#x2C6;&#x161;đ?&#x2018;&#x201C;Ě đ?&#x2018;? đ?&#x2018;?â&#x201E;&#x17D;2

(5)

6

The ultimate momentum of the cross- section via the Whitney rectangle method: đ??´đ?&#x2018; đ??šđ?&#x2018;Ś

đ?&#x2018;&#x20AC;đ?&#x2018;&#x; = đ??´đ?&#x2018; đ??šđ?&#x2018;Ś (đ?&#x2018;&#x2018; â&#x2C6;&#x2019; 0.6đ?&#x2018;&#x201C; Ě đ?&#x2018;?)

(6)

đ?&#x2018;?

The ultimate momentum on the cross- section: đ?&#x2018;&#x192;

đ?&#x2018;&#x20AC;đ?&#x2018;˘ = 2 Ă&#x2014; 15

(7)

The created momentum in the x distance from the base to the concentrated load: đ?&#x2018;&#x192;

đ?&#x2018;&#x20AC;đ?&#x2018;˘ = 2 Ă&#x2014; đ?&#x2018;&#x2039;

(8)

Considering the above- mentioned equations and the results of the experiment, the below table can be established and it is possible to compare the results of the experiment with the applied theories in the concrete lesson and conduct the required analysis.

Table 5. Comparison of the Results of the Experiment with the Existing Theories.

Sample No.

Compressive Crack Ultimate Momentum Maximum of Created force (KN) Momentum of the Cross- Section Momentum at the Cross(t.m.)

Section (t.m.)

Created Momentum at the Rupture of the Cross- Section (t.m.)

1

26.0

0.065

1.52

0.1950

0.1950

2

12.4

0.065

1.54

0.0930

0.0775

3

11.8

0.062

1.50

0.0885

0.0885

4

10.8

0.064

1.55

0.0810

0.0810

In beams number 3 and 4 the crack was at the place of bar cut and because in this beams the length of the tensile bar was less than the distance between the two point forces, the momentum which caused the beam to break is the maximum momentum forced on the beam which was almost 40% more than the crack momentum of the cross- section. In beam number 2, the tolerated momentum MMSE Journal. Open Access www.mmse.xyz

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was 5% more than beam number 3 and 15% more than beam number 4 which is in accordance with the existing theories and in beam number 2 the momentum is less than the tolerated maximum momentum at the place of the crack which indicates the fact that the length of the applied bar was shorter that what could prevent the beam from breaking.

The results show that increasing the length of the bar results in increase of tolerable momentum by the cross- section. Even in beams number 3 and 4, where the crack was between the two forced loads and occurred at the bar cut, the tolerated momentum in beam number 3 was almost 10% more than beam number 4; whereas, according to the existing methods, the tolerated momentum should be equal at the two cross- sections. While the tolerated momentum in beam number 4 was 8% more than the experimented samples in the flexural experiment of the simple concrete samples, which carries the point that the presence of longitudinal bar influenced the increase in the capacity of the freight of the cross- section.

Through comparing the tolerated momentum by the cross- section and the crack momentum and the ultimate momentum which resulted from the theory, we can reach the conclusion that the momentum which caused the crack in the cross- sections without longitudinal bar (breaking at the longitudinal bar cut) is more than the crack momentum; whereas, it should be equal to the crack momentum.

Although this increase in strength can be due to the safety factors used in the equations and applying these factors is because of problems which exist in performance, such as less strength of concrete compared to the calculated amount. But in this experiment, due to thoroughness in performance and application of compressive strength which resulted from the experiment, the existing error is insignificant and has caused the tolerated momentum to be more than the crack momentum. In beam number 1 the crack was at the point where the bar was longitudinal but the tolerated momentum was far less than the ultimate momentum calculated by the Whitney rectangle method and is in no accordance with the above theory.

Summary. The purpose of this study was to make an effort to determinate bond capacity in reinforced concrete beam and its influence on the flexural strength. The result gained from this study are as follows:  In beams that the length of the tensile bar was less than the distance between the two point forces, the crack took place at bar cut place and because in this beams, the momentum which caused the beam to break is the maximum momentum forced on the beam which was almost 40% more than the crack momentum of the cross- section.  In beam number 2 (the flexural bar of 30cm), the tolerated momentum was 5% more than beam number 3 (the flexural bar of 20cm) and 15% more than beam number 4 (the flexural bar of 15cm) which is in accordance with the existing theories and in beam number 2 the momentum is less than the tolerated maximum momentum at the place of the crack which indicates the fact that the length of the applied bar was shorter that what could prevent the beam from breaking.  Increasing the length of the bar results in increase of tolerable momentum by the cross- section.  The momentum which caused the crack in the cross-sections without longitudinal bar (breaking at the longitudinal bar cut) is more than the crack momentum; whereas, it should be equal to the crack momentum. Although this increase in strength can be due to the safety factors used in the equations and applying these factors is because of problems, which exist in performance, such as less strength of concrete compared to the calculated amount.

References [1] Darwin, D., Zuo, J., Tholen, M.L., and Idun, E.K., Develpomnet length criteria for conventional and high relative rib area reinforcing bars, ACI Structural Journal, No. 3, 93, 347-359, 1993.

[2] Orangun, C.O., and Breen, J. E., Strength of anchored bars: A re-evaluation of test data on development length and splices, Research Report No. 154-3F, Center for Highway Research, University of texas at Austin, Austin, Tex., 78, 1975.

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[3] Orangun, C. O., and Breen, J. E., Reevaluation of test data on development length and splices, ACI Journal, Proceedings, No. 3, 74, 114-122, 1977.

[4] Zuo, J., and Darwin, D., Splice strength of conventional and high relative rib area bars and high strength concrete, ACI Structural Journal, No. 4, 97, 630-641, 2000.

[5] Rehm G, Uber die grundlagen des verbudzwischen stahl undbeton, Heft 138, Deutscher Ausschuss fur Stahlbeton, Berlin, 1961.

[6] Goto Y., Cracks formed in concrete around deformed bars in concrete. ACI Journal 68(2) 244251, 1971.

[7] Mathey RG and W Watstein D, Investigation of bond in beam and pull out specimens with high yield strength deformed bars. ACI Journal T. No.57-50 1071-1089, 1961.

[8] Ferguson PM, Robert I and Thompson JN, Development length of high strength reinforcing bars in bond. ACI Journal T. No.59-17 887-922, 1962.

[9] Lutz LA and Gergely P, Mechanics of bond and slip of deformed bars in concrete. ACI Materials Journal T. No. 64-62 711-721, 1967.

[10] Nilson AH, Internal measurement of bond slips. ACI Journal 69(7) 439-441, 1972.

[11] Jiang DH, Shah SP and Andonian AT. Study of the transfer of tensile forces by bond. ACI Journal T. No.81-24 251-258, 1984.

[12] Ueda T, Lin I and Hawkins NM, Beam bar anchorage in exterior column-beam connections. ACI Structural Journal T. No. 83-41 412-422, 1986.

[13] Soroushian P, Pull out behavior of hooked bars in exterior beam-column connections. ACI Structural Journal 85 269-276, 1988.

[14] Soroushian P and Choi KB, Local bond of deformed bars with different diameters in confined concrete. ACI Structural Journal 86(02) 217-222, 1989.

[15] Soroushian P, Choi KB, Park GH and Aslani F, Bond of deformed bars to concrete: effects of confinement and strength of concrete. ACI Materials Journal 88(3) 227-232, 1991.

[16] Abrishami HH and Mitchel D, Simulation of uniform bond stress. ACI Materials Journal T. No. 89-M18 89(2) 161-168, 1992.

[17] Malvar LJ, Bond of reinforcement under controlled confinement. ACI Materials Journal 89(6) 593-601, 1992.

[18] Yankelevsky DZ, Adin MA and Farhey DN, Mathematical mode0l for bond slip behavior under cyclic loading. ACI Structural Journal 89(6) 692-698, 1992.

[19] Bortolotti , Strength of concrete subjected to pull out load. ASCE Materials Journal 15(5) 491495, 2003.

[20] Harajli MH, Hamad BS and Rteil AA, Effect of confinement on bond strength between steel bars and concrete. ACI Structural Journal 101(5) 595-603, 2004. DOI: 10.14359/13381 [21] Somayaji S and Shah SP, Bond stress versus slip relationship and cracking response of tension members. ACI Journal 78(3) 217–225, 1981.

[22] Azizinamini A, Stark M, Roller JJ and Ghosk SK, Bond performance of reinforcing bars embedded in HSC. ACI Structural Journal 90(5) 554–561, 1993.

[23] Azizinamini A, Pavel R, Hatfield E and Ghosh SK, Behavior of spliced reinforcing bars embedded in HSC. ACI Structural Journal 96(5) 826–835, 1999a.

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[24] Azizinamini A, Darwin D, Eligehausen R, Pavel R and Ghosh SK, Proposed modification to ACI 318-95 tension development and lap splice for high strength concrete. ACI Structural Journal 96(6) 922â&#x20AC;&#x201C;926, 1999b.

CEB-FIP Report, Bond of reinforcement in concrete: state of the art report. FIB Bulletin-10, Switzerland, 2000.

Cite the paper Mohammad Rashidi & Hana Takhtfiroozeh (2016). Determination of Bond Capacity in Reinforced Concrete Beam and Its Influence on the Flexural Strength. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.18300.95361

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Prediction of Rubber Element Useful Life under the Long-Term Cyclic Loads Dyrda V.I.1, Loginova A.A.2, Shevchenko V.G.3 1 – M.S. Polyakov Institute of Geotechnical Mechanics, NAS of Ukraine, Professor, Doctor of Technical Sciences (D. Sc.) 2 – National Mining University, Doctoral Student 3 – M.S. Polyakov Institute of Geotechnical Mechanics, NAS of Ukraine, Senior Researcher, Doctor of Technical Sciences (D.Sc.) DOI 10.13140/RG.2.2.25883.46884

Keywords: rubber element useful life, failure of vibroinsulators, cyclic loads.

ABSTRACT. Problems associated with the change of physical and mechanical properties and structure of resilient rubber elements of the VR type under long-term cyclic loads are considered integrally.

A general algorithm was developed for predicting useful life for the resilient rubber elements, which is based on: interrelated equations of equilibrium and simultaneousness of deformations for determining deflected mode of the vibroinsulators; equations of local useful life; equations of heat conductivity for determining temperature field in the rubber mass; criterion equation of destruction, which connects parameters of the system destruction with hours of the system service till its failure.

The algorithm for predicting useful life for the elements of the VR type takes into account changes of the element physical and mechanical properties.

The resilient rubber elements of the VR type [1] are considered, general view of which is shown on the fig. 1. Mainly, in the process of exploitation, the VR vibroinsulators experience cyclic compressive deformation [2, 6-7].

Algorithm for predicting useful life of the VR elements. A procedure for predicting useful life assumes a necessity to solve interrelated equations of equilibrium and simultaneousness of deformations for determining deflected mode of the vibroinsulators; equations of local useful life; equations of heat conductivity for determining temperature field in the rubber mass; criterion equation of fracture, which connects parameters Fig. 1. is Vibration insulator of the VR type. of the system fracture with hours of the system service till its failure.

Let’s consider these equations in more details: 

equations of equilibrium

2 u 

1 grad divu  0 1  2

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(1)

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where  – is the Laplace operator; 2

u – is a displacement vector;  – is the Poisson's ratio.

The boundary conditions are as follows:

u r  0; u z  0 – in the lower boundary of the rubber mass; u r  0; u z  FA0 sin t – in the upper boundary of the rubber mass; where ur, uz –are the radial and axial displacements, accordingly; 

equation of stationary heat conductivity k div gradT  D  0 ,

(2)

where k – is a heat conductivity; D – is a cycle average dissipative function expressed as: 

D 





2 3K   2G    2 2  2 2G  z      r  2  rz  2 2   3      





z



r

(3)



where z, , r, rz – are components of deformation tensor calculated by displacements ur, uz in the formulas of elasticity theory:

G 

Gq 2

; 3K  

2G1   1  2

where G – is a dynamic modulus of elasticity;

 – is a the energy dissipation factor.

The boundary conditions in the assumed convective heat exchange with the environment on the vibroinsulator surface are as follows: k grad T n  1 (T  T0 ) – on the lateral surface; k grad T n   2 (T  T0 ) – on the supporting surfaces; T  T0 – in initial moment of time

where T0 – is ambient temperature.

This scheme of calculation is based on the energy fracture criterion, which is justified in [1, 3].

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(4)

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Change of rubber properties in the process of long-term cyclic loading.

Such changes can be caused by time and environment influence: heat, oils, acids, solar radiation.

Sometimes, exactly they become the key reason of the system failure, as the system parameters can exceed the tolerable values.

The fig. 2 and fig. 3 show experimental time dependences between dynamic modulus of elasticity and dissipation factor.

As it is stated in the [4], the dynamic modulus of elasticity is changed by the exponential law; and functional dependence E(t) can be expressed as:

Ed (t )  Ed1  Ed1  Ed 2 1  exp( k e t )

(6)

where Ed1 – is an initial value of the dynamic modulus of elasticity; Ed2 – is the modulus final value.

In the considered vibroinsulator: Ed1 = 47,8 MPa; Ed2 = 82 MPa, velocity constant kЕ = 1,110-4 h-1 [3].

Fig. 2. Time dependence of dynamic module of compression.

Functional dependence (t) is practically linear [4]

 (t )   н  k t where н – is an initial value of coefficient of power adsorption ; k – is a velocity constant.

In the considered vibroinsulator: 0 = 0,31; k = 0,08310-8 с-1 [3].

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(7)

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Prediction of the element useful life with taking into account change-in-time of the rubber mechanical properties. Analytical expressions are of the form [1,3]:

 U p dt    ijij dt    q dt

t*

t*

0

0

(8)

t*

 U

p

dt U *p

0

N* 

Fig. 3. is Time dependence of the dissipation factor. where t* 

2



U *p

0,5 E *  2 1  T  f x, y, z 

,

(9)

N * – is hours till vibroinsulator failure (destruction of central area of the rubber);

N * – is number of cycles till the vibroinsulator failure;

E * – is an absolute value of the complex modulus of elasticity; ε – is relative deformation of the vibroinsulator; U *p – is a maximum (critical) value of energy density, which destructs the rubber;

Ψ – is an energy dissipation factor;

T – is a coefficient, which shows which part of the energy dissipating in the rubber is used for heat buildup; f x, y, z  is a function, which characterizes distribution of the stress field and deformation field in the loaded vibroinsulator.

In the case under the consideration, the VR vibroinsulator loading is characterized by stationary temperature field caused by dissipative self-heating, and parameter T , in the first approaching, can be assumed as independent from the loading conditions and temperature of external environment; this parameter can be also assumed as permanent by rubber volume [1,3].

With these assumptions, it is possible to write down the criterion equation (9) for the central area of the vibroinsulator in more simplified form:

N* 

U *p 0,5 E *  2 p



N 

U p 0,5 E   2 p

,

(10)

where  p  1  T – is a coefficient, which characterizes those part of the energy, which is directly used for destructing the rubber structure, for averagely filled rubber of the A-1 grade (soot of 220 grade, raw rubber of the SKI-3 grade) р = 0,6 [3].

Or, if to take into account:

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p 

U *p

(11)

U g*

we can get: N* 

U g*

(12)

0,5 E *  2

where U g* – is a critical (tolerable) value of the energy density, which dissipates in the vibroinsulator under the loads.

If parameters E * and  of the rubber depend on time of the vibroinsulator loading, the expression (12) can be rewritten in the following way

N* 

U g*

(13)

0,5 E * (t )  2 (t )

Or

2   2 2

t*

 E * (t ) (t )dt  U

* g

.

(14)

0

If to take into account the evolutional equations (6) and (7) for the dynamic modulus of elasticity Ed(t) and energy dissipation factor  (t), the expression (14) will be of the form:

E gk k t *  

E gk k E 2

 k E gk  E gн  k E2

t *2 

k ( E gk  E gн ) k E2

 exp( k E t*)  1 

 exp( k E t*)(k E t * 1)  1 

4

 2

(15)

U g*

As values of the last two members of the equation (15) are too small, we can ignore them and get the following equation:

E d 2 k t * 

E d 2 k E *2 4 t  2 U g* 2  

(16)

Experimental data and analysis of the equation (16) show that changing of the energy dissipation factor is the key aspect impacting on the vibroinsulator useful life.

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U *p 

 p 2 Ed 2 ( N * ) 2



0

N *  k N *2



(17)

Summary. Comparison of the calculation data with experimental data, which have been received in process of long-term exploitation of the RV rubber vibroinsulators, shows their satisfactory matching.

References [1] Dyrda V.I. (1988) The strength and the destruction of elastomeric structures in extreme conditions, Kiev, 239 p.

[2] Loginova A.A., Dyrda V.I. and Shevchenko V.G. (2015) Study parameters anti-vibration mounts under cyclic loading. Geo-technіcal Mechanіcs, Dnipropetrovs'k, Vol. 126, Pp. 249-259.

[3] Bulat A.F., Dyrda V.I., Zviagilskiy E.L. and Kobets A.S. (2012) Applied mechanics of elastichereditary environments, Vol. 2. Metody calculation elastomeric parts, Kiev, 616 p.

[4] Loginova A.A., Dyrda V.I. and Shevchenko V.G. (2015) Calculation of vibration isolation systems of mining machines in view of the aging, Geo-technіcal Mechanіcs, Dnipropetrovs'k, Vol. 125, Pp. 249-259.

[5] Massimo Viscardi, Maurizio Arena (2016), Experimental Characterization of Innovative Viscoelastic Foams, Mechanics, Materials Science & Engineering Journal, Vol.4, Magnolithe GmbH, Austria , DOI: 10.13140/RG.2.1.5150.6325 [6] A. Touache, S. Thibaud, J. Chambert, P. Picart (2016), Characterization and Thermo – Elasto – Viscoplastic Modelling of Cunip Copper Alloy in Blanking Process, Mechanics, Materials Science & Engineering Journal, Vol.3, Magnolithe GmbH, Austria , DOI: 10.13140/RG.2.1.3289.0645 [7] Qian Li, Jian-cai Zhao, Bo Zhao (2009), Fatigue life prediction of a rubber mount based on test of material properties and finite element analysis, Engineering Failure Analysis, Volume 16, Issue 7, DOI: 10.1016/j.engfailanal.2009.03.008 Cite the paper Dyrda V.I., Loginova A.A. & Shevchenko V.G (2016). Prediction of Rubber Element Useful Life under the Long-Term Cyclic Loads. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.25883.46884

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Calculation of Strength and Stiffness of Sports Equipment for Games in a Radial Basketball V. P. Ovchinnikov1, A. A. Nesmeyanov2, A. N. Chuiko3 1 – Russian State Pedagogical University, A. I. Herzen, Saint-Petersburg, Russia 2 – President of the Federation of Piterbasket, Saint-Petersburg, Russia 3 – Kharkiv State University of Food Technology and Trade, Kharkiv, Ukraine DOI 10.13140/RG.2.2.18653.20966

Keywords: radial basketball, stand to play basketball in the radial, stiffness, strength, resistance, stress–strain state, solid modelling.

ABSTRACT. Discusses the main problems of strength of Sports Equipment for Games in a Radial Basketball. Based on structural analysis for stiffness, strength and stability of the proposed scheme two-piece installation that fixes all shortcomings noted by the developers. Formulated proposals for the modernization of sports equipment. Only after such studies, one can speak about the ways of its modernization. The main bearing element of the structure, defining the key performance indicators, is a stand. An analysis was conducted of its stiffness and strength.

Introduction. Radial basketball sports ball game, the presentation of which took place in December 2002 in St. Petersburg. For the first, time the rules of the game published in 2002. Playing on the Playground, made in the form of a circle 18 meters in diameter with three concentric circles bounding the zone for 3 seconds (R-3 m), two-point (e-6 m) and three-pointers (R-9 m)[1]. In the centre of the play area Desk with attached shields with baskets in the form of an equilateral triangle 9 (Fig.1).

Fig. 1. The circuit court for radial basketball.

Device for playing basketball in the radial consists of a strut Assembly mounted on the base, made in the form of a circle. It fixes 3 of the shield with a basketball basket with the possibility of installation on the same or on different levels. The base of the strut is provided with a ball bearing fixed on its circumference, in the centre of the base console is installed, on which are strung hollow containers MMSE Journal. Open Access www.mmse.xyz

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with caps and strut, which includes a clutch shiftable along it and fixation with metal retainers (Fig.2).

The proposed utility model relates to sports equipment and can be used during sports games in a radial basketball [2].

Fig. 2. Diagram of the shield with rings to play in a radial basketball.

The terms "load", "force", "force" are frequently used, even among experts far from the calculations on the stiffness and strength. Hence, the need for systemic presentation of these issues, especially the notion of "force" and "displacement" in mechanics and concepts "voltage" and "deformation" in the strength of materials and theory of elasticity are the main, primary concepts. Certainly the concept of "power", according to which "Value, which is a quantitative measure of the mechanical interaction of material bodies, is called force in mechanics" [1]. Strength is the magnitude of the vector. Its effect on the body is determined by:1) the numerical value or modulus of the force; 2) direction of force; 3) point of application of force. In resistance of materials, for ease of analysis, the concept of inner power (the power factor). Distinguish between longitudinal force Nz, the transverse forces Qx and Qy, the torque Mz, the bending moments Mx and My. The index when strength factors correspond to the coordinate axes with respect to which the relevant factor. The numerical value of inner strength is from the equilibrium condition, i.e. its value is equal to the sum of all external forces located on one side of the considered cross-section. Further, to simplify, as well as in elasticity theory, we assume that the equilibrium condition is satisfied, and the inner power factor is expressed as the external force.

Currently, when calculating the strength and stiffness that meet modern scientific and technical level, the popularity and widespread CAD/CAE (Computer Aided Designer/Computer Aided Engineering) system [2, 3]. Particularly fruitful was the use of specialized programs to assess stress–strain state (SSS) of technical systems based on this method, mechanical–mathematical modelling as finite element method (FEM). FEA – international standard for solving problems of solid mechanics by means of numerical algorithms. Suffice it to say that no bridge, no plane, etc. is not certified by international organizations if they are designed without the use of this method. In this study, we applied a widely used technique in the software package Solid Works/COSMOS Works – integrated program that includes the module solid modelling CAD and a module of finite element analysis CAE [4]. The main feature and advantage of this software is its orientation to the design of complex systems with a virtually unlimited number of items and their calculation on strength and rigidity, including under dynamic loads.

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The results of the study. The aim of our study was to analyse the design for rigidity, strength and stability of the installation. Only after such studies, one can speak about the ways of its modernization.

The main bearing element of the structure, defining the key performance indicators, is a stand.

The calculation of the strength and stiffness of cylindrical model. Fig. 3 shows the horizontal displacement of the end of the bar. To the right in Fig. 3 shows a histogram of the displacements of all points of the model. Module sensing results can be received and recorded in the Protocol analyse the result at any point in the model [5].Two points on the intersection of the pipe with the bar these movements (table. in fig. 3) is equal to UX=58, 3 mm and UX=59,16 mm.

Fig. 3. The displacement field of the upper end of the stand.

In the same way it is possible to bring the voltage at any point in the system. The fig. 4 shows a field of von Mizes stress Mo the lower end of the strut, and Fig. 5 normal stress SY in the same area. A comparison of these stress fields confirms the assumption expressed above that in this problem the stresses SY are the major components in von Mizes stresses, SM, and further we will analyze only the stress SY.

Fig. 4. A field of von Mizes stress SM at the lower end of the stand.

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Fig. 5. Field normal stress SY at the lower end of the stand.

For completeness, in Fig. 6 shows the stress field SY at the upper end of the stand.

Fig. 6. The stress field SY at the upper end of the stand.

Analysis of the stress field shows: 1) The Results obtained using FEM, fixed voltage, taking into account their concentration. The program differential tension on the tensile (+) and compressive (â&#x20AC;&#x201C;sign).

2) Results obtained using the finite element method and the results of the analytical solutions match very close.

3) Operating voltage in this problem is significantly below the allowable limit, i.e. the problem of strength for this structure is not decisive.

The calculations show that the results of analytical solution and finite element analysis are the same, which actually is a test solution to move on to more complex tasks.

Calculation for strength and rigidity two-piece stand.

Alternatively, stand design for play in the radial hoops with chains, consider a model in which the rack is made two-piece: the upper cylindrical part of length l=1500 mm and radius R=50 mm and a conical bottom part length l=2000 mm and base radius R=250 mm. Such constructive scheme follows the scheme with the chains, but visually looks "graceful". In Fig. 7 shows the total displacement field on the deformed model. In Fig. 8 shows the summary of travels with the results of sensing at the point MMSE Journal. Open Access www.mmse.xyz

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of intersection with the cross member and at its end. Here, we present fields of total displacements UR, knowing that at the point of intersection with a cross these displacements UR and UX differ slightly. The obtained value of UR=19,98 mm is very close to the obtained above analytical way, the value Δс=17,67 mm and smaller than obtained in the cylindrical model (fig. 3) 3 times.

Fig. 7. The summary of movement

Fig. 8. The summary of the movements with the results of the sensing the point of intersection with the cross member and at its end.

Thus, with relatively simple constructive solution is obtained the desired result – reducing displacement of the end of the bar 3 times. Complete this section of the research, the analysis of stress values. In Fig. 9 shows a field of normal stress SY at the lower end of the strut, and Fig. 10 – at the upper end of the bar. Note that at the upper end of the counter voltage, with respect to the cylindrical model has not changed, and at the bottom decreased almost an order of magnitude, due to the increased moment resistance due to the increase of the radius of the base.

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Fig. 9. Field normal stress SY at the lower end of the stand.

Fig. 10. Field normal stress SY at the upper end of the stand.

It is natural to check how affects the change, such as displacement, due to the changes of pipe wall thickness. In Fig. 11 shows the displacement field for the model, in which the thickness of the upper cylindrical part is increased to 6 mm.

Fig. 11. The displacement field for the model thickness of cylindrical part is increased to 6 mm.

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The obtained value of UR=13,76 mm is almost one and a half times less than that obtained earlier (fig. 8) the value UR=19,98 mm. Thus again it is shown that by using different design solutions to result in reduced displacement of the end of the bar. A preliminary analysis of stand design for play in the radial basketball shows that its stability is provided by a large weight. Therefore, the structural weight reduction will inevitably lead to the decline of sustainability. In the table of basic design parameters indicated that the balances have size 15•10•3 see In this case, the volume of the counterweight V=450 cm3 and density ρ=7,85·10-3 kg/cm3 the weight of one bar is equal to G=3,53 kg. Weight of the bars 18 G=63.5 kg, and the weight of the bars 30 G=106 kg. The Declared weight of the counterweight 200 kg is not obtained. At the same time the weight of 200 kg can be distributed rationally and stability of the installation will be provided constructively.

Summary. 1. Developed a methodology for estimating the basic parameters of the setup for the game in a radial basketball, by solid modelling and subsequent finite element analysis.

2. The proposed two-piece design scheme stands for the radial of basketball, which resolves nearly all shortcomings noted by the developers of the installation.

3. According to the experiment, operating experience and the above calculations, you should specify the main parameters of the installation: the horizontal movement of the upper end of the strut and the allowable weight "hangs" on the ring of the player. This will be the original data. Then all the installation parameters can be recalculated.

4. Base design of the installation must be specified after the adoption of the concept design and reception of the main carrier element.

5. When designing shields and rings can be used by existing development in the classic basketball.

6. Having a great weight, the design has some tolerance. Any improvements aimed at reducing weight will reduce the stability margin and to a possible injury.

