Journal of Mechanical Engineering 2013 4

http://www.sv-jme.eu

59 (2013) 4

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

203

David Heath, Brane Širok, Marko Hočevar, Boštjan Pečnik: The Use of the Cavitation Effect in the Mitigation of CaCO3 Deposits

216

Ani Luo, Heping Liu, Cheng Li, Yongfan Wang: Study of a Flowerlike Deployable Structure

223

Tomaž Katrašnik, Henrik Aleš Schuemie, Johann C. Wurzenberger: On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling

237

Nikolaj Mole, Gašper Cafuta, Boris Štok: A Method for Optimal Blank Shape Determination in Sheet Metal Forming Based on Numerical Simulations

251

Xiaoni Qi, Yongqi Liu, Zhenyan Liu: Exergy Based Performance Analysis of a Shower Cooling Tower

260

Goran Radoičić, Miomir Jovanović: Experimental Identification of Overall Structural Damping of System

269

Han Lianfu, Tang Wenyan, Wang Jun, Fu Changfeng: Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center

Journal of Mechanical Engineering - Strojniški vestnik

Contents

4 year 2013 volume 59 no.

Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis.

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Editor in Chief Vincenc Butala University of Ljubljana Faculty of Mechanical Engineering, Slovenia Technical Editor Pika Škraba University of Ljubljana Faculty of Mechanical Engineering, Slovenia Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu, http://www.sv-jme.eu Print DZS, printed in 450 copies Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia

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59 (2013) 4

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Strojniški vestnik Journal of Mechanical Engineering

Journal of Mechanical Engineering - Strojniški vestnik

s

year

no. 4 2013 59

volume

Cover: Cavitation near the solid surface causes local pressure shock as a result of cavitation cloud collapse, which causes material destruction in the form of cavitation erosion.The figure shows the effect of ultrasonic cavitation on a lime stone coated surface. The figure shows vectors of velocity of fluid flow in the vicinity of the test sample.

Image Courtesy: Laboratory for Water and Turbine Machines, Faculty of Mechanical Engineering, University of Ljubljana.

International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 59, (2013), number 4 Ljubljana, April 2013 ISSN 0039-2480 Published monthly

Papers David Heath, Brane Širok, Marko Hočevar, Boštjan Pečnik: The Use of the Cavitation Effect in the Mitigation of CaCO3 Deposits Ani Luo, Heping Liu, Cheng Li, Yongfan Wang: Study of a Flowerlike Deployable Structure Tomaž Katrašnik, Henrik Aleš Schuemie, Johann C. Wurzenberger: On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling Nikolaj Mole, Gašper Cafuta, Boris Štok: A Method for Optimal Blank Shape Determination in Sheet Metal Forming Based on Numerical Simulations Xiaoni Qi, Yongqi Liu, Zhenyan Liu: Exergy Based Performance Analysis of a Shower Cooling Tower Goran Radoičić, Miomir Jovanović: Experimental Identification of Overall Structural Damping of System Han Lianfu, Tang Wenyan, Wang Jun, Fu Changfeng: Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center

203 216 223 237 251 260 269

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 203-215 © 2013 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2012.732 (1.02) Review Scientific Paper

Received for review: 2012-08-13 Received revised form: 2013-01-11 Accepted for publication: 2013-02-12

The Use of the Cavitation Effect in the Mitigation of CaCO3 Deposits Heath, D. – Širok, B. – Hočevar, M. – Pečnik, B. David Heath1 – Brane Širok2 – Marko Hočevar2,* – Boštjan Pečnik1,3 1 MPŠ

Jožef Stefan International Postgraduate School, Slovenia of Ljubljana, Faculty of Mechanical Engineering, Slovenia 3Gorenje d.d., Slovenia

2 University

Hard mineral scale fouling results in significant economic losses both industrially and domestically. Recently, attempts have been made to use ultrasound to mitigate scale formation and its removal based on the phenomena of cavitation. Cavitation erosion is the removal of material from a solid surface by pressure shock waves associated with the formation and collapse of bubbles. This paper reviews the literature on cavitation erosion of brittle crystalline materials in an attempt to better understand the relationship between the material properties of CaCO3 scale deposits and its potential removal by cavitation. The study finds that from a materials perspective cavitation erosion is intimately associated with both a material’s bulk properties and importantly to its microstructure. The situation is further complicated because the macro and micro-properties of CaCO3 scale are dependent on many factors relating to its depositional environment. The type of scale formed will affect how it is removed by cavitation.

Keywords: cavitation, cavitation erosion, ultrasound, scale, calcium carbonate

0 INTRODUCTION The build-up of CaCO3 (lime) scale is a major operational headache that affects a range of industrial processes including membrane filtration (desalination), drilling, and especially those that involve heat transfer. In a heat exchanger the effects are twofold: first, even a small coating of scale can significantly reduce heat transfer efficiency and second, by acting as an insulator, it will cause a heating element to get much hotter internally leading to component failure. Scale can also drastically reduce the flow in pipes. In the domestic setting, in areas affected by hard water, home appliances like boilers, washing machines, and solar collectors are prone to scale build-up and, just as in industry, it can reduce performance and working life. Additional consequences are increased service charges, frequent equipment replacement, and higher energy bills - up to 25% more [1]. The costs incurred in preventing and remedying scale both industrially and domestically are considerable. According to Müller-Steinhagen et al. [2], the total annual cost due to fouling for highly industrialized countries, such as the United States or the United Kingdom are about 0.25% of their Gross National Product (GNP). These figures may only be indicative, but they underline the significant impact that scale has on global energy efficiency [3]. The percentage of energy losses due to fouling is at least 2% globally. Another way of estimating the size of the problem is by looking at how much is spent on preventing scale. Scale inhibitors are the third largest product segment of the water treatment chemicals

market (corrosion inhibitor market being the largest) and by 2016 will be worth an estimated $ 3.6 billion [4].

Scale management ideally is about the prevention of scale and currently the most efficient way to prevent scaling is by using chemical inhibitors or descalers. Chemicals, however, are costly and their environmental impact is questionable, which has led to the search for both cheaper and “greener” alternatives [5]. They include “natural” chemical agents, surface treatments including coatings, and magnetic and electronic conditioning, all of which show varying degrees of efficacy (Table 1). Once formed, the removal of a tenacious scale involves either chemical (acid) treatment to dissolve the scale or some form of mechanical removal [6]. The main benefit of using a mechanical method is that it avoids the use of chemicals, but where this has to be done by hand, it can be time consuming and costly. An alternative approach is the possible use of ultrasound.

Ultrasonic cleaning is a well-established and efficient technology particularly for dislodging solid residues and removal of biological fouling. It is also the bestknown application of acoustic cavitation. Cavitation is the phenomena of the rapid formation and implosion of bubbles in an area of low-pressure in liquids by means of mechanical forces [7] and [ 8].

The aim of this paper is to present a literature review in relation to existing studies involving ultrasonically induced cavitation for the mitigation and, importantly, the removal of tenacious CaCO3 deposits. Fortunately, a large corpus of literature now exists on both scale formation and cavitation

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, marko.hocevar@fs.uni-lj.si

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 203-215

phenomena including mechanisms of cavitation erosion. From an initial survey of the literature it becomes readily apparent that cavitation erosion is intimately associated with material bulk properties and microstructure and therefore this topic is best researched from a materials science perspective. It is essential to review published studies that develop our understanding of the bulk macro- and micro-structural surface properties of scale and the cavitation erosion of brittle crystalline materials.

Table 1. The effectiveness of different scale treatments. Adapted from [9] Scale treatment Ion exchange Acid dosing Chemical inhibitors Metal ions Magnetic conditioners Electronic conditioners Electrolytic Ultrasound Surface modifications

Effectiveness (% reduction) 100 100 100 80 80 80 40 30 90

liquid/surface boundary or the presence of particles can also promote crystal growth particularly if there are surface defects and areas of turbulent flow. This is known as heterogeneous nucleation [11]. As the micro-crystals continue to grow, they fuse to form adherent macro-crystals. Growth then continues via the further adsorption of scaling ions to eventually form a uniform coating with subsequent deposition covering the rough scaled surface [12].

Of significance to this study is that calcium carbonate precipitates in different polymorphic crystalline forms, which, albeit chemically identical, have different crystal lattice structures and macroand micro-structural properties. Three anhydrous crystalline polymorphs occur: vaterite, aragonite, and calcite and three hydrated phases, including amorphous calcium carbonate (ACC), monohydrate calcium carbonate (MCC; monohydrocalcite) and hexahydrate calcium carbonate (HCC; ikaite) [13]. The hydrated forms are unstable and most studies focus on the three anhydrous crystalline forms, of which calcite is the most stable, aragonite is metastable, and vaterite is the least stable form. Calcite and aragonite are the most common polymorphs (Fig.1).

1 SCALE FORMATION Calcium carbonate is atypical in that it is a reverse solubility salt, i.e. solubility decreases with increasing temperatures and hence is a fouling nuisance in heating equipment in areas of high water hardness.

Generally, exceeding the solubility of a particular salt will initiate scale formation, albeit the actual process is complex and depends on many factors including supersaturation level, fluid composition, temperature, flow regime, and surface properties like surface energy and roughness [9] and [10]. When CO2 dissolves in water, it forms a weak carbonic acid, which dissociates into hydrogen carbonate, hydrogen, and carbonate ions. Calcium ions then react with the bicarbonate to form calcium carbonate. The overall reaction can be written as follows:

Ca2+ + 2HCO3- → CaCO3(s) + CO2(g) + H2O The physical process involves ion clustering, nucleation, crystal growth, agglomeration, and deposition. Initially there is an incubation or induction period, where seed crystals form. Briefly, cationic and anionic species collide to produce ion pairs in solution. These pairs then go on to create microaggregates or unstable clusters, which become nucleation centres or seed crystals in a process known as homogenous nucleation. Alternatively, a 204

a) b) Fig. 1. Scanning electron micrographs of CaCO3 scale deposits showing the characteristic a) calcite (5 hrs, T = 18 °C, u =2 m/s, pH = 10.5, S = 15.5) and b) aragonite (60 min, T = 45 °C, u = 0.41 m/s, pH = 9.5, S = 12). Reprinted with permission from [14]. Copyright 1997 American Chemical Society

Factors affecting the rate of scale deposition and the type of polymorph include pH, temperature, flow dynamics, surface properties, presence of other ions, and time. The higher the pH, the greater is the tendency for scale to form. According to Andritsos and Karabelas [15], an increase in the pH from 8.8 to

Heath, D. – Širok, B. – Hočevar, M. – Pečnik, B.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 203-215

10 over a 2 hour period, will increase scale deposition from 2 to 12 mg/cm2. Aragonite content also increases with increasing pH. As stated, CaCO3 shows reverse solubility and, as water temperature increases, carbon dioxide is removed, which causes a shift in the equilibrium to the right. This results in an increase in pH and the precipitation of CaCO3. Additionally, Xia et al. [16] find that at lower temperatures, CaCO3 scale is denser and adheres more strongly to the substrate, while at higher temperatures it exhibits a characteristic dendritic crystal structure. This is explained by the fact that below 30 °C, calcite is essentially the only polymorph present and is mainly in a prismatic form, while above 35 °C, aragonite is stable and creates dendrite formations (Fig. 2). a)

b)

Under low flow conditions a less dense scale will typically prevail, whereas higher flows result in a smoother surface. This is because the low density, poor coherence, and lower tenacity of aragonite mean it will be more easily removed. Scale properties are also influenced by the substrate surface in terms of size and crystal arrangement. For example, Malayeri et al. [3] find that highly alloyed austenitic stainless steel produces larger thicker harder and more adherent scale crystals. Time is also an important factor and with age the relative amount of calcite will increase.

It has long been known that the presence of other ions can affect scale formation from supersaturated solutions. Magnesium ions show the most pronounced affect [20] and [21]. Their presence not only delays and inhibits bulk precipitation and surface deposition processes, but also, to a different degree, favours the formation of the aragonite polymorph [20] and [21]. Low concentrations of magnesium ions in solution favour calcite formation, while aragonite is formed at high magnesium concentrations. It is the (Mg2+):(Ca2+) ratio and the initial ionic product of carbonate in solution that controls the time of nucleation and the mineralogy of the precipitate when Mg2+ ions are present [20]. The exact mechanism is not fully understood but it is thought that Mg2+ ions poison calcite crystal growth by becoming incorporated into the CaCO3 lattice at Ca sites thus affecting its morphology and growth [21] and [22].

2 ULTRASONIC SCALE REMOVAL

Fig. 2. The relative concentrations of calcite, aragonite and vaterite with a) temperature and b) time. Adapted from [17]

Vaterite can also be present at higher temperatures but it rapidly converts to calcite. The induction period increases with decreasing initial temperature because of the slower growth rate of CaCO3, which favours formation of calcite. During the post-induction period the rate of CaCO3 crystal growth is higher and favours the formation of aragonite [18] and [19].

Ultrasonic cleaning is a technology that uses high frequency sound waves and combines the phenomenon of cavitation, which dislodges contaminants, and agitation (micro-streaming), which accelerates the dissolution of contaminants by supplying fresh solution to the surface being cleaned.

Cavitation describes the life-cycle of a transient cavity or bubble from formation via inception and nucleation, its expansion, and its eventual implosion.

The result is extremely high pressure shock-waves (Table 2) and temperatures (approx. 5,000 K). Its observable effect, when near a solid boundary, is material damage known as cavitation erosion.

Acoustically driven cavitation is the most important source of the cavitation effect for practical purposes, although it is also possible to utilise hydrodynamic cavitation in scale prevention [23]. Cavitation occurs as an acoustic wave (20 to 100 kHz) passes through a fluid. This then creates oscillating pressure with resultant phases of compression and rarefaction.

During rarefaction, molecules are increasingly pulled

The Use of the Cavitation Effect in the Mitigation of CaCO3 Deposits

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 203-215

apart, eventually forming a cavity or void [24]. Fig. 3 shows how a bubble grows and collapses in an ultrasonic field. In reality, pre-existing nucleation sites require less energy to develop into bubbles.

These sites are typically micro-heterogeneities, such as tiny gas bubbles, gas bubbles trapped in pockets and in crevices, trace impurities, and solid particles [25]. In most situations ample nuclei exist for bubble formation [26]. Two distinct types of bubble oscillations exist. The first are “stable” bubbles that oscillate for many periods of the sound field, while the second are “transient” (inertial) unstable bubbles that exist for less than a single compression half-cycle and increase in size 2 to 3 times their resonant size before imploding [27]. This transformation of a low energy density sound wave into a high energy density one means that a collapsing bubble will concentrate the energy into very small volumes producing extreme pressures (shock waves), temperatures, and even light in a process known as sonoluminescence [26]. Fig. 3 gives an overview of ultrasonic cavitation.

bubbles collapse spherically and their effect cannot be considered in isolation.

Bubbles that interact with a surface (rigid boundary) and with other bubbles will collapse in a non-spherical or asymmetrical manner, which in turn results in a re-entrant jet that is capable of delivering a directional and much higher local impact. Many authors believe that the shock waves produced by single bubble collapse and jet formation cannot account for much of the observed macroscopic damage and they focus on the combined effect of multiple bubble systems (clouds) and the so-called “splashing effect” as a major damage mechanism, [29] to [33].