References [1] Nesmeyanov A. A., Makeev B. L., Dolgov, I. B., Danilova G. V., Efimov A. Yu., Nesmeyanov N. A. Ovchinnikov V. P. Stand for peterbaznica: the patent for invention RUS 2517543 23.10.2012.

[2] Nesmeyanov A. A., Nesmeyanov D. A., P. A. Nesmeyanov, Nesmeyanov N. A. Kazemov A.A., Korablev S. V., Ovchinnikov P. V., Cherkesov L. Z. Device for playing basketball in the radial (Piterbasket) in the period of preschool education and primary school: useful model patent RUS 83932 10.03.2009.

[3] S. M. Targ, a Short course of theoretical mechanics: Textbook. For technical colleges. – 10th ed.

Rev. and extra – M.: Higher. Wk., 1986. – 416 p [4] Pavlov S. I. System of high performance computing in 2010-2011: overview of achievements and analysis of the market. Part I. // CAD/CAM/CAE Observer, 2011, vol. 5, pp. 74 – 84.

[5] Jenkins B. (B. Jenkins) the Creation of opportunities for computer modelling of physical processes and engineering // CAD/CAM/CAE Observer, 2010, № 1, 44 – 48.

[6] Lyamovskii A.A. Solid Works/COSMOS Works / 2011, M.: DMK Press, - 432 p.

[7] Chuiko A. M., Lewandowski R.A., Ugrin M., Belikov A. B. the Terms fixation and stabilization from the standpoint of biomechanical analysis. A young scientist. 2013, no. 9.

Cite the paper V. P. Ovchinnikov, A. A. Nesmeyanov & A. N. Chuiko (2016). Calculation of Strength and Stiffness of Sports Equipment for Games in a Radial Basketball. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.18653.20966

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Development of Force Monitoring Transducers Using Novel MicroElectromechanical Sensor (MEMS) Dimitar Chakarov1, Vladimir Stavrov2, Detelina Ignatova1, Assen Shulev1, Mihail Tsveov1, Rumen Krastev1, Ivo Vuchkov1 1 – Institute of Mechanics, BAS, Sofia, Bulgaria 2 – AMG Technology Ltd., Botevgrad, Bulgaria DOI 10.13140/RG.2.2.20337.48487

Keywords: mechanical, transducer, force, monitoring, MEMS, piezoresistive, sensor, experimental, data.

ABSTRACT. MEMS piezoresistive sensors are a favourable and attractive option for strain detection due to a number of key advantages such as high sensitivity, low noise, good scaling characteristics, low cost and their ability to have the detection electronics circuit farther away from the sensor or on the same sensing board. This paper represents the results obtained at characterization of novel transducers to be employed into force monitoring systems. Each transducer comprises a coherently designed novel mechanical transducer and a positional MEMS sensor with very high accuracy.

The exploited positional MEMS microsensor and the mechanical transducer are presented in this paper. The particular MEMS sensor provides a voltage output signal having sensitivity in the range of 240 µV/µm at 1V DC voltage supply.

The range of operation of the mechanical transducer is optimized to fit the 500 µm travel range of the microsensor. A finite element model is constructed to simulate the system structure using the commercial FE package. Two prototypes of force transducers are described and manner of used silicon MEMS sensor attachment is demonstrated. An experimental set-up and experimentally measured load curve are presented in the paper. Diagrams force/voltage for two prototypes at different supply voltage 1V and 2V are revealed.

Introduction. A load cell is a transducer that converts load acting on it into an analog electrical signal. Typically, this conversion is achieved by strain gauges which are bonded into the load cell beam and wired into a Wheatstone bridge configuration. Strain gauge load cells dominate the weighing industry [1].

High-performance strain sensing systems, consisting of sensors and interface electronics, are highly desirable for advanced industrial applications, such as point-stress and torque sensing, and strain mapping. Conventional strain sensors made from metal foils suffer from limited sensitivity, large temperature dependence and high power consumption. Further, the metal-foil strain gauges offer flexibility and a potential for use in this format, but they suffer low gauge factor (GF) and limited scalability to large areas due to lack of strategies for multiplexed addressing [2]. Therefore, they are inadequate for high performance and low power consumption applications [3] and hence other strain sensing methods, based on the Micro Electro Mechanical Systems (MEMS) technology, have been proposed [4, 5].

New advances in the field of Micro Electro Mechanical Systems (MEMS) have broadened considerably the applications of these devices. MEMS technology has also enabled the miniaturization of the devices, and a typical MEMS sensor is at least one order of magnitude smaller compared to a conventional metal-foil strain sensor that is used to measure the same quantity.

Consequently, MEMS devices can be batch-fabricated, which offers a high potential for cost reduction. Moreover, proper design can solve problems related to power consumption, while providing improved performance characteristics, such as accuracy, sensitivity and resolution. Finite MMSE Journal. Open Access www.mmse.xyz

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Element Analysis (FEA) provides a reliable tool to carry out the required parametric studies in order to optimize the sensor performance [6].

Several physical sensing principles have been explored in MEMS strain sensors including the modulation of optical, capacitive, piezoelectric, and piezoresistive properties or frequency shift [7, 8].

More particularly, MEMS piezoresistive strain sensors are more favourable and attractive due to a number of key advantages such as high sensitivity [1], low noise, better scaling characteristics, low cost and their ability to have the detection electronics circuit farther away from the sensor or on the same sensing board. Moreover, they have high potential for monolithic integration with low-power CMOS electronics. Furthermore, piezoresistive strain sensors need less complicated conditioning circuits [9].

MEMS load sensors capable of both steady-state and dynamic measurements are generally designed as compliant structures. The device geometry and operating voltages can be optimized for maximum force resolution and range, subject to a number of manufacturing and electromechanical constraints [10].

In present paper, two force measuring systems including mechanical transducer and a new silicon MEMS position sensor with sidewall piezoresistors [11] has been studied. Thus, the performance of the entire force monitoring systems, such as high class electronic scales, can be strongly improved by replacing the currently employed force transducers.

MEMS position sensor for detection of 500µm range displacement.

This approach is based on elaboration of contact MEMS device for displacement detection in the range of 500 µm [12]. The envisaged position microsensor comprises of an anchored (1) and a moveable (2) part. Both parts are connected with a monolithic flexure (3), shown in Fig. 1 (а). The monolithic flexure (3) comprises two pairs of differential springs and two detecting cantilevers (4), which are also attached to point C of the springs.

52

51 0.12

4

data linear fit U = 0.00024 D - 0.002

0.1

4

2

U [V]

0.08

3

1

0.06 0.04 0.02

53

54 0 0

50

100

150

(а)

200

250 300 D [um]

350

400

450

500

550

(b)

Fig. 1. (a): Optical micrograph of a positional microsensor with a single anchored (1) and a single actuated (2) part and a flexure (3) comprising two cantilevers (4) oriented in X direction; four sidewall piezoresistors (51–54), sensitive in Y direction are built-in at the fixed ends of cantilevers; (b): Plot of the sensor signal showing 240µV/µm V sensitivity & 500µm travel range.

The cantilevers are oriented in X direction and the move of part (2) in Y direction is transduced to a bending of cantilevers’ (4) by differential springs. In the sidewalls of the fixed end of the both cantilevers (4) four piezoresistors are embedded (51 - 54). In more details, each cantilever possesses MMSE Journal. Open Access www.mmse.xyz

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a pair of two piezoresistors, 51 -52 and 53 - 54 respectively, which are electrically connected in two voltage dividers. Both voltage dividers are connected in a full bridge configuration, to amplify each other, when actuated part (2) moves in Y (horizontal) direction. This particular sensor has demonstrated a displacement sensitivity of 240 µV/µm at 1V DC voltage supply of the bridge as plotted in Fig. 1(b), and the travel range is limited to approx. 650 µm.

Since differential springs displayed in Fig. 1(a), are mechanically instable in compression mode, this particular flexure (3) can be used in tensile mode, only. Additionally, the devices with sidewall embedded piezoresistors exhibit very low noise and extremely low (i.e., non-detected, at all) temperature dependence [13]. By means of a modification of the flexure layout the sensitivity and the travel range can be tuned to meet optimization criteria [14].

As far, the detecting cantilevers (4) with sidewall piezoresistors ensure sensor signal having above 1,000,000 of intervals in the full scale range, there is a room to achieve a ppm (part per million) resolution of the force transducers, if optimized mechanical transducers are developed.

Since the silicon flexures are extremely fragile, auxiliary mechanical parts having package features for fixing and protecting the both MEMS parts and providing a relative displacement in the range of from 50µm to 1.5mm, have been developed. They could be made of different materials and the mechanical properties could be tuned to measure in desired force range. Based on experimentally measured results, a method of force monitoring with ppm-accuracy, independently on ambient conditions, has been proposed [14]. Thus, the performance of the entire force monitoring systems, such as high class electronic scales, can be strongly improved by replacing the currently employed force transducers.

Development of a mechanical transducers for force monitoring using MEMS sensor.

The goal of present study is to develop a new high performing mechanical transducers applicable in force monitoring systems. To monitor the forces within specified limits it is necessary to develop a flexible transducer mechanism, which transforms the force-load to a displacement of the MEMS sensor, and the stiffness of the transducer determines the range of the measured forces. In order to develop the targeted high performing force monitoring systems, the design approach exploiting flexure mechanisms [15] has been proposed. Respectively, a flexible transducer mechanism, which transforms the applied load to an elastic strain to be detected by the position microsensor, has been created. There were developed two types mechanical transducers.

A. Mechanical transducer type "O-ring", as illustrated in Fig.2 (a). Tensile load is applied to one side of the "O-ring" and a support reaction occurs at the opposite side. The MEMS position sensor is placed and fixed in the middle of the ring along the axis of the applied force, thus it monitors the displacement directly. The constructed 3D CAD model of the mechanical transducer is shown in Fig. 2(b). Admissible deformations of MEMS sensor are determined - 0.5 mm, and the upper limit of the measured force is specified: -1000 N. To meet the specified range of monitored forces, it is necessary to provide a relevant stiffness of the mechanical transducer. Commercial CAD system exploiting finite element modelling (FEM) has been used for caring out a simulation of a static load. Computer simulations were conducted, assuming that the load has been attached at the upper end of the transmission mechanism and the lower end is immobilized. Load with a static force of 1000 N was simulated, using a model with varying thickness of the transducer plate. After a series of experiments, plate with thickness of 11 mm has been selected to achieve a suitable stiffness. Steel alloy with Young’s modulus of E = 210 GPa and Yield strength of σ = 0.620 GPa has been chosen as a raw material. The screen plot with calculated results for effective displacements is shown in Fig. 3 (a). At so selected stiffness of the transducer, a load of 1000 N generates a displacement of 0.300 mm, which is to be measured by the MEMS position sensor. The screen plot with effective stress is shown in Fig. 3 (b).

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F

F (Đ°)

(b)

Fig. 2. Mechanical transducer as an "O-ring" type: (a) kinematic scheme; (b) CAD model.

(Đ°)

(b)

Fig. 3. Simulations of mechanical transducer mechanism: (a) screen plot with effective displacements, (a) screen plot with effective stress.

A prototype of the mechanical transducer has been manufactured with the help of wire electrodischarge machining. Photos of both sides of the force transducer prototype are shown in Fig 4. Both parts of the piezoresistive MEMS positional sensor are firmly bonded to chip-carriers which are further fixed by screws to both loaded sides of the "O-ring", as shown in Fig.4. (a). The maximum travel range of the transducer is constrained to 0.300 mm, by cutting into the housing a gap with same clearance - the gap G in Fig. 4 (b), thus, limiting the monitored force-load to 1000N. This constrain prevents mechanical overload and damage to the MEMS position sensor, as well as keeps the tensile and bending stresses in the transducer bellow yielding stress with a safety factor of 1.2.

B. Mechanical transducer type double symmetric parallelogram mechanism has been created.

To meet the specified range of monitored forces, it is necessary to achieve the relevant stiffness of mechanical transducer. Since, the admissible displacement of the position microsensor is 0.5 mm and the upper limit of the measured force has been specified to 100 N, the low-stiffness mechanical transducer mechanism, as illustrated by structure scheme in Fig. 5 (a) and by 3D CAD model shown in Fig. 5 (b), has been designed.

The transducer mechanism is made of a solid plate titanium alloy and it was processed to obtain the elastic joints of the mechanism. The elastic joints are double-notched elastic beam type and they possess a number of advantages compared to contact ones, as they are free from backlash, friction, and hysteresis. This geometry allows the desired low stiffness in one directions of joint bending and high stiffness in the remaining non-motional direction to be simultaneously achieved. The position MMSE Journal. Open Access www.mmse.xyz

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microsensor is attached at the middle of the mechanism. In this way there is transmission ratio between the displacements along the two axes of the mechanical transducer.

(а)

(b)

Fig. 4. Photo of a prototype of the mechanical transducer– (a) front side with mounted MEMS position sensor, (b) rear side. n α h

l

H

N

(а)

(b)

Fig. 5. Mechanical transducer mechanism: (a) kinematic scheme; (b) CAD model.

The height of the device is denoted by H, and the width of the device in the sensor area is denoted by N (Fig. 5a). Each symmetric branch of the transmission is considered as a rectangle with two sides n and h. According to this geometry transmission ratio k of symmetric parallelogram mechanism is determined as:

k

n N h   h H n

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Commercial CAD system exploiting finite element modeling (FEM) has been used for caring out an experiment of a static load. The constructed 3D CAD model of the mechanical transducer shown in Fig. 5(b) has been involved for computer simulations. The titanium alloy (Ti-8Al-1Mo-1V) with Young’s modulus of E = 120 GPa and Yield strength of σ = 0.910 GPa has been chosen as a raw material.

The screen with calculated results for effective displacements is shown in Fig. 6. The screen with effective stress is not shown here, but normal stress in the mostly-loaded areas did not exceed the allowable stress.. Computer simulations were conducted, when the load has been attached at the upper end of the transmission mechanism and the lower end has been rigidly immobilized. Load with a static force of 100 N was simulated, using a model with varying thickness of the titanium alloy plate.

The displacements H of the force application point, the deflection N between the attachment points of the position microsensor, and calculated transmission ratio k are reported in Table 1.

Table 1. Calculated displacements and transmission ratio for mechanical transducer at load of 100N. Plate

9 [mm]

10 [mm]

11 [mm]

12 [mm]

13 [mm]

N [mm]

1.138

0.879

0.707

0.598

0.498

H [mm]

0.695

0.537

0.432

0.365

0.304

k

1.637

1.637

1.6365

1.638

1.638

thickness

The plate thickness of t = 11 mm has been selected for prototyping of the force transducer, for which additional simulations during load changes were carried out. A load respectively of 90N, 70N, 50N and 30N is applied. The calculated displacements H , N and transmission ratio k are shown in Table 2.

Table 2. Calculated displacements and transmission ratio for different loads.

Load

90 [N]

70[N]

50[N]

30[N]

N [mm]

0.636

0.494

0.353

0.212

H [mm]

0.389

0.302

0.216

0.130

k

1.635

1.636

1.634

1.631

As a result of the experiment it has been concluded that the transmission ratio is maintained constant.

The range of operation of the force transducer depends on the width of the gap that limits the transducer displacement noted by “G” in Fig.4(b). A gap of 0.500mm has been selected for which the displayed force transducer will work in the range of 70N. The upper limit of the measured force specified as 70 N provides load margin which avoid position microsensor damaging.

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Fig. 6. Screens with effective displacements.

A prototype of the mechanical transducer has been manufactured with the help of wire electrodischarge machining. A photo of the force transducer prototype is shown on Fig 7. It is enabled with piezoresistive position microsensor, shown in the zoomed image, having a travel range of 650µm limited by the width of the gap G of 500µm, also shown in additional zoomed image.

G

Fig. 7. Photo of the force transducer prototype having travel range of 500µm limited by the width of the gap G. Experimental set-up and experimental data.

The mechanical transducer prototypes are tested for load measurements by a precise loading test machine TIRAtest 2300. The system exploits a load cell with a measuring range of 10kN, a digital multi-meter PeakTech 3415 connected to a computer and the prototype of force transducer, connected on both sides with loading machine traverses.

A. Mechanical transducer "O-ring" type.

The MEMS sensor is supplied with a stabilized DC voltage supply – one test has been carried out at 1V and other test - at 2V. The output voltage of the sensor is recorded using a digital multi-meter MMSE Journal. Open Access www.mmse.xyz

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Protek D470 connected to the computer. Fig. 8 shows the experimental set-up used for load monitoring.

Fig. 8. Photo of the experimental set-up for mechanical transducer "O-ring" type.

The testing machine loads the prototype at a constant speed by measuring both: the loading force and the output voltage of the position sensor. To obtain the force/voltage ratio, both registered signals are time synchronized.

Measurements are made of two identical prototypes marked as P1 and P2, at power supply of 1V and 2V and loading/unloading from 0 to 1000 N and vice versa. There are three trials made to each prototype under loading and unloading at a speed of mobile traverse of 0.48 mm/min. The obtained force/voltage diagrams of these experiment are shown in Fig. 9. The resulting diagrams are linear, which can be expressed by F = aU + b, where F is loading force and U is output voltage.

1000 900 800 700

F, N

600 500 400 1V, P1 2V, P1 2V, P2 1V, P2

300 200 100 0 0.000

0.050

0.100

U, V

0.150

0.200

0.250

Fig. 9: Diagrams force/voltage for two prototypes P1 and P2 at supply voltage of 1V and 2V. The slope a of each diagram depends on the supply voltage. Both transducers have similar slopes at the same supply voltage, but each transducer has a different offset b. This offset depends on the sensor preload. The average values of coefficients a and offset voltage Uo=-b/a, at a given supply for each prototype are as follows U=1V: a1= 21767 N/V, U01=0,0660 V, a2=22047N/V, U02=0,0315 V; U=2V: a1=11027N/V, U01= 0,1318 V, a2=11226N/V, U02=0,0624 V. The charts show that the unloading curve does not tally with loading chart but the hysteresis loop is fairly small. Hysteresis observed can MMSE Journal. Open Access www.mmse.xyz

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be due to imperfections of the measuring system, i.e. miss-synchronization between the two measurements channels when switching load/unload measurements, etc.

B. Mechanical transducer type double symmetric parallelogram mechanism.

In this experiment the MEMS sensor is supplied with a stabilized DC voltage supply of 1V and in a similar manner the output voltage of the sensor is recorded using a digital multi-metre Protek D470 connected to the computer. Fig. 10 a) shows the experimental set-up used. There are four trials made under loading from 0 till 70 N and unloading at a speed of mobile traverse of 0.48 mm/min. The obtained force/voltage diagram of one trial is shown in Fig.10 b).

70.0 60.0

F, N

50.0 40.0 30.0 20.0 10.0 0.0 0.0200

0.0400

0.0600

0.0800

0.1000

U, V

(a)

( b)

Fig. 10. (a) Experimental set-up and (b) Diagrams force/voltage for mechanical transducer type B at supply voltage of 1V. The resulting diagram is linear and the hysteresis get is fairly small. The average values of slope a and offset voltage Uo=-b/a, at a given supply voltage of 1V have the values: a= 1119 N/V and U0= 0,0271V. Development of a control unit for prototypes of mechanical transducers with original MEMS sensor.

This control unit is computerized system, designed so that it is compatible with the original MEMS sensors. It includes high-precision amplifier, that cover the functional characteristics of all transducers. It includes also user display, setup keyboard, a channel for communication with host computer for real time data transfer. Control unit can be used like smart transmitter. The overall look of control unit for transducer prototypes is shown in Fig.11 (a). Presented experimental setup has been used for calibration of the control system. The prototypes were loaded with constant speed and both: loading force and measured force were recorded using â&#x20AC;&#x153;smart transmitterâ&#x20AC;? mode. The resulting diagrams after calibration are strictly linear. The relationship - loading force / measured force for transducer "O-ring" type is shown on Fig. 11 (b). It can be expressed by equality: Fm = 1,0012F + 0,0085.

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100 80

F x10[N]

Mechanics, Materials Science & Engineering, September 2016 – ISSN 2412-5954

Fm = 1,0012F + 0,0085

60 40 20

Fmx10[N]

0 0.0

(a)

50.0

100.0

(b)

Fig. 11. (a) Photo of the control unit;(b) Relationship- loading force / measured force for transducer "O-ring" type.

Summary. This paper represents the results obtained at characterization of novel transducers to be employed into force monitoring systems. Each transducer comprises a coherently designed mechanical transducer and a MEMS positional sensor with very high accuracy.

The MEMS positional micro sensor and the design of two type mechanical transducers are presented in the paper. The range of operation of the mechanical transducer is optimized to fit the 500µm travel range of the positional micro sensor. Respectively, the flexures’ stiffness corresponds to achieve the maximum displacement at the upper limit of the measured load. A finite element model is constructed to simulate the system structure using a commercial FE package. The force transducers range of operation is limited by the width G of a gap, which also, avoids damages of the mechanical transducer and MEMS positional sensor. Prototypes of two transducers are described and manner of usage of the silicon MEMS position sensor attachment is demonstrated. An experimental set-up for measuring the load curves are reported in the paper. Diagrams of force vs. sensor output voltage of the prototypes at different supply voltage 1V and 2V are demonstrated and discussed.

As a result of the experiment it has been concluded that the force/voltage ratio is constant.

Piezoresistive MEMS position sensors can be successfully used for strain detection at force loading of the mechanical transducer. Respectively, other than force values like: 3D strain, acceleration, torque, temperature and other environmental parameters, as well as their combinations can be accurately monitored, when suitable transducers are designed.

Acknowledgments. Authors gratefully acknowledge the financial supports of this work by grant 6IF02-13/15.12.2012 of the Bulgarian National Innovation Fund.

References [1] Kumar, Sh., K P Venkatesh, S. S. Baskar, S. P. Madhavi. System integration design in MEMS–A case study of micromachined load cell. – Sadhana, Vol. 34, 2009, No. 4, pp. 663–675.

[2] Won, S. M., Hoon-Sik Kim, N. Lu, Dae-Gon Kim, Cesar Del Solar, T. Duenas, A. Ameen, J. A. Rogers. Piezoresistive Strain Sensors and Multiplexed Arrays Using Assemblies of SingleCrystalline Silicon Nanoribbons on Plastic Substrates. - IEEE Transactions on electron devices, Vol. 58, No. 11, 2011,pp. 4074- 4078.

[3] Nagy, M.; C. Apanius; J. A. Siekkinen. A user friendly, high-sensitivity strain gauge. - Sensors, Vol. 18, 2001, pp. 20-27.

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[4] Hrovat, M., D. Belavic, Z. Samardzija, J. Holc. An investigation of thick-film resistor, fired at different temperatures, for strain sensors. - International Spring Seminar on Electronics Technology, 2001, pp. 32-36.

[5] Mohammed A., W. Moussa, E. Lou. Mechanical Strain Measurements Using Semiconductor Piezo-resistive Material.The 4th IEEE International Conference on MEMS, Nano and Smart Systems, and The 6th IEEE International Workshop on System-on-Chip for Real-Time Applications 2006, pp.5-6.

[6] Mohammed, A., W. A. Moussa, E. Lou. High-Performance Piezoresistive MEMS Strain Sensor with Low Thermal Sensitivity. - Sensors, Vol. 11, 2011, pp. 1819-1846; DOI: 10.3390/s110201819].

[7] Cao, L., T. Kim, S. Mantell, D. Polla. Simulation and fabrication of piezoresistive membrane type MEMS strain sensors. - Sensors and Actuators A: Physical, Vol. 80, 2000, pp. 273-279.

[8] Han, B., J. Ou. Embedded piezoresistive cement-based stress/strain sensor. - Sensors and Actuators A: Physical, Vol. 138, 2007, pp. 294-298.

[9] Fraden, J. Handbook of modern sensor: physics, designs, and applications. 2nd ed. AIP PressSpringer: New York, 1996.

[10] Torrents, A, K. Azgin, S. W. Godfrey, E. S. Topalli, T. Akin, L. Valdevit. MEMS resonant load cells for micro-mechanical test frames: feasibility study and optimal design. - J. Micromech.

Microeng., Vol. 20, No. 12, 2010, pp. 1-17. DOI: 10.1088/0960-1317/20/12/125004.

[11] Stavrov, V., Tomerov, E., Stavreva, G., Hardalov, C., Shulev, A., “Lateral Displacement MEMS Sensor,” Proc. Eurosensors XXIV, Linz, Austria, 2010, pp. 649-652 [12] Stavrov, V., Todorov, V., Shulev, A., Hardalov. C., “MEMS Sensors for mm-Range Displacement Measurements with Sub-nm-Resolution”, Proc. of SPIE Conf. Microtechn, Grenoble, France, 2013, v. 8763 87632G-1-6A. [13] Todorov, V., Stavreva, G., Stavrov, V., “Contact mode MEMS position sensors with piezoresistive detection” Proc. XXVIII EUROSENSORS 2014, Brescia, Italy.

[14] Stavrov V., Shulev A., Chakarov D., Stavreva G., Force monitoring transducers with more than 100,000 scale intervals, Proc. of SPIE Conf. Microtechn, Barcelona, Spain, 2015, v. 9517 95171Q1-6.

[15] Pham, H., Chen, I., “Stiffness modeling of flexure parallel mechanism”, Precision. Engineering 29, 2005, pp. 467–478.

Cite the paper Dimitar Chakarov, Vladimir Stavrov, Detelina Ignatova, Assen Shulev, Mihail Tsveov, Rumen Krastev & Ivo Vuchkov (2016). Development of Force Monitoring Transducers Using Novel Micro-Electromechanical Sensor (MEMS). Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.20337.48487

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Analytical Simulation of Dynamical Process in One-Dimension Task Kravets V.V.1, Kravets T.V.2, Fedoriachenko S.A.1, Loginova A.A.1 1 â&#x20AC;&#x201C; National Mining University, Dnipropetrovsk, Ukraine 2 â&#x20AC;&#x201C; National University of Railway Transport, Dnipropetrovsk, Ukraine DOI 10.13140/RG.2.2.20337.34347

Keywords: kinematics forced vibration, mathematical model, analytical solution, dynamic design.

ABSTRACT. We consider a one-dimensional dynamical system, characterized by the kinetic energy of the translational motion, the potential elastic energy and dissipative function. The system is subjected to external kinematic effects of a harmonic oscillation. It constitutes a mathematical model of the dynamic process. It is proposed to form an ordered analytical representation of a dynamic process, characterized by conservatism regarding the index of the root of the characteristic equation. The structure of the analytical solutions is determined by three components caused by the initial phase of the system state, kinematic influence of the initial time and the time-varying harmonic influence. The dynamic process is the superposition of the transition (the first two components) and establish a process corresponding to each of the roots. The qualitative analysis of the dynamic process provides, depending on the distribution of the roots of the characteristic equation including resonance and beat. Analytically solved parametric synthesis (dynamic design) providing the desired distribution of the roots in the complex plane, i.e. the required quality of the dynamic process.

Introduction. The vibrations play an important role in the technics and are the subject of numerous works, generalized in the fundamental work [1]. The main tool for the study of oscillations in multidimensional nonlinear dynamical systems are physical and mathematical experiment. Particular scientific interest presents the individual analytical methods for solving the problem of non-linearity, which include the method of phase plane, a small parameter, harmonic balance, successive approximation [2, 3]. The problem of multi-dimensional linear problems is solved by classical mathematical methods using Laplace transform, residue theorem [4,5]. In this paper, a new form of analytic representation of dynamic processes, which used in [6] is applied to the problem of vibration isolation of kinematic external influence. The dynamical scheme of the vibroinsulation task while kinematical influence is provided on the fig. 1.

Fig. 1. The dynamical scheme of the kinematical influence.

Where đ?&#x2018;&#x161; â&#x20AC;&#x201C; mass; đ?&#x153;&#x2021; â&#x20AC;&#x201C; dumping coefficient; MMSE Journal. Open Access www.mmse.xyz

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đ?&#x2018;? â&#x20AC;&#x201C; rigidity; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; , đ??żđ?&#x2018;&#x203A; â&#x20AC;&#x201C; geometric parameters in natural configuration; đ?&#x2018;Ľ(đ?&#x2018;Ą) â&#x20AC;&#x201C; dynamical process; đ?&#x2018;&#x17D;(đ?&#x2018;Ą) â&#x20AC;&#x201C; kinematical influence.

External kinematic effects rely specified in the form of harmonic oscillations: đ?&#x2018;&#x17D;(đ?&#x2018;Ą) = đ?&#x2018;&#x17D;0 đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą where đ?&#x153;&#x2C6; â&#x20AC;&#x201C; frequency of forced vibrations; đ?&#x2018;&#x17D;0 â&#x20AC;&#x201C; amplitude of forced oscillations.

Required to find an analytical representation of a dynamic process, depending on the set and varying system parameters for harmonic kinematic exposure. Mathematical modelling.

A mathematical model of the problem is given in [6] and is as follows:

đ?&#x2018;ĽĚ&#x2C6; (đ?&#x2018;Ą) +

đ?&#x153;&#x2021; đ?&#x2018;? đ?&#x153;&#x2021; đ?&#x2018;? đ?&#x2018;ĽĚ&#x2021; (đ?&#x2018;Ą) + đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;ĽĚ&#x2021; (đ?&#x2018;Ą) + đ?&#x2018;Ľ(đ?&#x2018;Ą) đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ?&#x2018;&#x161;

Matrix model in normal form. We introduce the variables:

{

đ?&#x2018;Ľ1 (đ?&#x2018;Ą) = đ?&#x2018;ĽĚ&#x2021; (đ?&#x2018;Ą) , Ń&#x201A;.Đľ. đ?&#x2018;ĽĚ&#x2021; 2 (đ?&#x2018;Ą) = đ?&#x2018;Ľ1 (đ?&#x2018;Ą) đ?&#x2018;Ľ2 (đ?&#x2018;Ą) = đ?&#x2018;Ľ(đ?&#x2018;Ą)

Where the corresponding coefficients: đ?&#x153;&#x2021;

đ?&#x2018;?

đ?&#x2018;&#x17D;21 = 1;

đ?&#x2018;&#x17D;22 = 0;

đ?&#x2018;&#x17D;11 = â&#x2C6;&#x2019; đ?&#x2018;&#x161; ; đ?&#x2018;&#x17D;12 = â&#x2C6;&#x2019; đ?&#x2018;&#x161;;

In addition, force function while external kinematical effect:

đ?&#x2018;&#x201C;1 (đ?&#x2018;Ą) =

đ?&#x153;&#x2021; đ?&#x2018;? đ?&#x2018;&#x17D;Ě&#x2021; (đ?&#x2018;Ą) + đ?&#x2018;&#x17D;(đ?&#x2018;Ą); đ?&#x2018;&#x161; đ?&#x2018;&#x161; đ?&#x2018;&#x201C;2 (đ?&#x2018;Ą) = 0.

Then the original differential equation of motion takes the following matrix notation in normal form: The analytical solution.

A mathematical model corresponds to the following analytical record of the dynamic process for any kinematic effects đ?&#x2018;&#x17D;(đ?&#x2018;Ą) during given initial conditions đ?&#x2018;ĽĚ&#x2021; 0 , đ?&#x2018;Ľ0 and different existing roots đ?&#x153;&#x2020;1 , đ?&#x153;&#x2020;2 of characteristic equation [2].

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∞

𝑒 𝜆1 𝑡 𝜆2 𝑥(𝑡) = | 𝜆1 − 𝜆2 1

∞

𝑒 𝜆1 𝑡 1 1 1 −𝑥̇ 0 |+ ∑ 𝑛 𝑓(0)(𝑛−1) − ∑ 𝑛 𝑓(𝑡)(𝑛−1) −𝑥0 𝜆1 − 𝜆2 𝜆1 𝜆1 − 𝜆2 𝜆1 1

𝑒

𝜆2 𝑡

𝜆 + | 1 𝜆2 − 𝜆1 1

𝜆2 𝑡

1

∞

∞

𝑒 1 1 1 −𝑥̇ 0 |+ ∑ 𝑛 𝑓(0)(𝑛−1) − ∑ 𝑛 𝑓(𝑡)(𝑛−1) −𝑥0 𝜆2 − 𝜆1 𝜆2 𝜆2 − 𝜆1 𝜆2 1

𝑎 −𝜆 Here the roots of the characteristic equation:| 11 𝑎21

1

𝑎12 | = 0. 𝑎22 − 𝜆

associated with the dynamic parameters of the system in the form: 𝜇

𝑐

𝜆1 + 𝜆2 = − 𝑚; 𝜆1 ∙ 𝜆2 = 𝑚.

Due to this, harmonic kinematic effects: 𝑎(𝑡) = 𝑎0 𝑠𝑖𝑛𝜈𝑡, 𝑎̇ (𝑡) = 𝑎0 𝜈𝑐𝑜𝑠𝜈𝑡, power function and its derivatives take the form of: 𝑓(𝑡) = −(𝜆1 + 𝜆2 )𝑎0 𝜈𝑐𝑜𝑠𝜈𝑡 + 𝜆1 𝜆2 𝑎0 𝑠𝑖𝑛𝜈𝑡; 𝑓̇(𝑡) = (𝜆1 + 𝜆2 )𝑎0 𝜈 2 𝑠𝑖𝑛𝜈𝑡 + 𝜆1 𝜆2 𝑎0 𝜈𝑐𝑜𝑠𝜈𝑡; 𝑓̈(𝑡) = (𝜆1 + 𝜆2 )𝑎0 𝜈 3 𝑐𝑜𝑠𝜈𝑡 − 𝜆1 𝜆2 𝑎0 𝜈 2 𝑠𝑖𝑛𝜈𝑡; 𝑓⃛(𝑡) = −(𝜆1 + 𝜆2 )𝑎0 𝜈 4 𝑠𝑖𝑛𝜈𝑡 − 𝜆1 𝜆2 𝑎0 𝜈 3 𝑐𝑜𝑠𝜈𝑡; 𝑓(𝑡)(4) = −(𝜆1 + 𝜆2 )𝑎0 𝜈 5 𝑐𝑜𝑠𝜈𝑡 + 𝜆1 𝜆2 𝑎0 𝜈 4 𝑠𝑖𝑛𝜈𝑡; 𝑓(𝑡)(5) = (𝜆1 + 𝜆2 )𝑎0 𝜈 6 𝑠𝑖𝑛𝜈𝑡 + 𝜆1 𝜆2 𝑎0 𝜈 5 𝑐𝑜𝑠𝜈𝑡; And so forth.

Respectively at initial time moment (𝑡 = 0), we’ll get: 𝑓(𝑡) = −(𝜆1 + 𝜆2 )𝑎0 𝜈; 𝑓̇(𝑡) = 𝜆1 𝜆2 𝑎0 𝜈; 𝑓̈ (𝑡) = (𝜆1 + 𝜆2 )𝑎0 𝜈 3 ; 𝑓⃛(𝑡) = −𝜆1 𝜆2 𝑎0 𝜈 3; 𝑓(𝑡)(4) = −(𝜆1 + 𝜆2 )𝑎0 𝜈 5; 𝑓(𝑡)(5) = 𝜆1 𝜆2 𝑎0 𝜈 5 𝑐𝑜𝑠𝜈𝑡; Etc. MMSE Journal. Open Access www.mmse.xyz

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Where follows: â&#x2C6;&#x17E;

â&#x2C6;&#x2018; 1

1 1 đ?&#x153;&#x2C6;2 đ?&#x153;&#x2C6;4 đ?&#x153;&#x2C6;6 (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) (đ?&#x153;&#x2020; )đ?&#x2018;&#x17D; đ?&#x2018;&#x201C;(0) = â&#x2C6;&#x2019; + đ?&#x153;&#x2020; đ?&#x153;&#x2C6; [1 â&#x2C6;&#x2019; + â&#x2C6;&#x2019; +â&#x2039;Ż]+ 2 0 đ?&#x153;&#x2020;1đ?&#x2018;&#x203A; đ?&#x153;&#x2020;1 1 đ?&#x153;&#x2020;12 đ?&#x153;&#x2020;14 đ?&#x153;&#x2020;16 +

1 đ?&#x153;&#x2C6;2 đ?&#x153;&#x2C6;4 đ?&#x153;&#x2C6;6 đ?&#x153;&#x2020; đ?&#x153;&#x2020; đ?&#x2018;&#x17D; đ?&#x153;&#x2C6; [1 â&#x2C6;&#x2019; + â&#x2C6;&#x2019; +â&#x2039;Ż] đ?&#x153;&#x2020;12 1 2 0 đ?&#x153;&#x2020;12 đ?&#x153;&#x2020;14 đ?&#x153;&#x2020;16

The resulting alternating series converges provided: đ?&#x153;&#x2C6; 2 ( ) <1 đ?&#x153;&#x2020;1 in addition, its sum finds in accordance with [4]:

1â&#x2C6;&#x2019;

đ?&#x153;&#x2C6;2 đ?&#x153;&#x2C6;4 đ?&#x153;&#x2C6;6 đ?&#x153;&#x2020;12 + â&#x2C6;&#x2019; + â&#x2039;Ż â&#x2020;&#x2019; đ?&#x153;&#x2020;12 đ?&#x153;&#x2020;14 đ?&#x153;&#x2020;16 đ?&#x153;&#x2020;12 + đ?&#x153;&#x2C6; 2

Where follows: â&#x2C6;&#x17E;

â&#x2C6;&#x2018; 1

1 đ?&#x153;&#x2020;12 (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) đ?&#x2018;&#x201C;(0) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D; đ?&#x153;&#x2C6; 0 đ?&#x153;&#x2020;1đ?&#x2018;&#x203A; đ?&#x153;&#x2020;12 + đ?&#x153;&#x2C6; 2

Similarly, we get: â&#x2C6;&#x17E;

1 đ?&#x153;&#x2020;22 (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) â&#x2C6;&#x2018; đ?&#x2018;&#x203A; đ?&#x2018;&#x201C;(0) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;0 đ?&#x153;&#x2C6; 2 đ?&#x153;&#x2020;2 đ?&#x153;&#x2020;2 + đ?&#x153;&#x2C6; 2 1

Etc. â&#x2C6;&#x17E;

1 đ?&#x153;&#x2020;12 đ?&#x153;&#x2C6; 2 (đ?&#x153;&#x2020;1 + đ?&#x153;&#x2020;2 ) + đ?&#x153;&#x2020;12 đ?&#x153;&#x2020;2 (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) â&#x2C6;&#x2018; đ?&#x2018;&#x203A; đ?&#x2018;&#x201C;(đ?&#x2018;Ą) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;0 đ?&#x153;&#x2C6; 2 đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x2C6;đ?&#x2018;Ą + đ?&#x2018;&#x17D;0 đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą đ?&#x153;&#x2020;1 đ?&#x153;&#x2020;1 + đ?&#x153;&#x2C6; 2 đ?&#x153;&#x2020;12 + đ?&#x153;&#x2C6; 2 1

â&#x2C6;&#x17E;

â&#x2C6;&#x2018; 1

1 đ?&#x153;&#x2020;22 đ?&#x153;&#x2C6; 2 (đ?&#x153;&#x2020;1 + đ?&#x153;&#x2020;2 ) + đ?&#x153;&#x2020;22 đ?&#x153;&#x2020;1 (đ?&#x2018;&#x203A;â&#x2C6;&#x2019;1) đ?&#x2018;&#x201C;(đ?&#x2018;Ą) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D; đ?&#x153;&#x2C6; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x2C6;đ?&#x2018;Ą + đ?&#x2018;&#x17D; đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą 0 0 đ?&#x153;&#x2020;đ?&#x2018;&#x203A;2 đ?&#x153;&#x2020;22 + đ?&#x153;&#x2C6; 2 đ?&#x153;&#x2020;22 + đ?&#x153;&#x2C6; 2

Dynamic process is determined by three components corresponding to each of the two roots of the characteristic equation: MMSE Journal. Open Access www.mmse.xyz

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2

đ?&#x2018;Ľ(đ?&#x2018;Ą) = â&#x2C6;&#x2018;[đ?&#x2018;Ľđ?&#x2018;Ľ(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x2018;Ą) + đ?&#x2018;Ľđ?&#x2018;&#x201C;(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) + đ?&#x2018;Ľ(đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą)] đ?&#x2018;&#x2DC;=1

Where đ?&#x2018;Ľđ?&#x2018;Ľ(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x2018;Ą) â&#x20AC;&#x201C; transient dynamic process due to perturbations of the initial phase of the system status: 2

â&#x2C6;&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľ(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x2018;Ą) = đ?&#x2018;&#x2DC;=1

đ?&#x2018;&#x2019; đ?&#x153;&#x2020;1 đ?&#x2018;Ą đ?&#x153;&#x2020;2 | đ?&#x153;&#x2020;1 â&#x2C6;&#x2019; đ?&#x153;&#x2020;2 1

đ?&#x2018;&#x2019; đ?&#x153;&#x2020;2 đ?&#x2018;Ą đ?&#x153;&#x2020;1 â&#x2C6;&#x2019;đ?&#x2018;ĽĚ&#x2021; 0 |+ | â&#x2C6;&#x2019;đ?&#x2018;Ľ0 đ?&#x153;&#x2020;2 â&#x2C6;&#x2019; đ?&#x153;&#x2020;1 1

â&#x2C6;&#x2019;đ?&#x2018;ĽĚ&#x2021; 0 | â&#x2C6;&#x2019;đ?&#x2018;Ľ0

đ?&#x2018;Ľđ?&#x2018;&#x201C;(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) â&#x20AC;&#x201C; unsteady dynamic process caused by harmonic external kinematic disturbance at the initial time 2

â&#x2C6;&#x2018; đ?&#x2018;Ľđ?&#x2018;&#x201C;(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) = â&#x2C6;&#x2019; đ?&#x2018;&#x2DC;=1

đ?&#x2018;&#x2019; đ?&#x153;&#x2020;1 đ?&#x2018;Ą đ?&#x153;&#x2020;12 đ?&#x2018;&#x2019; đ?&#x153;&#x2020;2 đ?&#x2018;Ą đ?&#x153;&#x2020;22 đ?&#x2018;&#x17D;0 đ?&#x153;&#x2C6; 2 â&#x2C6;&#x2019; đ?&#x2018;&#x17D; đ?&#x153;&#x2C6; đ?&#x153;&#x2020;1 â&#x2C6;&#x2019; đ?&#x153;&#x2020;2 đ?&#x153;&#x2020;1 + đ?&#x153;&#x2C6; 2 đ?&#x153;&#x2020;2 â&#x2C6;&#x2019; đ?&#x153;&#x2020;1 0 đ?&#x153;&#x2020;22 + đ?&#x153;&#x2C6; 2

đ?&#x2018;Ľ(đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) â&#x20AC;&#x201C; stationary dynamic process caused by harmonic external kinematic exposure at the current time.

2

â&#x2C6;&#x2018; đ?&#x2018;Ľ(đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) = đ?&#x2018;&#x2DC;=1

(đ?&#x153;&#x2020;12

đ?&#x2018;&#x17D;0 {đ?&#x153;&#x2C6; 3 (đ?&#x153;&#x2020;1 + đ?&#x153;&#x2020;2 )đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x2C6;đ?&#x2018;Ą + [đ?&#x153;&#x2C6; 2 (đ?&#x153;&#x2020;12 + đ?&#x153;&#x2020;22 + đ?&#x153;&#x2020;1 đ?&#x153;&#x2020;2 ) + đ?&#x153;&#x2020;12 đ?&#x153;&#x2020;22 ]đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą} + đ?&#x153;&#x2C6; 2 )(đ?&#x153;&#x2020;22 + đ?&#x153;&#x2C6; 2 )

The stability of the dynamic process.

Stability conditions for a dynamic process determined by the method of Lyapunov [2-3]: đ?&#x153;&#x2020;1 < 1, đ?&#x153;&#x2020;2 < 1; and taking into account the region of convergence obtained for the harmonic effects of a power series [4]: đ?&#x153;&#x2C6; 2

đ?&#x153;&#x2C6; 2

(đ?&#x153;&#x2020; ) < 1, (đ?&#x153;&#x2020; ) < 1.

1

2

Analytical modeling.

In the case of real and different roots đ?&#x153;&#x2020;1 and đ?&#x153;&#x2020;2 , satisfying the conditions of stability, transient time dynamic process tends asymptotically to zero at: lim đ?&#x2018;Ľđ?&#x2018;Ľ(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x2018;Ą) = 0

lim đ?&#x2018;Ľđ?&#x2018;&#x201C;(0) (đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x2018;Ą) = 0

đ?&#x2018;&#x203A;â&#x2020;&#x2019;â&#x2C6;&#x17E;

đ?&#x2018;&#x203A;â&#x2020;&#x2019;â&#x2C6;&#x17E;

Fixed dynamic process should be presented in the form of: MMSE Journal. Open Access www.mmse.xyz

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â&#x2C6;&#x2018; đ?&#x2018;Ľ(đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; , đ?&#x153;&#x2C6;, đ?&#x2018;Ą) = đ??´đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x153;&#x2C6;đ?&#x2018;Ą + đ?&#x153;&#x2018;) đ?&#x2018;&#x2DC;=1

where đ??´ â&#x20AC;&#x201C; amplitude and Ď&#x2020;-phase forced harmonic oscillations are known formulas: đ?&#x2018;&#x17D;

đ??´ = â&#x2C6;&#x161;đ?&#x2018;&#x17D;2 + đ?&#x2018;? 2 , đ?&#x2018;Ąđ?&#x2018;&#x201D;đ?&#x153;&#x2018; = đ?&#x2018;? đ?&#x153;&#x2C6; 3 (đ?&#x153;&#x2020; +đ?&#x153;&#x2020; )

1 2 where đ?&#x2018;&#x17D; = đ?&#x2018;&#x17D;0 (đ?&#x153;&#x2020;2 +đ?&#x153;&#x2C6;2 )(đ?&#x153;&#x2020; 2 +đ?&#x153;&#x2C6; 2 ), 1

2

đ?&#x2018;? = đ?&#x2018;&#x17D;0

đ?&#x153;&#x2C6; 2 (đ?&#x153;&#x2020;21 +đ?&#x153;&#x2020;22 +đ?&#x153;&#x2020;1 đ?&#x153;&#x2020;2 )+đ?&#x153;&#x2020;1 đ?&#x153;&#x2020;2 (đ?&#x153;&#x2020;21 +đ?&#x153;&#x2C6; 2 )(đ?&#x153;&#x2020;22 +đ?&#x153;&#x2C6; 2 )

Then appears a possible formulation of the problem of finding the optimal distribution of the roots of the characteristic equation, providing extreme value of the amplitude of forced harmonic oscillations and subsequent parametric synthesis of a dynamic system for optimal distribution of the roots, using variable parameters Îź, c.

The occasion of the imaginary roots đ?&#x153;&#x2020;1,2 <1 corresponds to a dynamic system stability boundary and the region of convergence of a power series with the harmonic action: đ?&#x153;&#x2C6; 2 ( ) <1 đ?&#x203A;˝ The components of the dynamic process of the following form is used: 2

â&#x2C6;&#x2018; đ?&#x2018;Ľđ?&#x2018;Ľ(0) (đ?&#x203A;˝, đ?&#x2018;Ą) = đ?&#x2018;Ľ0 đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x203A;˝đ?&#x2018;Ą + đ?&#x2018;ĽĚ&#x2021; 0 đ?&#x2018;&#x2DC;=1 2

â&#x2C6;&#x2018; đ?&#x2018;Ľđ?&#x2018;&#x201C;(0) (đ?&#x203A;˝, đ?&#x153;&#x2C6;, đ?&#x2018;Ą) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;0 đ?&#x2018;&#x2DC;=1

đ?&#x203A;˝2

đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x203A;˝đ?&#x2018;Ą đ?&#x203A;˝

đ?&#x153;&#x2C6;đ?&#x203A;˝ đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x203A;˝đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x153;&#x2C6;2

2

đ?&#x203A;˝2 â&#x2C6;&#x2018; đ?&#x2018;Ľ(đ?&#x203A;˝, đ?&#x153;&#x2C6;, đ?&#x2018;Ą) = â&#x2C6;&#x2019;đ?&#x2018;&#x17D;0 2 đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą đ?&#x203A;˝ â&#x2C6;&#x2019; đ?&#x153;&#x2C6;2

đ?&#x2018;&#x2DC;=1

Dynamic process is a superposition of his own harmonic oscillations with a frequency of β and forced harmonic oscillations with a frequency đ?&#x153;&#x2C6;: đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ??ľđ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x203A;˝đ?&#x2018;Ą + đ?&#x203A;ž) + đ??´đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x153;&#x2C6;đ?&#x2018;Ą + đ?&#x153;&#x2018;). đ?&#x203A;˝2

đ?&#x2018;?

where đ?&#x153;&#x2018; = 0; đ??´ = đ?&#x2018;&#x17D;0 đ?&#x203A;˝2 â&#x2C6;&#x2019;đ?&#x153;&#x2C6;2; đ??ľ = â&#x2C6;&#x161;đ?&#x2018;? 2 + đ?&#x2018;&#x2018; 2 ; đ?&#x2018;Ąđ?&#x2018;&#x201D;đ?&#x203A;ž = đ?&#x2018;&#x2018;, here Ń = đ?&#x2018;Ľ0 ; đ?&#x2018;&#x2018; =

đ?&#x2018;ĽĚ&#x2021; 0 đ?&#x203A;˝

đ?&#x153;&#x2C6;đ?&#x203A;˝

â&#x2C6;&#x2019; đ?&#x2018;&#x17D;0 đ?&#x203A;˝2 â&#x2C6;&#x2019;đ?&#x153;&#x2C6;2.

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Then analytical modeling of the dynamic process is conveniently carried out by vector (phase) diagrams [7], introduced by rotating vectors corresponding to forced vibrations: đ??´Ě&#x2026; = đ??´đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x153;&#x2C6; and natural oscillations: đ??ľĚ&#x2026; = đ??ľđ?&#x2018;&#x2019;Ě&#x2026;đ?&#x203A;˝ where đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x153;&#x2C6; , đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x203A;˝ â&#x20AC;&#x201C; unit vectors defined as: đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x153;&#x2C6; = đ?&#x2018;&#x2013;Ě&#x2026;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x153;&#x2C6;đ?&#x2018;Ą + đ?&#x2018;&#x2014;Ě&#x2026;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x153;&#x2C6;đ?&#x2018;Ą đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x203A;˝ = đ?&#x2018;&#x2013;Ě&#x2026;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; (đ?&#x203A;˝đ?&#x2018;Ą + đ?&#x203A;ž) + đ?&#x2018;&#x2014;Ě&#x2026;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x203A;˝đ?&#x2018;Ą + đ?&#x203A;ž) Vector resultant of the dynamic process is defined as the vector sum of: đ?&#x2018;&#x2020;đ?&#x2018; Ě&#x2026; = đ??´đ?&#x2018;&#x2019;Ě&#x2026;đ?&#x153;&#x2C6; + đ??ľđ?&#x2018;&#x2019;Ě&#x2026;đ?&#x203A;˝ where đ?&#x2018;&#x2020; â&#x20AC;&#x201C; the resulting vector unit; đ?&#x2018; Ě&#x2026; â&#x20AC;&#x201C; unit vector: đ?&#x2018; Ě&#x2026; = đ?&#x2018;&#x2013;Ě&#x2026;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; (đ?&#x153;&#x201D;đ?&#x2018;Ą + đ?&#x153;&#x201C;) + đ?&#x2018;&#x2014;Ě&#x2026;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;(đ?&#x153;&#x201D;đ?&#x2018;Ą + đ?&#x153;&#x201C;) where đ?&#x153;&#x201D;, đ?&#x153;&#x201C; â&#x20AC;&#x201C; frequency and phase of the resulting dynamic process.

The phase state of a dynamic system is located on the resulting vector using the scalar product of the form: đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;&#x2014;Ě&#x2026; â&#x2C6;&#x2122; đ?&#x2018; Ě&#x2026;đ?&#x2018;&#x2020; By this method, it is possible to analytically model the dynamic process in a wide range of natural frequencies, including the heartbeat mode (βâ&#x2030;&#x2C6;ν), the resonance mode (đ?&#x203A;˝ = đ?&#x153;&#x2C6;).

Summary. For analytical modeling of the dynamic process in the problem of vibration isolation uses a new form of analytical solutions of inhomogeneous systems of linear differential equations, wherein the ordering and recording relatively conservative index characteristic equation root. When a harmonic external kinematic effects of established conditions for the stability of a dynamic process linking the external oscillation frequency and magnitude of the characteristic equation roots. Analysis of stable oscillatory dynamic process proposed to carry out the method of vector diagrams. Tasks dynamic design system analytically solvable by the distribution of the roots.

References [1] Lobas, L.G., Lobas, Lyudm.G. Theoretical mechanics. Kiev: DETUT, 2009 â&#x20AC;&#x201C; p.407. (in Russian)

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[2] de Oliveira, L.R. & de Melo, G.P. J Braz. Soc. Mech. Sci. Eng. (2016) 38: 59. doi:10.1007/s40430015-0413-6 [3] Ovchinnikov P.P. Visha matematika [Higher mathematics], part 2, Kyiv, Technika Publ., 2000, p.797 (in Ukrainian).

[4] Andrianov, I.V., Danishevsâ&#x20AC;&#x2122;kyy, V.V. & Kalamkarov, A.L. Nonlinear Dyn (2013) 72: 37. doi:10.1007/s11071-012-0688-4 [5] Ervin, E.K. & Wickert, J.A. Nonlinear Dyn (2007) 50: 701. doi:10.1007/s11071-006-9180-3 [6] Kravets Victor V., Bass Konstantin M., Kravets Tamila V. & Tokar Lyudmila A. (2016). Analytical Modeling of Transient Process In Terms of One-Dimensional Problem of Dynamics With Kinematic Action. Mechanics, Materials Science & Engineering, Vol 2. doi:10.13140/RG.2.1.4017.0005 Cite the paper Kravets Victor V., Kravets Tamila V., Fedoriachenko Serhii A. & Loginova Anastasia A. (2016). Analytical Simulation of Dynamical Process in One-Dimension Task. Mechanics, Materials Science & Engineering, Vol 6. doi:10.13140/RG.2.2.20337.34347

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VII. Environmental Safety M M S E J o u r n a l V o l . 6

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The Impact of Vehicular Emissions on Air Quality in Uyo, Nigeria Aondona Paul Ihom1, a, Ogbonnaya Ekwe Agwu1, b, John Akpan John1, c 1 – Department of Mechanical Engineering, University of Uyo, Uyo, Nigeria a – paulihom@yahoo.co.uk b – o.agwu4@yahoo.com c – mailjohnjohn001@gmail.com DOI 10.13140/RG.2.1.1813.7845

Keywords: air quality, pollution, vehicular emission, roads, health implications.