Table 2. Examples of cavitation load pressure Source

Pressure [GPa]

[34] [35]

[37] [37]

10 0.2 4 (200 µm) 4.8 to 8.1 (200 µm) 0.1 0.15

[38]

2.5

[39] [40]

1.7 1 1.85 to 2.24 (200 µm) 285 to 336 (10 µm) 4.3 (70 µm) 720 (4 µm) 0.55 0.1 4 to 16 MPa (10 mm from bubble) 4 to 6 8 (at bubble wall) 1.3 ± 0.3 (68 μm from bubble)

[36]

[41]

Fig. 3. A generalised view of bubble dynamics in an ultrasonic field

In terms of pressure as a damage mechanism, the forces impacting a surface are a result of spherical bubble collapse, impact of micro-jets [28], splash effect following the impact of a jet [29] and the combined effect of multi-bubble systems known as “clouds” [30]. The interaction of these shock waves subjects a material to repeat localized loading.

The actual mechanisms producing shock waves of sufficient energy to cause material damage are a continual point of debate. Table 2 gives some examples of cavitation pressure loadings reported in the literature. Comparing these values is, however, difficult because some are estimates while others are measured using different methods and different ways of producing the cavitation phenomenon. In addition, the extreme pressure produced in the centre of the bubble decreases dramatically over very short distances. Spherical bubble collapse was the focus of initial investigations, but this is now known to be only a part of a more complex phenomenoa, since not all 206

[42] [43] [44] [45] [46] [47]

Bubble collapse single bubble micro-jet

Cavitation source spark induced ultrasound hydrodynamic, vibratory

single bubble micro-jet

single bubble

simulated and hydrodynamic hydrodynamic hydrodynamic

single bubble multi -bubble

hydrodynamic, vibratory ultrasound ultrasound ultrasound

single bubble

ultrasound hydrodynamic

[48]

7

single bubble

[49]

1.5

hemi-spherical cloud (760 ultrasound μm)

Developing an understanding of cloud phenomena is difficult because whereas individually collapsing bubbles can be observed experimentally, shock waves arising from a concerted collapse of bubbles in a cloud system involve more complex dynamics [50] and [51]. In a bubble cloud, the entire bubble population dominates the cavitation effect and the collapse of the cloud creates a shock wave that radiates out into the fluid, the magnitude of which is attenuated as it travels

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toward a solid surface. Williams et al. [52] suggest that jet formation may be enhanced by bubble-shockwave interaction. Brujan et al. [47] believe that collapsing clouds are potentially the most destructive form of cavitation as they produce short duration pressure pulses of large magnitude (1.5 GPa). Van Terwisga et al. [53] argue that the acoustic power released from a synchronized bubble cloud collapse is greater than that released via a micro jet. These shock waves are of interest in those cases where bubbles collapse at a distance from a boundary so that no jets form [53] and [54]. More recently, Brujan et al. [55] investigated secondary shock waves, which are a consequence of the free collapse of a bubble within the cloud by the ambient pressure in the fluid. In the second case, it is a consequence of the interaction of the cloud-collapseinduced shock wave with micro-bubbles situated close to the collapse site of the cloud. The latter can be very powerful, resulting in a secondary shock wave emission with a maximum amplitude of about 0.5 GPa. Despite this, Brujan [56] finds that, whereas small jet velocities do not have the potential to produce erosion of metals, they can play an important role in the fragmentation of brittle materials, since the yield strength is much lower than for ductile materials.

Also of interest is a study by Barrow et al. [57] using atomic force microscopy, where the authors suggest that rapidly growing cavities have the potential to damage adjacent structures by the development of tension rather than by the evolution of a (positive) pressure shock. This type of damage together with micro-streaming could have a significant effect on the removal of the poorly adhering dendritic crystal structure, i.e. aragonite.

3 INFLUENCES OF MATERIAL PROPERTIES ON CAVITATION EROSION From the published studies it can be seen that cavitation erosion is a function of material properties, including macro-properties, such as: tensile strength, strain energy, resilience, ductility, hardness, fatigue resistance and its microstructure [58]. To date, no overall single or combination of material properties can adequately predict cavitation erosion [59] and [60]. What is certain is that cavitation erosion is a time dependent phenomenon classically represented by an S-shaped erosion curve (Fig. 4) of material loss or rate of material loss against time [61]. The four distinct stages identifiable are the following: an incubation period, an acceleration period, a deceleration period, and finally a period of steady state erosion. The

S-curve is a generalisation and the actual shape is specific to a material’s properties.

The incubation period represents the accumulation of plastic deformation and internal stresses under the surface that precedes significant material loss [62] and [63]. It is related to the accumulation of fatigue damage but many other factors including, for example, grain size, hardness, and plasticity are also relevant. In ductile materials, cavitation shock waves impacting the surface will cause work hardening. Eventually, micro-cracks appear around grain boundaries and surface heterogeneities and begin to propagate outwards. In metals, these include notches, tears, undercuts, welding defects, inclusions or at heterogeneous areas of the material such as at the directionality of metal flow and decarburized sections [64]. Eventually the material will no longer be able to withstand such shocks and material will break loose [65] and [66]. This represents the transition from the incubation to the acceleration period. Initially the rate of material loss is exponential but becomes linear as the volume loss rate reaches a maximal value.

a)

b) Fig. 4. General cavitation erosion s-curve of material loss with time (a) showing four distinct stages (b). The rate that material is lost is initially slow but then steadily increases up to a point of maximum damage after which it (the damage rate) decreases until a steady-state condition is achieved. Adapted from [67]

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A lessening in the rate of this damage typifies the deceleration period. The volume loss rate decreases and removal of new material uncovers major areas with no accumulated internal stresses. Eventually, the erosion reaches a steady state with an almost constant volume loss rate since the surface, that is already strongly fractured, behaves as a protective cushion.

In brittle materials, zero or only a limited amount of plastic deformation occurs. Instead, micro-fractures will appear at the onset and cracks will rapidly propagate without increasing the overall applied stress. The erosion curve can still show the four characteristic stages of cavitation erosion since the incubation period reflects the accumulation of microcracks up to a point where material will break free. In addition, when erosion begins it becomes much more intensive than in ductile materials. The slope of the acceleration part of the cavitation erosion curve will be much steeper so that, in the end, the cumulated mass loss will be greater than in ductile materials given the same exposure time and cavitation intensity.

Fatjo et al. [68] find that for silicon nitride and zirconia pseudoplastic deformation pits are visible, but, unlike the work hardening seen in metals, there is a softening effect on the surface and the observable pits are linked to crack creation. The authors also describe how in ceramics these erosion pits have micro-cracks extension at the boundaries and cracking is mainly intergranular, although some cracks are visible within large or elongated grains. According to Karunamurthy et al. [69], material is removed as cracking eventually leads to dislodging of partial grains, grains, and eventually whole grain clusters. The transition from the incubation period to the acceleration period will occur when erosion of the entire surface is at a depth approximately equal to the average grain size [70].

Lu et al. [71] state that micro-structures including grain size, phases at grain boundaries, cavities or pores, and defects, such as cracks, inclusions, single agglomerates of large grains, and scratches from machining, promote fracture between grains. These structures are important in controlling the cavitation erosion behaviour of ceramic materials. Grain size is also important; increasing grain size leads to a greater erosion rate and vice versa [63].

Cavitation erosion is a natural process but only a few studies exist that investigate the cavitation of carbonate rocks. Carbonate rocks are made up of calcite crystals and hence provide valuable information about how cavitation can affect hard scale deposits. For instance, Momber, [72] having studied the cavitation erosion of limestone and marble using a flow cavitation chamber, finds that the most important 208

parameters influencing cavitation erosion are exposure time, cavitation intensity, and materials properties.

Grabco et al. [73] find that, in addition to mineral composition and mechanical properties like hardness, strength, fracture, and stability, microstructure is an important factor in the erosion resistance of brittle minerals. Calcite is unusual in that the presence of cleavage plane slip bands allows a limited amount of plastic deformation, while aragonite, having a single plane cleavage, does not. Generally, in such material, plastic deformation is relatively small and the formation of micro-cracks is the most important dissipation mechanism of inelastic deformation and brittle failure. This is enhanced by the presence of structural homogeneities like pores and interfaces, and pre-existing flaws are important sites where damage initiates while additional energy is absorbed due to crack surface friction (frictional resistance to sliding), crack bridging, and micro-crack formation [74].

There exist two possible mechanisms of brittle fracture: transcrystalline and intercrystalline cleavage.

Lubarda et al. [75] find that, in most cases, it is inelastic deformation that occurs in rocks. As with ceramics, micro-cracks begin and propagate at the surface of pores and grain boundaries, and most evidence is for the damage being intercrystalline rather than transcrystalline fracture. In the case of limestone and marble, Momber [72] observes two erosion zones of high and low intensity in his samples of carbonaceous rocks. In the high intensity zone there is considerable erosion, while in the low intensity zone there are only a few single erosion events, which is a result of the flow conditions in the cavitation chamber. For marble and limestone, the transition or threshold time (critical exposure time) from there being just a few singular erosion events to producing a large cavity (damage) was 30 and 90 s, respectively. Low intensity cavitation of the limestone exposed coarse grain conglomerates 800-1000 µm in diameter. At higher intensity, individual grains 100 µm in diameter (approximately two times cleavage spacing of the calcite crystals) were exposed and removed. For marble, the author observes exposed individual calcite crystals (800 µm) at the surface and transcrystalline cleavage (200 µm cleavage planes) in the calcite minerals, indicating that marble is eroded in a two-step mode.

The author proposes that fracture toughness and, secondly, structural homogeneity (pre-existing flaws) are good indicators of cavitation erosion, as erosion mainly occurs in an intergranular or intercrystalline mode and erosion resistance decreases with grain size.

Wagterveld et al. [19] investigating cavitation erosion of suspended calcite crystals find that crystal breakage

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can only occur if the energy input is higher than the energy required for maximum elastic deformation.

For transcystralline cleavage, the strength of the shock wave pressure pulses would have to be significant given the hardness of calcite (Vickers hardness of 1.2 to 1.5 GPa), with aragonite being slightly harder [76] and [77]. This is feasible given the strength of cavitation pressure pulses (> 1.5 GPa) reported in the literature. There is evidence suggesting that the strength of the crystal bonds may be higher than actually necessary to cause transcrystalline crystal cleavage. However, Wagterveld et al. [19], observing suspended calcite crystals, find that less energy is required to break off small fragments of corners by attrition compared to large fragments as total crystal breakage. The authors also find a discrepancy between theoretical surface fracture energies, defined by material bond strength, and their experimentally observed surface fracture energies, which they relate to so-called “Griffith cracks”.

These exist as microscopic flaws in the crystal grains.

The numbers of these cracks are known to increase with volume, causing large grains to fracture more easily. This may also explain why cavitation erosion resistance increases with decreasing crystal grain size.

Fatjo et al. [78] observe in ceramic materials that, while cracking is mainly intergranular, some cracks are also visible within larger or elongated grains.

Wagterveld et al. [19] find no evidence of re-entrant jet impacts.

Amongst those studies that look at removing scale using cavitation erosion specifically, most are involved with the precipitation of scale in the bulk fluid rather than how cavitation actually affects surface deposits. In addition, the effects of ultrasound on preventing mineral scale from forming are not fully understood, despite several studies that have looked directly at preventing scale deposits [79] to [82]. In most cases they do not directly discuss the effects of cavitation erosion.

Dalas [79] was one of the first to apply ultrasound to preventing scale and reports that ultrasound retards CaCO3 precipitation (reduction in scale by 62 to 76%) but has no effect on morphology. According to Li et al. [81], during cavitation, scale is “shattered and abscised”. The authors find that temperature is an important factor when removing scale with ultrasound, e.g. 40 °C is preferable for preventing scale, whereas 50 °C is better for removing scale.

They also conclude that a stronger acoustic intensity is better for removing scale deposits and vice versa a lower acoustic intensity is better for mitigating scale

precipitation. Similarly, Feng et al. [83] also find that distance to the transducer is a factor.

Nishida [84], performing a sonochemical study, describes how ultrasonic irradiation and not cavitation increases the precipitation rate of CaCO3 by enhancing nucleation. They also find that physical mixing by macro-streaming, rather than either a chemical effect or micro-streaming, is responsible for the observed increase in the rate of precipitation.

Mihai et al. [85] also observed that ultrasound increases the precipitation rate (nucleation) by 40%, but reduces “association” (sic), i.e. crystal growth and agglomeration of the precipitated particles. He et al.

[86] make similar observations but suggest that the reduction in average measured particle size is due to rapid nucleation resulting from the supersaturation of Ca2+ around the cavitating bubbles. Santos et al. [87] have investigated ultrasound as a means to break up the particle size of a CaCO3 powder as a method for increasing mineral carbonation. The authors found a significant reduction in particle size. More recently, Price et al. [88], investigating the composition of CaCO3 polymorphs precipitated using ultrasound, did not observe the retardation effect reported by Dalas [79] and state that this contradiction is a result of different reaction conditions. Price et al. [88] also find that the yield of CaCO3 increases in the presence of an ultrasonic field until a maximum is reached, which is consistent with Nishida [84].

There is also the potential for cavitation to break up agglomerates. Marković et al. [89] investigated acoustic cavitation and fracture de-agglomeration of barium titanate powder and report that cavitation pressure must be larger than the cohesive strength by which primary particles are bound together. The authors find little evidence of changes in crystal structure, but were able to break up agglomerates and found that damage is maximised at crystal boundaries. Price et al. [88], however, unlike Dalas [79], did find that under specific reaction conditions different polymorphs of calcite are formed during irradiation with ultrasound, but that ultrasound has only a small effect on the polymorphs produced.

The most important factor is initial concentration (supersaturation). This was not the case for Kojima et al. [90] who, by applying larger amplitudes, could promote the formation of the less stable vaterite phase and who suggest that morphological control of CaCO3 is possible. Mateescu et al. [91] find that an ultrasonic field accelerates the formation of stable phases (calcite) from metastable ones (aragonite). Santos et al. [92] successfully use ultrasound combined with the addition of Mg2+ to promote aragonite synthesis

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and suggest it as a potential method for industrial production of pure aragonite powder.

Since, in many cases, scale forms a coherent layer, information about damage from repeated shock impact can be gained from studies investigating surface treatments and coatings as strategies for improving cavitation erosion resistance. Such studies, besides investigating polymers, composites, and fillers, also investigate hard brittle coatings made of ceramics [93]. On their own, these coatings are resilient to cavitation but, despite some initial success, fail over time [94]. Buravova and Gordopolov [95] observe that pressure loading on the outer surface, albeit of insufficient strength to fracture the surface, i.e. below the yield stress, can still cause damage.

Failure mechanisms include debonding, delamination, and spallation [96] to [99]. Damage occurs when a reflected shock wave is generated when the incident shock hits the material boundary. The interaction and focusing of the incident and reflected shock waves cause subsurface cracks to form parallel to the interface. Delamination then occurs as these cracks extend along a plane parallel to the film/substrate interface, while spalling occurs when the subsequent propagation travels up through the film to the surface.

Spalling is a common damage mechanism seen in hard coatings [100]. Ledrappier et al. [101] observe that such damage mechanisms depend on impact parameters and coating characteristics. Additional factors include residual stresses, substrate properties and substrate coating adherence issues, porosity and interfacial defects, and other imperfections [102].

Such sites (imperfections) can also act as nucleation sites for cavitation, thus further accelerating material removal. Loh et al. [103] find that spalling only occurs at defective regions of a coating, while nondefective regions are apparently immune to spalling.