ABSTRACT. Roadways, especially road intersections, contribute in no small way to the degradation of air quality in modern cities. This is largely due to toxic emissions from the ever-increasing number of vehicles plying these roads. For rapidly growing cities like Uyo, Nigeria, periodic and quantitative analysis of vehicular emissions may provide knowledge needed to stave off disastrous air pollution. Consequently, this study characterised vehicular emissions in four different locations in Uyo metropolis. Particular locations of the study were: Ekpri Nsukara Junction by Nwaniba Road (Station 1), Uyo City Centre (Station 2), Ikot Ekpene Road by Udi Street (Station 3) and Edet Akpan Avenue by Oron Road (Station 4). Attair 5X Multigas Detector was used to identify and measure the air pollutant concentrations in the four stations of study from July to September 2015. The average concentration of CO obtained in stations 1, 2, 3 and 4 were 8.5 ppm, 3.58 ppm, 11.08 ppm and 3.50 ppm respectively. The highest average concentration of CO was obtained in station 3. Also, the mean concentration of NOx and SOx was less than 0.01 ppm in all four stations. These and the other pollutants measured, H2S, NH3, CO2 and SPM, were found to be well within the habitable range of pollutant concentration as stipulated by the World Health Organization and other relevant bodies.

1. Introduction. In December 2015, a red alert, signifying very unhealthy air conditions, was triggered in Beijing, China. As part of the emergency response plan, the government reduced the number of cars on the road by half whilst consistently monitoring the air quality [1]. Three years earlier, [2] recommended reduction in vehicular emissions as a means of tackling poor air quality in China was made. Therefore, the very high levels of air pollution in places like Beijing, are partly attributable to emissions from the large number of vehicles plying the roads. Not just in Beijing, vehicular emissions are a big problem in most cities of the world today. This is because these emissions lower air quality with resultant health implications on animal and human life [3]. In the south-south Nigeria city of Uyo, vehicular traffic has rapidly increased over the last five years.

Whereas the number of vehicles in Uyo is nowhere near those in Beijing, it is instructive to carry out a study of local air quality with a view to assessing the pollution levels, as it would amount to ‘fixing the car whilst the engine is yet running’.

Several papers have been published corroborating the fact that roadways are one of the most important sites for the emanation of dangerous pollutants emitted by vehicles ([2], [5]–[8]). Significant reduction in the levels of vehicular emissions and hence air pollution are recorded when vehicle speed is reduced. [9], [10]. The impact of vehicular emissions on air quality in the Indian city of Calcutta was assessed by [11]. Vehicular emission samples collected in the city were reported to contain suspended particulate matter (SPM), NOx, SO2, CO and lead (Pb) with CO at a concentration of around 7000 microgram/m3 (µg/m3) as the leading pollutant [12]. The work also evaluated the effects of traffic and vehicle characteristics on vehicle emissions near road intersections. They noted that vehicular emissions depend a great deal on engine loading. Oxides of nitrogen (NO x) were found to increase with increasing engine loading. Similarly, [13], observed low CO emissions with the increase of engine capacity. [14] carried out a study very much like present work but in selected heavy traffic MMSE Journal. Open Access www.mmse.xyz

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areas in Jos, Nigeria. Of the pollutants considered, CO was found to be present in unhealthy levels.

[15] sought to correlate fuel quality improvement with air quality. The study lasted for 10 years tracing the trends in fuel quality improvements in Bangalore city of India and the resultant effect on air quality in the city. The paper reported that particulate matter and sulphur dioxide concentrations decreased as a consequence of the incremental fuel quality norms implemented during the study period. This occurred despite the near 50% increase in traffic loads.

The effect of outdoor air pollution on human health is destructive. Increase in pulmonary diseases, lower respiratory infections, lung and urinary tract cancer have all been attributed to breathing contaminated air [16].

Vehicular emissions are due to the complete or incomplete combustion of hydrocarbon fuel in vehicle engines. The combustion of hydrocarbons yields different results depending on the nature of impurities and the extent of combustion. Regardless of fuel composition and purity, the combustion reaction conforms to the general equation: Fuel + Oxidiser â&#x2020;&#x2019; Products of Combustion + Energy The oxidiser is commonly the oxygen part of air. Where there is just enough quantity of oxygen to complete the reaction, the process is said to involve a stoichiometric amount of oxygen. Otherwise, there may be excess or inadequate amount of oxygen for the combustion process. Often, there is excess oxygen and the reaction is limited by the amount of fuel available. On the other hand, with insufficient supply of oxygen, combustion is terminated by the inadequacy of the oxidiser. The products of combustion are different in both cases.

Complete combustion occurs where there is at least the stoichiometric amount of oxygen required for the process. The products of complete combustion of hydrocarbon fuels are carbon dioxide and water [4].

Incomplete combustion is the result of inadequate supply of the oxidizer (oxygen). Carbon monoxide and carbon are produced instead of carbon dioxide. The products of combustion also include, among others, oxides of nitrogen and oxides of sulphur.

Consequently, the constituents of vehicular emissions are many and varied, some more dangerous than others. These contribute in no small way to the reduction of air quality in large and growing cities of the world. Little wonder that in 2014, the World Health Organisation (WHO) reported several cases of death related to air pollution globally [3].

Such reports are often based on data obtained from the most advanced countries of the world. The extent of damage done in less affluent and developing countries are largely not considered. This void needs periodic completion especially as rural-urban migration is high in such cities. The main objective of this study then, is to determine the levels of concentration of the different air pollutants resulting from vehicular emissions in Uyo, Nigeria and compare them to safe standards so that wellinformed and appropriate action or attention is given as quickly as possible before very dangerous levels are reached. Four key high traffic areas in Uyo metropolis were identified and the research work was confined to these four stations.

2. Materials and Method.

Materials and Equipment. Equipment for detecting and characterising pollutants in vehicular emissions have become commonplace even if expensive. The Attair 5X Multigas detector, a portable device owned by the Research and Development Laboratory of the Ministry of Science and Technology, Uyo, was used in detecting the amounts of NH3, H2S, NOx, SOx, CO and CO2 in four high traffic areas of the city. The study areas were: Ekpri Nsukara Junction by Nwaniba Road, Uyo Town Centre, Ikot-Ekpene Road by Udi Street and Edet Akpan Avenue by Oron Road.

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Method. The catalytic sensor of the Multigas detector when turned on and exposed to the environment is capable of detecting and displaying readings of the gaseous pollutant concentration in the surroundings. Readings were taken in the monitoring stations over a period of three months (July â&#x20AC;&#x201C; September, 2015). Twice during every month of study, the air quality index parameters were measured during periods of peak vehicular activity in each station.

The amount of suspended particulate matter in these study areas were determined using a less sophisticated process: In the petri dish, filter paper was dried in an oven at 105 °C for an hour, allowed to cool in a desiccator for half an hour before weighing and the result recorded as W1.

The filter paper in the petri dish was placed uncovered at a strategic position at a point of interest for a period of 2 hours. At the expiration of the period, it was covered and taken to the laboratory. The set-up was dried in the oven at the same temperature (105oC) for 15minutes and then removed from the oven and allowed to cool in the desiccator for 30mins before weighing and the result recorded as W2. Using the formula below, the concentration of suspended particulate matter was calculated.

đ?&#x2018;&#x2020;đ?&#x2018;&#x192;đ?&#x2018;&#x20AC; =

(đ?&#x2018;&#x160;2 â&#x2C6;&#x2019;đ?&#x2018;&#x160;1 )Ă&#x2014;106 đ?&#x2018;&#x2030;

(1)

Îźg/m3

where SPM refers to the concentration of suspended particulate matter in Âľg/m3; W1 and W2 â&#x20AC;&#x201C; are respectively, the initial and final weight of filter paper in grams; V â&#x20AC;&#x201C; is volume of petri dish in m3.

3. Results and Discussion.

Results. The results of the study are presented below:

Average Concentration of NOx and SOx in the monitored stations over 3 months

Concentration (ppm)

0.03 0.025 0.02 0.015 0.01 0.005 0 Station 1

Station 2

Station 3 NOx

Station 4

SOx

Fig. 1. Average Concentration of NOx and SOx in the monitored stations.

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Average concentration of CO, H2S and NH3 in the monitored stations over 3 months

Concentration (ppm)

30 25 20 15 10 5 0 Station 1

Station 2 CO

Station 3

Hydrogen Sulphide

Station 4

WHO/NAAQS 2010

Ammonia

Fig. 2. Average Concentration of CO, H2S and NH3 in the monitored stations.

Average concentration of CO2 in the monitored stations for the period of study

Concentration (ppm)

600 500 400 300 200 100 0 Station 1

Station 2

Station 3

Station 4

Fig. 3. Average Concentration of CO2 in the monitored stations.

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Concentration (µg/m3)

Average concentration of SPM in the monitored stations for the period of study

200 180 160 140 120 100 80 60 40 20 0 Station 1

Station 2

Station 3

Station 4

WHO/NAAQS 2010

Fig. 4. Average Concentration of SPM in the monitored stations.

Discussion. Seven different air quality index parameters were examined across the selected four stations. In all charts and tables, the amounts of NOX, SOX, CO, CO2, H2S, NH3 were measured in part per million (ppm) whereas the values of suspended particulate matter (SPM) was rendered in micrograms per cubic metre (µg/m3).

From Fig. 1, NOx and SOx are found to have similar concentrations across all four stations studied.

For both pollutants, the highest average concentrations occur in stations 1 and 3. The top NOx concentration was 0.020 ppm measured in stations 1 and 3. This is 20% lower than the value (0.025 ppm) considered inhabitable by the National Ambient Air Quality Standards (NAAQS). The peak concentration of SOx across the stations (0.01 ppm) is similar to the average values (Fig. 1) and falls well within acceptable levels as further illustrated in Fig. 1.

Fig. 2 displays the average concentrations of three more pollutants: CO, H2S and NH3. H2S and NH3 are found in small amounts in stations 1 and 3, but are virtually non-existent in the other two stations.

The highest concentration of CO (18 ppm) occurred in station 3. This single peak value is a little worrisome as the NAAQS benchmark is 20 ppm. However, further readings over the course of three months presented an average CO concentration of about 11 ppm – a lot less disturbing Fig.. The presence of NH3 and H2S may be as a result of the waste bins containing biodegradables close to where the readings were taken. Nevertheless, the levels of all three pollutants recorded were found to be well below dangerous values. The highest average concentrations occur in stations 1 and 3.

Compared to the National Ambient Air Quality Standards (NAAQS), these peak concentrations are well within acceptable levels as further illustrated in Fig. 1.

Fig. 2 displays the average concentrations of three more pollutants: CO, H2S and NH3. H2S and NH3 are found in small amounts in stations 1 and 3 but are virtually non-existent in the other two stations.

The presence of NH3 and H2S may be as a result of the waste bins containing biodegradables close to where the readings were taken. Nevertheless, the levels of all three pollutants recorded were found to be well below dangerous values.

The concentration range of CO2 is between 276 and 384 ppm as shown in Fig. 3. CO2 is largely implicated in global warming which is why it was evaluated. Compared to the NAAQS, the highest value of 384 ppm observed in station 1 on day 3 is 36 % less than the acceptable limit of 600 ppm (WHO and NAAQS, 2010).

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The results for suspended particulate matter were scattered over the duration of the study. This was partly due to the method of measuring this index, a procedure that did not take into account the prevailing wind direction. The average values as shown in Fig. 4 as well as the peak value recorded were all within a very safe range compared to the NAAQS. Summary. One of the major sources of air pollution in growing cities like Uyo, Nigeria is vehicular emissions. Understanding and evaluating the nature and concentration of vehicular emissions is therefore important in predicting city air quality. This work determined, using the Multi-gas detector, the different types and concentration of each pollutant in four separate study areas in Uyo over a period of three months. The air pollution index parameters considered were: oxides of nitrogen (NOX), oxides of sulphur (SOX), carbon monoxide (CO), carbon dioxide (CO2), hydrogen sulphide (H2S), ammonia (NH3) and solid suspended particulate matter (SPM). The study was carried out across the four areas, which are the busiest road intersections in Uyo. Across the four locations, stations 1 and 3 consistently present significantly higher levels of pollutant concentration than stations 2 and 4. However, it appears there may not be worrying concern of immediate nature concerning air pollution in the city as emission levels are found to be below the hazardous range set by the World Health Organisation and other relevant bodies.

Acknowledgement. The authors wish to acknowledge with thanks the contributions of the staff of Ministry of Science and Technology Uyo Akwa Ibom State who was assigned to us during the course of this study. We appreciate you for painstakingly ensuring that all the needed measurements were taken and at the right time.

References [1] The Guardian, (2015). Beijing's smog 'red alert' enters third day as toxic haze shrouds city.

Available at; http://www.theguardian.com/world/2015/dec/21/beijings-smog-red-alert-enters-thirdday-as-toxic-haze-shrouds-city (Accessed January 18, 2016).

[2] Wang, S. and Hao, J. (2012). Air Quality Management in China: Issues, Challenges and Options.

Journal of Environmental Sciences, 24(1): 2 – 13.

[3] Ihom, P. (2014). Environmental Pollution Prevention and Control: The Current Perspective (A review). Journal of Multidisciplinary Engineering Science and Technology, 1(5): 93-99.

[4] Ashley, S. (1979). Thermodynamic Analysis of Combustion Engines. 2nd ed., New York: Wiley, p. 85.

[5] Skiba, Y. N and Davydova-Belitskaya V. (2003). On the Estimation of Impact of Vehicular Emissions. Ecological Modelling, (166), 169 – 184. DOI: 10.1016/S0304-3800(03)00133-9 [6] Clifford, M. J., Clarke, R. and Raffat, S. B. (1996). Local Aspects of Vehicular Pollution.

Atmospheric Environment, 31 (2): 271 – 276.

[7] Clarke, A. G., Chan, J. M., Pipitsangchan, S. and Azadi-Bougar, G. A. (1996). Vehicular Particulate Emissions and Roadside Air Pollution. The Science of the Total Environment, 189(190): 417 – 422.

[8] Mustafa, S., Mohammed, A., Vougias, S. (1993). Analysis of Pollutant Emissions and Concentrations at Urban Intersections. In: Institute of Transportation Engineers, Compendium of Technical Papers, 2, pp. 3-5. New Scientist (1999). No. 2173, February 13, 1999.

[9] Gonclaves, M., Jimenez-Guerrero, P., Lopez, E. and Baldasano, J. (2008). Air Quality Models Sensitivity to On-Road Traffic Speed Representation: Effects on air Quality of 80 km/hr Speed Limit in the Barcelona Metropolitan Area. Atmospheric Environment, (42), 8389 – 8402.

[10] Dijkema, M., Van der Zee, S., Brunekreef, B. and Van-Strien, R. (2008). Air Quality Effects of an Urban Highway Speed Limit Reduction. Atmospheric Environment, (42), 9098 – 9105.

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[11] Ghose, M. K., Pual, R., and Banerjee, S. K. (2004). Assessment of the Impacts of Vehicular Emissions on Urban Air Quality and its Management in Indian Context. Environmental Science & Policy, (7), 345-351.

[12] Pandian, S., Gokhale, S. and Ghoshal, A. K. (2009). Evaluating Effects of Traffic and Vehicle Characteristics on Vehicular Emissions Near Traffic Intersections. Transportation Research Part D, (14): 180-196.

[13] Gokhale, S. and Khare, M. (2005). A Hybrid Model for Predicting Carbon monoxide from Vehicular Exhausts in Urban Environments. Atmospheric Environment, (39), 4025 – 4040.

[14] Ola, S. A, Salami, S. J., and Ihom, A. P. (2013). The levels of Toxic Gases; Carbon-monoxide, Hydrogen Sulphide and Particulate Matter to index Pollution in Jos Metropolis, Nigeria. Journal of Atmospheric Pollution, 1 (1): 8-11.

[15] Sabapathy, A. (2008). Air Quality Outcomes of Fuel Quality and Vehicular Technology Improvements in Bangalose City, India. Transportation Research Part D, (13), 449– 454.

[16] WHO, (2014). Ambient (Outdoor) Air Quality and Health. http://www.who.int/mediacentre/factsheets/fs313/en/ (Accessed January 18, 2016).

Available

at

Cite the paper Aondona Paul Ihom, Ogbonnaya Ekwe Agwu & John Akpan John (2016). The Impact of Vehicular Emissions on Air Quality in Uyo, Nigeria. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.1.1813.7845

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Utilization of Point Clouds Characteristics in Interpretation and Evaluation Geophysical Resistivity Surveying of Unstable Running Block Marcel Brejcha1, Petr Zbíral2, Hana Staňková2, Pavel Černota2 1 – Data System s.r.o., Dr. Vrbenského 1, 415 01 Teplice, Czech Republic 2 – Technical Univerzity of Ostrava, Faculty of Mining and Geology, Institute of Geodesy and Mine Surveying, 17, listopadu 15/2172, 708 33 Ostrava-Poruba, Czech Republic DOI 10.13140/RG.2.1.1577.2401

Keywords: 3D model, point clouds, active rock collapse, point cloud, electrical resistivity tomography.

ABSTRACT. Close to human residences the places often abound where anthropogenic activity and external factors cause their changes. The changes can often influence their inhabitants’ life thanks to incipient dangerous places. The project of successful design of measures to ensure stability of unstable running blocks depends on chosen approaches and primary resource preparation. Utilization of modern technologies in their taking and processing is required nowadays. The paper describes the taking and processing of data for project of solution „Stabilization of unstable running block“ in a municipal settled part with efficient utilization of unusual method of processing of geophysical resistivity method.

1. Introduction. In the paper described way of mapping and interpretation of geophysical survey is the new approach example of resource preparation for project processing of active rock collapse and its stabilization in the sandstone massif, which outcrops delimite the Jizera river valley. In last ten years Mladá Boleslav the city contends with the demonstration of rock collapse in above mentioned locality. Many active measures were done including breaks of rocky blocks and installation of protective solid safety nets. Specification of requiring resources from mapping and geophysical survey was provided in the case of rocky collapse solution in abandoned workings in Rožátov (Fig. 1).

2. Definition of technical requirements for surveying activity. Active rock collapse problems are based particularly on geomorphological pertinence and the characteristics of locality.

Geomorphology of the region and locality is the result of tectonic development and resulting weathering and its source. Natural processes in the locality are stressed by anthropogenic activity, historically long-term settled area. Along the river Jizera there is the washtub-shaped valley counterbored to calcific sandstones, which make bank cliffs, usually in two levels, in the surrounding of the soluted locality with strong profile on the left side – a cut-bank of the river stream.

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Fig. 1. Area Rozatov.

2.1 Geological connections and description of the area. Locality Rožatov is a part of the left bank, about 7 km long rocky salient, which starts about 2 km northerly and ends in the south of the historical part of Mladá Boleslav – the Old Town, a hump of Mladá Boleslav castle. Natural morphology of the most substantial part – rocky stage, is in many places anthropogenically modelled – mining of thickbedded sandstones (south of locality Radouč and Rožatov) and the integration of historical and modern building on the upper plateau of the stages or in their foothills, usually with the cleaning of fallen cambering materials in pediment. It causes the highlighting and uncovering of the whole profile naturally made, almost vertical, and up to 25m high slope (absolute height of the upper edge of the slope over the horizontal part of pediment).

Locality Rožatov is a part of the left bank, about 7 km long rocky salient, which starts about 2 km northerly and ends in the south of the historical part of Mladá Boleslav – the Old Town, a hump of Mladá Boleslav castle. Natural morphology of the most substantial part – rocky stage, is in many places anthropogenically modelled – mining of thick-bedded sandstones (south of locality Radouč and Rožatov) and the integration of historical and modern building on the upper plateau of the stages or in their foothills, usually with the cleaning of fallen cambering materials in pediment. It causes the highlighting and uncovering of the whole profile naturally made, almost vertical, and up to 25m high slope (absolute height of the upper edge of the slope over the horizontal part of pediment).

The important road is founded on the lower and younger level of the rock-cut terrace – Ptácká street, and the old and new-build constructions along it very often reach the river flats of Jizera (particularly gardens and meadows).

The dividing of the slope structures and their forms of erosions and cambering has come from the geomorphological classification.

In connection with the past mining of thick-bedded sandstones in this locality the natural conditions are strongly anthropogenically changed. Pediment is leveled and maintained, area of accumulation in the foothill of the quarry face is missing. The origin statement is observable compared with the continuation of the older stage to north, along the Ptácká street.

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The upper konvex part over the rocky stage is the result of natural processes and anthropogenic activity. The direct margin over the edge of the face is made of weathering cover from petrean loam.

Its basement complex is represented by slab-disintegrated incoherent sandstones with the transformation to less weathered thick-bedded sandstones. The phenomenon, which is typical for the soluted quarry face, is the breaking of triangular blocks with different bulk on the cross of the joints and opened bedding joints. On the quarry face there are observed frequent breaks with cascade character, i.e. after the fall of a small block the stability of upper-laying block becomes weaker and after its fall the block in the upper level releases again, etc.

A few places with potential danger of rocky collapse are identified in the solved quarry face along the joint systems in the direction of NNW – SSE. The total capacity of rock, which can be touched by this danger, is more than over 100 m3.

2.2 Geodetic surveying and it’s interpretation Based on the facts the basic requirements for geodetic surveying realized in the locality have been set by these criterions: – Detailed 3D model particularly of vertical parts of the rocky massif face; – Surveying of visible bassets of massif; – Possibility of specific identification a evaluation the joint system; – Link to binding reference systems; – Precision of surveying of detailed points with code 3; – Possibility of interpretation of the electric resistivity tomography results; – Economic efficiency (declining of investor´s costs); – Possibility of realization of subsequent project activities in common CAD program systems.

3 Mapping of solved locality. Locality mapping has been done in these phases: 1. Building and stabilizing of survey points no. 4001 to 4007 by the method GNSS surveying in RTK mode. Surveying network has been completed with points no. 5001 to 5005 localized directly in the quarry face (chosen important, unequivocally identified and colorfully different apical formation of rocky massif). Points no. 4002, 4004 až 4007 were colorfully signed for easy unequivocal identification during later processing of photogrammetric surveying. After reciprocal surveying of lengths and directions all the network has been leveled by the method of least squares.

2. Photogrammetrical data capture UAV (by pilotless aircraft) with parameters needed for 3D modelling with high-resolution (Fig. 2 and Fig. 3). Photogrammetric work were realized in two phases: taking surveying pictures by the vertically oriented camera (DJI_0001 – DJI0043 (Fig. 2) and by squarely on vertical face of rock massif oriented camera (DJI_0052 – DJI_0059 (Fig. 3)).

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Fig. 2. Point cloud and cameras position.

Fig. 3. Camera location and image overlap.

Proccessing of photogrammetric surveying including final issues for subsequent processing in Bentley Microstation and AutoCAD was provided by AGISOFT PhotoScan Professional software.

The basic requirement for 3D processing was the final precision of surveying represented by the standard error of image correlation realized by control points with relation to binding reference systems S-JTSK and Bpv (Tab. 1).

From the table it is evident, that in case of convenient decomposition of control points in the vertical part of the quarry face, the results with required conditions can be achieved, even in big amount of projections and difficult terrain settings. Further deployment of UAV technology in quarry enviroment can be found in [1], [2] and [3].

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Table 1. Control Points

In SW Agisoft PhotScan the point clouds were exported in las, dxf and jpeg georeferenced ortophoto map, for subsequent processing. Administration of heavy data load has been proceed by [4]. The drawing of wire model was added with geophysical control points surveyed by the polar method with electric resistivity tomography and with the longitudinal profile axis were constructed from them (Fig. 4). For subsequent project documentation, the borderline map of plots of land was constructed by the data resource INSPIRE ČUZK (Fig. 4). Detailed methodics is described in [5].

Evaluation of joint system in the locality was made by the measuring of direction and inclination from the space model by the authorial methodology, which allows the measuring the direction and inclination from the image reconstruction to the cloud of oriented 3D points also in physically unavailable parts of the joint or rocky salients, as in Rožátov locality. The issue of surveying was in Bentley Microstation provided the joint system drawing in wire model (Fig. 5) and then the rose diagram of direction and inclination of the joints in the locality (Fig. 6).

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Fig. 4. ERT profiles.

Fig. 5. Joint system drawing.

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Fig. 6. Rose diagram of inclination of the joints direction.

4. Setting of the geophysical method Electrical Resistivity Tomography. During the realization of geomorphological analysis of solved unstable rocky block the geophysical method Electical Resistivity Tomography (ERT) was used. The base of surveying by ERT is founded on the calculation of decomposition of resistivity under the earth. The initial preliminary parameter is the measured electrical potential caused by the passing of direct current between the pair of electrodes. The disposition of electrodes and other configuration is influenced by local lithological structure and geological characteristics of solved formation. ERT primary results capture is followed by processing of survey points by two-dimensional tomographical inversion with their subsequent modification by the statistic methods. The result is the two-dimensional model of resistivity decomposition under the earth in the axis of measured profile (Fig. 7). Detailed ERT method description is listed in [6].

Fig. 7. Two-dimensional interpretation of surveying results by ERT of solved rock block.

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Utilization of ERT for identification of subareal structures supposes preliminary knowledge of the solved locality. The important atribute of succesful utilization of this method is the creation of comparison-reference model from the data captured on the base of surveying of the structural elements in the rock face, from the boreholes and probes. That is why the first ERT surveyed profile was done close to the rock face.

4.1 ERT results used together with data procured by UAV technology. Spatial objects, which are represented by point clouds, can be analysed simply with high precision. It was achieved because new technologies use instruments for processing huge point clouds with high users comfort. At processing of primary data procured by image correlation, the characteristics of point clouds have been used to get detailed data about surface qualities of the solved rock massif. For interpretation and surveying of the formations like joints, cracks and other structures there have been used Carlson Point Cloud 2016 system, which has a lot of functions available for point cloud analyses. That is why it was important to find how to use ERT results effectively.

4.2 Transformation of resistivity values surveyed by ERT to point cloud structure. ERT surveying was realized by specialized company, which during the work acted upon the geodetically established profiles. Then voxelisation of surveyed values was provided and distribution of electrical resistivity in the rock massif to the form of grids with measurement 0.2 m (Fig.8). Subsequently their export to data files in ASCII format was done. This text file contained spatial coordinates of individual exported points from grid structured expressed by S-JTSK system and relevant electrical resistivity (Fig. 9).

Fig. 8. Surveyed data in ASCII format.

Fig. 9. Interpretation of electrical resistivity progress in the rock massif in longitudinal profiles.

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Data structure describing point clouds offers, besides the information about spatial position of individual points, information about other parameters, for example values R, G, B, describing real colour of particular points, or intensity of laser ray reflexion [7]. The parameter of reflexion intensity was used in additional processing of surveyed values by ERT method. Electrical resistivity values in every point were set to the point cloud as substitution for values of laser reflexion intensity.

Sequentially, after retrieval to programme system Carlson Point Cloud 2016, structure borderlines were made in the form of vectors. These profiles represented by the spatially localised points were processed as the point clouds. This way of display was used to create three profiles, surveyed by ERT method with represented intensity of laser ray reflexion in colour hypsometry. Their interpretation in Autodesk ReCap system is in Fig. 10 and Fig. 11.

Fig. 10. Display of measured electrical resistivity as a substitution for ray reflexion intensity in Autodesk ReCap 360.

Such prepared data were added to the point cloud taken by image correlation method from the UAV technology resources (Fig. 11). This way created spatial model has allowed quality base for detailed analyses of the condition of the solved unstable rock massif.

Fig. 11. Data combination of rock block surface and measured results ERT in Autodesk ReCap 360.

Summary. The application of described process of mapping by the photogrammetric method added by surveying of control points of geotechnical research has, compared with earlier research workings MMSE Journal. Open Access www.mmse.xyz

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in solved areas, unequivocally better basis for a project documentation. Compared with the wire model, or digital model of terrain, made from data by polar method surveying of detailed points, the point cloud taken by photogrammetry offers more complex information about the progress of geological failures and joint system.

CAD software systems for processing of point clouds guarantee the direct measurability of other important geological information including bedding, direction and inclination. Combined with the possibility of interpretation of geotechnical research results by electrical resistivity tomography into the homogenous point cloud together with the digital terrain model, this technique offers the most complex resources for the structure evaluation of the rock mass, risks of rock collapse and definition and proposals of available arrangements.

Unequivocal benefit for an investor is the time effectivity of this mapping method. Compared with earlier used polar mapping method and subsequent difficult processing of results, this method can save up to 40 % on time, especially in the phase of detailed terrain mapping. The integration of geotechnical research results allows the complex and effective evaluation of the locality and precise spatial orientation of problem places. The benefit of this method is illustrated by the fact, that its utilization is planned in the preparation of arrangements against active rock collapse in other three localities in the Central Bohemian Region.

References [1] Gasinec, J.; Gasincova, S.; Trembeczka, E. Robust Orthogonal Fitting of Plane. Inżynieria Mineralna, 2014, 15.1: 7–13.

[2] Pukanska, K.; Labant, S.; Bartoš, K.; Gašinek, J.; Zuzik, J.: Determination of deformation of high-capacity tank using terrestrial laser scanning. Acta Montanistica Slovaca, 2014, 19.1: 41–46.

[3] Salvini, R., Riccucci, S., Gullì, D., Giovannini, R., Vanneschi, C., & Francioni, M. (2015).

Geological application of uav photogrammetry and terrestrial laser scanning in marble quarrying (Apuan alps, Italy). In Engineering Geology for Society and Territory - Volume 5: Urban Geology, Sustainable Planning and Landscape Exploitation (pp. 979-983). Department of Environment, Earth and Physical Sciences and Centre of Geotechnologies CGT, University of Siena, Via Vetri Vecchi 34, San Giovanni Valdarno, Italy. http://doi.org/10.1007/978-3-319-09048-1_188 [4] Bartoněk, D., Bureš, J.;Opatřilová, I.: Workflow for Analysis of Enormous Amounts of Geographical Data. Advanced Science Letters. 2015, 21(12), 3680-3683. DOI: 10.1166/asl.2015.6540.

ISSN 19366612.

Dostupné také z: http://openurl.ingenta.com/content/xref?genre=article [5] Poláček, J., Souček, P. (2012). Implementing INSPIRE for the Czech.Geoinformatics FCE CTU, 8, 9-16.

[6] Daily, W., Ramirez, A., Binley, A., & LeBrecque, D. (2004/05/01). Electrical resistance tomography. The Leading Edge, 23(5), 438-442. http://doi.org/10.1190/1.1729225 [7] Pacina, J., Brejcha, M. (2014). Digital terrain models. Ústí nad Labem: J.E. Univerzity in v Ústí nad Labem, Faculty of environment, 2014. ISBN 978-80-7414-815-6.

Cite the paper Marcel Brejcha, Petr Zbíral, Hana Staňková & Pavel Černota (2016). Utilization of Point Clouds Characteristics in Interpretation and Evaluation Geophysical Resistivity Surveying of Unstable Running Block. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.1.1577.2401

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Atmosphere Re-Entry Simulation Using Direct Simulation Monte Carlo (DSMC) Method Francesco Pellicani1,a 1 – Department of Civil and Industrial Engineering , Faculty of Engineering, Università di Pisa, Pisa, Italy / Institute of Mechanical Engineering, Faculty of the Science and Technique of Engineering, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland a – francescopellicani17@gmail.com DOI 10.13140/RG.2.2.31445.83684

Keywords: space, re-entry, Monte Carlo, simulation, thermal protection system.

ABSTRACT. Hypersonic re-entry vehicles aerothermodynamic investigations provide fundamental information to other important disciplines like materials and structures, assisting the development of thermal protection systems (TPS) efficient and with a low weight. In the transitional flow regime, where thermal and chemical equilibrium is almost absent, a new numerical method for such studies has been introduced, the direct simulation Monte Carlo (DSMC) numerical technique. The acceptance and applicability of the DSMC method have increased significantly in the 50 years since its invention thanks to the increase in computer speed and to the parallel computing. Anyway, further verification and validation efforts are needed to lead to its greater acceptance. In this study, the Monte Carlo simulator OpenFOAM and Sparta have been studied and benchmarked against numerical and theoretical data for inert and chemically reactive flows and the same will be done against experimental data in the near future. The results show the validity of the data found with the DSMC. The best setting of the fundamental parameters used by a DSMC simulator are presented for each software and they are compared with the guidelines deriving from the theory behind the Monte Carlo method. In particular, the number of particles per cell was found to be the most relevant parameter to achieve valid and optimized results. It is shown how a simulation with a mean value of one particle per cell gives sufficiently good results with very low computational resources. This achievement aims to reconsider the correct investigation method in the transitional regime where both the direct simulation Monte Carlo (DSMC) and the computational fluid-dynamics (CFD) can work, but with a different computational effort.

Introduction. When an atmosphere has to be penetrated, whether it be the Earth's or that of another planet, the conditions encountered are always extreme because of the heat, pressure and chemical activity which are encountered. It is evident that spacecrafts have to be protected during atmosphere entry. This is achieved by installing a TPS (Thermal Protection System) on the vehicle. Since the 1950s, estimation of the heating environment experienced by atmospheric entry vehicles was achieved using analytical formula that rely on theoretical and empirical correlations. It is clear that such a simplified hypothesis induces large error margins and consequently a higher safety factor has to be maintained when designing the TPS. That is why there is the need to improve this field of space engineering to build a model as closest as possible to what happens in reality in order to obtain exact previsions of what a thermal shield will encounter during its lifetime. This is done passing from the use of analytical formulas to numerical simulations, which now are the main tool of investigation in this field. Space vehicles entering the atmosphere undergo not only different velocity regimes, hypersonic, supersonic and subsonic, but also different flow regimes, free molecular flow, transition, and continuum. Each of these flow regimes must be considered during the vehicle aerothermodynamic design. At the highest altitudes, the interaction of the vehicle with the atmospheric air is characterized by free molecular flow. In this regime, the air molecules collide and interact with the vehicle's surface. However, collisions of reflected particles from the surface with freestream particles are not likely to occur. As the vehicle enters deeper into the Earth's atmosphere, the mean free path decreases and collisions between particles reflected from the vehicle's surface and the incoming freestream particles can no longer be ignored. As a result, the flow in this condition MMSE Journal. Open Access www.mmse.xyz

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defines the transition flow regime, i.e., transition between the free molecular flow regime and the continuum flow regime. In the transition flow regime, the contribution of aerodynamic forces and heat flux to the vehicle surface start to increase rapidly with decreasing altitude, causing large changes in the aerodynamic characteristics of the vehicle when compared with those observed in the free molecular flow. At such altitudes, strong shock waves are formed in front of the vehicle and new flow features such as thermodynamic and chemical nonequilibrium become important for the correct prediction of heating rates and pressure loads acting on the vehicle's surface. As the vehicle continues to enter into the atmosphere, it finally reaches a dense atmosphere characterized by the continuum flow regime. In this regime, the flow around the vehicle is examined by means of a macroscopic model that considers the air as a continuum and the description of the flow is made in terms of spatial and temporal variations of macroscopic properties, such as velocity, pressure, density and temperature. The purpose of this research is to provide a valid method which allows the calculation of the conditions encountered by a spacecraft entering the atmosphere without inertness in the transitional regime. This kind of regime is borderline for the Navier-Stockes equations applicability.

In pursuit of this goal, flows are investigated by employing the direct simulation Monte Carlo (DSMC) method instead of the classic Computational Fluid Dynamics (CFD). The use of this kind of method in the study of hypersonic spacecrafts atmosphere entry at high altitude has been recently implemented in some software, so it is quite a new technique used to deal with these kinds of phenomena. The direct simulation Monte Carlo method (DSMC) was almost exclusively developed

by G. A. Bird [1] between 1960 and 1980 and has become one of the most important numerical techniques for solving rarefied gas flows in the transition regime. The first DSMC solvers were developed years later initially by Bird himself and then by other researchers. Only recently, after extensive development, the software is able to provide realistic results. However the diffusion of this method is still low. In addition, there is not a massive validation of the DSMC technique in the reentry study, which guarantees the production of realistic simulations each time it will be used, even if some results are produced in this way [2].

Therefore the preliminary target of this study is to find the best setting to be used for the DSMC to have accurate simulations, but at the same time efficient in terms of time, power and resources consumption. In particular, two types of software are taken into consideration, OpenFOAM and Sparta. The subject of the study is the Hayabusa capsule, a probe launched by the Japan Aerospace Exploration Agency (JAXA). The choice of Hayabusa is dictated by the collection of some aerodynamic data made at point H3 of its re-entry trajectory. This point, at an altitude of 78,8 km with respect to the Earth's surface, is entirely in the transitional regime so it provides some experimental data to be compared with the values obtained from the DSMC. In this way, it is possible also to obtain the second target of this study, that is to validate the program and guarantee the production of correct results, considering the absence of experience in this field of application.

DSMC method. The DSMC method has its basis on physical concepts of rarefied gases and on the physical assumptions that form the basis for the derivation of the Boltzmann equation [3]. However, the DSMC method is not derived directly from the Boltzmann equation. As both, the DSMC method and the Boltzmann equation are based on classical kinetic theory, then the DSMC method is subject to the same restrictions of the Boltzmann equation, i.e., restrictions related to diluted gases and the assumption of molecular chaos. The DSMC method models the flow as a collection of particles or molecules. Each particle is classified with a position, velocity and internal energy [4]. The state of the particle is stored and modified with the time as the particles move, collide and interact with the surface in the simulated physical domain. The assumption of dilute gas (where the mean molecular diameter is much smaller than the mean molecular space in the gas) allows the molecular motion to be decoupled from the molecular collisions. The particles movement is modeled deterministically, while collisions are treated statistically. Since it is impossible to simulate the real number of particles in the computational domain, a small number of representative particles are used and each one represents a large number of real particles (nEquivalentParticles). Simulations can contain from thousands to millions of DSMC particles simulators in rarefied flow problems. A computational grid, MMSE Journal. Open Access www.mmse.xyz

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representing the physical space to be investigated, is necessary in order to use this method. Each cell provides a convenient reference for the sampling of the macroscopic gas properties and for the choice of the potential collision pairs. The DSMC algorithm can be briefly divided into four individual main steps [5]: 1) move particles over the time step. 2) apply boundary conditions such as introducing new particles at inflow boundaries and removing particles at outflow boundaries. 3) organize particles into cells and perform collisions. 4) sample average particle information.

The fundamental parameters in a DSMC simulation are the cell size, the equivalent number of particles and the time step. The linear dimensions of the cells should be small in comparison with the length of the macroscopic flow gradients normal to the streamwise directions. In order to accurately model collisions by using a statistical approach, the cell size should be of the order or smaller than the local mean free path in the direction of primary gradients [1] [6-7]. This is because in certain regions, such as in the vicinity of the surfaces, the cell size must be small enough to adequately capture the steep macroscopic gradients and the flow field physics near the wall. The mean free path for the Earth's atmosphere at the H3 point according to the U.S. Standard Atmosphere is 3.762e-3 m.

An additional requirement of the DSMC method is related to the minimum number of simulated particles in the cells. As mentioned earlier, the DSMC method uses a cell-based system for the sampling of the macroscopic properties and for the selection of collision partners. As the collision rate is a function of the number of particles in the cells, it is desirable that each cell has the largest possible number of particles. However, the possible number of collision partners is a function of the number of particles in each cell. In this scenario, the greater the number of particles, the greater is the number of possible collision pairs. As a result, it is necessary to determine the optimum number of particles in each cell; enough to promote statistical accuracy while maintaining realistic computational expenditure. Thus, it is desirable that each cell has a minimum number around 20 to 30 particles [1]. Another requirement of the DSMC method is the setting of an appropriate time step.

The trajectories of the particles in physical space are calculated under the assumption of the decoupling between the particle motion and the intermolecular collisions. The time step should be chosen to be sufficiently small in comparison with the local mean collision time to allow the uncoupling between the movement and collisions of a set of particles. In addition, if the time step is too large, particles can cross many cells in one time step and consequently the results may be inaccurate. On the other hand, too small a time step will result in inefficient computation [8-9]. The local mean collision time for the problem analyzed is 3.161e-7 s. Considering the data reported previously, the theoretical initial parameters for the simulation considered are: 1) cell size: 0.001 m.

2) equivalent particles per cell: 20. 3) time-step: 1e-7 s. These parameters are the ones that mainly rule the simulations. The success of the calculation and also the accuracy of the results depend on them [10]. In order to examine how these parameters change the results and which are the best ones, a sensitivity study is performed varying the cell size, the number of particles per cell and the timestep.

Sensitivity study and simulation optimization. The principal parameters in the sensitivity study are changed in a schematic way so that it is possible to obtain such a law for variations of the parameters.

The data of each simulation are compared with the mean value of all the simulations to establish the most accurate [2]. This analysis is also important to understand what is the most efficient way to do these simulations, in order to find the best way to achieve accurate results, but also an acceptable computational time, considering that this calculation is very long and resource consuming. From this preliminary study, efforts have been made to find the correct final time for the simulations, thus the minimum time for which the variations in the results are not significant. This is another thing to take into account to establish the most efficient way for the calculation. This study uses simulations without reactions to obtain results more quickly.

In the sensitivity analysis only some data given directly from the software are considered and they are taken from the stagnation line. The pressure, overall temperature and Mach number data for the sensitivity analysis are chosen to be analyzed. In particular the significant data considered are as follow: (a) the position where the Mach number is equal to 1 which should give the exact position of MMSE Journal. Open Access www.mmse.xyz

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the shock (Position(x) Ma=1). Indeed, in the conditions analyzed, it is evident the shock is diffused, there is not a precise discontinuity point, because of the rarefaction of the gas; (b) the part of the pressure slope where the gradient is higher than one (Shock Layer p). This is a feature of the shock in a rarefied gas. In fact in a dense fluid, the increase of the pressure in the shock should be vertical, so it happens in a single point (discontinuity), as anticipated; (c) the maximum temperature (max T) and the part of the temperature slope where the temperature is higher than 95% of the maximum temperature reached along the streamline (Shock Layer T). These give an indication about the accuracy of the simulation, comparing the absolute value reached by the temperature with similar data deriving from literature, and about the thickness of the shock respectively. For each simulations the time required by the super computer to complete the calculations and the computational resources, in terms of RAM Gb, used to evaluate the most efficient simulation are recorded. For each series of simulations the variations of only one of the principal parameters are considered while the others are maintained constant at the reference value, the value proposed by Bird. As an example case only the sensitivity analysis of the number of particles per cell factor for the Sparta software is reported. The other factors are studied in a similar way. Also the sensitivity analysis of the other software analyzed, OpenFOAM, is omitted in this article because it is less interesting. In fact OpenFOAM reveals a certain solidity of the code considering that the variation of the principal parameters, at least in the range analyzed, does not produce appreciable differences in the results. Also the computational resources are quite similar for the entire set of simulations and the end time analysis provide the same result of the Sparta one.

The number of particles suggested by Bird is 20/30 per cell. Starting from this number as the reference value, a correct sensitivity analysis should have a simulation with both a higher and a lower number of particles, considering the same factor in both cases. However, a problem arose with Sparta to run a simulation with a high number of particles, because of some problems linked to the subdivision of the domain for launching the calculation in a cluster. For this reason, the standard number of particles per cell was drastically reduced. Besides, the same increasing factor was not used for each case. The particle factor is analyzed using a reference value of 1 particle per cell. Then the other cases investigated have a higher number of particles in two cases, respectively of a factor of 10 and 50 compared to the reference value. Instead, one has a lower number of particles by a factor of ten as reported in Table 1. The values considered related to each case are reported in the following Table 2.

In case III p S, as can be seen, there are no results because the simulation was not able to run due to the problem explained previously.

Table 1. Parameter chosen for the Sparta sensitivity analysis for the particle factor.

Simulation Particles per Cell nEquivalentParticles Factor IpS 1 1e16 Ref. II p S 10 1e15 10 III p S 50 5e14 50 IV p S 0.1 1e17 1/10

Table 2. Values obtained in the Sparta sensitivity analysis for the particle factor.

Simulation Position(x) Ma=1 max T, (K) Shock Layer T Shock Layer p (m) (m) (m) IpS 3.550e-2 7.632e4 1.000e-2 5.500e-2 II p S 3.550e-2 7.329e4 2.000e-2 4.000e-2 III p S / / / / IV p S 2.650e-2 1.046e5 2.000e-3 3.500e-2 The sensitivity analysis provides a complete picture about how the main parameters change the results. The number of particles per cell is the most sensitive element for the Sparta simulations. The MMSE Journal. Open Access www.mmse.xyz

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higher the number of particles per cell, the better the results are. That is not so visible in the absolute numbers reported in Table 3 (where "abs" is for "absolute" and "Diff." for "difference"), but it is clear from the quality of the curves (see Fig. 1 and 2). In fact, the data from I p S and II p S are quite similar in the absolute value, but the graphs of the II p S simulation are smoother and more homogeneous with respect to the I p S ones. The difference is much more evident also in the IV p S. However, if an attempt is made to increase the number of particles per cell again, a limit will be encountered due to the lack of computational power for a single computer. However, considering the results and the quality of the curves, it is possible to say that a number of particles per cell equal to 10 is enough to obtain excellent results. Besides, if the main interest is only in the absolute values of the results and an interpolation method is used to obtain graphs from them, also a simulation with 1 particle per cell produces acceptable results. An in depth examination of the other factors reveals that the time step factor does not make a big difference to the absolute results. The cell size, like the time step, does not significantly change the results analyzed. However, the absolute value of the pressure at the capsule surface is very different for the three simulations considered for the sensitivity analysis of the cell size factor. The one which produces correct results, or at least, aligned with the value of the other simulations, is 0.005 m. To sum up, it is possible to conclude that the best setting is when the highest number of particles per cell is present, but 10 particles per cell is already more than enough. Instead, it is impossible to derive a proper law of how the results vary with the time step and, considering that the quality of the results vary very little, it is possible to affirm a best time step does not exist among those considered. From this sensitivity study the correct value of the cell size is 0.005 m, that is 5 times bigger then the one deriving from Bird's study. According to these final considerations, the best simulation is the one with 10 particles per cell and a cell size of 0.005 m, where the time step is irrelevant in the range analyzed.

Table 3. Comparison among the values analyzed in the Sparta sensitivity analysis for the particle factor.

Simulation IpS II p S IV p S

Difference abs Diff. % Diff. abs Diff. % Diff. abs Diff. % Diff.

Position(x) Ma=1 (m) 5.000e-4 0.211 5.000e-4 0.211 -8.500e-3 3.721

max T -4.194e3 5.503 -1.155e3 1.584 -3.251e4 31.112

Shock Layer T 7.000e-3 70.020 -7.000e-3 15.003 1.500e-3 750.653

Shock Layer p -1.800e-3 21.211 -3.000e-3 4.292 2.000e-3 3.081

Fig. 1. Mach number comparison among the simulations considered in the Sparta sensitivity analysis for the particle factor.

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Fig. 2. Overall temperature comparison among the simulations considered in the Sparta sensitivity analysis of the particle factor.

The computational time and also the resources vary considerably with the number of particles per cell. It varies from 1h to infinite in the simulations analyzed. The computational resources used for the other parameters sensitivity analysis are almost equal. From the computational time and resources analysis it is possible to draw the following conclusions. If excellent results are the main objective, time will not be important, and the number of particles per cell used will be equal to 10. However, if the main aim is to discover the most efficient simulation, the choice will be the one with a number of particles per cell equal to 1, because the absolute results of this simulation are very close to the one with 10 particles per cell, even if there are a lot of fluctuations. However this aspect could be correct with an interpolation of the data.

The last part of the sensitivity analysis is about establishing the minimum end time for the simulations to have steady results. As a preliminary value an end time of 0.005 s has been used. For this study the end time will be decreased and the same parameters of the previous study will be checked. The aim is to discover the point from which the variations in the results are sensitive, so higher than the 5% with respect to the 0.005 s results. The conclusion which come out from this study is that the minimum end time for such a simulation is 0.001 s, which means about 5 times the time which the flow takes to cross the characteristic length of the problem.

Validation. In the simulation without reactions it is possible to make a comparison with the theoretical data obtained using the perfect gas approximation. Considering all the properties of a perfect gas, the one concerning the volume is not properly respected while the others are all matching with the case studied. In fact, even if the density considered is very low, it becomes higher in front of the capsule because of the collisions of the particles with the surface. As a consequence, the pressure is quite high in this area. Therefore, theoretical results which do not match perfectly with simulation data can be expected because some hypothesis are not respected, even if not such a big difference is expected in the case of correct simulations. The results obtained from the shock layer relations, using as upstream values the ones present at the point considered, H3, are the following visible in Table 4 under the heading "Theoretical". The comparison with the simulation data is shown in the same Table 4. As can be seen, the data from both simulations do not perfectly match the theory ones, as expected.

However they are in the same order of magnitude and the difference is not significant. Moreover, the theoretical pressure is referred to the value at the shock, but the simulation datum considered is the value of the pressure just before the capsule surface. In such rarefied conditions, it is difficult to evaluate the end of the shock to establish what pressure value to take into account, because the shock layer is wider with respect to the one in the continuous regime. In fact, the increasing part of the pressure slope is not vertical as in dense fluids. Nevertheless it is possible to see from Fig. 5 that the pressure slope continues to increase until it reaches the surface of the capsule, where certainly the discontinuity zone constituted by the shock layer is already finished. Therefore, the pressure values MMSE Journal. Open Access www.mmse.xyz

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taken into account deriving from the DSMC are overestimated. This means that the pressure value matches the theoretical data better. It is possible to conclude that the results obtained by the simulations in the case of absence of reactions can be trusted and the results match what is expected.

Table 4. Pressure and overall temperature values after the shock in the theoretical case and in the Sparta and OpenFOAM simulation.

Data

Theoretical

OpenFOAM

Sparta

Pressure (Pa)

2.165e3

2.440e3

2.484e3

Overall T (K)

6.807e4

6.613e4

7.329e4

Another source of comparison is found in a simulation done with a CFD software which performs simulations using Navier-Stockes equations, Eilmer3. In fact, the case analyzed is in a limbo between the continuum regime and the free molecular regime. In particular, H3 is in the regime called slip regime, where the fluid is slightly rarefied. Anyway, when the fluid arrives in the zones near the capsule, it starts to become more and more dense because of the obstacle represented by the capsule which stops the motion of the particles and makes them concentrate there. In fact, the density grows approaching the surface and the mean free path decreases. As a consequence, the Knudeus number in the proximity of the surface, found using as a characteristic length of the problem the radius of the circular part of the Hayabusa capsule, which is about 0.18 m, shows that the problem belongs completely to the continuum regime. Therefore the Eilmer3 software, which uses the Navier-Stockes equations, works in a proper regime. Eilmer3 gives reliable results, in fact it is already widely used and tested in fluid dynamic simulations. From Fig. 3 to 5 the comparison between the data of the two software is shown. It is possible to observe that the Sparta curves are very similar to the Eilmer3 ones in shape, even if they are less refined. However it is important to remember that here the Sparta simulations are run with a number of particles per cell equal to one, which is not the best number discovered in the sensitivity analysis. It is possible to notice that the shock in the Sparta simulations is wider with respect to the Eilmer3 one and this difference is of about 2 cm. The difference in the width of the shock layer is probably due to some incorrect results obtained from the Sparta software because the best simulation setting is not used. Nevertheless the difference can be also caused by Eilmer3 and in particular by its mesh. In fact the results from a CFD simulation are very dependent on the quality of the mesh. In particular they cannot describe correctly the strong variations in the physical parameters present in the shock layer if a very refined mesh is not present there. Therefore a CFD mesh must be built recursively, trying, each time, to refine more the zone where the gradient of the thermodynamic parameters are stronger, according to the results from the previous simulation.

This iteration work is not done in the present study for reasons of time, but it could curtail the distance between the width of the shock layer obtained from the two simulations. However the absolute value of the pressure and temperatures considered are very close in the Sparta and Eilmer3 simulations.

This confirms that the results produced by the DSMC are reliable.

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Fig. 3. Translation Temperature Comparison between Sparta and Eilmer3.

Fig. 4. Vibration Temperature Comparison between Sparta and Eilmer3 Simulation.

Conclusion. The aim of studying and finding the best setting of both the OpenFOAM and Sparta DSMC solver, is achieved and shown. Now the software are more easily accessible for anyone interested in using them. Validation and verification studies of the new DSMC codes have been undertaken for both inert and chemically-reacting, hypersonic rarefied flows. They have been assessed against other numerical and analytical solutions for equilibrium conditions. The results for inert flows showed close agreement for temperatures and pressure with analytical counterparts. The simulations with chemically-reacting flows demonstrated excellent agreement compared with the results from an alternative code, Eilmer3, except for the greater thickness of the shock layer with respect to the term of comparison. The inclusion of chemical reactions in the dsmcFoam calculations resulted in an alteration of the flow structure with a reduced shock stand-off distance, a significant reduction in the overall temperature in the shock layer, and a substantial decrease in the predicted convective heat flux to the vehicle surface when compared with the inert gas case, as expected, because of the activation of endothermic dissociation reactions.

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Fig. 5. Pressure comparison between Sparta and Eilmer3 simulation.

Finally, the results presented in this research clearly show a new tool of investigation in the field of fluid-dynamics is available and reliable, that is the direct simulation Monte Carlo method. It provides an alternative to the classic CFD in the analysis of slightly rarefied fluids, with extremely low resources consumption, but at the same time producing acceptable results. In fact, even if the computational resources used for a single DSMC can be comparable to the ones of CFD, the mesh creation is simpler with respect to the latter and an iteration in the simulations to model the mesh zones with higher thermodynamic gradients is not necessary. In addition, the DSMC is the best method to provide results in the high rarefied flow.

Future work. The last step to officially validate the DSMC solver is the comparison with the experimental data. Considering that the data collected by the Hayabusa capsule at the point analyzed are radiative spectra, a further study is necessary to extract them from the data obtained from the DSMC. In the near future the research contained in this thesis could be the catalyst for further investigations starting from numerical trials of the methodology shown. The most useful for the re-entry analysis is the following. It consists in the implementation, inside the DSMC solver, of surface chemistry analysis, outgassing and surface ablation from which the direct evaluation of convective and radiate heating is possible. Another possible way to obtain the same result is to couple already existing software, used to simulate surface chemistry, outgassing and ablation with Sparta or OpenFOAM DSMC solver which provide them all with the necessary starting data for the work.

Summary. The knowledge of the conditions encountered during an atmospheric re-entry is crucial to build a reliable Thermal Protection System (TPS). In the transition regime, where the NavierStockes equations are no longer valid, but also in every other regime, a new numerical method can be used, the Direct Simulation Monte Carlo Method (DSMC). This new technique is the subject of this study which is structured as follows: 1) The DSMC and its theory are presented.

2) A sensitivity analysis is conducted on the principal parameters of a DSMC that are the number of particles per cell, the time step and the cell size. This analysis determines how the parameters setting changes the simulation results and what the best setting is in terms of realistic results and computational resources consumption.

3) A validation process is made through the comparison of the data deriving from the DSMC with the theoretical data and the data coming from another fluid-dynamic simulation made using the software Eilmer3.

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References [1] Bird, G.A. (1994). Molecular gas dynamics and the direct simulation of gas flows. Clarendon Press.

[2] Palharini, R. C. (2014). Atmospheric Reentry Modelling Using an Open-Source DSMC Code.

University of Strathclyde PhD thesis.

[3] Cercignani, C. (2000). Rarefied gas dynamics: from basic concepts to actual calculations.

Cambridge University Press.