Acoustic pressure also differs according to material properties that affect the speed of sound waves as they travel through different materials. This in itself can lead to a damaging shearing force [104].

Such a force acting on a scale deposit is therefore another damage mechanism causing material (scale) to become loose and break free.

Cavitation erosion creates free particles in solution, making particulate abrasion a possibility since cavitation is accompanied by acoustic streaming [105]. Krella et al. [97] provide evidence of possible abrasion/erosion that results from collisions of the hard coating micro-particles torn off during cavitation, while Chen et al. [106] report enhanced cavitation erosion in the presence of particles. Zeiger et al. [107], investigating the molecular crystals (aspirin), rule 210

out particle-particle interaction as a viable damage mechanism.

Removal of material could result from cavitation affecting changes in the chemical equilibrium that exist close to a surface deposit. Stephanis et al. [108] find that for gypsum, another common scale forming mineral, dissolution from the solid occurs since solvent is being continuously renewed at the point of jet impact due to asymmetrical bubble collapse. Vinten’ko and Gumnitskii [109], investigating the dissolution of the mineral Langbeinite (K2Mg2(SO4)3), draw a similar conclusion and also suggest that cavitation enhances the dissolution of solids. In the case of CaCO3 this would benefit from further investigation.

Another minor removal mechanism of interest is possibility the direct volatilisation of CaCO3 to CaO and CO2. Impact simulations of carbonaceous material suggest that approximately 10 GPa is the minimum shock pressure required for partial release of CO2 from CaCO3, while 60 to 70 GPa is necessary for complete conversion. Recent studies continue to show lower shock pressure required for incipient devolatilization of calcite, e.g. <24.9 ± 2.6 GPa [110] and 18 GPa [111], which is significantly lower than reported in previous experimental studies. Grady [112] referenced in [113] states that at a “Static high pressure, high temperature experiments up to 3.5 GPa on calcite show that below 4 MPa it decomposes into solid CaO and gaseous CO2 without melting”.

Pressure pulses from cavitation, especially when bubbles collapse close to a surface, may be sufficient to cause partial devolatilization since pressures of 10 GPa are potentially feasible at liquid-solid interfaces [114]. The presence of a back reaction resulting in the re-precipitation of secondary carbonates may also greatly diminish this effect. Overall, too little is known about the extent of devoltaliszation of CaCO3 under cavitating conditions.

4 CONCLUSIONS Our objective in writing this paper was to review the major aspects of the possible cavitation erosion mechanisms affecting tenacious CaCO3 mineral scale as part of a larger study examining the influence of cavitation on different mineral scale coated metal surfaces. Our eventual aim is to formulate a phenomenological model of cavitation erosion, which will link parameters of cavitation intensiveness with damage of the mineral scale surface. This literature study shows how scale formation and cavitation erosion are complex phenomena dependent on many factors. The density, hardness, and tenacity of the

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scale will depend on its depositional environment.

In the case of CaCO3, this will affect the material and surface properties of the resultant scale layer.

Of the different mechanism of cavitation, bubble cloud collapse and the splashing effect following jet formation are considered to be the most important.

Cavitation erosion suggests that brittle material fracture occurs with only a limited amount of plastic deformation and that the formation of micro-cracks is the most important erosion mechanism, enhanced by the presence of structural homogeneities. A better understanding of the damage mechanism of cavitation will be useful not only for those looking at improving the efficiency of ultrasonic removal of tenacious mineral scale deposits, but will also prove useful in the field of sonochemistry, medicine (shock wave lithotripsy), and those studying the natural cavitation processes of carbonate materials.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 216-222 © 2013 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2012.482 (1.01) Original Scientific Paper

Received for review: 2012-03-36 Received revised form: 2013-01-28 Accepted for publication: 2013-02-18

Study of a Flowerlike Deployable Structure Luo, A. – Liu, H. – Li, C. – Wang, Y. Ani Luo – Heping Liu – Cheng Li – Yongfan Wang

College of Mechanical and Electrical Engineering, Harbin Engineering University, China A deployable structure is a kind of mechanism that can be folded and deployed automatically. It is able to form required shape or curved surface after deployment. In this paper, a flowerlike deployable structure, which forms a circle plane after deployment, was studied. First, the required circle plane was decomposed to determine the shapes of the members. Then the relation expressions were set up, which include the structural dimensions of the members and how to calculate the volume of the mechanism in the folded state. The number of the petals was optimized with the aim of minimizing the volume of the structure in its folded state. In order to determine the connecting forms among the petals and in one petal, the degrees of freedom of the mechanism were studied. On the basis of these analysis results, the flowerlike deployable structure was designed in detail. Using mechanical topology analysis, the mechanism was simplified. Based on the screw theory, the vector form output matrix of the mechanism was built. After analysing its mobility, the driving inputs positions were obtained. Finally, a virtual prototype technology was used to verify the feasibility of the design.

Keywords: deployable structure, degrees of freedom, topology, screw theory

0 INTRODUCTIONS Umbrellas, fans and tents, which have similar features in common allowing them to be folded and deployed, are widely used in daily life [1] and [2]. When they are folded, their volumes are small and they can be stored and carried easily. When they are deployed, large surface areas or volumes are obtained. These kinds of structures are called deployable structures [3]. These characteristics of deployable structures have been used to solve difficult problems in space exploration equipment, such as satellite antennas and solar arrays [4] and [5]. Under working conditions, space exploration equipment is required to have large surface areas. However when they are put into launchers, their volumes have to be small. Therefore, deployable structures have been researched and applied in the area of spaceflight, and due to this important application, deployable structures have undergone rapid development [6].

In this paper, one kind of flowerlike deployable structure was studied. The deployable structure is a mechanism that is deployed to form a circle plane.

Its shape is similar to a kind of flower whose petals surround the flower core. The movement of the mechanism is such that its petals rotate away from the fixed flower core to a state of complete deployment.

Moreover, movements of the petals all are same and synchronous. According to the above requirements, we designed a flowerlike deployable structure as described below.

1 DETERMINING THE NUMBER OF PETALS Because the flowerlike deployable structure is required to form a circle plane after its complete 216

deployment, the shapes of its petals and rigid planes (in Fig. 1, surfaces 1, 2 and 3 refer to rigid planes, and the rigid planes compose a petal) on every petal can be obtained by separating the circle plane. The shapes of the petals and rigid planes refer to the reference [7]. When the deployable structure is folded, its petals are needed to surround the central member. So the its central member is a regular polygon and the shapes of all petals of the mechanism should be the same. The circle plane is separated as shown in Fig. 1.

Fig. 1. Separating the circle plane

When the deployable structure is folded, its section shape is shown as in Fig. 2a. The thickness of the rigid planes can be neglected, because the length of the diameter of the circle plane formed after deploying the mechanism completely is three magnitudes larger than the thickness of the rigid planes. L1 and L2 (shown in Fig. 1) must be equal to the side length of the central regular polygon b, which simplifies the folded deployable structure as shown in Fig. 2b. The

*Corr. Author’s Address: College of Mechanical and Electrical Engineering, Harbin Engineering University, 145 Nantong Street, Harbin, China, luoani@hrbeu.edu.cn

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 216-222

relation of the structural dimensions of the mechanism is analysed in order to obtain the folded shape of the deployable structure shown in Fig. 2. The relationship is shown by the following

( p - 1) b

(

cos 180

s

)

( ( ( s )))

r − b / 2 sin 180 2

2

≤ 1, (1)

The length AF shown in Fig. 1 is the maximum axial height of the mechanism in the folded state. With an analysis of the geometrical dimensions, AF can be solved as: 180° 2

AF = 4 sin 2 × r − b 2 . (3) s Eq. (1) is transformed to get:

where p is number of rigid planes on a petal of the deployable structure, b side length of the regular polygon, r radius of the circle plane formed after deployment of the mechanism and s number of its petals.

r

b≤

( p − 1)

2

1 + 180° 180 ° 4 sin 2 cos 2 s s

. (4)

So, the maximum of b can be calculated from Eq. (4). The folded deployable structure can be put into a cylindrical case. The inner radius of the case is equal to the length of IO, and its height is equal to the length of AF. We let the volume of the deployable structure in the folded state equal that of the case. Therefore, the volume V of the deployable structure in the folded state is approximately solved as:

(

V = π × r 2 − 2 × ( p − 1) × r × b + +

a)

( 2 p − 1) ( 2 p − 3)

× 4 sin 2

4

  b2 b + × 180°  4 sin 2  s  2

180° 2 × r − b2 . s

(5)

b) Fig. 2. Simplified section of the deployable structure in the folded state

The section shape of the deployable structure in folded state is simplified as shown in Fig. 2b. In the Fig 2b, OJ is the distance from a tip of the central polygon to its center and OI is the distance from the point remotest to the center to the central point. The angle of OI and OJ is (180°/s), and the structural dimensions of the deployable structure are: IO = r − 2 ( p − 1) × r × b + 2

( 2 p − 1) ( 2 p − 3) 4

b2 b + . (2) 180° 4 sin 2 s

Fig. 3. V vs. b

2

As shown in Eq. (5), b and s influence V. If we let p equal 3 then s respectively equals to 3~8. Fig.

Study of a Flowerlike Deployable Structure

217

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 216-222

3 shows the resulting V vs. b for the simulation. As shown in Fig. 3, V is inversely proportional to b. After further analysis, when p is equal to other constant, V is still inversely proportional to b..

V is the main characteristic of the deployable structure and is required to be minimized. With the aim of minimizing V, using the above result, s can be determined as shown below. Let r equal 1 m. When p equals 3, after calculating with Eq. (4), s, b and V are summarized in Table 1. When p is equal to other values, values of s, b and V are similar to those in Tab.1. Therefore, the conditions when p is equal to another value are omitted here.

From Table 1, it is known that when s is equal to 6, b is at its maximum value, and V is at its least. So s is determined finally to equal 6..

Table 1. s, maximum b and V s

Maximum of b [m]

V [m3]

3 4 5 6 7 8

0.25 0.34 0.395 0.397 0.391 0.385

0.581 0.552 0.495 0.463 0.514 0.56

2 CALCULATING DEGREES OF FREEDOM The flowerlike deployable structure is a multibody system and has complicated movement. In order to reduce the output energy and mass of the driving members and control the mechanism more easily, it is necessary to minimize the degrees of freedom (DOF) of the mechanism. Because DOF of a closed mechanism chain is less than an open one, links are needed to connect neighbour petals to make the mechanism into several close mechanism chains.

To make movement of the deployable structure less complicated, let number of the links only equal 1 or 2,

and the kinematic pairs connecting the links and petals only be a revolute pair or spherical pair.

DOF of the space mechanism can be solved with the equation: j

F = 6 ( n − 1) − ∑ ui , (6)

i =1

where n is a number of members of the mechanism, j a number of its kinematic pairs, ui a number of constraints of the i kinematic pair [8] to [10].

Let F be equal to s, and Eq. 6 is used to calculate the DOF of the flowerlike deployable structure with different connecting forms between the petals. The number of members and kinematic pairs and resulting DOF are summarized in Table 2.

Based on Table 2, the numbers of connecting links, kinematic pairs and the type of the kinematic pairs have a direct influence on p. In the 2nd structural form, the number of members is too little so that ratio of volumes of the deployable structure in folding and deploying states is too large, which doesn’t fit requirement of the deployable structure. There are too many kinematic pairs in the 5th structure, and its movement is complicated, which makes it difficult to control. The 6th and 7th structure forms are obviously impossible. So only the 1st, 3rd and 4th structure are feasible. In the three feasible structures, the 1st structure has the least members, so it is adopted here.

3 TOPOLOGY ANALYSIS OF THE MECHANISM Using the results of the above analysis, we designed a flowerlike deployable structure. A virtual prototype of the mechanism is shown in Fig. 4. It is composed of a hexagonal central plane and six petals. Each petal has three rigid planes, including a root rigid plane, middle rigid plane, and top rigid plane as shown in Fig. 4. The root rigid plane is a right-angled triangle. The surface of the middle rigid plane is trapezium. The top rigid

Table 2. The number of members and DOF Serial number of structural forms 1 2 3 4 5 6 7

218

Number of links conncecting the neighbouring petals 1

2

Type and number of kinematic pairs connecting between links and petals A spherical pair and a revolute pair Two spherical pairs Two revolute pairs Three revolute pairs Two revolute pairs and a spherical pairs A revolute pair and two spherical pairs Three spherical pairs

Number of members of the mechanism sp+s sp+s sp+s sp+2s sp+2s sp+2s sp+2s

Luo, A. – Liu, H. – Li, C. – Wang, Y.

Number of revolute pairs

Number of spherical pairs

sp+s sp sp+2s sp+3s sp+2s sp+s sp

s 2s 0 0 s 2s 3s

Degrees of freedom of the whole mechanism sp–2s sp sp–4s sp–3s sp–s sp+s sp+3s

Number of rigid planes on a petal p=3 p=1 p=5 p=4 p=2 p=0 p = –2

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 216-222

plane has an amorphous formation. The rigid planes and the central hexagon plane are connected to each other via revolute pairs. A bar connects the two top rigid planes on the neighbouring petals together. There are a revolute pair on an end of the bar and a spherical pair on its other end.

Using mechanical topology analysis, we can simplify the mechanism and the result is shown in Fig. 5. The meanings of the below symbols, figures and terms can be found in reference [11].

As shown in Fig. 5, the mechanism is composed of several close mechanism chains. In order to analyse the mechanism further, one must decompose the mechanism into several minimum close mechanism chains (the minimum close mechanism chains are called simply SLC). Because the mechanism is symmetrical, it can be decomposed into six identical SLC as shown in Fig. 6. The composition of the SLC can be expressed as:

Fig. 6. A close mechanism chain (SLC)

The SLC is changed into an open mechanism chain (SOC) as shown in Fig. 7. The composition of the SOC can be expressed as: SOC {− R1 ⊥ R2 / / R3 ⊥ R4

}.

⊥ R5 − R6 − R7 / / R8 / / R9 ⊥ R10 −

SLC{–R1 ⊥ R2 // R3 ⊥ R4 – S1 – R8 // R9 ⊥ R10–}, where Ri denotes the ith revolute pair and sj denotes the jth spherical pair. In Fig. 6, the bar between R1 and R10 is cut and the spherical pair S1 can be substituted with three revolute pairs.

Fig. 7. The open mechanism chain (SOC)

Fig. 4. The flowerlike deployable mechanism

The below analysis will be carried out using Descartes coordinates. Using screw theory, we make a list of the independent outputs of the output eigenmatrix of every kinematic pair in turn, and these are then added together to get the vector form of the output matrix of the SOC as: t1( ⊥ R3 )  t0     t0 MS =  + + 1    r1(/ / R1 )1  r1(/ / R2 ) 2  r (/ / R2 / / R3 )

{

t    t0 +  + 1 1  r (/ / R4 ) 4  r (/ / R5 ) 0

{

}

 t1( ⊥ R7 ) + 1  r (/ / R7 ) 

{

Fig. 5. The flowerlike deployable structure after simplified

}

}

  t ( ⊥ R6 )  +   r1(/ / R6 ) 5  1

{

 t3   =  .

3   7  r  S

}

  +  3

  +  6

(6)

In the SOC, R1 and R2 lie on a rigid plane and their axis lines are vertical each other, so the two axis

Study of a Flowerlike Deployable Structure

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lines always intersect at one point and the outputs of the two revolute pairs are independent of each other.