[4] Scanlon, T. J. Roohi, E. White, C. Darbandi, M. and Reese, J. M. (2010). An open source, parallel, DSMC code for rarefied gas flows in arbitrary geometries. Computer and Fluids, 39:2078–2089, DOI: 10.1016/j.compfluid.2010.07.014 [5] Boyd, I. D. (2009). Direct Simulation Monte Carlo for Atmospheric Entry. Part 2. Code Development and Application Results. Hypersonic Entry and Cruise Vehicles. Von Karman Institute for Fluid Dynamics.

[6] Alexander, F. J. Garcia, A. L. and Alder, B. J. (1998). Cell size dependence of transport coefficients in stochastic particle algorithms. Physics of Fluids, 10(6):1540–1542, DOI: 10.1063/1.869674 [7] Alexander, F. J. Garcia, A. L. and Alder. B. J. (2000). Erratum: Cell size dependence of transport coefficients in stochastic particle algorithms. Physics of Fluids, 12(3):731, DOI: 10.1063/1.869674 [8] Garcia, A. G. and Wagner, W. A. (2000). Time step truncation error in direct simulation Monte Carlo. Physics of Fluids, 12(10):2621–2633.

[9] Hadjiconstantinou, N. G. (2000). Analysis of discretization in the direct simulation Monte Carlo.

Physics of Fluids, 12(10):2634–2638.

[10] Rieffel, M. A. (1999). A method for estimating the computational resquirement of DSMC simulations. Journal of Computational Physics, 149:95–113, DOI: 10.1006/jcph.1998.6140

Cite the paper Francesco Pellicani (2016). Atmosphere Re-Entry Simulation Using Direct Simulation Monte Carlo (DSMC) Method. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.31445.83684

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VII. Information Technologies M M S E J o u r n a l V o l . 6

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Comparison of Modeling and Simulation Results Management Microclimate of the Greenhouse by Fuzzy Logic Between a Wetland and Arid Region Didi Faouzi 1, N. Bibi-Triki2, B. Draoui3, A. Abène 4 1 – Faculty of Science and Technology, Department of Physics, University of Abu-Bakr Belkaïd, PB 119, Tlemcen, Algeria 2 – Materials and Renewable Energy Research Unit M.E.R.U University of Abu-Bakr Belkaïd, PB 119, Tlemcen, Algeria 3 – Energy Laboratory in Drylands University of Bechar, BP 417, 08000 Bechar Algeria 4 – Euro-Mediterranean Institute of Environment and Renewable Energies (123ER) University of Valenciennes, France DOI 10.13140/RG.2.2.28996.42880

Keywords: modeling, fuzzy logic controller, optimization, simulation, greenhouse, microclimate.

ABSTRACT. Currently the climate computer offers many benefits and solves problems related to the regulation, monitoring and controls. Greenhouse growers remain vigilant and attentive, facing this technological development. They ensure competitiveness and optimize their investments / production cost which continues to grow.

The application of artificial intelligence in the industry known for considerable growth, which is not the case in the field of agricultural greenhouses, where enforcement remains timid. It is from this fact, we undertake research work in this area and conduct a simulation based on meteorological data through MATLAB Simulink to finally analyze the thermal behavior -greenhouse microclimate energy.

In this paper, we present comparison of modelling and simulation management of the greenhouse microclimate by fuzzy logic between a wetland (Dar El Beida Algeria) and the other arid (Biskra Algeria).

I. Introduction. Agricultural greenhouse originally designed as a simple enclosure limited by a transparent wall, as is the case for conventional tunnel greenhouses and largely answered chapel in several countries including those of the Mediterranean basin. They amplify certain characteristics of the surrounding environment, thus involving variations of internal energy and significant heat loss due to the low inertia [1] of the clamp system.

The first objective is to improve the thermal capacity of the greenhouse (greenhouse).

This is, to characterize the behavior of the complex system that is the greenhouse with its various compartments [2] (ground, culture, cover, indoor and outdoor environment). To develop nonstationary mathematical models usable for simulation, optimization [3] and the establishment of laws and control of simple and effective regulation.

These models must reproduce the essential properties of the mechanisms and interactions between different compartments. They must be both specific enough to obey the dynamic and real behavior of the greenhouse system, and small to be easily adaptable to the phases of the simulation.

Good modulation instructions depending on the requirements of the plants to grow under shelter and outdoor climatic conditions, result in a more rational and efficient use of inputs and equip the best production performance.

The greenhouse climate is modified by artificial [4] actuators, thus providing the best conditions in the immediate environment of energy costs and it requires a controller, which minimizes the power consumption while keeping the state variables as close as possible optimal [5] harvest.

Many facilities have been designed to regulate and monitor climate variables in an agricultural greenhouse [6], such as: temperature, humidity, CO2 concentration, irrigation, the ventilation [7], etc.

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The possibilities offered by greenhouse climate computers have solved the problems relating to the regulation and respect of climate instructions required by protected cultivation.

In this paper, or using fuzzy logic [8-9], which is a powerful way to optimize and facilitate the global management of modern greenhouse, while providing through simulation interesting and encouraging which results in an optimization of favorable state variable values for the growth and development of protected cultivation.

II. State Equation II.1 Energy balance of the greenhouse. The analytical energy balance equation of the greenhouse: Stored energy change = Gain from internal sources+ Gain from the sun - Losses due to conduction through the cover - Losses due to long wave radiation - Unrealized losses (evaporation) - Losses due to the exchange of air .

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â?&#x17E; đ?&#x153;&#x2020;đ??žđ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??´đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; [đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą (đ?&#x2018;&#x2021;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; )] . . . â&#x2C6;&#x2019; đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; +đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;,đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; +đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą â?&#x; đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą +đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;,đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x;

đ??śđ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ??żđ?&#x2018;&#x201A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ??¸đ?&#x2018;&#x2020;

â?&#x17E; â&#x201E;&#x17D;đ?&#x2018;&#x;,đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;Ś (1 â&#x2C6;&#x2019; đ?&#x153;&#x20AC;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; )(đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2021;đ?&#x2018; đ?&#x2018;&#x2DC;đ?&#x2018;Ś ) â&#x2C6;&#x2019; 08đ?&#x153;&#x20AC;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; â&#x201E;&#x17D;đ?&#x2018;&#x;,đ?&#x2018;?đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; (đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2021;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; ) â&#x20AC;Ś _ â?&#x; đ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x201D;đ?&#x2018;¤đ?&#x2018;&#x17D;đ?&#x2018;Łđ?&#x2018;&#x2019; đ??żđ?&#x2018;&#x201A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ??¸đ?&#x2018;&#x2020;

â?&#x17E; đ??´đ?&#x2018;&#x201C;đ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x153;&#x201A;đ?&#x2018;˘đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;§đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; â?&#x;

Î&#x201D;đ?&#x2018;&#x2026;đ?&#x2018;&#x203A;đ?&#x2018;&#x2019;đ?&#x2018;Ą Î&#x201D;+đ?&#x203A;ž

â&#x20AC;Śâ&#x2C6;&#x2019;

đ??¸đ?&#x2018;Łđ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Ąđ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ??żđ?&#x2018;&#x201A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ??¸đ?&#x2018;&#x2020; đ??śđ?&#x2018;&#x201A;đ?&#x2018; đ?&#x2018;&#x2021;

â?&#x17E; 1 [đ?&#x2018;&#x; đ??´ + đ?&#x2018;&#x192;đ?&#x2018;&#x201C;đ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x; (đ?&#x2018;&#x2C6;â&#x201E;&#x201C;)đ?&#x2018;?đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2013;đ?&#x2018;&#x161;đ?&#x2018;&#x2019;đ?&#x2018;Ąđ?&#x2018;&#x2019;đ?&#x2018;&#x; ] (đ?&#x2018;&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2021;đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą ) â&#x20AC;Ś +đ?&#x2018;&#x; â?&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;,đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; +đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x;

(1)

đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;đ?&#x2018;˘đ?&#x2018;?đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

đ??żđ?&#x2018;&#x201A;đ?&#x2018;&#x2020;đ?&#x2018;&#x2020;đ??¸đ?&#x2018;&#x2020; đ??śđ?&#x2018;&#x201A;đ?&#x2018;

â&#x2C6;&#x2019;â?&#x17E; đ?&#x153;&#x2020;Î&#x161;đ??´ â?&#x; đ?&#x2018;&#x203A;đ?&#x2018;&#x2019;đ?&#x2018;Ą (đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą (đ?&#x2018;&#x2021;đ?&#x2018;¤đ?&#x2018;? [đ?&#x2018;&#x2021;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; , đ?&#x2018;&#x;â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; ]) â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; ) đ??šđ?&#x2018;&#x153;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;

where đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; , đ?&#x2018;&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;Ł,đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą â&#x20AC;&#x201C; heat transfer coefficient inside and outside by convection (W/m2.k); đ?&#x2018;&#x2019;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą â&#x20AC;&#x201C; indicates the report saturated with the relative humidity in the sub-model of combustion (Kg steam / kg air); đ?&#x2019;Źheaters â&#x20AC;&#x201C; is the heat provided by the heating system (W).

II.2 The mass transfer in the greenhouse. The mass balance for moisture in the greenhouse can be written as following eq. (2) :

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đ?&#x153;&#x152;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; đ?&#x2018;&#x2030;đ?&#x2018;&#x201D;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;â&#x201E;&#x17D;đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018; đ?&#x2018;&#x2019; 1

đ??´ đ?&#x153;&#x201A; đ?&#x153;&#x2020; đ?&#x2018;&#x201C;đ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x2018;˘đ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;&#x2013;đ?&#x2018;§đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; â?&#x;

Î&#x201D;đ?&#x2018;&#x2026;đ?&#x2018;&#x203A;đ?&#x2018;&#x2019;đ?&#x2018;Ą Î&#x201D;+đ?&#x203A;ž

đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;Ą

= â&#x2C6;&#x2019;đ?&#x2018;&#x2030;Ě&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x201C; â&#x2C6;&#x2014; đ?&#x153;&#x152;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; (đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ??ťđ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą ) â&#x2C6;&#x2019; đ?&#x2018;&#x2030;Ě&#x2021;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ą â&#x2C6;&#x2014; đ?&#x153;&#x152;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; (đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ??ťđ?&#x2018;?đ?&#x2018;&#x17D;đ?&#x2018;&#x2018; ) +

â&#x2C6;&#x2019;đ??ž â?&#x;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??´đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; [đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą (đ?&#x2018;&#x2021;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x; )] +

đ??¸đ?&#x2018;Łđ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Ąđ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ?&#x2019;Ź

â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x17D;đ?&#x2018;Ą (đ?&#x2018;&#x2021; ) đ??žđ??´ â?&#x; đ?&#x2018;&#x203A;đ?&#x2018;&#x2019;đ?&#x2018;Ą (đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą (đ?&#x2018;&#x2021;đ?&#x2018;¤đ?&#x2018;? [đ?&#x2018;&#x2021;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x;, đ?&#x2018;&#x;â&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; ]) â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018;&#x17D;đ?&#x2018;&#x2013;đ?&#x2018;&#x; ) + đ?&#x2018;&#x;đ?&#x153;&#x2122;đ?&#x2018;&#x2019; â?&#x; đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą đ?&#x2018;&#x2019;đ?&#x2018;Ľâ&#x201E;&#x17D;đ?&#x2018;&#x17D;đ?&#x2018;˘đ?&#x2018; đ?&#x2018;Ą â&#x201E;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x161;đ?&#x2018;?đ?&#x2018;˘đ?&#x2018; đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

đ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;

(2)

đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x161;đ?&#x2018;?đ?&#x2018;˘đ?&#x2018; đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

where đ?&#x2018;&#x2030;Ě&#x2021;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x201C; â&#x20AC;&#x201C; the speed of air infiltration (m/s); đ?&#x2018;&#x2030;đ?&#x2018;&#x201D;đ?&#x2018;&#x;đ?&#x2018;&#x2019;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;â&#x201E;&#x17D;đ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018; đ?&#x2018;&#x2019; â&#x20AC;&#x201C; the total volume of agricultural greenhouse (m3); đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A; , đ??ťđ?&#x2018;&#x153;đ?&#x2018;˘đ?&#x2018;Ą â&#x20AC;&#x201C; indoor and outdoor humidity (KJ / kg); đ?&#x2018;&#x2030;Ě&#x2021;đ?&#x2018;Łđ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ą â&#x20AC;&#x201C; ventilation rate (m3 air / s).

And for the humidity balance: Rates of change in absolute humidity = Infiltration + Ventilation * (humidity difference with the outside) + Misting + Cooling + AND - Condensation.

The status of humidity function is eq (3): đ?&#x2018;&#x2018;đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A; đ?&#x2018;&#x2018;đ?&#x2018;Ą

= â&#x2C6;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;&#x2030;đ?&#x2018;? (đ??ťđ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ??ťđ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą ) + đ??žđ?&#x2018;&#x201C;đ?&#x2018;&#x153;đ?&#x2018;&#x201D;đ?&#x2018;&#x201D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018; (đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą,đ?&#x2018;¤đ?&#x2018;&#x2019;đ?&#x2018;Ąđ?&#x2018;?đ?&#x2018;˘đ?&#x2018;&#x2122;đ?&#x2018;? ) â&#x2C6;&#x2019; đ??žđ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; (đ?&#x2018;&#x2030;đ?&#x2018;&#x192; â&#x2C6;&#x2019; đ?&#x2018;&#x2030;đ?&#x2018;&#x192;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ą ) + â?&#x; đ?&#x2018;&#x2030;đ?&#x2018;&#x2019;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;&#x201C;đ?&#x2018;&#x2013;đ?&#x2018;&#x2122;đ?&#x2018;Ąđ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

đ??¸ â?&#x;

(3)

đ??¸đ?&#x2018;Łđ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Ąđ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

đ??¸ â?&#x;

â&#x20AC;&#x201C; The amount of heat provided by evapotranspiration (W).

đ??¸đ?&#x2018;Łđ?&#x2018;&#x17D;đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;Ąđ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;?đ?&#x2018;&#x2013;đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;&#x153;đ?&#x2018;&#x203A;

III. Fuzzy controller modeling Fuzzy logic is widely used in the machine control. The term "fuzzy" refers to the fact that the logic can deal with concepts that can not be expressed as the "true" or "false" but rather as "partially true" [10]. While alternative approaches such as genetic algorithms and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution can be cast in terms that human operators can understand, so that their experience can be used in the design of the control device. This makes it easier to mechanize the tasks have already been performed successfully by man (https://en.wikipedia.org/wiki/Fuzzy_control_system).

III.1. Fuzzy inference method Mamdani. Fuzzy inference Mamdani type, as defined for Toolbox fuzzy logic, expects the output membership functions to be fuzzy sets. After the aggregation process, there is a fuzzy set for each output variable to defuzzification. It is possible, and in some cases much more efficient to use a single peak as output membership function, rather than a distributed fuzzy set.

This is sometimes known as singleton output membership function, and we can think like a fuzzy set of pre defuzzification. It improves the efficiency of defuzzification because it greatly simplifies the calculation required by the more general method Mamdani that has the center of gravity of a twodimensional function [9] To calculate the output of the SIF in view of inputs, six steps should be followed : ď&#x192;&#x2DC;

The determination of a set of fuzzy rules.

ď&#x192;&#x2DC;

Fuzzification inputs using the input membership functions.

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 rule.

By combining Fuzzificaion entries according to the fuzzy rules to establish a resistance to the

 Find the consequence of rule by combining the resistance to the rule and the output membership function. 

By combining the consequences to get a distribution outlet.



Defuzzification the output distribution.

III. 2. Fuzzy sets. The input variables in a fuzzy control system are generally mapped by sets of membership functions similar to it, called "fuzzy set". The process of converting a crisp input value to a fuzzy value is called "fuzzy logic". A control system may also have different types of switch, or "ON-OFF", inputs and analog inputs and during switching inputs will always be a truth-value of 1 or 0, but the system can handle as simplified fuzzy functions happen to be one value or another. Given "mappings" of input variables membership functions and truth-values, the microcontroller then makes decisions for action based on a set of "rules".

2.1 Membership functions

Fig.1 Representation rules of membership.

2.2 Rules of decisions  If

(Ti is TVCOLD) then (FOG1FAN1 is OFF)(FOG2FAN2 is OFF)(FOG3FAN3 is OFF)(NV is OFF)(Heater1 is ON)(Heater2 is ON)(Heater3 is ON) (1) 

If (Ti is TCOLD) then (FOG1FAN1 is OFF)(FOG2FAN2 is OFF)(FOG3FAN3 is OFF)(NV is OFF)(Heater1 is ON)(Heater2 is ON)(Heater3 is OFF) (1) 

If (Ti is TCOOL) then (FOG1FAN1 is OFF)(FOG2FAN2 is OFF)(FOG3FAN3 is OFF)(NV is OFF)(Heater1 is ON)(Heater2 is OFF)(Heater3 is OFF) (1) 

If (Ti is TSH) then (FOG1FAN1 is OFF)(FOG2FAN2 is OFF)(FOG3FAN3 is OFF)(NV is ON)(Heater1 is OFF)(Heater2 is OFF)(Heater3 is OFF) (1)  If

(Ti is TH) then (FOG1FAN1 is ON)(FOG2FAN2 is OFF)(FOG3FAN3 is OFF)(NV is OFF)(Heater1 is OFF)(Heater2 is OFF)(Heater3 is OFF) (1) 

If (Ti is TVH) then (FOG1FAN1 is ON)(FOG2FAN2 is ON)(FOG3FAN3 is OFF)(NV is OFF)(Heater1 is OFF)(Heater2 is OFF)(Heater3 is OFF) (1)

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ď&#x192;ź If

(Ti is TEH) then (FOG1FAN1 is ON)(FOG2FAN2 is ON)(FOG3FAN3 is ON)(NV is OFF)(Heater1 is OFF)(Heater2 is OFF)(Heater3 is OFF) (1) IV. Simulation and model validation.

Our model is based on the greenhouse GUESS model that is set for a multi greenhouse chapel, which each module is 8.5 m wide, 34 m deep and ridge height of 4.5 m. Infiltration rate is 1.1 air changes per hour, and a U value of 5.76 W / m2.K was used. The model of the plant was set for Douglas seedling plants were started at 0.57 g dry weight, and harvested 1.67 g dry weight; a new growing season was recorded at harvest.

A set of hourly data for 2015 (1 January to 31 December) weather station of Dar El Beida Algeria and Biskra Algeria [6], was used to validate our model as a CSV file that consists of four columns (global solar radiation, temperature, humidity and wind speed).

The model of the greenhouse was coded using the full version of Windows MATLAB R2012b (8.0.0.783), 64bit (win64) with Simulink. The simulation was performed on a Toshiba laptop. The laptop is equipped with a hard drive 700 GB and 5 GB of RAM. Simulink model of the parties were made in "Accelerator" mode that has first generated a compact representation of Code C of the diagram, then compiled and executed.

IV.1 Greenhouse Climate Model

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Fig.2 SIMULINK representation of the greenhouse climate model.

VI.2. Fuzzy logic controller simulation model of the greenhouse

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Fig. 3. SIMULINK representation of the fuzzy logic controller model.

Discussions on above Fig.s (2-3) While a complete list of equations may show the relationships between quantities, it provides no indication of how these equations are to be solved numerically on the computer, let alone how they are to be expressed and organized as part of the overall model software. Esoteric mathematical equations must be translated into computer code, which upon compilation and execution translates MMSE Journal. Open Access www.mmse.xyz

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raw input data into meaningful output. Nothing is said about the different pre and post processing steps which must taken to go from raw input data to meaningful output graphs.

In a block diagram, each machine or block is described by three sets of variables: The inputs, the state variables that describe the condition of the machine and the output which depend directly upon the state. At each time step, the machine or block can be called upon perform to following commands:  Initialize/reset outputs and states  Calculate state derivatives  Integrate state derivatives to calculate future state  Calculate outputs based upon current state V. Results The simulation results clearly visualize the actual thermo-energy behavior of agricultural greenhouse, applying the model of artificial intelligence, namely the application of fuzzy logic in arid and wetland region (http://www.wunderground.com/cgibin/findweather/getForecast?qery²).

Results Simulation for the wetland region (Dar El Beida): 4

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Fig. 5. Histogram of the indoor distribution.

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Temperature

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Fig. 6. The evolution of humidity and temperature.

Results of the arid region simulation (Biskra):

4 9 x 10

Indoor Temperature Distribution

8 7 6

freq

.

5 4 3

2 1 0 5

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Fig. 7. Histogram shows the distribution of indoor temperature.

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Temperatures 30

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Relative Humidity % of 100 100 80 60 40 20 outdoor 0

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Fig. 8. The evolution of humidity and temperature (interior/exterior) Discussions on above Fig.s (5 - 8): It is found in the wetland region (Dar El Beida) that most of the internal temperature values are in the range 14 ° C to 22 ° C for autumn winter period and in the range 20 ° C to 26 ° C for the spring summer period in a large variation the temperature during the winter period is autumn due the heat loss at night , the compensation is insufficient by heating and expensive for this improved thermal insulation of the cover wall is necessary.

The improvement of the thermal isolation of the cover may be carried out in practice by the addition of an air bubble plastic layer assembled to the face interior of wall..

During the period spring summer the temperature is within the desired range.

The relative humidity is almost in the interval desired during all the year except at the few days of half of the summer because of the important vaporization used for the compensation of the temperature .

But Conversely in the arid region It is found that most of the internal temperature values are in the range 15 ° C to 25 ° C for the autumn winter period, and in the range 20 ° C to 28 ° C for the spring summer period in a large variation the temperature during the winter autumn period is due to heat loss during the night, clearing heating is insufficient and expensive for this improved thermal insulation of the covering wall is necessary.

The improved thermal insulation of the cover may be carried out in practice by the addition of an plastic air bubble layer mounted to the inside wall face.

During the period spring summer the temperature is almost within the desired range except for half of the summer where the temperature is a little increase.

The use of cooling systems and spray is necessary to lower the temperature in the interval longed for.

However, this solution is insufficient and expensive, for this purpose we should improve the characteristics of the coverage of the agricultural greenhouse for example thermal insulation or blanket double wall, which demonstrates improved efficiency of heating and cooling etc.

The relative humidity generally stays close to the optimum for all the year except in summer when the humidity drops below threshold due to significant vaporization used for temperature compensation, to resolve this problem adding a screen on the roof of the greenhouse and improving irrigation can compensate the lack of relative humidity in the arid region.

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Summary. However, our objective is achieved to the extent that it has been shown through modeling and control by the use of fuzzy logic, this area is very difficult because it is a multi-control variable, which the greenhouse is a biophysical system where parameters are highly correlated as shown by the results. This technique of fuzzy logic that has been adapted to the greenhouse to a promising future for the climate control and management of the greenhouse. For greenhouse growers, it is a preferred approach for structuring and knowledge aggregation and as a means of identification of gaps in the understanding of mechanisms and interactions that occur in the system - greenhouse.

Fuzzy logic is a branch of artificial intelligence, which must point out its advantages and disadvantages. Its use has led to quite satisfactory results of the control and regulation perspective.

We remain optimistic in the near future, as to the operation of artificial intelligence, including the use of fuzzy logic, which indicates:  For the control and regulation of the greenhouse microclimate.  By the conservation of energy.  For the efficiency of energy use in the greenhouses operation.  For improved productivity of crops under greenhouses.  In a significant reduction of human intervention.

Acknowledgement This research was supported/partially supported by [N. Bibi-Triki, B. Draoui, A. Abène]. I thank our colleagues who provided insight and expertise that greatly assisted the research and we thank “anonymous” reviewers for their so-called insights.

References [1] Bendimerad, S., T. Mahdjoub, N. Bibi-Triki, M.Z. Bessenouci and B. Draoui et al., 2014.

Simulation and Interpretation of the BIBI Ratio CB (.), as a Function of Thermal Parameters of the Low Inertia Polyethylene Wall of Greenhouses. Physics Procedia, 55: 157-164. DOI: 10.1016/j.phpro.2014.07.023 [2] Bibi-Triki, N., S. Bendimemerad, A. Chermitti, T. Mahdjoub and B. Draoui et al., 2011.

Modeling, characterization and analysis of the dynamic behavior of heat transfers through polyethylene and glass walls of greenhouses. Physics Procedia, 21: 67-74. DOI: 10.1016/j.phpro.2011.10.011 [3] Faouzi Didi , N. Bibi Triki and A. Chermitti, 2016. Optimizing the greenhouse micro-climate management by the introduction of artificial intelligence using fuzzy logic. Int. J. Computer Eng. Technology, 7: 78-92 , Volume 7, Issue 3, May-June 2016, pp. 78–92, Article ID: IJCET_07_03_007.

[4] El Aoud, M.M. and M. Maher, 2014. Intelligent control for a greenhouse climate. Int. J. Advances Eng. Technology, 7: 1191-1205. http://www.e-ijaet.org/media/8I22-IJAET0722618_v7_iss4_11911205.pdf [5] Abdelhafid Hasni, B., T. Draoui, Boulard, R. Taibi and A. Hezzab, 2008. Evolutionary algorithms in the optimization of greenhouse climate model parameters. Int. Rev. Comput. Software.

[6] Draoui, B., F. Bounaama, T. Boulard and N. Bibi-Triki, 2013. In-situ modelisation of a greenhouse climate including sensible heat, water vapour and CO2 balances. EPS Web Conferences, 45: 01023-01023. DOI: 10.1051/epjconf/20134501023 [7] Hasni, A., B. Draoui, T. Boulard, R. Taibi and B. Dennai, 2009. A particle swarm optimization of natural ventilation parameters in a greenhouse with continuous roof vents. Sensor Transducers J., 102: 84-93.

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[8] Bouaama, F., K. Lammari and B. Draoui, 2008. Greenhouse air temperature control using fuzzy PID+I and Neuron fuzzy hybrid system controller. Proceedings of the International Review of Automatic Control (IRE.A.CO).

[9] Dhamakale, S.D. and S.B. Patil, 2011. Fuzzy logic approach with microcontroller for climate controlling in green house. Int. J. Emerg. Technol., 2: 17-19.

[10] Gurbaoui, M., A. Ed-Dahhak, Y. Elafou, A. Lachhab and L. Belkoura et al., 2013.

Implementation of direct fuzzy controller in greenhouse based on labview. Int. J. Electr. Electron.

Eng. Stud., 1: 1-13.

Cite the paper Didi Faouzi, N. Bibi-Triki, B. Draoui & A. Abène (2016). Comparison of Modeling and Simulation Results Management Microclimate of the Greenhouse by Fuzzy Logic Between a Wetland and Arid Region. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.28996.42880

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IX. Economics & Management M M S E J o u r n a l V o l . 6

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The Role of Education in Formation of Knowledge Economy Tetiana Chumachenko1, Olena Hladun1 1 - State Higher Educational Institution “National Mining University”, Dnipropetrovsk, Ukraine DOI 10.13140/RG.2.1.4242.7125

Keywords: knowledge economy, knowledge, education system, innovative economy, competence, professional qualities, creativity.

ABSTRACT. The article describes conceptual basis of education system for knowledge economy. Using analytical method, the theoretical aspects of the role of education in formation of innovative economy were studied. It sets out the tasks education system facing based on the knowledge of the new format. It determines key skills required for preparation of highly skilled specialists for labour market – professional qualities and competence, the basis of effective management and entrepreneurial skills.

Introduction Modern trends in world economic development stipulate necessity of strengthening an innovative component and a systematic approach to training specialists on the knowledge of new format, corresponding to the modern requirements and capable to provide strong economic growth.

For this reason, it is important to improve the efficiency of markets and the level of education [1], forming a system of educational services for knowledge economy.