The axis line of R3 is parallel to R2, so rotation of R3 is independent relatively of R1 and R2. Therefore the revolute pair R3 produces a dependent translation.

Rotation of R4 is not relative to R1 ~ R3, so its rotating output is independent. Rotating outputs of R5 and R6 have a relationship with the above revolute pairs and they will produce two independent translation outputs.

Because there are six independent outputs, all outputs of R7 ~ R10 are dependent.

Independent rotations of the output matrix MS of the SOC are r1 (// R1), r1 (// R2) and r1 (// R4), so the number ξSR of its independent rotation outputs is 3. Independent translation outputs of the matrix are t1( ⊥ R3), t1( ⊥ R5) and t1( ⊥ R6), and the number ξSR of its independent translation outputs is 3 too. Therefore order ξ of the output matrix Ms can be calculated as:

ξ = ξS = ξSR + ξSP = 3 + 3 = 6.

m

F = ∑ fi − ξ , (8) i =1

where F = DOF, fi = DOF of the ith kinematic pair and m is a number of kinematic pairs in the mechanism.

We use Eq. (8) to solve for DOF of the SLC as shown in Fig. 6. Because F > 0, namely ∑ fi ＞ ξ，the SOC (shown in Fig. 7) that the SLC is transformed to form is a redundance mechanism and its independent outputs can be completed only with R1 ~ R7. The SOC is simplified to get another SOC only composed of R1 ~ R7. The DOF of the SOC is solved as:

}

Then R1 is rigidized and the SOC is changed as shown in Fig. 8. We then list the independent outputs of output eigenmatrix of every kinematic pair in turn, and further obtain the vector form of the output matrix of the SOC shown in Fig. 8 as:  t1( ⊥ R3 )    t0   + MS =  + 1   r1(/ / R2 ) 2  r (/ / R2 / / R3 )  3

{

{

 t1( ⊥ R7 ) + 1  r (/ / R7 ) 

{

7

}

 t3   =  .

3   7  r  S

}

   6 (10)

The order of the Ms is calculated as:

The DOF result indicates that the SLC as shown in Fig. 6 only needs an input. Because R8 ~ R10 and R1 ~ R7 lie on different petals, in order to reduce movement coupling among the petals, the input will be placed at any of R1 ~ R7. Because the flowerlike deployable structure is composed of six identical SLC, only 6 inputs are required to control the mechanism completely.

}

  t1( ⊥ R6 )    t0 t0  + 1  1  + 1  r (/ / R4 ) 4  r (/ / R5 ) 5  r (/ / R6 )

F = ∑ fi − ξ = 7 − 6 = 1. (9) i =1

220

{

SOC* − R1 ⊥ R2 / / R3 ⊥ R4 ⊥ R5 − R6 − R7 − .

Fig. 8. The open mechanism chain after rigidizing R1

DOF of the mechanism can be calculated using the equation:

(7)

4 DETERMINING THE POSITION WHERE INPUTS ARE LOCATED

The position of the input will be determined below. Before the analysis is carried out, it is estimated whether there are passive kinematic pairs in the mechanism shown in Fig. 7. After rigidizing R8 ~ R10, the composition of the SOC can be expressed as:

ξ1* = ξS = ξSR + ξSP = 3 + 3 = 6.

The order of the SOC is same as in Fig. 7. The analysis method in reference [2] confirms that R1 is not a passive kinematic pair and that input can be located there.

The above analysis method is used further to determine whether R2 ~ R7 are passive kinematic pairs, and the result is summarized in Table 3.

As shown in Table 3, there are no passive kinematic pairs in the SOC composed of R1 ~ R7.

Luo, A. – Liu, H. – Li, C. – Wang, Y.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 216-222

Because R5, R6 and R7 are used to substitute for the spherical pair, inputs are not able to be located there.

So inputs only are at R1 ~ R4. In the whole structure shown in Fig. 5, R1 ~ R3 in a SLC are expressed as R8 ~ R10 in the one next to it. If an input is placed at either R1, R2 or R3, it will influence movements of the two SLC, which makes movement analysis and control of the mechanism complicated. Because R4 only belongs to the SLC of the whole mechanism, and can not influence movements of other SLC, the input

beginning and end points of the curves are all 0, and there are no sudden changes in acceleration. So, there are no impulsions and vibrations in the movement of the mechanism.

should be located at R4.

a)

5 SIMULATION CHECK By analysing the results of the input position, we can set the virtual prototype of the flowerlike deployable structure and further simulate it. The movement of the inputs is designed to coincide with the sine law.

From the boundary conditions of the movement of the mechanism, the movement equation of the inputs can be solved as:

c) Fig. 9. Movement of the flowerlike deployable structure

θ = 18.5 × sin (π × t / 8 + 0.23) .

b)

The virtual prototype of the flowerlike deployable structure moves from the folding state to deploying state as shown in Fig. 9, which proves that the flowerlike deployable structure can be folded and deployed automatically and that the design scheme is reasonable. Let θ1 equal the angle of rigid plane 3 and the central polygon in Fig. 1, θ2 be equal to the angle of rigid plane 2 and rigid plane 3, and θ3 equal the angle of rigid plane 1 and rigid plane 2. Figs. 10 and 11 show the resulting θi and ωi (i = 1, 2, 3) vs. time for the simulation. As shown in Fig. 10, the rigid planes can rotate continuously to the desired positions. In Fig. 11, the angular velocities change continuously. The

6 CONCLUSION A deployable structure is a kind of mechanism that is deployed completely to form the desired surface or spatial structure. In this paper, a design method of for such a deployable structure is depicted. After analysis of the shapes of the mechanism in folding and deploying states, its volume in the folded state is solved and its number of petals is further determined.

In calculating DOF, we select the structure connecting the petals and the number of the rigid planes. Using mechanical topology analysis and the screw theory,

Table 3. Determination of passive kinematic pairs Rigidized kinematic pair R1 R2 R3 R4 R5 R6 R7

Open mechanism chain gotten after rigidization

{ } SOC {- R ⊥ R ⊥ R ⊥ R - R - R -} SOC {- R ⊥ R ⊥ R ⊥ R - R - R -} SOC {- R ⊥ R \\ R ⊥ R - R - R -} SOC {- R ⊥ R \\ R ⊥ R ⊥ R - R -} SOC {- R ⊥ R \\ R ⊥ R ⊥ R - R -} SOC {- R ⊥ R \\ R ⊥ R ⊥ R - R -} SOC* - R2 \ \ R3 ⊥ R4 ⊥ R5 - R6 - R7 *

*

*

* * *

Order(ξi*)

Whether the rigidized kinematic pair is a passive kinematic pair?

Whether an input can be positioned on it?

6

No

Yes

1

3

4

5

6

7

6

No

Yes

1

2

4

5

6

7

6

No

Yes

1

2

3

5

6

7

6

No

Yes

1

2

3

4

6

7

6

No

Yes

1

2

3

4

5

7

6

No

Yes

1

2

3

4

5

6

6

No

Yes

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the position of the inputs is determined. From the analysis in this paper, the below conclusions are obtained: (1) Because the volume of the deployable space structure in the folded state must be reduced, requiring that the structural dimensions of the mechanism be optimized with the aim of obtaining the minimum volume is reasonable.

This has been verified in the paper; (2) In order to make control of the mechanism easier, it is necessary to reduce the number of inputs into the mechanism, namely DOF. Therefore, the number of elements and type of kinematic pair in the mechanism can be determined via analysis of its DOF; (3) Screw theory and mechanical topology analysis are combined, and thus the positions of the inputs can be determined correctly. This was also verified.

These methods have been used in this paper, and a feasible deployable structure has been obtained.

Therefore, the methods can be applied in the design of this kind of deployable structure.

Fig. 10. Angle curves

Fig. 11. Angular velocity curves

222

7 ACKNOWLEDGMENT This work was supported in member by the Special Fund from the Central Collegiate Basic Scientific Research Bursary (HEUCF110703) and the Heilongjiang Postdoctoral Science Foundation (323630297).

8 REFERENCES [1] Puig, L., Barton, A., Rando, N. (2010). A review on large deployable structures for astrophysics missions. Acta Astronautica, vol. 67, no. 1-2, p. 12-26, DOI:10.1016/j.actaastro.2010.02.021.

[2] Brown, M.A. (2011). A deployable mast for solar sails in the range of 100-1000 m. Advances in Space Research, vol. 48, no. 11, p. 1747-1753, DOI:10.1016/j. asr.2011.01.014.

[3] Meguroa, A., Shintateb, K., Usuib, M., Tsujihatab, A. (2009). In-orbit deployment characteristics of large deployable antenna reflector onboard Engineering Test Satellite VIII. Acta Astronautica, vol. 65, no. 9-10, p.

1306-1316, DOI:10.1016/j.actaastro.2009.03.052.

[4] Soykasap, Ö. (2009). Deployment analysis of a self-deployable composite boom. Composite Structures, vol.89, p. 374-381, DOI:10.1016/j. compstruct.2008.08.012.

[5] Fazli, N., Abedian, A. (2011). Design of tensegrity structures for supporting deployable mesh antennas.

Scientia Iranica, vol. 18, no. 5, p. 1078-1086, DOI:10.1016/j.scient.2011.08.006.

[6] Fanchini, G., Gagliostro, D. (2011). The e-st@r CubeSat: Antennas system. Acta Astronautica, vol. 69, no.11-12, p. 1089-1095, DOI:10.1016/j. actaastro.2011.06.009.

[7] Guest, S.D. (1994). Deployable structures: concepts and analysis. PhD thesis. The University of Cambridge, Cambridge.

[8] Hopkins, J.B., Culpepper, M.L. (2011). Synthesis of precision serial flexure systems using freedom and constraint topologies. Precision Engineering, vol. 35, no. 4, p. 638-649, DOI:10.1016/j. precisioneng.2011.04.006.

[9] Gallardo, J., Lesso, R., Rico, J.M., Alici, G. (2011).

The kinematics of modular spatial hyper-redundant manipulators formed from RPS-type limbs. Robotics and Autonomous Systems, vol. 59, no. 1, p.12-21, DOI:10.1016/j.robot.2010.09.005.

[10] Yu, J., Li, S., Su, H.-J., Culpepper, M.L. (2011). Screw Theory Based Methodology for the Deterministic Type Synthesis of Flexure Mechanisms. Journal of Mechanisms and Robotics, vol. 3, no. 3, p. 1-14, DOI:10.1115/1.4004123.

[11] Tingli, Y. (2004). Structural Topology of Robot.

Machine press, Beijing. (in Chinese)

Luo, A. – Liu, H. – Li, C. – Wang, Y.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 223-236 © 2013 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2012.668 (1.01) Original Scientific Paper

Received for review: 2012-06-19 Received revised form: 2012-12-28 Accepted for publication: 2013-01-18

On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling Katrašnik, T. – Schuemie, H.A. – Wurzenberger, J.C. Tomaž Katrašnik1,* – Henrik Aleš Schuemie2 – Johann C. Wurzenberger2 1 University of Ljubljana, Faculty of Mechanical Engineering, Slovenia 2 AVL List GmbH, Advanced Simulation Technologies, Austria

The development of real time capable 0-dimensional internal combustion engine models places high demands on convergence, stability, and computational speed of the applied numerical methods. The cylinder model represents the crucial element in attaining high computational speed and accuracy of results. A basic example comprising a single cylinder connected to two plenums is analysed with different numerical schemes in order to reveal methods effectively associating accuracy requirements with computational time constraints. The integration performance to solve a system of coupled ODEs was compared for explicit Euler and explicit fourth order Runge-Kutta schemes, as well as for multi-step methods including backward differentiation formulas and Adams-Moulton formulas. The performed analysis emphasizes two major points. First, the numerical accuracy of integration schemes differs significantly at equal computational effort revealing the necessity of selecting an adequate scheme for a specific task. Second, the comparison of integral engine parameters (e.g. indicated mean effective pressure, mean engine torque), calculated by different methods, with a numerically assumed exact solution should not be used as an estimate for the convergence and stability of the applied numerical approach, since good agreement in integral parameters does not imply good agreement in cycle resolved traces of thermodynamic variables. This paper provides clear guidelines for selecting the appropriate numerical integration methods with respect to the intended application. Analyses are also based on innovative test examples. Finally, a comparison of numerical and experimental in-cylinder pressure traces is shown for a series production engine confirming the applicability and accuracy of the cylinder model.

Keywords: internal combustion engine modeling, integration schemes, convergence, stability, computational speed

0 INTRODUCTION The steadily increasing demands on vehicle fuel economy, emissions, and performance are a major driving force in developing and optimising future powertrains. In order to meet customers’ and legislative requirements it is of particular importance not only to simulate and optimise different powertrain elements, but also to develop simulation models capable of modeling the performance of the powertrain system as a whole. The latter is of particular importance for developing and testing components in a virtual environment, as well as for accompanying the conventional development steps from design phases in the office to the application and validation phases on the test bed. Test bed validation, development and calibration of components in the hardware-in-the-loop (HiL) environment and modelbased engine controlling require real-time capable simulation models. Moreover, computational speed is also of considerable importance when simulating complete emission test cycles where engine and vehicle dynamics are simulated on time scales of 103 s.

Recently, 0-dimensional (0-D) crank-angle (CA) resolved engine simulation models, e.g. [1] to [4], have emerged as a promising approach for real-time engine simulations of internal combustion engines (ICE) and hybrid powertrain of heavy-duty engines [5]. Unlike

higher resolution 1-D, e.g. [6] to [9], and 3-D models, e.g. [10] and [11], 0-D models can comply with realtime constraints and provide a higher level of detail compared to mean-value engine models, e.g. [12] and [13]. The latter models reach their limits when future control functions need to be taken into account, which incorporate CA resolved events such as cylinder pressure based control functions, electronic control unit controlled fuel injections, and variable valve timing. Mean-value engine models are not able to model these phenomena predictably since they rely on experimental data that are used to tune or train the model as presented in [14] for camshaft variability.

Besides computational speed, the convergence and stability of numerical solution algorithms are the prerequisites for the successful application of 0-D CA resolved engine simulation models. This topic has not yet been systematically analysed in the literature and therefore the presented paper aims to provide a study of the convergence, stability, and computational speed of different explicit and implicit integration schemes.

The cylinder model represents the crucial element to attaining high computational speed and accuracy of results, since the time variation of the thermodynamic variables is most pronounced in the cylinder due to piston kinematics and combustion. It is common practice to validate the accuracy of simulation models by comparing numerical and experimental results.

This indeed gives an indication of the consistency

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, SI-1000 Ljubljana, Slovenia, tomaz.katrasnik@fs.uni-lj.si

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of the virtual model and the hardware being simulated. However, it is well known that 0-D models incorporate many parameters included in combustion and heat transfer models that can be tuned with the aim of achieving agreement between numerical and experimental results. If numerical stability and convergence of the model are not validated separately, the agreement between numerical and experimental results alone cannot be understood as sufficient validation. This is particularly true as long as model parameters are tuned with the objective of achieving this agreement, since numerical errors might thereby be tuned out of the model. Agreement between numerical and experimental results does therefore not necessarily imply convergence and stability of the solution, since the results may be non-converged and even locally non-stable. The drawbacks of such models are often revealed during transient engine simulation where significant discrepancies may also arise between the numerical and experimental results, since it is generally not possible to tune the model for the complete transient operating region.

These phenomena are particularly discernible with turbocharged engines exhibiting turbocharger lag, as evident in [2] where very large integration increments are applied.

parameters (results for IMEP are shown, since other integral engine parameters are directly dependent on IMEP) calculated by the different methods and the numerically assumed exact solution might often easily comply with these accuracy requirements. However, the traces of thermodynamic parameters may be nonconverged and even locally non-stable. The same consideration can also be applied to the comparison of numerical and experimental data. It is thus shown that the agreement of integral engine parameters should not be the only measure for assessing the accuracy of the model.

This paper presents an analysis of convergence, stability, and computational speed of integration schemes for 0-D modelling of ICE cylinders in a simple model. A single four-stroke cylinder (Fig. 1) connected via two valves to two plenums (intake and exhaust) of infinite inertia (all thermodynamic variables are constant) is used. The model considers fluxes through the attached valves, the flux of the injected fuel, combustion of fuel, heat transfers to the combustion chamber walls and the walls of the ports, and the power given to the piston. This simplified test case is selected since it excludes other engine components, thus enabling the investigation of numerical phenomena relevant only to the low and high pressure phase of the in-cylinder process.

2 GOVERNING EQUATIONS The framework of equations was laid out in a general way enabling consideration of an arbitrary number of species:

W = [ w1  wN ] , (1) T

where wi represents the mass fraction of species i. In the present analysis, the species vector, W, represents the burned fuel (FB), combustion products (CP), and fuel vapor (FV). The species concentration of air is derived by:

wair = 1 − wCP − wFV . (2) Fig. 1. Schematic of the considered four-stroke cylinder connected to the intake and exhaust plenum including state variables and fluxes

Moreover, integral engine parameters (e.g. indicated mean effective pressure (IMEP), brake mean effective pressure, mean engine torque, effective fuel consumption) are often used to validate numerical models. The accuracy threshold for validating high fidelity ICE models is typically 1%. The presented paper shows that the agreement of integral engine 224

wFB gives the concentration of the fuel that was burned, i.e. converted from FV to FB, whereas wCP gives the concentration of the corresponding combustion products, i.e. burned fuel plus air that was used in combustion. Therefore, it is possible to evaluate the air-fuel ratio of CP for the particular fuel, which is mandatory for evaluating adequate gas properties.

The governing equations were derived considering the dependencies of gas properties,

Katrašnik, T. – Schuemie, H.A. – Wurzenberger, J.C.

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i.e. internal energy (u), the specific gas constant (R), and all partial derivatives of both properties, on temperature (T), pressure (p), and species concentrations (wFB , wCP , wFV . wair). Gas properties are given by input maps taking into account chemical equilibrium considerations. A database was prepared for different fuels and fuel blends. In the analysed case, the properties of Diesel fuel are used.

A revised ideal gas equation [15]: pV = m

Zℜ T = mRT , (3) M

adequately captures the deviations in the real gas from the ideal one in the range of temperatures and pressures characteristic for the in-cylinder processes of ICEs; V is the volume, m is the mass, Z is the compressibility factor, ℜ is the universal gas constant, and M the molecular weight.

The mass balance equation is: dm = dξ

nvalves

∑ j

dm j dξ

+

dminj dξ

dQ dQC dQHT = + , (8) dξ dξ dξ

dm j dminj dH nvalves = ∑ hGE + hinj , (9) dξ dξ dξ j

and the index GE is defined by Eq. (6). After algebraic manipulation of Eq. (3) in differential form, and insertion together with Eq. (4) into Eq. (7), Eq. (7) was algebraically reformulated to be explicit in temperature. This derivation was done in analogy to [16] by considering a more general dependency of the internal energy and the gas constant on species concentration is taken into account leading to: dT B  dQ dH dV dφ =  + + ( Km − 1) p − dξ m  d ξ d ξ d φ dξ − ( u + KTRm )

, (4)

 dm ∂R ∂u  dwk  − m  KmT +  , (10)  dξ ∂wk ∂wk  dξ  

where: where ξ is the independent variable representing either time (t) or CA rotation (φ). The change in mass depends on the fluxes through the attached valves and the amount of injected fuel.

The rate of change of species concentration comprises the species concentration variation due to mass transfer and species concentration variation due to chemical kinetics, i.e. combustion. It follows that:

nvalves

+

∑ (w j

k , GE

dwk dwk ,C = + dξ dξ dm j dminj − wk ,cyl + wk ,inj − wk , cyl dξ dξ , (5) m

)

(

)

where index GE denotes:

cylinder for dm ≤ 0 , (6) GE =   plenum for dm > 0

K=

∂u ∂p

1  p ∂R  V 1 −   R ∂p 

, (11)

and

B=

1 . (12)  T ∂R  ∂u KmR 1 + +  R ∂T  ∂T

The combustion is modeled according to the single-zone Vibe model [17]. The heat transfer coefficient to the walls of the combustion chamber is modeled according to the model proposed by Hohenberg [18]. The mass flow rates through the intake and exhaust valves are calculated by the orifice equation (see e.g. Ref. [19]).

3 SOLVING THE SYSTEM OF COUPLED ODEs

where the positive value of dm indicates that mass flows into the cylinder as given in Eq. (4).

The first law of thermodynamics for an open system is:

where

dU dQ dV dH = −p + , (7) dξ dξ dξ dξ

3.1 Solution Vector The 0-D ICE cylinder model is described by the system of coupled ODEs described above. An overall solution vector can be written in the following form:

y = [ m, T , w1  wN ] . (13) T

It is common practice to solve the system of coupled equations (i.e. Eqs. (4), (10) and N Eqs. (5)) On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling

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as an initial value problem approaching a convergent solution by advancing an independent variable. This is given by: dy = f (ξ , y ) , dξ

y (ξ 0 ) = y0 ,

y ∈ R N + 2 , (14)

where ξ represents either time (t) or CA rotation (φ).

3.2 Integrations Schemes The convergence, stability, and computational speed of integration schemes for 0-D modeling of the ICE cylinder (Eq. (14)) are analysed for the following integration schemes: Explicit Euler scheme [20], Explicit fourth order Runge-Kutta scheme [20], Backward Differentiation scheme [21], and AdamsMoulton scheme [21], where the first two schemes are constant step schemes and the last two are variableorder, variable-step schemes.

The Euler scheme features a very high computational speed for a selected integration increment and it is therefore already applied in realtime 0-D ICE models presented in literature, e.g. [2].

However, according to [20], it is not recommended for any practical use. The Runge-Kutta fourth order scheme is known as a robust, widely applied scientific and engineering tool. The Backward Differentiation scheme and Adams-Moulton scheme are included in the CVODE solver package [21]. Variable-order, variable-step multistep CVODE methods are organized to monitor internal consistency. Numerical errors are controlled by adaptive step sizes within a global integration increment. The CVODE package offers the Adams-Moulton formulas for non-stiff problems.

For stiff problems, Backward Differentiation schemes are available in the CVODE package. The governing equations for solving thermodynamic variables in the cylinder are nonstiff if physically meaningful input data are used. Thus, all four schemes could be applied in the analysis and compared for performance and robustness. The CVODE solver offers the choice of either functional iteration, suitable only for non-stiff systems, and various versions of Newton iteration applicable to both stiff and non-stiff systems. Both iteration methods were therefore applied in the present analysis and thus the performance of all combinations of non-stiff and stiff numerical schemes and iteration methods were examined. Six solvers are applied to solve cylinder balance equations. The following abbreviations are used in the text below and in the Figures: • Euler, • RK4 - Runge-Kutta fourth order, 226

•

BDF, NWT - Backward Differentiation scheme, Newton iteration, • BDF, FCN - Backward Differentiation scheme, functional iteration, • AM, NWT - Adams-Moulton scheme, Newton iteration, • AM, FCN - Adams-Moulton scheme, functional iteration.

The Newton iteration requires the solution of linear systems. The selection of the method used to solve a linear system does not noticeably influence the results, since a system of only 5 equations describing a single cylinder engine is considered. In all CVODE schemes, equal methods to solve the linear system and equal iteration parameters were applied.

The Euler and RK4 schemes are integrated on a CA rotation basis. With this approach, similar numerical accuracy can be achieved for different engine operating conditions. However, computational time increases with increasing engine speed. CVODE schemes were integrated on a time basis. The tradeoff between numerical accuracy and computational speed was analysed by varying integration increments of Euler and RK4 schemes, and by varying tolerances and global integration increments of CVODE solvers.

The main mechanism influencing numerical accuracy and computational speed of CVODE solvers is the definition of tolerances, whereas the global integration increment does not significantly influence the results.

This is because of the fact that the solver internally reduces its integration increments until local error tests are satisfied. The tolerances of the CVODE solver were given by relative and absolute tolerance values. The first were given as scalar values and the latter were provided in vector form. Components of the solution vector, Eq. (13), evaluated for cylinders of automotive engines, feature very distinct orders of magnitude when evaluated in SI units, i.e.

O(m) = 10-3, O(T) = 103 and O(wi) = 1. The absolute tolerance vector was therefore defined by:

TOLabs

O ( m )    = O (T )  × TOLrel , (15)   O ( wi ) 

where Eq. (15) represents an efficient handling of tolerances, since only TOLrel is varied to analyse the trade-off between numerical accuracy and computational speed. Computational speed is inversely proportional to TOLrel.

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Table 1. Basic cylinder data and parameters Bore [m]

0.085

Stroke [m] Connection rod length [m] Compression ratio [-] Piston pin offset [m] IVO [deg. CA] IVC [deg. CA] EVO [deg. CA] EVC [deg. CA] Number of valves [-] Vibe parameter m [ -] Vibe parameter A [-] Combustion duration [deg. CA]

0.094 0.1505 17 0.002 17 BTDC = 703 33 ABDC = 213 57 BBDC = 483 25 ATDC = 25 2 0.7 6.901 70

4 RESULTS The assembly of the considered single cylinder engine is shown in Fig. 1. An internal mixture preparation compression ignition, i.e. Diesel, engine (see Table 1) was chosen. Besides temperature and

concentration variation, the in-cylinder mass also varies during combustion due to direct fuel injection, which therefore represents a more demanding test for the accuracy of numerical schemes. Two different engine speeds are analysed to explore the characteristics of the different integration schemes.

The selected speeds cover the lower and upper speed boundary of automotive engines. In order to ensure a reliable comparison between the different integration schemes, special emphasis is placed on a converged solution of cylinder traces. Thus, for Euler and RK4 schemes, the simulation was cyclically repeated until the differences in the last two cycles were below the round-off of machine arithmetic for all analysed parameters. Likewise, for the CVODE schemes, the simulation was cyclically repeated until the differences in the last two cycles are within the specified tolerance limits and it can therefore be assumed that a steadystate solution was reached.

Traces of selected thermodynamic variables were analysed first to expose specific characteristics of the particular schemes. Results of these schemes

Fig. 2. Pressure traces for an engine speed of 1500 rpm and an IMEP of 1.92 MPa; a) entire cylinder cycle, b) Î&#x201D;p for the entire cylinder cycle, c) and d) low pressure phase of the cylinder cycle On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling

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were compared to the numerically assumed exact solution (denoted “exact solution” in the subsequent text) calculated using the BDF, NWT method, with TOLrel = 10–12 being an unreasonably small tolerance resulting in excessive computational times. For a comparison of different numerical integration schemes applied to different engine operating points, it is necessary to condense the cycle resolved results to a clearly traceable set of lumped parameters. Therefore a non-dimensional root mean square (NDRMS) and a real time (RT) factor were introduced. NDRMS describes a relative difference in a considered variable (i.e. pressure, temperature, mass, etc.) calculated with a chosen integration scheme and the exact solution throughout the complete engine cycle. The NDRMS is defined by: 1

NDRMS ( X ) =

ndata

ndata

∑(X i =1

i

− X i , exact )

X max, exact

2

, (16)

where X denotes an arbitrary cylinder variable, ndata denotes number of the data within one engine cycle, exact is the exact solution, and Xmax,exact represents the maximum value of the considered variable. The NDRMS value is always calculated for the complete cylinder cycle comprising both the gas exchange and the high pressure phase. In order to define a measure for the computational effort of the different numerical schemes an RT factor is applied. RT is given by:

RT =

tcomputation t physical

, (17)

where tcomputation is computational time obtained on a standard linux machine with an Intel Xeon 3.00 GHz processor, and tphysical represents the considered physical time of the investigated engine cycle. It is obvious that efficient integration schemes simultaneously feature both a low NDRMS value and a low RT value.

4.1 Analysis of Cycle Resolved Results The accuracy of integration schemes by means of pressure traces is analysed first. Pressure is not directly calculated by the integration scheme (Eq. (13)) and therefore it represents a good indication of numerical accuracy, since it is influenced by all components of the state vector via Eq. (3). The pressure is also the driving force for mass, species, and enthalpy transfer during the gas exchange phase, and therefore is the 228

most significant parameter for determining numerical accuracy. Additionally, the pressure trace determines the indicated work per cycle via the indicated mean effective pressure (IMEP), which is related to the engine power (P). It is given by:

(18) is the indicated work per cycle, and

where

Vdisplacement represents the engine displacement. IMEP is numerically integrated by a trapezoidal scheme considering the pressure at the beginning and at the end of the integration step.

Table 2. Boundary conditions for Test case 1: engine speed 1500 rpm and IMEP = 1.92 MPa Plenum Intake Exhaust

T [K] 320 320

p [Pa] 220000 200000

wFB [-] 0 0

wCP [-] 0 0

wFV [-] 0 0

Fig. 2 shows pressure traces for an engine speed of 1500 rpm, IMEP = 1.92 MPa. The boundary conditions in the intake and exhaust plenum, which correspond to a turbocharged engine, are given in Table 2. In all graphs, 0 deg. CA corresponds to the top dead center (TDC) of the gas exchange phase.

The pressure traces are plotted for different numerical schemes featuring a low RT factor (e.g. RT<0.1 for engine speed 1500 rpm) and for the exact solution.

The time steps of the two explicit schemes (Euler, RK4) are chosen to keep the computational effort at a comparable level (RK4 with 1 and 4 deg. CA and Euler with 0.25 and 1 deg. CA). Additionally, the results of the Euler scheme with 2 deg. CA are shown to point out the significantly reduced accuracy. The results of the AM, FCN scheme with TOLrel =10–5 (Eq. (15)) and a global time increment corresponding to 1 deg. CA are shown, since it features both low RT and the best NDRMS vs. RT ratio out of the schemes given by the CVODE package for this example.

Fig. 2a shows that integration schemes featuring low RT values and consequently lower numerical accuracy provide a reasonable agreement for the overall in-cylinder pressure trace. A more profound analysis is therefore necessary to reveal the characteristics of the particular schemes. Figs. 2c and d show the low pressure, i.e. gas exchange, phase of the cylinder cycle. It can be observed that results of the RK4/1 deg. CA scheme and of the AM, FCN

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/ TOLrel = 10–5 scheme coincide very well with the exact solution (within the line thickness), whereas all other schemes produce significant deviations. These deviations strongly influence mass, species, and enthalpy transfer during gas exchange and therefore imply nonconverged solutions. This non-convergence additionally compromises the accuracy in the subsequent integrations steps. Stability issues are observed for all Euler schemes and for the RK4/4 deg.