Analysis of the global education market development showed that education is undergoing a major transformation under the direct and indirect exposure of the global trends [2]. The term “knowledge economy” is increasingly being used not only in relation to developed countries. Although there are different interpretations of the definition of knowledge economy [3.4], the researchers agree on one thing – without knowledge, their accumulation, development is impossible. Knowledge economy includes mechanism of production knowledge and technology, concentrated primarily in universities in form of basic and applied science, research and development, as well as effective protection of intellectual property system. At the same time, this knowledge is not regarded as a finished product, but as a process, improving and undergoing evolutionary changes. It is impossible to implement this knowledge in economy at once without taking into account the level of economic development in general, and without developed communications system, including formation of a systematic approach to knowledge transfer - technology transfer. These provisions define the contemporary trends in the education market, and attitude of the governments to education.

The purpose of the study is to define the conceptual basis of the educational system of for the knowledge economy, based on the knowledge of new format, which is formed as an innovative system where competitive advantages are created through investment in human capital by training qualified specialists for high-tech economy and possess management skills and creative thinking, having developed entrepreneurial skills.

Today, competition is shifting towards creation and development of knowledge. The company's future is determined by the ability to create and accumulate knowledge. The nature of knowledge required in contemporary labour market has changed. If earlier knowledge should be more formal and materialized in the form of clear instructions, standards and technologies, today the speed of creating and updating of knowledge has sharply increased, and their volume increased significantly.

The business related to intellectual work: software development, development of science and technology, are developed actively. In these areas, the ideas are valued above all else. Many countries are already beginning to realize that the future prosperity of their economies depends on the MMSE Journal. Open Access www.mmse.xyz

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innovation flow, and the ability to think creatively is becoming more important than formal education.

Creativity - today is not a luxury, but a means of survival. Companies entice each other's employees who are able to think outside the box or create inside their structures the conditions in form of courses, seminars and trainings to enhance the creative potential of their employees. The involvement in the process of higher education institutions can become a part of marketing strategy of universities and considered as a part of commercialization of higher education institutions through the development of special education programs. Success depends on how and how quickly educational institutions are able to adapt to the labour market realities, to prepare specialists of new level, who are able to think strategically innovation. Universities should not just give a set of specific knowledge through their educational programs, but to teach students to formulate their idea, to bring this idea to practical implementation, covering thus the whole complex of relations: from accumulation of knowledge to production and consumption, thereby contributing to economic development. It puts forward the relevant requirements to the whole education system.

The impact of technology on economy was studied by many economists, who argued that the technology is an important determinant of society, as well as each step is a method of production; it is obvious that with each new stage the humanity rose to a new stage of development [5]. Namely, the method of production is considered to be a factor determining the institutions of human society, and not vice versa, and new technologies and new knowledge is the source of economic development is [6].

Francis Bacon was the first who put forward the idea that in order to improve the living conditions of any society must it is necessary to develop a science [7].

Analysing the work of economists, the famous theoretician of economics Erik S. Reinert, showed that Schumpeter and Marx, being supporters of various concepts of economic development, argued that the capital is fruitless without investment opportunities, and only new technologies and innovations can provide them. The synergy effect has great importance, because the development of processing industry enables a country to modernize its agriculture [8], and technological progress, is the primary source of growth and transformation of society [9].

At the same time, it is possible to see clearly a correlation between scientific discoveries and innovations, between the development of theory and practice. According to Eric Reinert [10], the term “innovation” and attitude of society to it has radically changed during the recent years. To create and introduce innovations is the pleasant obligation of humanity. However, this process is not possible without accumulation of new knowledge, and how much wisdom the humanity has accumulated, it continues year after year to expand the endless boundaries of knowledge [11]. Carlos Moedas, Commissioner for Research, Science and Innovations of EU, speaking at the World Economic Forum in Davos said: “Most politicians in the world do not realize that there would be no growth, no jobs and no development if there is no science” [12].

Today innovations are the basis of competitive advantages of national economies, which are formed due to the introduction in the results of scientific and research activities production aimed not only at optimizing production processes, but also on the formation of non-economic - environmental, social standards of economic activities of enterprises and their implementation in economy [13].

Thus, in the bases of economic growth as a joint synergy product, along with a purely economic factor, new knowledge are lying. It is through knowledge, transmitted through innovations and used in production process, it is possible to reduce significantly the costs that contribute to the growing impact that, in fact, is a source of economic growth.

Among the basic rules of economic development, two of them directly related to scientific activities: recognition of importance of training / education and promotion of valuable knowledge. The attitude to science and education depends on the industry development, since the production process is considered as a multiplier, which requires knowledge, mechanization, technology, division of labour, increasing returns, and at the same time it creates them itself [14].

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As it was rightly noted by American economist James Bright, the only one process that combines science, technology, economics, entrepreneurship and management is innovation. An innovative economy is the basis of formation of the knowledge economy. In the works of G. Becker [15] P. Druker [16], F. Makhlup [17] the definition of the term “knowledge economy” as a new type of economy is provided for the first time. Not only economists, but also philosophers and sociologists work on the development of the ideology of the knowledge economy. However, American economist F. Makhlup for the first time equated knowledge to the sector of economy, the development of which determines the development of economy as a whole, the, having introduced the term “knowledge industry” and regarding knowledge as the process of transferring them.

However, the invention and innovation, as well as accumulated knowledge lying in the basis of the modern education could never be recreated on free markets without government interference.

The objectives of the state, tending to development of higher education, are understood as the labour market needs changes with the development of economy. University graduates more easily adapt to society than those who only finished school, they actively participate in social activities, commit less crime, better train their children and rarely get sick because they adhere to a healthy lifestyle. These are the additional benefits that even more important for the state than the additional income from higher education for every individual throughout his life.

However, in the transition to mass higher education, the entire educational system has a risk to develop faster and cheaper forms of education, which will be much less useful for students and society. Mass leads to reduction of the quality of education, levelling the role of universities in forming of society, ignoring what has been considered for a long time as the most important goals of education: strengthening of moral qualities of students and their preparation for life as active, informed citizens.

Today there is a necessity of reforming the classical model of education. But in order the process start actively, it is necessary first of all to focus on the question, what is the purpose of education in a historical and contemporary perspective, to identify correctly the disadvantages of the existing university education models, and what prevents more personalized, direct and free approach to education?

The main mission of higher education remains unchangeable - to prepare highly qualified specialists able to implement their knowledge and competence, creative approach to any process and able to work on the final result.

In terms of marketing the conceptual model, the sphere of market relations in the system of higher education is presented by relationship between universities engaged in basic and further training of young specialists, the labour market, which is the consumer of young professionals and young professionals themselves as a specific item. University, entering the market of educational products and services with its educational programs, at the same time produces specialists of various levels and profiles. Quality of training programs determines the demand for graduates of certain schools from the labour market.

However, university by offering to all students of the same course in the same faculty the same educational program, releases different specialists as a result. They differ in degree of assimilation of the said program, in quality of knowledge, in personal orientation on various aspects of t profession, etc. Therefore, the university is not the manufacturer of graduates, but educational programs for the labour market, in the form in which they are mastered by its graduates, that is, entering the labour market with the results of its educational activities, mediated by the knowledge, professionalism and skills of graduates. Graduates offer their labour force to enterprises, those, in turn, assess the skills of this labour force in form of starting salaries and other conditions of employment.

University is interested not only in the fact that its educational programs meet consistent requirements of the labour market as much as possible, but also that the graduates assimilate this program as fully as possible. The labour market, in turn, defines the basic standards of the quality of education in form MMSE Journal. Open Access www.mmse.xyz

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of changes in demand for these or other specialists, clarifying of knowledge and skills, which the applicants for a certain job must have, and assesses the qualifications of this labor force in form of salary. On the other hand, the prospect of employment is an important motive that contributes to the choice of educational products by the main consumer - the graduates of the secondary schools. [18] The protracted economic crisis in Ukraine showed that employers increasingly need creative, flexible and adaptable to changes workers. Can the Ukrainian higher educational system, even in light of the recent legislative changes in the field of educational services, meet the requirements of the labour market? Probably not, because the existing education system no longer corresponds to the real needs of economy. In addition, the climate of motivational to knowledge is deteriorated in Ukraine. The interest of students to the process of education itself reduces and educational programs almost do not provide personal development. The same trend is observed in relation to scientific research.

Solving these problems is possible through increasing the creative activity of students, the development of programs to reveal their creativity, develop the potential of functional knowledge and to teach students not only to think creatively, but also to realize their ideas through the baggage of theoretical knowledge. Thus, this knowledge is converted into competence, and the presence of a variety of educational programs, the variability of their use for training throughout life, makes it possible to increase the human capital of the state.

The attitude of the state to formation of higher education system varies depending on development of economic processes. Economic globalization forces the governments to invest heavily in training of new work forces, especially in developing countries. Governments of the countries, where there is a serious economic downturn, vice versa, begin to reduce funding in higher education, making it more and more elitist. For someone who is looking for an opportunity to get higher education, it means that he would bear all costs and risks himself.

The potential students still adhere to the stereotype that if to study hard, enrol to university and get a diploma, it is possible find a quiet job with good earnings and be in easy circumstances until the end of life. This explains the fact that 2/3 of secondary school graduates each year tend to enrol in higher education institutions in Ukraine. Now the situation has changed. Although a degree of higher education still pays off, the analysis of general trends in the salary market showed that the is no longer guarantee that education can provide a high-paying job. Most often, work and salary are not equivalent of received education. The number of unemployed university graduates is steadily growing in Ukraine. Thus, according to the State Statistics Service, the unemployment among young people under 25 years makes 23% more than 13% of university graduates remain unemployed, at least 20% do not work in their specialty. [19] There is a general depreciation of traditional academic education, so-called inflation of diplomas. The situation is partly rescued by unification of educational programs of universities in accordance with international standards. Recognition of the national educational programs opens the borders of the labour market for university graduates.

Mass higher education has led to that in the recent years the level of professionalism of graduates has dropped to 15%. In traditional system of higher education, students study a very narrow set of compulsory subjects, because it was believed that the knowledge of these subjects were prospective in terms of employment, and our politicians even now believe that in future it will be useful for economy. However, according to businesspersons, university graduates do not have propensity to innovations, cannot work as a team and do not know how to communicate effectively even with each other. Today, the professionalism of university graduates is determined not only by academic knowledge, but is achieved through joining them with practical skills, participation in selfimplemented projects, in a word, what is primarily required by employers.

In order to meet the requirements of modern times, educational system should be integrated into the complex of â&#x20AC;&#x153;education-research-innovationâ&#x20AC;?, and the function of universities should be modernized and adjusted accounting for emergence of new circumstances and needs that require the flexibility of education system.

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According to Ken Robinson, the human development cannot be linear [20], and the main purpose of education is to teach creativity, using non-standard approaches in the process of acquiring knowledge.

The main factor hindering the development of personal qualities is obsession with standardized tests and, due to financing, the need to teach for tests. The existing educational system, supported by the development of interactive technologies, unfortunately, teaches us to think alone, whereas the cooperation of enthusiastic people together, namely, leads to growth and progress.

Higher education institutions, giving professional knowledge, unfortunately, do not teach the culture of collective relations. The number of bachelors are not so important as how well students develop cognitive skills such as critical thinking and ability to solve a problem.

Recently, the approach of school graduates to selection of universities has changed. Along with increasing of popularity of higher education [21], they are more focused on selection of training programs, basing primarily on the interest to profession. Being guided by the algorithm “selection of a certain specialty - the choice of the university”, the applicants, choosing an educational institution, have given the dominant role to the quality of educational services. It was confirmed by the research carried out among the students of the Higher Education Institution “National Mining University” [22].

The absolute majority of students (90%) consciously treated to a choice of profession, choosing namely profession, but not institution, therewith 36.5% of them were guided by the interest in their future profession, and 21.6% believed that the profession would give them the opportunity of good earnings in future. Only 19% of respondents chose university. The vast majority of students during the process of education formed a positive opinion about the chosen specialty, which has either improved after enrolment (40.2%), or the good attitude has not changed (34%).

Slightly more than a half of the respondents intend to work in the specialty after graduation, and 39% at the time of the survey were undecided on this issue. Only 10% of students after graduation do not plan to tie their future work with the chosen specialty.

The studies have confirmed that the disadvantage of education system existing in Ukrainian is the lack of a full-fledged production practices. 35.1% of students consider themselves well prepared theoretically, but suffer from the lack of practical skills. The students of the fifth year of study of engineering, computer and economic specialties of the University in assessing of the knowledge quality noted the importance of the following factors: the educational programs offered by the institution, including their content and providing of training, the impact of training of the teaching staff in the skills of graduates and conducting production practices.

The evaluation of each criteria was determined as the quotient from dividing the perception rating on expectation rating. The quality of education was evaluated on a scale from 0 to 1, according to the methodology [23]. According to students, the quality educational services of the university is quite high and lies in the ranges from 0,7 to 0,86 (see. Fig. 1).

Content of education process 0,799 Conducting of production practices 0,830

Provision of education process 0,859 Professional and pedagogical skills 0,70

Fig. 1. Comprehensive assessment of quality of educational services MMSE Journal. Open Access www.mmse.xyz

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The students of economic specialties mostly suffer from the lack of the practical skills (see. Fig. 2).

Fig. 2. The results of the assessment of educational services quality by the students of different specialties Highly appreciated quality of education is supported by the results of the annual research carried out by the International Chair of UNESCO [24]. According to the rating indicators, the National Mining University is surely among the TOP-10 of educational institutions of Ukraine. Two-thirds of the students surveyed have noticed a large number of unnecessary, from their point of view, things that prevent them to focus on the major disciplines. However, in terms of university education, it is impossible to prepare a versatile personality, capable to non-standard solutions, having moved to the dry teaching and learning focusing on several academic disciplines, displacing creative and humanitarian disciplines from the university. It is also impossible to imagine a qualitative education without science. The accumulation of knowledge is mostly caused by the need for research. Checking of their results leads to development of innovation and production, and as a result of the University improves educational programs, preparing qualified specialists for economy.

The analysis of the major rankings of university activities has shown that the importance of scientific criteria in them is not less than 20%. By increasing the level of competitiveness of the countries, education and training of the workforce through the regular access to new knowledge and technologies may be crucial [25].

Thus, the main tasks facing the education system in Ukraine nowadays are not only administration, but also the content of educational process, which aims to improve the quality of education. The creation of infrastructure and facilities that can provide accelerated implementation of the acquired knowledge in professional activity is also very important. Taking into account the experience of many foreign universities, it is necessary to expand the training of specialists in the field of innovation management - the organizers, who have the skills of finding prospective scientific and technical ideas and practical implementation of innovation, the specialists who are able not only to acquire new knowledge and technology, but also to implement them.

University graduates must meet not only of the labour market requirements, but also the social demands of society - to be creative, flexible, possess the skills of critical thinking, to understand their goals and goals of their society that are able to form a holistic understanding of the world as a system.

The use of another resource of development - cooperation with high-tech companies is strategically important for universities, improving the educational component, extending the possibility of obtaining practical skills of students, and not only in the professional sphere. This will give an

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opportunity to combine education, science and the economy, forming the basis for the knowledge economy.

The need to perform these tasks is caused by the fact that our national education system is the part of a global educational space. Without updating the content and application of innovative technologies in education it is not possible to prepare professionals who meet the requirements for professional activity in the market conditions. The main scientific and methodical task of universities is to provide mechanisms for formation of professional competencies in students. The structure of professional competence must include the skills such as “teamwork”, “ability to acquire new knowledge”, “knowledge and willingness to use innovative ideas”, “studying throughout life” and others.

Basing on the fact that the knowledge economy is a new type of economy, which development basis are the knowledge that produce innovations, it is necessary to reform the education system that will allow to prepare the specialists, capable to generate new knowledge and technologies. In this type of economy, science and knowledge are the capital investments for all participants of educational process - universities with their staff and scientific potential, students, the quality of education of which provides high-level knowledge and allows meeting the new requirements of the labor market and employers interested in professional staff. However, the demand on knowledge should be formed not only by innovative companies, but also, by the state [26]. Failure to recognize this fact can have serious consequences.

References [1] V. Simonenko “Not ready to breakthrough” // Weekly journal “2000”, Vol. 8(No. 595).

[2] The shape of things to come: higher education global trends and emerging opportunities to 2020.

[Online]. Available: www.britishcouncil.org/higher-education.

[3] O.V. Starovojt. Economy of knowledge in the strategy of innovational development of science / О.V. Starovoit. [Online]. Available: http://osvitata.com/osvita-ta-ekonomika/ekonomika-znan-ustrategi-innovatsiynogo-rozvitku-osviti.html.

[4] L.І. Fedulova. Economy of knowledge: tutorial / L.І. Fedulova; Institution of economy and forecasting of the National Academy of Sciences of Ukraine. – Kiev, 2009. – 600 p.

[5] Erik Reinert. Karl Bücher and the Geographical Dimensions of Techno-Economic Change // Karl Bücher Theory – History – Anthropology – Non-Market Economies / Jurgen Backhaus (ed.).

Marburg, 2000. P. 177–222.

[6] UNCTAD, United Nations Conference on Trade and Development (2006). The Least Developed Countries Report 2006.

Developing Productive Capacities, Geneva.

URL: http://www.unctad.org/en/docs/ldc2006_en.pdf.

[7] F. Bekon. Essays: in 2 Vol. Moscow: Mysl, 1978. Vol. 2. p. 78.

[8] E. Reinert. How reach countries became rich and why poor countries remain poor. – Moscow: Editorial House of the State University – Higher School of Economy, 2011, p.51.

[9] Johan Akerman. Politik och Ekonomi i Atomalderns Varld. Stockholm, 1954. P.26–27.

[10] Erik Reinert, Arno Mong Daastol. Exploring the Genesis of Economic Innovations: The Religious Gestalt-Switch and the Duty to Invent as Preconditions for Economic Growth // European Journal of Law and Economics. 1997. Vol.4. No. 2/3. P.233–283.

[11] Koyre Alexandre . From the Closed World to the Infinite Universe. Baltimore, 1957.

[12] Results of economic forum in Davos 2016 [Online].

Available: http://hyser.com.ua/economics/mezhdunarodnyj-forum-v-davose-podvedenie-itogov.-57015 [13] T.N. Chumachenko. The role of science in innovational development of Ukraine. Т.N. Chumachenko // Innovations in business establishment and management: materials of the V MMSE Journal. Open Access www.mmse.xyz

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International Science Conference of lecturers, staff and postgraduates: Digest of articles. Moscow, 15-17 of October, 2014.- Moscow: RUDN, 2014, p.112-118.

[14] E. Reinert. How reach countries became rich and why poor countries remain poor. – Moscow: Editorial House of the State University – Higher School of Economy, 2011, p. 63.

[15] G.S. Bekker. Human behavior. Economic crusade. Selected papers on economic theory: [translated from English] / G.S. Bekker; cons., scientific afterwards editorial R.S. Kapeliushnikov ; forewards M.I. Levin. – Moscow: GUVSHE, 2003. − 672 p.

[16] Piter F. Drucker. Gap era: the reference points for our changing society / F.P. Drucker. – Moscow: Williams, 2007. – 336 p.

[17] F. Machlup. Production and dissemination of knowledge in the United States / F. Machlup. – Moscow: Progress, 1966. – 462 p.

[18] Social monitoring results “Educational process in the National Mining University by students’ eyes” / L.О. Kolisnyk, M.V. Mosyondz. – Dnipro: State Higher Educational Institution “NMU”, 2015. – 35 p.

[19] Higher education into digital era: crisis and opportunities. Richard Trovatten 19th of July, 2013 [Online].

Available: http://theoryandpractice.ru/projects/richard [20].

Available: http://www.ukrstat.gov.ua [21] Ken Robinson. Vocation. How important to find a vocation to change everything – M.: Mann, Ivanov and Ferber, 2009. ISBN 978-5-91657-123-3.

[22] T. Chumachenko, K. Frolovа, M.Shuvalova, Marketing researches of educational process./ T. N.Chumachenko// Majesty of Marketing: Materials of the International conference for the students and junior research staff.– Dnipropetrovsk, SHEI “National Mining University”, December 2013.– p. 50-52.

[23] L.G. Milyaieva. Marketing research on the market of educational services of country towns // “Marketing in Russia and overseas” - 2005.- No 5., p. 6 [24] 200 of the best higher educational institutions of Ukraine [Online]. Available: http:// kpi.ua/web_unesco#sthash.1oiKf6Cz.dpuf.

[25] The Global Competitiveness Report 2015-2016 http://reports.weforum.org/global-competitiveness-report-2015-2016

[Online].

Available:

[26] S. Osipov. Closing speech on scientific session of the General Meeting of Russian Academy of Sciences // Bulletin of Russian Academy of Sciences. 2003. Vol. 73. No 5. p. 461 Cite the paper Tetiana Chumachenko & Olena Hladun (2016). The Role of Education in Formation of Knowledge Economy. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.1.4242.7125

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Selection of the Reconstruction Options for Industrial Power Supply System under Uncertainty Conditions on the Basis of the Game Theory Criteria Alina Iuldasheva1,a, Aleksei Malafeev2,b 1 – PhD Student of Department of Industrial Electric Power Supply, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation 2 – Candidate of Engineering Sciences of Department of Industrial Electric Power Supply, Nosov Magnitogorsk State Technical University, Magnitogorsk, Russian Federation a – alinayuldasheva1@gmail.com b – malapheev_av@mail.ru DOI 10.13140/RG.2.2.34252.41609

Keywords: production risks, game theory, decision criteria, reliability assesment, uncertainty, damage.

ABSTRACT. Research objective: The reliable power supply at a reasonable cost is a fundamental for the development of any country. Special attention should be paid to the power supply system of industrial enterprises. In the designing, the operation and the mode planning of this systems it is required to account not only the power supply reliability, but also the risks associated with operation interruptions. The task of risk assessment is complicated because of such characteristic feature of industrial power supply system as the uncertainty of information of possible emergency modes, operational loads, etc.

Methods. The combination of two methods: the sequential network reduction and the Newton's method is proposed for the calculation of equivalent reliability indices of complex systems. On the basis of reliability calculation the damage from power supply interruption is determined. The game theory criteria are proposed to use for the decision making in case of uncertainty. The Wald, Minimin, Hurwicz, Bayes, Hodge-Lehmann, Savage, Laplace, Multiplication, GermeierHurwitz criteria are calculated and analyzed.

Scientific novelty and practical significance. Proposed algorithm for reliability evaluation allows to determine the probability of no-failure, failure intensity and recovery time. The algorithm can be used to evaluate the reliability of an existing distribution system and to provide useful planning information regarding improvements to existing systems and the design of new distribution systems. This algorithm within the PC “KATRAN” is implemented in the exploitation at the Iron and Steel Works in Russia. Application of game theory criteria allows toselect the optimal strategy for the power supply system development and to compare the different variants of normal and repair maintenance schemes of network in uncertainty conditions.

Introduction.In terms of modern market economy, the financial impact of unreliable power supply is of great importance. Particular attention should be paid to the reliability of power supply systems of the energy-intensive industries such as Iron and steel works. Power supply system of a large Iron and steel work has a number of features: the high level of redundancy, chosen on the design stage; the significant transformer power that considers development of production; the combination of explicit and implicit redundancy at all voltage levels; insufficient statistical information on certain elements outages in 35-220 kV networks. So the accounting of reliability of power supply systems is nessesary in both the design and the operation of power supply systems of industrial enterprises.

The operation of power supply system of large industrial enterprises with their own power plants and meshed distribution networks of 110-220 kV, is accompanied by operational risks caused by the technical condition of electric power grid and substation equipment, the work of relay protection, the mode management solutions for grid and stations, the investment and energy saving policy. If the risks associated with equipment failures can be assessed by methods of reliability theory, the risks

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associated with the decision-making of operational and administrative technical personnel should be assessed taking into account the specific psychological aspects.

In recent years, the economic crisis at the Iron and steel works has led to an extreme tightening of energy-saving policy, and therefore the solutions for energy efficiency are often made without taking into account the reliability of power plants. Reason for that is the relatively low frequency of outages of any particular consumer. The volume of statistical information on emergency outages is extremely insufficient to determine the outage probability and the damage. Therefore, the approach allowing to account the uncertainty should be used for quantitative risk assessment.

1. Review of the literature related to the risk evaluation.With the rapid increase of energy demand, correct risk evaluation of power supply systems is of immediate interest. Power systems behave probabilistically because of random nature of load variations and element outages so the risk assessment in real time is challenging. An effective risk assessment model should provide quantitative risk indices to represent system reliability [1]. Usually, only failure statistics are used in power system risk assessment; but the overall system risk is also related to components’ operation conditions.

Significant works have been dedicated to probabilistic risk evaluation of power supply system and substation configurations. Commonly used framework for power system risk assessment was reviewed in [1-3], where the method, use and economic cost were considered in detail. But in this traditional methodology failure risks of elements, such as circuit breakers and transformers, were not studied. Generally, the elements risk assessment in substations is made independently [4–6]. As a result, there is a lack of a mechanism to transform element operation conditions into failure risks in the traditional framework.

The risk assessment model of a combinative system of a transmission network and substations was presented in [4]. Proposed method allows to evaluate system risks considering both transmission networks and substations by assessing new load limitations at load points for every failure state. As an improvement, substations are no longer observed as a transmission node and substation configurations and individual elements, such as breakers and transformers, are linked to system risks by analysing the statistical data of substation elements. However, the component failure data are still based on historical statistics and the effect of online element operation conditions cannot be integrated in risk evaluations. A multi-objective risk assessment framework was presented in [7] and probabilistic indices for assessing real-time power system security levels were derived. But the operation risks of elements still were not considered. Failure probability model, which can demonstrate the influence of surroundings on failure probabilities, based upon the Evidential Reasoning (ER) theory was developed in [8] for overhead lines. Nevertheless element outage rates were set as a fixed value, which were not connected to operating conditions of elements. Contingency identification method of components was presented in [9] and based on the ER theory and the functional group decomposition principle. In that work, element conditions, such as operating conditions and monitoring data of power transformers were not considered, and elements were just observed as part of transmission lines. The ER algorithm [10] is developed for combining evidence with a firm mathematical foundation, which can be employed to aggregate diagnosis information and deal with uncertainties. Proposed in [11] method employs ER for component risk assessment and the Monte Carlo (MC) simulation for system state selections and considers not only historical failure statistics of transmission systems but also operation failure risks of system components. The ER approach is used to evaluate element conditions and connect such conditions to failure rates using upto-date element operation data (online and offline data).

Different approaches for risk management in renewable energy projects are presented in [12]. In this work two ways of risks evaluation are proposed: qualitative and quantitative. Qualitative approaches deal with the evaluation of single risk issues, while quantitative approaches deal with the evaluation of all risk combined. In “qualitative” evaluations information is relatively descriptive and mainly based on expertise, so the results is presented in descriptive (risk register) or graphical (risk mapping) formats. The risks were identified through a Delphi process and assessed in terms of probability, MMSE Journal. Open Access www.mmse.xyz

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impact and affected parameter (CAPEX, OPEX, revenues, etc.). The analysis of risks in power supply systems with renewable energy sources is presented out in [13]. The operational risk of consumers power supply (ORCS) is used to account the effect of the random nature of weather conditions on the electricity generation. Proposed method of risk calculation is based on the economic theory of portfolio analysis. The ORCS determines the probability of consumers required power limitations due to the random nature of weather changes. The ORCS calculation is required when choosing the optimal combination of renewable energy sources for the local power grid.

In the paper [14] risk evaluation method is proposed for power grid renovation project in power market. It presents the risk variables affecting performance of power supply company in power grid investment activity, and builds the probability distributing functions according to the variables' physical characteristic, which changes the method fixing power sale quantity, power sale/purchase price, power supply reliability and loss rate in traditional power grid renovation project technology and economics evaluation. The method builds risk evaluation model with increment principle and quantifies power grid investment risk.

In [15-18] authors have developed risk-based measures for various types of security assessment and various components of the power system. These measures of risk consider only a predefined set of contingencies, but do take into account the probability of these events, the uncertainty affecting the future load and other system parameters. Paper [19] states that power system and risk-based approach to security assessment provides considerably more information to base operating decisions, then the traditional N-1 security criterion. Authors argues that risk should be evaluated in terms of expected outage costs to the consumers and the risk calculation should factor in the real probabilities of outages leading to load interruptions. This paper illustrates how to compare the cost and benefit of relaxing operating limits with the adaptive deterministic security boundaries application.

In addition to the N-1 criterion, probabilistic risk indices have been developed and used for power system planning [1, 20, 21]. However, the this risk indices used in selecting planning alternatives do not include long-term voltage instability risk. In current planning practice, the deterministic longterm voltage stability analysis [22] for planning alternatives is performed only for N-1 contingencies without considering probabilities of occurrence of contingencies. Some probabilistic voltage stability risk indices have been presented in the past years: system-wide risk indices [15, 23, 24] and local risk indices [25]. The system-wide risk index functions are used to judge whether or not the voltage instability risk of a system in an objective year meets the requirement in planning, while the local risk functions are used to identify weak location, where a strengthening is needed to increase system voltage stability. However these risk indices cannot be used for the planning purpose, because they do not consider various possible pre-contingency states and have to be recalculated once load level changes. The paper [26] analyses the long-term voltage stability for the power system planning. The combination of system-wide and local risk index functions with load level changes are proposed in [26] to evaluate voltage instability risk for planning alternatives during a whole planning period. The presented risk index functions are calculated by integrating a quadratic optimisation model into one single Monte Carlo simulation process. The optimisation model can find the maximum load level for voltage stability and the Monte Carlo simulation does not need to be repeated during the whole planning period for a given planning alternative.

A lot of works is dedicated to risk assesment, however non of them could allow to evaluate risks of industrial power supply system with accounting of all it's characteristic features. Thus, in conditions of uncertianty the task of development the method for risk assesment for the complex industrial power supply system is very important today.

2. The methodology for reliability evaluation of industrial enterprise power supply system. The review of reliability evaluation methods for the power supply system had shown that their application to the complex power supply system of large industrial enterprise is difficult. The existing methods suggest different criteria for reliability evaluation and individual detailed analysis of each scheme.

Such methods as table-logical and logic-probabilistic inapplicable for this task because of the extreme MMSE Journal. Open Access www.mmse.xyz

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complexity of table of expected logical connections or resulting fault tree. Thus, the development of method for reliability evaluation of large industrial enterprises and its application in the task of reliability analysis for the power supply systems is of immediate interest.

The the method of sequential network reduction [21] was proposed for equivalent reliability indices calculation for the meshed network. According to it the block diagram, which represents an analog of real elements connections of power supply scheme: transformers (T), circuit breakers (CB), overhead lines (PL), generators (G) and cables, is composed. Possibility of power flow direction accounting on the network elements is implemented in the algorithm; thus the part of scheme which is not involved to the electricity transmission to particular consumer will be excluded from the equivalent reliability indices calculation.

Approach based on the sequential network reduction method [27] for calculation of power supply systems modes is proposed to determine the reliability indices. Block diagram representing an analog of real elements connections of power supply scheme (transformers (T), circuit breakers (CB), overhead lines (PL), generators (G)) is composed on the basis of the power supply scheme. Each element of the block diagram is represented as a multi-beam star, which form is determined by the number of element connections. The simplification algorithm is based on the sequential elimination of elements with the replacement of the n-beam star (fig. 1) by the n-gon (fig. 2) with diagonals polygon; this operation reduces the number of elements at the each stage of transformation by one.

Fig. 1.Fragment of scheme before excluding node “0”.

Fig. 2. Fragment of scheme after excluding node “0”.

For the excluded element a set of equations, linking the probability of no-failure of the star p1, p2, p3, p4 and the sides and diagonals of the polygon p12, p13, p14, p23, p24, p34, is composed. The set of equations for the considered example has following form:

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 p1 p2  p12  p14 p24  p13 p23  p14 p24 p12  p13 p23 p12  0;  p p  p  p p  p p  p p p  p p p  0; 23 12 13 24 34 12 13 23 24 34 23  2 3  p3 p4  p34  p24 p23  p13 p14  p24 p23 p34  p13 p14 p34  0;   p1 p4  p14  p13 p34  p12 p24  p13 p34 p14  p12 p24 p14  0;  p1 p3  p13  p12 p23  p14 p34  p12 p23 p13  p14 p34 p13  0;   p2 p4  p24  p23 p34  p12 p14  p23 p34 p24  p12 p14 p24  0.

(1)

The Newton's method [28] was chosen for solving set of equations (1) as the most efficient numerical iterative method for finding roots of systems of nonlinear equations. Obtained values of the probability of no-failure combined with existing ones in the scheme according to the rules of seriesparallel reduction[29]: for series connection of 2 elements: pekv =p1p2; for parallel connection: pekv = p1+p2–p1p2.

Before the reliability indices evaluation the steady-state mode calculation is carried out. The results of mode calculation are used for the consideration of power flow direction. This procedure allows to exclude from the scheme for reliability indices calculation the part of scheme which is not participate in electricity transmission to the selected consumer. On the basis of the developed calculation algorithm in the framework of the program complex (PC) KATRAN [30] the block “Reliability” was created. It allows to calculate probability of no-failure pekv the failure flow parameter ωekv and the recovery time TRekv. The calculation algorithm in more detail is described in [31], [32].

3. Application of the reliability evaluation methodology. According to the modernization project of industrial enterprise the reconstruction of production workshop is planned, that will result to the increment of load. To meet the new power needs of workshop there is a project of construction of the power plant which will provide an extra total capacity of 37 MW. The fragment of the power supply system scheme of workshop is presented on Fig. 3. For the approbation of proposed algorithm the calculation of the power supply reliability indices of the workshop in different operating conditions is carried out.

The calculation results are presented in Table 1.

Table 1. Results of calculation of reliability indexes.

Type of accident Normal operating mode I Short circuit on PL "Ss-85 – Ss-62", planned maintenance of PL "Ss-60 – Ss62" II Planned maintenance on PL "Ss-85 – Ss-62", short circuit on PL "Ss-60 – Ss62" III Short circuit on PL "Power Plant – G2", planned maintenance on PL "Ss-60 – G-2"

Probability of nofailure, pekv 0.9976

Failure intensity, ωekv, (1/h) 0.047

Restoration time,TRekv, (h) 0.0511

0.99294

0.071

0.0994

0.99189

0.075

0.1081

0.99437

0.059

0.0954

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Fig. 3. The fragment of the power supply system scheme of production workshop.

Results had shown that the considered fragment has a high reliability in normal operating mode, but in some modes, for example when the emergency (short circuit) happens in the process of repair mode in close areas the recovery time increased by 2 times what leads to the significant economic and technological damages. In order to reduce the possible damage caused by power supply interruption the measure for reliability improvement is proposed â&#x20AC;&#x201C; installation of redundancy feeders. Fig.s 4, 5, 6 and Table 2 show the proposed redundancy options at voltage of 110 kV.

Fig. 4. Redundancy option A1.

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Fig. 5. Redundancy option A2.

Fig. 6. Redundancy option A3.

Table 2. Characteristics of the proposed redundancy options.

Strategy

Redundancy option

L, km

K, (mln.RUB)

A1

PL "Ss-62-PowerPlant"

2.8

2.76

A2

HVL branch line PL "PowerPlant -G-25МW" – Ss-62

0.1

0.06

A3

PL "Ss-62 – Ss-30"

3.6

2.10

4. Damage from power supply interruption. In modern industrial enterprises the equipment of the first category of the power supply reliability is dominate, as a consequence in tasks of mode planning the damage control is important. So in planning of industrial power supply system modes it is

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proposed to make decision not only on the basis of the equivalent reliability indices assessment, but also taking into account the expected damage from unreliability.

To assess the effects of the most serious outages the VaR-method (Value at Risk) is used [7], the value of the membership function (confidence level) μ = 0.02. Damage is represented as a unilateral fuzzy number with a boundary in the form of the Cauchy curve. The value of the technological damage DTECH for μ – level:

DTECH  DB  ( D AV  DB )

1



1 ,

(2)

where DAV – average damage; DB – boundary damage.

This approach is similar to the popular delta-normal method, which is based on the use of normal distribution fractile and volatility of risk factors, which is acts as its coefficient of variation [8]. The value of μ could also be determined by the experts method.

In case of sufficient statistical information about outages in power supply system for particular enterprise the average damage DAV could be determined on the basis of calculated values of recovery time:

DAV  D0 PTR ,

(3)

where D0 – damage, depending on the type of production, RUB/kW∙h (for the enterprises of Iron and steel industry D0=18.3 RUB/kW∙h); TR – recovery time, h; ΔP – consumer`s load limitiation, kW. The boundary damage in this case can be calculated approximately on the basis of the coefficient of variation CV of outage time resulting from processing information about emergency events in the power supply system: DB  DAV (1  u0,95CV ) ,

(4)

where u0,95 – the quantile of the normal distribution relevant to the confidence probability of 0,95 ( u0,95 = 1,67).

The characteristic feature of the large industrial enterprises is the installation of power plants on their territory. In this regard in case of stopping the power plant equipment, additionally to the technological damage there will occur the damage caused by underproduction of electric power DSt by the enterprise power plants, due to the need to purchase electricity at a higher price: DSt  (C g  C p )PSt TR ,

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(5)

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where (Cg – Cp) – the difference between the cost of electricity generated Cg by enterprise’s power plants and purchased electricity Cp, (RUB/kW·h); ΔPst – underproduction of power by local power plant during TR , kW. For considered redundancy options (Table 2, Fig. 4, 5, 6) damages for different types of possible accidents are calculated. Type of units installed at the power plant is taken into account for damage calculation. There are two gas reciprocating units installed according to the reconstruction project.

For example, a steam-turbine plant can stop only in case of auxiliary services supply interruption or in case of loss of stability. But when the the gas reciprocating units are installed the short circuit in network can lead to their shoutdown by relay protection and that will result to technological damage and the damage caused by underproduction of electricity.

The results of damage calculation are presented in Table 3.

Table 3. Damage calculation.

II

1.2 Р 1.4 Р

Type of Workshop acciden load value t Initial load P III

1.2 Р 1.4 Р

ТR, h 1 8 1 8 1 8

DTECH

DTECH

DTECH

Initial load P

DSt (G-2)

1.4 Р

DTECH+ DSt (G-2)

1.2 Р

DSt (G-2)

I

1 8 1 8 1 8 1 8 1 8 1 8

No damage

Initial load P

ТR, h

The damage caused by power supply interruption, (mln.RUB) A1 A2 A3 0.82 0.82 6.53 6.53 0.98 0.98 7.83 7.83 1.14 1.14 9.14 9.14 0.89 0.82 7.10 6.53 1.06 0.98 8.44 7.83 1.22 1.14 9.78 9.14 The damage caused by power supply interruption, (mln.RUB) A1 A2 A3 0.05 0.05 0.05 0.4 0.4 0.4 0.05 0.05 0.05 0.4 0.4 0.4 0.05 0.05 0.05 0.4 0.4 0.4

DSt (G-2)

Type of Workshop acciden load value t

5. The optimal strategy selection in uncertainty conditions. The task of the optimal strategy selection in mode planning for power supply systems can interfered to game with nature. There are two types of basic tasks in the games with nature: 1) the problem of decision-making in risk conditions, when the probabilities with which nature takes every possible state are known; 2) the

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problem of decision-making under conditions of uncertainty, when it is not possible to get the information about the probabilities of possible nature state.

The study of games with nature, as well as strategic should begin with composing of a payoff matrix, which is essentially the most time consuming step in the decision making process.

There are several criteria for the optimal strategy selection in theory of games with nature.

1. The Wald criterion (maximin) is a pessimistic approach. The strategy is chosen accordingly to the condition max  aij  min  and coincides with the lower price of the game. This criterion appeals to the i  j  cautious player who looks for ensurance that in the event of an unfavourable outcome, there is at least a known minimum payoff. This approach may be justified because the minimum payoffs may have a higher probability of occurrence or the lowest payoff may lead to an extremely unfavourable outcome.

Thus, the measure of efficiency Wi of strategy Ai according to the Wald criterion is minimum gain of player A: Wi  min aij . Price of the game according to the Wald criterion: W  max Wi .

1 j n

1i m

2. The criterion of Minimum is an pessimistic and it is selected from the condition of min min aij  . i  j  3. The Hurwitz criterion adheres to the intermediate position, taking into account the possibility of the worst as well as the best behavior of nature. It recommends a strategy defined by the formulas below.

-

for positive-flow payoffs (profits, revenues): H ( Ai )  A(row max)  (1  A)(row min) .

-

(6)

for negative-flow payoffs (costs, losses):

H ( Ai )  A(row min)  (1  A)(row max) .

(7)

where A – the optimistic coefficient (ranges from 0 to 1). The value of A depends on the player's responsibility: the higher it is, the closer A to the 1. A cautious player will set A = 1, which reduces the Hurwicz criterion to the Wald criterion. An adventurous player will set A = 0, so the Hurwicz criterion may be replaced by maximax criterion.

The optimal strategy will have the maximum value Max(H(Ai)) for positive-flow payoffs, and minimum value Min(H(Ai)) for negative-flow payoffs.

4. The Bayes criterion. At the primary stage of calculation the measure of efficiency for the each n

strategy Bi is determined as: B( Ai )   Q j a ij where Q1, ..., Qn - distribution of probabilities of nature j 1

states. The optimal strategy will have the maximum value Max(B(Ai)) for profits, and minimum value Min(B(Ai)) for losses.

5. The Hodge-Lehmann criterion is based on the Wald and the Bayes criterion. The efficiency indicator for each strategy HL(Ai):

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HL( Ai )  (1  q)Wi  qBi ,

(8)

where q – quantitative measure of the player confidence degree to a given distribution of probabilities qi = p (Pj).

Price of the game on the criterion of the Hodge-Lehmann: Max(HL(Ai)) for profits, Min(HL(Ai)) for losses.

6. According to the Savage criterion (minimax) the strategy that does not allow excessively high losses is optimal. The regret matrix is used. It elements reflects the player losses in case when for each nature state the best strategy will not be chosen. Decision rule is defined as: 1. Transform the payoff matrix X ={Xij} into an regret matrix R ={Rij}.

-

for positive-flow payoffs (profits, income):

Rij  (column j max)  X ij .

(9)

for negative-flow payoffs (costs) where Rij is the payoff (reward) for row i and column j of the payoff matrix R:

Rij  X ij  (column j min) .

(10)

2. The efficiency indicator for each strategy is determined as the maximum from regret matrix Max(R(Ai)).

3. The optimal is the strategy with minimum efficiency indicator.

7. The Laplace criterion is stated that to none of the possible nature states Sj, j = 1, ..., n, can not be given the preference. All of the nature states are considered as of equal probability. This principle is called the principle of Laplace "insufficient reason". Therefore, if there are n outcomes, the probability of each is 1/n. The strategy price is determined as:

L( Ai ) 

1 n

n



j 1

a ij .

(11)

Strategy with the “best” L(Ai) is optimal. “Best” means max for positive-flow payoffs (profits, revenues) and min for negative-flow payoffs (costs).

8. The Multiplication criterion. The measure of efficiency Pi of strategy:

P( Ai ) 

n



j 1

Qi aij .

(12)

Price of the game: Max(P(Ai)) for profits, Min(P(Ai)) for losses. For the multiplication criterion the positivity of all the probabilities of nature state and of all player wins is essential.

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9. The Germeier-Hurwicz criterion as well as the Wald criterion is a criterion highest pessimism of player, but in contrast to the Wald criterion player makes decision with highest possible circumspection, and takes into account the probabilities of the nature states Q. The measure of efficiency:

GH ( Ai )  Min(Q j aij ) (1  A)  Max(Q j aij ) A .

(13)

Price of the game determined as maximum of measure of efficiency - Max(GH(Ai)).

6. Application of the game theory criteria for the selection of strategy for power supply system development. To determine the optimal variant of redundancy for the above considered power supply scheme the payoff matrix is made. Elements of matrix represent the cost of measures to create a reserve for each strategy taking into account the damage. As uncertain information were considered: the restoration time TR, load growth and the type of accident. The possible types of accidents are modelled using a scenario approach. Thus, the payoff matrix of the game include 3 uncertain values (Table 4). As optimality criterion for strategy selection the costs (including damage value) are selected.

Table 4. The payoff matrix.

Type of accident

Workshop load value Initial load P

I

1.2 Р 1.4 Р Initial load P

II

1.2 Р 1.4 Р Initial load P

III

1.2 Р 1.4 Р

ТR, (h) 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8

The cost of measures to create a redundancy for strategies, (mln.RUB) A1 A2 A3 2.76 0.87 0.87 2.76 6.58 6.58 2.76 1.04 1.04 2.76 7.89 7.89 2.76 1.20 1.20 2.76 9.19 9.19 2.76 0.95 0.87 2.76 7.16 6.58 2.76 1.11 1.04 2.76 8.50 7.89 2.76 1.28 1.20 2.76 9.84 9.19 2.81 0.11 2.15 3.16 0.46 2.50 2.81 0.11 2.15 3.16 0.46 2.50 2.81 0.11 2.15 3.16 0.46 2.50

Graphically considered strategy shown in Fig. 7.

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C, mln.RUB 10 8 Strategy A1 Strategy A2 Strategy A3

6 4 2

Nature state

0 0

5

10

15

20

Fig. 7. The cost of measures to create a redundancy for the considered strategies.

It can be seen from the Fig. 7 that none of the strategies is dominant, therefore the mentioned above criteria are used for the optimal strategy selection.

1. Wald’s criterion. The Wald’s criterion is a way which the pessimistic player will choose. The decision maker prefers the highest value of bad conditions. However, according to Wald's criterion, he should select the maximum of the row minima. So the strategy A1 is selected as optimal (Table 5).

Table 5. The computation results for Wald’s criterion.

Strategy

Minimum value, (mln.RUB)

А1

2.76

А2

0.11

A3

0.87

2. The strategy A2 is selected as minimal of the row minima (Table 5) under Minimin criterion.

3. According to the Hurwicz’s criterion, the player is between pessimistic and optimistic choice.

Hurwicz criterion value is calculated according to (7). Minimum and maximum values of each strategy has been multiplied by optimistic coefficient (A = 0.6). The lowest calculated value is selected, so the optimal strategy is A1 (H(Ai) = 2.92 (mln.RUB)).

Table 6. The computation results of Hurwicz’s criterion.

Strategy

Minimum value, Maximum (mln.RUB) value,(mln.RUB)

Hurwicz criterion value, (mln.RUB)

А1

2.76

3.16

2.92

А2

0.11

9.84

4

A3

0.87

9.19

4.20

4. The characteristic feature of selection the optimal strategy by the Bayes criterion is the consideration of nature states probabilities. In considered example it will be the distribution of probabilities of operating with different power load Р (Table 7). Elements of payoff matrix (Table 8) are multiply by the probabilities (Table 7) and the measure of efficiency for each strategy will be determined as a sum of values. The strategy A1 with a minimum sum is optimal (Table 9).

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Table 7. The probability of operating with different power load Р.

Q1 (for Initial load P)

Q2 (for 1,2Р)

Q3 (for 1,4Р)

0.45

0.3

0.25

Table 8. The payoff matrix costs for average value of recovery time.

Strategy

Р 2.76 3.73 3.73

А1 А2 А3

I 1.2P 2.76 4.46 4.46

1.4Р 2.76 5.20 5.20

II 1.2P 2.76 4.81 4.46

Р 2.76 4.05 3.73

1.4Р 2.76 5.56 5.20

Р 2.98 0.28 2.32

III 1.2P 2.98 0.28 2.32

1.4Р 2.98 0.28 2.32

Table 9. The computation results of Bayes criterion.

Stra tegy А1 А2 А3

Value of costs multiplied on probabilities, (mln.RUB) I II III Q1.P

Q2.1.2 Р

Q3.1.4 Р

Q1.P

Q2.1.2 Р

Q3.1.4 Р

Q1.P

Q2.1.2 Р

Q3.1.4 Р

1.24 1.68 1.68

0.83 1.34 1.34

0.69 1.30 1.30

1.24 1.82 1.68

0.83 1.44 1.34

0.69 1.39 1.30

1.34 0.13 1.04

0.90 0.08 0.70

0.75 0.07 0.58

Bayes criterion value, (mln.RUB ) 8.50 9.26 10.95

5. To determine the optimal strategy by the Hodge-Lehmann criterion the values of the Wald (Table 5) and the Bayes (Table 9) criteria are used. For each strategy themeasure of efficiency was calculated as (8) for different values of q (Table 10). The strategy with minimum H(Ai) selected as optimal.

Table 10. The computation results for Hodge-Lehmann criterion.

Hi , (mln.RUB) Stra tegy

W, (mln.RU B)

B, (mln.RU B)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

HodgeLehmann criterion, (mln.RU B)

7.9 7.3 5.6 5.0 4.4 3.9 3.3 6.78 6.21 7.93 3 5 3 6 8 1 3 8.2 7.3 4.6 3.7 2.8 1.9 1.0 А2 0.11 9.14 6.43 5.53 8.24 4 3 2 2 2 1 1 9.9 8.9 5.9 4.9 3.9 2.8 1.8 А3 0.87 10.95 7.93 6.92 9.94 4 4 1 1 0 9 8 6. The Savage criterion focuses on avoiding the worst possible consequences that could result when making a decision. It views actual losses and missed opportunities as equally comparable. For decision making the payoff matrix (Table 4) is сonverted to the regret matrix (Table 11), using formula (10), and the minimax rule applied to the regret matrix. The optimal strategy is A1. А1

2.76

8.50

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Table 11. The fragment of Regret matrix and computation results for Savage criterion.

Regret matrix, (mln.RUB) Savage Str I II III criterion, ate (mln.RUB Р 1.2P 1.4Р Р … 1.4Р Р … gy ) 1 8 1 8 1 8 1 8 1 8 1 8 1.7 1.5 1.8 1.5 2.7 А1 1.89 0 0 0 0 0 2.70 2.7 2 6 9 6 0 3.8 5.1 6.4 0.0 4.1 0.0 6.8 А2 0 0 0 0 0 6.87 2 3 3 5 9 6 7 3.8 5.1 6.4 3.8 6.4 2.0 А3 0 0 0 0 0 2.04 6.43 2 3 3 2 3 4 7. The Laplace criterion can be interpreted as a transition model between the probability/risk model of decision theory and game theory in that it suggests that in the absence of any probabilities which could potentially differentiate the payoffs, equal probabilities should be assigned. The value of game for each strategy according to (11), was calculated. The optimal strategy according to this criterion is with the minimum measure of efficiency value A1.

Table 12. The computation results for Laplace criterion.

Laplace criterion, Strategy Sum (mln.RUB) А1 51.02 2.84 А2 57.32 3.18 А3 67.47 3.75 8. The Multiplication criterion. The measure of efficiency P(Ai) for each strategy is calculated according to (12) (Table 13). Strategy A2 corresponding to the minimal measure of efficiency is optimal.

Table 13. The computation results for Multiplication criterion.

Value of costs multiplied on probabilities Strateg I II III . . . . . y Q1 Q2 1.2 Q3 1.4 Q2 1.2 Q3 1.4 Q2.1.2 Q1.P Q1.P P Р Р Р Р Р 1.2 А1 4 0.83 0.69 1.24 0.83 0.69 1.34 0.90 1.6 А2 8 1.34 1.30 1.82 1.44 1.39 0.13 0.08 1.6 А3 8 1.34 1.30 1.68 1.34 1.30 1.04 0.70

Q3.1.4 Р

Multiplicatio n criterion, (mln.RUB)

0.75

0.45

0.07

0.01

0.58

3.59

9. The Germeier-Hurwicz criterion as well as the Wald criterion is a criterion highest pessimism of player A, but in contrast to the Wald criterion player A makes decision with highest possible circumspection. The measure of efficiency GH(Ai) for each strategy is calculated according to (13) (Table 14). Strategy A1 corresponding to the minimal measure of efficiency is optimal.

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Table 14. The computation results for Germeier-Hurwicz criterion.

I Stra tegy

II

Q1.

P

Q2.1.2 Р

Q3.1.

4Р

А1

1.2 4

0.83

А2

1.6 8

А3

1.6 8

III Mi n

Max

Germeier -Hurwicz criterion, (mln.RU B)

0.75

0.6 9

1.34

1.08

0.08

0.07

0.0 7

1.77

1.09

0.70

0.58

0.5 8

1.68

1.24

Q1 P

Q2.1. Q3.1.

2Р 4Р

Q1 P

Q2.1. Q3.1.

2Р 4Р

0.69

1.24

0.83

0.69

1.34

0.90

1.34

1.30

1.82

1.44

1.39

0.13

1.34

1.30

1.68

1.34

1.30

1.04

.

.

Results of calculation of all listed above criteria and corresponding to them optimal strategies are presented in summary Table 15.

Table 15. The optimal strategies and corresponding measures of efficiency.

Strateg y

Criteria values, (mln.RUB) Wal d

Mini min

Hurwic z

Bayes

HodgeLehmann

Savage

Laplace

Multipli cation

Germeier - Hurwitz

А1

2.76

2.76

2.96

8.50

7.93

2.70

2.83

0.45

1.08

А2

0.11

0.11

4.87

9.14

8.24

6.87

3.15

0.01

1.09

А3

0.87

0.87

5.03

10.95

9.94

6.43

3.75

3.59

1.24

Optima l strategy

А1

А2

А1

А1

А1

А1

А1

А2

А1

Value of game

2.76

0.11

2.96

8.50

7.93

2.70

2.83

0.01

1.08

Thus, the A3 strategy is not optimal strategy on any of the criteria. The A2 strategy is the optimal strategy for the minimin criterion and multiplication criterion. According to the most of criteria (7 out of 9) A1 strategy is optimal, despite the fact that this redundancy option correspond to the highest capital costs.

Summary. In the article the methods of calculation of equivalent reliability indices and damage from the power supply interruption for the large industrial enterprice in different operating modes are presented. Different options of normal and repair schemes could be compared with the help of proposed algorithm on the basis of calculated reliability indices. Accounting of the power flow direction allows to calculate the planned operating conditions, in view of the operational configuration schemes and taking into account changes in load.

The method based on the use of the game theory criteria is used to determine the optimal strategy for the power supply system development in conditions of initial information uncertainty. The options for redundancy and different types of emergency modes were analyzed. On the basis of game theory criteria the optimal option for redundancy with the purpose of reliability increment was chosen. The MMSE Journal. Open Access www.mmse.xyz

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developed method allows to select options for the power supply system development, taking into account the operation of its power plants, the values of operational risks, the reliability indices.

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Programme ‘The complex of automated modal analysis KATRAN 9.0’, The bulletin. ‘Programme for the computer, database, TIMS’, 2015 [31] A. Iuldasheva, A. Malafeev, Reliability Evaluation for Electric Power Supply Management, Proc. Int. Sc. Symposium ‘Electrical power engineering 2014’, Varna, Bulgaria, 2014, pp. 10-12 [32] A. Malafeev, A. Iuldashevа, Accounting for power flow direction in the problem analysis of structural reliability of power supply systems // Proc. of Higher Education. Russian Electromechanics, 2015, № 2, pp. 36-40 Cite the paper Alina Iuldasheva & Aleksei Malafeev (2016). Selection of the Reconstruction Options for Industrial Power Supply System under Uncertainty Conditions on the Basis of the Game Theory Criteria. Mechanics, Materials Science & Engineering Vol.6, doi: 10.13140/RG.2.2.34252.41609

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