CA scheme. It can also be observed that the Euler/0.25 deg. CA scheme featuring similar computational times as RK4/1 deg. The CA scheme performs worse during the gas exchange period. This is mainly attributed to the low order of the Euler scheme. The difference is even more discernible in Fig. 2b, which shows the absolute differences between the pressure traces of the applied numerical schemes and the exact solution (Δp = | pscheme – pexact |). It can be observed that Δp reaches its maximum value in the vicinity of the firing TDC, i.e. 360 deg. CA, since deviations in thermodynamic variables at intake valve closing (IVC) result in increasingly pronounced pressure deviation during the compression stroke. A comparison of the results in Figs. 2a and b leads to the conclusion that pressure traces as shown in Fig. 2a cannot be used to derive reliable measures of the quality of different numerical schemes. Out of 6 applied schemes, the RK4/1 deg. CA scheme shows the smallest deviation from the exact solution in the vicinity of the TDC, whereas the RK4/4 deg. CA and the Euler schemes with 1 and 2 deg. CA on the other end show significant and non-negligible errors.

intake valve opening (IVO) for the exact solution, since the in-cylinder pressure is larger than the pressure in the exhaust plenum (p > pE) during the exhaust valve opened period and the in-cylinder pressure is smaller than the pressure in the intake plenum (p < pI) during the initial sequence of the intake valve opened period (as shown in Figs. 2c and d) resulting in an inflow of fresh air at IVO. wCP decreases from IVO up to 180 deg. CA, when a flow reversal across the intake valve occurs, since p > pI as shown in Fig. 2c and d. On the other hand, mCP decreases until exhaust valve closing (EVC) and is constant thereafter until 180 deg. CA, since only fresh air flows into the cylinder due to the intake boundary condition specification. mCP, which is not directly calculated by the integration scheme and is equal to a product of two variables in the state vector (m and wCP), therefore represents a suitable criterion for testing the convergence of integration schemes. It can be seen from the figure that RK4/1 deg. CA and AM, FCN / TOLrel = 10–5 schemes provide convergent results, whereas both Euler schemes fail to provide convergent results.

Fig. 4. Mass flux through the exhaust valve for a motored cylinder at an engine speed of 1500 rpm

Fig. 3. Mass of combustion products (mCP = wCP m) in the cylinder for an engine speed of 1500 rpm and IMEP = 1.92 MPa

Deficiencies in convergence of particular schemes are also analysed in Fig. 3, showing the mass of combustion products (mCP = wCP m) in the cylinder for the Test case 1 (Table 2). wCP starts decreasing at

Fig. 4 shows mass flux through the exhaust valve for a motored cylinder (without combustion) at an engine speed of 1500 rpm. These operating conditions are characteristic for vehicle deceleration where the engine represents an energy sink. Modeling of such operating conditions is a demanding test for the stability of integration schemes. In order to point out the differences between the various integration schemes the mass flux through the exhaust valve is examined. Only Euler and RK4 schemes are examined.

The integration schemes out of the CVODE package are not considered here, since adequate specifications of error tolerances and solver parameters prevent production of results exhibiting stability issues. The

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exhaust valve opens at 483 deg. CA resulting in an outflow from the cylinder. Afterwards, the cylinder approaches the bottom dead center (BDC), 540 deg.

CA, and a flow reversal occurs, since no combustion occurred in the high pressure phase and some mass has flown out of the cylinder resulting in p > pE when the piston approaches the BDC. Immediately after the BDC the flow direction changes again. The pressure in the cylinder is similar to the pressure in the exhaust plenum in the vicinity of the BDC and the derivative of the cylinder volume is small. It is discernible from Fig. 4 that the first order Euler scheme produces unphysical oscillations in mass flux due to coupled mass flow and pressure fluctuations as analysed in Fig. 2. Decreasing the time step of the Euler scheme from 1 deg. CA to 0.25 deg. CA diminishes mass flow oscillations, however, it does not suppress them.

Fig. 4 shows that the RK4 scheme does not produce oscillatory results. It can also be observed that the mass flow traces of the RK4/4 deg. CA and Euler/1 deg. CA schemes deviate significantly.

4.2 Test Case 1: Engine Speed 1500 rpm The trade-off between numerical accuracy and computational speed is first analysed for Test case 1 (Table 2). The integration increments of the Euler scheme were 0.1, 0.25, 0.5, and 1 deg. CA and those of the RK4 scheme were 0.25, 0.5, 1, 2, and 4 deg.

CA. TOLrel values of the CVODE schemes were increased by a factor of 10 in the range from 10–7 to 10–5 for Newton iteration schemes and in the range from 10–7 to 10–3 for functional iteration schemes.

Time increment of the CVODE schemes corresponds to 1 deg. CA. Additionally, for the BDF, FCN scheme results for the time increment corresponding to 4 deg. CA was analysed to investigate the influence of the global integration increment on the accuracy of results. In order to preserve readability of the figures only results for the BDF, NWT are shown. It might be noted that the results of the BDF, FCN scheme feature very similar trends as those of the BDF, NWT scheme. The TOLrel of CVODE schemes thus ranges from 10–7 to 10–5 for BDF, NWT scheme, from 10–7 to 10–5 for AM, NWT scheme, and from 10–7 to 10–3 for AM, FCN scheme. In the subsequent figures, markers on a line indicating a particular integration scheme represent different integration increments or different tolerance settings. Larger integration increments or smaller tolerance settings feature lower RT factors for a particular scheme.

Fig. 5 summarizes the dependency of the NDRMS on the RT factor for in-cylinder 230

theromynamic variables (Figs. 5a to e). Additionally, Fig. 5f shows the deviation of the IMEP calculated by a particular scheme and the exact solution (ΔIMEP = IMEP(exact)–1) as a function of the RT factor, where IMEP is defined by Eq. (18) and represents an integral engine parameter. Efficient integration schemes simultaneously feature both a low NDRMS value and a low RT value and thus tend towards the lower left end of the diagrams as indicated in Fig. 5a. It can be observed that all analysed cylinder data feature very similar trends. It can be seen from Fig. 5, with some local deviations, that an increased RT value (increased computational time) leads to increased numerical accuracy for a particular scheme.

However, it can also be observed that the difference in NDRMS values of different integration schemes could reach up to two orders of magnitude at the same RT factor. This fact underlines the necessity of selecting the appropriate numerical scheme for the specific simulation task.

It can be seen that all CVODE schemes feature a pronounced increase in NDRMS values for decreasing RT values. Fig. 5 shows that CVODE schemes reach similar minimum RT values (i.e. upper computational speed), when applying tolerance and iteration parameter settings, which prevent the occurrence of non-stabilities. On the other hand, explicit schemes without error control do not include any mechanism for controlling numerical accuracy. These schemes thus allow one to attain very small RT values, however, numerical accuracy is reduced significantly.

A thorough analysis of convergence and stability is therefore mandatory, especially for explicit methods, in order to avoid evaluation of non-converged simulation results.

Based on the results shown in Figs. 2 and 3 and NDRMS values shown in Fig. 5, basic guidelines for determining NDRMS values that indicate converged solutions can be defined. It is shown in Figs. 2 and 3 that RK4/1 deg. CA and AM, FCN/TOLrel = 10–5 schemes provide results that coincide very well with the exact solution. These results could thus be assumed as converged considering numerical accuracy requirements in the initial stage of the engine development process or in the HiL tests. From Fig. 5 it can be concluded that both schemes, i.e. RK4/1 deg.

CA and AM FCN/ TOLrel = 10–5, feature NDRMS values approximately 10–4 and lower. This value might thus be considered as a guideline for a converged solution taking into account numerical accuracy requirements needed in the engine development process. On the contrary, schemes featuring NDRMS values considerably larger than 10–4 exhibit larger

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Fig. 5. Dependencies of the NDRMS on the RT factor for an engine speed of 1500 rpm and an IMEP of 1.92 MPa; a) pressure, b) temperature, c) in-cylinder mass, d) mass fraction of combustion products, e) mass fraction of burned fuel, f) dependency of the ΔIMEP on the RT factor

deviations from the exact solutions indicating nonconverged solutions.

Furthermore, Fig. 5f shows that the ΔIMEP is below 1% (a typical accuracy threshold for validating high fidelity ICE models) for all analysed integration

schemes and all integration increments or specified tolerances. Considering the deficiencies of particular schemes shown in Figs. 2 to 4, it can be concluded that agreement of integral engine parameters should not be the only measure for assessing the accuracy of

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the model, since non-converged and even locally nonstable solutions might comply with these accuracy requirements.

Fig. 5 shows, with some local deviations, that RK4 schemes feature the lowest NDRMS values for the analysed RT range. It can further be concluded that the RK4/1 deg. CA scheme complies with the convergence and stability requirements and simultaneously features a very low RT value for this engine operating point. Fig. 5 also shows that the results of the RK4 scheme featuring a steep NDRMS increase in state variables for integration increments larger than 1 deg. CA. It is discernible from Fig. 5 that the Euler scheme features a worse trade-off between NDRMS and RT compared to the RK4 scheme for NDRMS values of 10–4 and lower, which is an acceptable level of numerical accuracy required in the engine development process, as discussed above. On the other hand, the Euler scheme performs better than the RK4 scheme for larger NDRMS values and thus small RT values. However it is discernible from Fig. 2 that these small RT values correspond to integration increments leading to numerical instabilities in both schemes.

It can also be observed in Fig. 5 that the functional iteration applied by the AM scheme results in better trade-off between NDRMS and RT compared to the application of the Newton iteration. This can be explained by the fact that the computationally fast functional iteration brings benefits over the computationally more demanding Newton iteration for this non-stiff example. Additionally, it can be observed that the BDF scheme might even perform better than the AM, NWT scheme although it is also applicable to stiff systems. Unlike for the AM scheme, only minor differences can be observed in the trade-

off between NDRMS and RT when applying different iteration methods to the BDF scheme. It is also shown for the BDF scheme that increasing the global integration increment from 1 to 4 deg. CA does not influence the trade-off between NDRMS and RT. This is in line with the above statement that the definition of tolerances represents the main mechanism influencing the numerical accuracy and computational speed of CVODE schemes. Due to these facts only the results of the BDF, NWT scheme are shown in the figures.

The analysis of different engine configurations has shown that the ratio of the effective valve flow area (μAgeom) and cylinder volume (V) most significantly influences the trade-off between NDRMS and RT values of a particular engine. This is justified by the fact that the gas exchange period generally represents the numerically most demanding phase, provided simple combustion models are used (e.g. Vibe, ROHR table, and the majority of empirical models that do not incorporate chemical kinetics or droplet evaporation).

This ratio together with the compression ratio strongly influences the ratio of the mass exchanged through the valves within one integration increment and the in-cylinder mass, and therefore the stability of the solution during the gas exchange period. The ratio of the mass exchanged through the valves within one integration increment and the in-cylinder mass thus most significantly determines the optimum integration increment of the explicit schemes when considering the trade-off between NDRMS and RT values.

Similarly, the solvers of the CVODE schemes adapt their integration increments to consider this effect.

Therefore, the ratio of RT factors of the explicit and CVODE schemes is almost unaffected by the variation of the ratio of μAgeom to V for a selected NDRMS value. However, the RT factors of all schemes increase

Fig. 6. Dependencies of the NDRMS on the RT factor for an engine speed of 6000 rpm and IMEP = 1.68 MPa; a) pressure, b) temperature

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at the same value of NDRMS if the ratio of μAgeom to V increases.

4.3 Test Case 2: Engine Speed 6000 rpm The computational time of CA based explicit schemes without error control directly correlates with the engine speed. In contrast, time based methods with error control and adaptive integration increments might not strictly follow this rule. Fig. 6 shows results for the Test case 2 (Table 3). nE = 6000 rpm corresponds to the upper speed range of the internal mixture preparation compression ignition engines applied in racing cars. For such operating conditions it is very common for turbocharged engines to feature exhaust pressure that is higher compared to the boost pressure as shown in Table 3.

Fig. 6 shows the results of NDRMS vs. RT for the cylinder pressure (Fig. 6a) and temperature (Fig. 6b), since it was shown in Fig. 5 that other state vector variables feature similar trends. It is discernible from Fig. 6 that the NDRMS vs. RT curves of the CVODE schemes move closer to the corresponding RK4 curve compared to the nE= 1500 rpm case shown in Fig. 5.

This is mainly a consequence of different integration variables, i.e. time or CA rotation, as pointed out above. Although dV/dt (Eq. (10)) increases with increasing engine speed, the time needed to integrate the coupled system of equations (Eq. (14)) with time based CVODE schemes does not increase proportionally to the engine speed.

Similarly to Fig. 5, it can be observed, with some local deviations, that RK4 schemes feature the lowest NDRMS values for the analysed RT range. It can be seen that the RK4/1 deg. CA scheme complies with convergence and stability requirements. The calculated RT factor of the RK4/1 deg. CA scheme is 0.19 for the applied platform. It could thus comply with real-time constraints for a very simple multicylinder engine configurations at nE = 6000 rpm.

Fig. 6 indicates that for very low values of NDRMS, CVODE schemes tend to show a better performance than RK4 schemes. However, at this engine speed the RT factor of CVODE schemes with small values of TOLrel approaches or already exceeds 1 for this simplified test case. It can also be concluded that it is not reasonable to apply integration schemes with very low tolerances, since other errors introduced by the application of the 0-D model are much larger and therefore eliminate the benefit of very low tolerances of the integration scheme.

It can be observed in Fig. 6 that the difference in the trade-off between NDRMS and RT value evaluated

for the AM, FCN, and AM, NWT scheme is reduced compared to the lower engine speed (Fig. 5). This can be explained by the fact that for higher engine speeds time derivatives of the variables increase and thus the computational overload of the more sophisticated gradient based Newton method is less pronounced than that of the functional iteration method. It might be concluded that BDF, NWT scheme performs comparable to both AM schemes for this engine speed.

Based on the above findings it could be assumed that time based CVODE schemes might show a better performance than RK4 at much higher engine speeds. Therefore an example with nE = 12000 rpm was analysed, although this engine speed represents a rather unrealistically high value for internal mixture preparation compression ignition engines. Again the RK4/1 deg. CA scheme featured the lowest RT values (RT = 0.38) for NDRMS values in the range of 10–4.

Table 3. Boundary conditions for Test case 2: engine speed 6000 rpm and IMEP = 1.68 MPa Plenum Intake Exhaust

T [K] 340 320

p [Pa] 200000 280000

wFB [-] 0 0

wCP [-] 0 0

wFV [-] 0 0

4.4 Comparison of Numerical and Experimental Results In addition to the thorough theoretical analyses presented above it is also meaningful to compare numerical and experimental in-cylinder pressure traces for a series production engine to confirm the applicability and accuracy of the cylinder model.

Therefore, the crank angle resolved cylinder model presented here is embedded into a mean value based gas path model of the engine manifolds [4]. As presented in [4], the overall model is applied to a turbocharged, intercooled 4-cylinder diesel engine with a cooled external EGR and variable geometry turbine. The combustion is modelled using rate-ofheat-release curves derived from experimental incylinder pressure traces. This ensures a consistent comparison of numerical and experimental cylinder pressure traces [4]. Fig. 7 shows numerical and experimental cylinder pressure traces for two selected full and part load operating conditions applying a RK4/1 deg. CA scheme to integrate the cylinder governing equations. It is clear that the model is capable of accurately predicting high- and low-pressure phase of the cylinder process. A good agreement can be seen for the early combustion characteristic for the full load condition and also for the late combustion characteristic for part load

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Fig. 7. Comparison of numerical and experimental cylinder pressure traces for different engine load points; a) 3800 rpm and 1.54 MPa BMEP, b) 1600 rpm and 0.33 MPa BMEP

conditions. Minor discrepancies in pressure traces can be observed for the exhaust stroke. The deviations are caused by the mean value gas path description of the engine manifolds neglecting gas-dynamic effects.

The considered engine features nearly negligible valve overlap and a high compression ratio. Thus the discrepancies during the exhaust stroke do not notably influence the in-cylinder state at EVC. Therefore, due to the application of consistent governing equations and adequate methods for their integration, numerical and experimental results of the intake stroke, compression, combustion, and expansion show very good agreement. Good agreement between the simulation and experimental results is furthermore confirmed by a very good agreement in simulated and experimental values of the peak cylinder pressure for 3 full load and 1 part load point as given in Table 4.

Table 4. Peak firing pressure in MPa Engine speed BMEP Simulation Experiment

1200 rpm 1.05 MPa

3000 rpm 1.94 MPa

3800 rpm 1.54 MPa

1600 rpm 0.33 MPa

10.29 10.24

16.33 16.19

16.37 16.24

5.25 5.29

5 CONCLUSIONS The trade-off between numerical accuracy and computational speed was analysed for different integration schemes applied to solve governing 234

equations of the 0-D internal combustion engine cylinder model. It was shown that the numerical accuracy of integration schemes differs significantly at equal computational effort. The analysis reveals that the RK4 scheme features the best ratio of numerical accuracy and computational speed in a realtime capable computation speed range. It was shown that explicit schemes without error control should apply CA rotation as an independent variable, since this approach ensures similar numerical accuracy for different engine operating conditions. It was also demonstrated that for a particular numerical accuracy threshold the difference in computational speed between explicit schemes applying CA rotation as an independent variable and time based methods with error control decreases with increasing engine speed.

Moreover, CVODE schemes show a comparable or better performance than the RK4 scheme for high engine speeds when very high accuracy of the results is required. However, it is generally not reasonable to apply integration schemes with very low tolerances, since other errors introduced by the application of the 0-D model are much larger and therefore eliminate the benefit of very low tolerances of the integration scheme. It was shown that the global integration increment of CVODE schemes does not significantly influence the performance of these schemes, being predominantly influenced by tolerance settings.

NDRMS values give valuable guidelines on the numerical accuracy of the integration schemes.

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However, they feature a limited capability of detecting local deficiencies produced by integration schemes.

The NDRMS vs. RT approach is therefore suitable for acquiring a basic insight into the performance of integration schemes, whereas cycle resolved results need to be subsequently analysed to study the convergence and stability of the solution in detail. It was also shown that agreement of integral engine parameters should not be the only measure for assessing the accuracy of the model. Very good agreement between numerical and experimental results additionally confirms the accuracy of the cylinder model and its applicability for predictive system level simulations.

6 NOMENCLATURE A H h M m N nvalves nE P p Q R ℜ T t TOL U u V wk y Z φ κ μ ξ

area [m2] enthalpy [J] specific enthalpy [m2/s2] molar mass [kg/kmol] mass [kg] number of species in the species vector [-] number of valves [-] engine speed [rpm] power [W] pressure [Pa] heat [J] specific gas constant [J/kgK] universal gas const. [J/kmolK] temperature [T] time [s] tolerance [-] internal energy [J] specific internal energy [m2/s2] volume [m3] species concentration [-] solution vector [-] compressibility factor [-] crank angle rotation [deg. CA] ratio of specific heats [-] flow coefficient [-] independent variable: t or φ

Subscripts: abs absolute air air C combustion cyl cylinder CP combustion products down downstream E exhaust plenum exact exact solution

FB fuel burned FV fuel vapor GE gas exchange geom geometric HT heat transfer I intake plenum inj fuel injection max maximum rel relative up upstream Abbreviations: A after B before BDC bottom dead center BMEP brake mean effective pressure CA crank angle EV exhaust valve EVC exhaust valve closes EVO exhaust valve opens IMEP indicated mean effective pressure IV inlet valve IVC inlet valve closes IVO inlet valve opens NDRMS non-dimensional root mean square RT real time factor TDC top dead center 7 REFERENCES [1] Paciti, G.C., Amphlett, S., Miller, P., Norris, R., Truscott, A. (2009). Real-time crank-resoved engine simulation for testing new engine managment systems. SAE International Journal of Passenger Cars- Mechanical Systems, vol. 1, no. 1, p. 801-809, DOI:10.4271/2008-01-1006.

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[14] Heywood, J.B. (1988). Internal Combustion Engine Fundamentals. McGraw-Hill, New York.

[15] Ghojel, J.I. (2010). Review of the development and applications of the Wiebe function: a tribute to the contribution of Ivan Wiebe to engine research.

International Journal of Engine Research, vol. 11, no.

4, p. 297-312, DOI:10.1243/14680874JER06510.

[16] Hohenberg, G. (1980). Experimentelle Erfassung der Wandwärme von Kolbenmotoren. Habilitationsschrift TU Graz, Graz.

[17] Pischinger, R., Krassnig, G., Taučar, G., Sams, T. (1989).

Thermodynamik der Verbrennungskraftmachine.

Springer-Verlag, Berlin.

[18] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992). Numerical Recipes in Fortran 77, 2nd ed. Cambridge University Press, Cambridge.

[19] Hindmarsh, A.C., Serban, R. (2006). User Documentation for cvode v2.5.0., UCRL-SM-208108, Center for Applied Scientific Computing Lawrence Livermore National Laboratory, Livermore.

Katrašnik, T. – Schuemie, H.A. – Wurzenberger, J.C.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 237-250 © 2013 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2012.989 (1.01) Original Scientific Paper

Received for review: 2012-12-04 Received revised form: 2013-01-25 Accepted for publication: 2013-02-12

A Method for Optimal Blank Shape Determination in Sheet Metal Forming Based on Numerical Simulations Mole, N. – Cafuta, G. – Štok, B. Nikolaj Mole1 – Gašper Cafuta2 – Boris Štok1,*

1 University

of Ljubljana, Faculty of Mechanical Engineering, Slovenia 2 Cimos d.d., Automotive industry, Slovenia

The main benefit of using optimally shaped blanks in sheet metal forming is to maximize the efficiency of the forming process and, since there is no need for additional cutting operations after the finished forming operation, this leads to a substantial reduction in overall production costs. This paper presents a numerical method for optimal blank shape determination, which is suitable in various sheet metal forming applications. The optimal blank shape is determined in an iterative way so that the edge geometry of the formed product fits its reference geometry as closely as possible. The iterative process starts with the blank shape from which the product is produced with its edge fitting its reference geometry just approximately. In subsequent iterations, the blank shape is continuously improved in accordance with the developed optimisation method. In order to determine the product edge geometry resulting from the given blank shape, a computer simulation of the forming process and related springback is performed at each iteration. Since its effectiveness greatly depends on the quality and physical objectivity of the computer simulation, the developed numerical blank shape optimisation procedure has also been validated experimentally by using the forming of a product with a rather complex edge geometry as the case study.

Keywords: sheet metal forming, blank shape optimisation, springback compensation

0 INTRODUCTION Nowadays most production processes are designed to increase economic efficiency while, at the same time, not influencing product quality, especially when dealing with large serial production. In order to improve the forming process from the economic point of view, the main objective is to minimize the number of operation phases involved in the forming process. The cutting phase, as the final operation, is the phase most often targeted for elimination. Along with less material required for product production, the elimination of the cutting phase may also reduce the occurrence of manufacturing defects during the forming process, such as tearing and wrinkling.

In some cases, however, the sheet cutting phase is unavoidable since excess material under the blankholder is required to achieve proper holding of the sheet metal [1].

When the cutting phase is eliminated, the accuracy of the product edge geometry should be provided by proper blank shape geometry. In most cases, the proper blank shape is determined experimentally by a trial and error procedure which requires several try-outs, causing the forming process design to be extremely time consuming and costly. Nowadays, this can be done virtually, based on computer simulations of the technological process under consideration, see, for example, [1] to [4]. Besides choosing a proper numerical approach and computational technique [5] and [6], advanced constitutive modelling [2], [4], [7] and [8] and proper material characterisation [4] and [7] to [9] is crucial for the computer simulation

to be physically objective and trustful. Although, in contrast to direct analysis, a Finite Element (FE) inverse analysis approach, see [10] to [14], could possibly be used in sheet metal forming simulation in order to reduce the computer time consumption, such an approach is not recommended because it results in less accurate strain–stress state determination. This, in turn, can considerably influence the subsequent springback analysis, which is a key element in tool design optimisation.

This paper describes a blank shape optimisation method, which allows determination of a blank shape such that the edge geometry deviation of the produced product with respect to the specified product geometry is obtained within a given tolerance. In this method, the optimal blank shape is determined in an iterative way, starting with an initial blank of approximately adequate shape, which over a series of subsequent iterations gets automatically adjusted to meet the tolerance criterion. Following the principal idea, which is a gradual changing of the given blank shape based on the computed product geometry and manifested edge deviation, the product geometry must be determined in each iteration by a corresponding forming process simulation, considering the blank shape as determined in the previous iteration.

The method is designed in such a way that it enables optimal blank shape determination of products having a general 3D shape. That the method is capable of tackling rather complex product geometries efficiently is demonstrated in Section 4, where results of the numerical optimisation process are also experimentally validated.

*Corr. Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia, boris.stok@fs.uni-lj.si

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1 REVIEW OF SOME APPROACHES TO BLANK SHAPE OPTIMISATION The problem of finding an appropriate blank shape, which would allow the production of a formed product with the edge geometry meeting the geometry tolerance requirements, as specified by the design, has been addressed by many authors. In the section, we give a brief review of different numerical approaches developed in this regard.

An optimisation method, where the initial blank shape approximation is determined by inverse FEM simulation and the blank shape is optimised iteratively afterwards using direct FEM computer simulations, is presented in [15] and [16] by Azaouzi et al. The applied blank shape modification consists of displacing, in each iteration, the blank edge nodes in the direction opposite to the geometry deviation obtained by comparing the computed and reference product geometry. After the new blank shape is determined, its domain is remeshed automatically using triangular FEs. Naceur et al. [17] presented a blank shape optimisation approach that is based on the coupling between the inverse approach used for the forming simulation and an evolutionary algorithm.

Their goal was to minimize the size of the blank shape and still ensure that the product is made without tearing the sheet metal. Park et al. [18] introduced a deformation path method. The Ideal Forming direct inverse design method, see [13] and [14], was used to determine initial blank shape, which was further improved by an iterative procedure. In [19], Yeh et al. use the inverse true strain method (TSM) to obtain an initial approximation of the blank shape, whereas for the optimal blank shape determination a method based on adaptive-network-based fuzzy inference system (ANFIS) is applied. To achieve satisfactory results on the optimised initial blank shape, at least for the case considered in the paper, 200 learning cycles, each requiring a complete computer simulation of the corresponding forming process, had to be used in building the appropriate ANFIS database. Another approach, where the abductive network is used to predict the optimal blank shape for forming an elliptic cup using FEM computer simulations, is presented by Lin and Kwan in [20]. In the optimisation procedure, the initially elliptic blank shape, with its geometry being specified in a polar coordinate system by four characteristic points, is subject to change by considering adequate variation of the respective points’ radial coordinates, while keeping their angular coordinates fixed. An iterative method, where the product geometry is also computed using inverse 238

FEM while the blank shape correction is made in each iteration manually, is introduced by Parsa and Pournia in [21]. Due to the application of the inverse analysis approach, the computational time is significantly reduced, but it is achieved at the expense of loss of accuracy in the product geometry determination.

The objective of the blank shape optimisation presented in [22] by Pegada et al. is to determine the blank shape that allows forming of a circular cup of uniform height, where the respective material properties are assumed to be orthotropic. In each iteration, considering the established deviation in the cup height, the blank shape is adapted proportionally, assuming a value of 0.75 as an adequate scaling factor to obtain satisfactory convergence to the method. In the method introduced by Son and Shim in [23], the blank shape correction is made in the direction opposite to the geometry deviation obtained by comparing the computed and reference product geometry. Furthermore, correction of each edge point is computed by applying a scaling factor between 0.5 and 0.9, its actual value being defined by a ratio of the initial velocity to the total deformation path length. In [24], Hamammi et al. propose a method where the blank shape correction is made in the direction described by the positions of each node lying on the blank edge at the beginning and at the end of the forming process. Another method, where the correction of the blank shape is also based on taking the displacement path of the product edge nodes into account, is proposed by Fazli and Arezoo in [25]. They proved that their method is slightly better in terms of accuracy and also in convergence than the previous two methods, [23] and [24].

The optimal blank shape can also be obtained by parameterisation of the blank geometry and using a sequential programming method for finding the optimal set of parameters, which is elaborated in [26]

by Sattari et al. Similarly, in [27], Padmanabhana et al. investigate blank shape optimisation using parameterisation of the blank shape carried out by using parametric NURBS curves and the blank shape correction based on the control points displacement.

The convergence of the latter method can be further improved by including the sensitivity analysis shown in [28] by Shim et al.

In principle, all those approaches to blank shape optimisation have in common a sequence of rather similar logical steps, which can be roughly summarised by the flow chart in Fig. 3. The approach we are going to propose in this paper does not differ in this regard.

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The iterative approach we present here is implemented in such a way that the user must provide an approximately determined blank shape, which is taken as the initial blank shape for the optimisation procedure. As demonstrated by the study case, elaborated in Section 4, the proposed numerical approach is efficient enough that it does not require a very accurate determination of the initial blank shape, which is the case in [15], [16], [18] and [19]. By using different strategies the user can still provide a better approximation for the initial blank shape, which is certainly advantageous. One thing that is common to all the methods presented in the review is that, in each iteration, a remeshing of the blank is required after blank shape correction. In our case, on the other hand, the mesh element topology is kept unchanged through the entire iterative procedure, while the corresponding FE mesh nodal points’ adjustment in the iteration is performed in accordance with the displacement field obtained by solving an associate linear elastic boundary value problem. In this boundary value problem, the blank with the actual shape geometry before correction is subjected to prescribed boundary displacements, the imposed boundary displacements taken equal the required blank edge geometry correction in the iteration. With the blank FE mesh correction dealt with in this way, any type of FEs can be used and no sophisticated remeshing methods are needed.

2 MATHEMATICAL, MODELLING AND PHYSICAL PRELIMINARIES In order to provide a corresponding mathematical framework for the numerical analysis, in this section we give some definitions and conventions on the geometry terminology used, which is followed by a determination of some related point topology quantities, such as surface and edge normal vectors’ determination, and, finally, an approximation for analytical surface reconstruction.

Some modelling and simulation assumptions are given in the last part of the section in order to focus our investigation on the key elements of the proposed numerical procedure for the blank shape optimisation described in Section 3.

2.1 Geometry and Point Topology Terminology In this paper, we adopt the same geometry terminology as introduced by Cafuta et al. in [29]. The product geometry prescribed by the design engineer is referred as the “reference geometry”, whereas the

product geometry computed in the simulation will be referred as the “actual geometry”. Considering that the forming process simulations are performed on the basis of FEM, we are actually dealing with discretised geometries. In accordance with the notation G , introduced for a general surface point topology definition, we will use notations G ref and G act to specify point topologies related to the reference and actual product geometry, respectively. Similarly, notation Gbl will be used to denote point topology related to the sheet blank geometry. Furthermore, Γ ref , Γ act and Γbl will be used to specify the associated edge point topologies notations. All those surface and edge topologies will exclusively refer to the sheet’s mid-surface. In addition, to support a numerical procedure of automatic sheet blank geometry adjustment, an auxiliary virtual surface Σ with its point topology notation being G will be generated, emanating from the reference product edge Γref considering its surface topological properties. Σ Σ The surface G having properties Γref ⊂ G may be considered as a target surface which is to be attained iteratively by the edge points Γ Pi k of the actual product Γ kact as closely as possible.

Since, in the numerical procedure, the unit normal vectors ni at point Pi perpendicular either to the surface or to the edge will also be referenced, it is reasonable to make a symbolic distinction between them. Accordingly, the notations ni and Σ ni will be used for the surface normal vectors, whereas Γ ni will be used for the edge normal vectors.

Similarly, to make a distinction between the points appertaining either to the product surfaces, G ref and G act , or to the auxiliary surface Σ G , the notations Pi and Σ Pi will be used with respective position vectors being Pi and Σ Pi . If point Pi is located on the edge Γ, it will be denoted as Γ Pi and its position specified by vector Γ Pi .

2.2 Surface Normal Vector Determination The accuracy of the surface normal vector determination at points of discretely given surface geometries plays an important, if not crucial, role in attaining convergence and speeding up the convergence rate of the entire optimisation procedure.

A general surface point topology G given, we apply in this paper a two-step procedure in which for a point of interest, say Pi , two auxiliary vectors, n′i and n′′i , are computed, respectively.

In the first step, the auxiliary vector n′i at point Pi is determined, considering FE surface discretisation.

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The following equation is used: Ni

n′i =

∑n

e i

∑n

e i

e =1 Ni e =1

, nie =

v1e × v e2 , (1) v1e × v e2

with N i being the number of neighbouring FEs connected to point Pi and nie being their respective normal vectors at that point. The latter are determined by considering the respective FE’s edge vectors v1e and v e2 emanating from point Pi (Fig. 1a).

Mathematical representation of the surface Si , Fi ( x, y, z ) = 0 , is built by considering coordinates of the respective points of the set Π i . To avoid round-off error due to possible computing with large numbers, it is advantageous to have the surface Si representation defined with respect to a particular local coordinate system, say ( x , y , z ) , having its origin at point Pi ≡ Pi1 (Fig. 1b). The base vectors e x , e y , e z , defining the respective local coordinate system ( x , y , z ) at point Pi in accordance with the above stated properties, are determined by expressions: e x = e z × n′i = r, e y = e z × e x , e z = n′i , (2)

where the base vector e z = (0,0,1) defines the global coordinate axis z.

The rotation of the global coordinate system (x, y, z) to the local coordinate system ( x , y , z ) is given by the rotational transformation matrix ℜ(r,θ ) :  axx  ℜ(r,θ ) =  c yx  bzx 

bxy a yy czy

cxz   byz  , (3) azz 

where the matrix coefficients are determined as:

a)

aii = ri 2 v(θ ) + cos(θ ) bij = ri rj v(θ ) − rk sin(θ ) cij = ri rj v(θ ) + rk sin(θ ) ; i ≠ j ≠ k ≠ i. (4)

b) Fig. 1. Surface normal vector determination: a) FE’s nodal normal vector n ie determination, and b) auxiliary vector n′′i determination

In the second step, the normal vector ni is determined considering the analytically defined smooth surface Si through point Pi . This surface is obtained by a corresponding interpolation through a point set Π i , built from points Pi j ( j = 1, 2,...,9) of the FE mesh. Those points are chosen from a sub-domain area surrounding the considered point Pi ≡ Pi1 on a closest point’s basis, which is applicable regardless of whether point Pi ≡ Pi1 belongs to the interior of the FE mesh or to its boundary.

240

Above, rx , ry and rz are the components of vector r = (rx , ry , rz ) and quantity v(θ ) is defined by v(θ ) = 1 − cos(θ ) , the angle θ denoting the angle between the base vectors e z and e z .

To enable analytical surface Si representation  f i ( x, y , z ) = 0 with respect to the local coordinate system ( x , y , z ) , the coordinates of the position vector Pi j are mapped to the local coordinate system in accordance with: P i j = ℜ(r,θ ) ( Pi j − Pi ) . (5)

 consisting of nine points With the points set Π i Pi (Fig. 1b), it is appropriate to approximate surface Si in accordance with: j

2

2

f i ( x , y , z ) = z − ∑∑ amn x m y n = 0 , (6) m =0 n =0

where nine coefficients amn are determined such that the surface Si is interpolated through the points Pi j .

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Finally, the auxiliary vector n′′i normal to the surface f i ( x , y , z ) = 0 at point Pi1 (0,0,0) can be determined by the gradient operator:

n′′i =

gradf i (0,0,0) . (7) gradf i (0,0,0)

By applying to the components of the above vector, given in the local coordinate system, the inverse mapping ℜ−1 (r,θ ) , the components of the normal vector ni to the surface Si at point Pi in the global coordinate system are obtained such that:

the edge normal vector Γ ni , defined in the global coordinate system, can be determined. The corresponding graphical representation, with vector Γ ni lying in the tangential plane to surface Si and pointing to its exterior, is given in Fig. 2.

If the point, at which the edge normal vector is to be computed, is positioned on the symmetry plane, in order to achieve geometric symmetry conditions, the tangential vector Γ t i is defined by a unit vector normal to the symmetry plane. The edge normal vector Γ ni is afterwards computed by the same procedure.

ni = ℜ−1 (r,θ ) n′′i . (8)

If the point, at which the normal vector is to be computed, is positioned on the symmetry plane, the normal vector ni is computed by the same procedure considering the mirror elements over the symmetry plane.

The abovedescribed procedure of the surface normal vectors’ determination is general. In Section 3, it will be carried out with reference to the reference and actual product geometry, G ref and G act , as well as to the auxiliary surface Σ G .

2.3 Edge Normal Vector Determination We will refer generally to a surface point topology G and associate edge point topology Γ also in the determination of the unit vectors normal to the edge of a bounded surface. At a point of interest on the boundary Γ, say Γ Pi , the edge normal vector Γ ni will be determined considering respective surface and boundary curvatures. While the surface curvature is characterised by the respective surface normal vector ni , with its determination being described in Section 2.2, the boundary curvature can be characterised by a unit vector Γ t i tangential to the surface boundary at point Γ Pi .

The tangential vector Γ t i can be correspondingly approximated considering actual discretisation of the boundary Γ in the closest vicinity of the point Γ Pi .

With points Γ Pi−1 and Γ Pi+1 being the points adjacent to point Γ Pi (see Fig. 2) the following equation may be applied:

Γ

P − ΓP t i = Γ i +1 Γ i −1 . (9) Pi +1 − Pi −1

2.4 Analytical Surface Reconstruction Assuming that the considered FE surface discretisations are all based on a quadrilateral surface element meshing (see Fig. 1), we will develop, in this section, an approximation to the analytical surface reconstruction of a single quadrilateral surface element, considering its particular topological properties.

Let a quadrilateral surface Si be discretely defined by its nodal points Pi j ( j = 1, 2,3, 4) and respective surface normal vectors nij at those points, determined following the procedure described in Section 2.2. This set of data represents twelve independent parameters, which can be used in analytical surface reconstruction of surface Si , Fi ( x, y, z ) = 0 . This can be achieved by considering the following functional form: Fi ( x, y, z ) = z − (a1i + a2i x + a3i y + a4i x y + + a5i x 2 + a6i y 2 + a7i x 2 y + a8i xy 2 +

Γ

Finally, by evaluating the vector product:

Γ Fig. 2. Edge normal vector n i determination

Γ

ni = Γ t i × ni , (10)

+ a9i x 2 y 2 + a10i x 3 + a11i y 3 + a12i x 3 y 3 ) = 0, (11)

with twelve coefficients ami (m = 1, 2,...,12) to be determined. By imposing that the above surface representation meets given requirements at four

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interpolation points Pi j the following system of linear equations is obtained: Fi ( x j , y j , z j ) =

0 , j i,x j i,z

∂Fi (xj , y j , z j ) = ∂x

n

∂Fi (xj , y j , z j ) = ∂y

nij, y

n

nij, z

, ; j = 1, 2,3, 4 , (12)

whose solution yields the value of the coefficients ami . Here, it should be emphasised that by fulfilling the nodal surface properties in the above way element after element C1 continuity at discrete points of the surface Si is achieved, which is significant for the quality of the overall surface approximation.

2.5 Modelling and Simulation Assumptions When the blank shape optimisation is based on the results of a corresponding computer simulation of the considered forming process, it is exceedingly important that the springback numerical estimation should be as reliable as possible. In this regard, the resulting stress-strain state in the formed product prior to the springback, as well as the established change in the sheet metal thickness itself, are crucial for determining the extent of the manifested springback.

To attain reliability of the computed results, the process parameters, such as the sheet material behaviour, tools’ kinematics, sheet blank shape geometry, the clearance between the punch and the die, the blank-holder force and tribological conditions between the surfaces in contact, should be considered as realistically as possible in the simulation. For the purpose of this investigation, let us assume that all above parameters, except sheet blank shape geometry, are defined adequately and after the computer simulation yield a product of a shape, whose deviation from the shape of the reference product, when measured in the direction perpendicular to the product surface, is within the prescribed tolerance (see Cafuta et al. [29]). A possible deviation in the circumference contour, larger than is allowed, is subject to a corresponding adjustment in the sheet blank shape geometry, which is actually the topic of this paper.

Accordingly, we also assume that all issues associated with FEM, such as adequate choice of the finite element type and the integration of the constitutive equations, are not disputable.

242

3 NUMERICAL IMPLEMENTATION OF THE PROPOSED BLANK SHAPE OPTIMISATION The proposed blank shape optimisation method consists basically of a sequence of steps executed iteratively following the flow chart in Fig. 3. Along with the initial blank shape geometry Gbl0 and reference product geometry G ref provided, a complete FE model with the tool geometry and forming process parameters (blank-holder forces, friction conditions, tool movement, etc.) must be specified in order to start the procedure. Since the initial blank shape geometry Gbl0 plays an important role in attaining convergence and computational efficiency of the described blank shape optimisation procedure, this geometry should be determined in a way that the edge geometry Γ 0act of the product, obtained by the corresponding forming process simulation under given process conditions, does not experience too excessive a deviation with respect to the reference edge geometry Γ ref . Before starting the procedure the auxiliary surface Σ G is determined as explained in Section 3.1.

Fig. 3. Flow chart of the iterative procedure

The iterative procedure essentially consists of performing in iteration, say kth iteration, three consecutive steps. In the first step, the complete technological process, consisting of the forming stage with the actual blank shape geometry Gblk considered

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and subsequent springback stage occurring after removal of the tools, is simulated. This is followed by computing the deviation between the actual product edge geometry Γ kact and its reference edge geometry Γ ref , measured with respect to the auxiliary surface Σ G . The edge deviation computation is performed in each iteration following the procedure described in Section 3.2. In the last step, the actual blank shape k geometry Gbl is either confirmed as appropriate or, if the considered edge deviation being found greater than the prescribed tolerance, its further correction and adjustment to Gblk +1 is required (see Section 3.3). Fulfilment of the tolerance requirements means stopping of the iteration loop, whereas non-fulfilment means that the iteration procedure goes on to the subsequent (k + 1)th iteration. A detailed description of how the above general procedure is managed is given in the following sections.

3.1 Determination of the Auxiliary Surface Σ G The appropriateness of the actual blank shape k geometry Gbl can be established by measuring the deviation of the actual product edge geometry Γ kact from the reference edge geometry Γ ref , which can be done in many ways. Here, the deviation will not be measured directly with respect to Γ ref , but indirectly by making use of an auxiliary virtual surface Σ G emanating from the reference product edge Γ ref . This surface can be generated considering surface topological properties of the reference product geometry G ref on its edge Γ ref . Lines li , aligned along the surface normal vectors ref ni at points refΓ Pi of the discretised edge, are used, accordingly, as the building basis for the auxiliary surface construction (Fig. 4). To enable numerical control of the actual product edge adequacy this surface is further discretised by quadrilateral sub-domains, yielding point topology Σ G . Having those properties, i.e. emanating from points refΓ Pi of the discretised reference product edge Γ ref ( Γ ref ⊂ Σ G ) and being at those points perpendicular to the reference product geometry G ref , the surface Σ G may be considered in the iteration process as the target surface, which is to be attained by the edge points actΓ Pi k of the actual product Gactk , as closely as possible.

For a determination of the auxiliary surface Σ G , which is computed before starting the optimisation procedure, computation of the surface normal vectors at points lying on the edge Γ ref of the reference product geometry G ref is required.

Σ Fig. 4. Auxiliary surface G determination

3.2 Determination of the Actual Product Edge Deviation In the k-th iteration, the deviation dik of individual edge point actΓ Pi k of the actual product geometry G kact is determined by identifying its distance to the auxiliary surface Σ G in the direction normal to the edge. The applied procedure may be divided into three parts. In the first part, the edge normal vector Γ k at point actΓ Pi k is determined in accordance act n i with the procedure, described in Section 2.3. With the surface normal vector act nik and tangential vector Γ k computed by Eqs. (8) and (9), respectively, act t i considering surface and edge topology properties of the actual product geometry Gactk , the vector actΓ nik is determined by Eq. (10).

Σ

Σ

Fig. 5. Finding surface element Si on G intersected by a line collinear with vector actΓ n ik

In the second part, intersection with the auxiliary surface Σ G is sought in the direction collinear with

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vector actΓ nik . As the auxiliary surface Σ G is described discretely by quadrilaterals (Fig. 2), it may be assumed that only one surface element on Σ G is intersected.

Being actually related to the edge point actΓ Pi k in the corresponding determination of the deviation dik , let Σ us denote this surface by S i (see Fig. 5) and its nodal Σ j points by Pi ( j = 1, 2,3, 4) .

To enable evaluation of the deviation dik Σ analytical surface reconstruction of surface S i , Σ Fi ( x, y, z ) = 0 , is required, which is carried out in the third part. Apart from the given position vectors Σ Pi j ( j = 1, 2,3, 4) , the respective surface normal vectors Σ nij ( j = 1, 2,3, 4) at those points are also needed in accordance with Section 2.4. The respective normal vectors’ determination follows the procedure described in Section 2.2.

Although the analytical surface reconstruction of Σ surface S i could be done using the global coordinate system ( x, y, z ) , as demonstrated in Section 2.4, it is Σ better to have the surface S i representation defined with respect to a local coordinate system, say ( x , y , z ) , which is built by taking the vector triple actΓ t ik , act nik , Γ k act n i as the system basis and setting the edge point Γ k as its origin (Fig. 6), actΓ P ik = (0, 0, 0) . In this act Pi way, occurrence of round-off error due to possible computing with large numbers is avoided and accuracy of the computation ensured.

After performing all coordinate transformations on the position vectors Σ Pi j ( j = 1, 2,3, 4) and surface normal vectors Σ nij ( j = 1, 2,3, 4) , as shown in Section 2.2, and obtaining vectors Σ P i j and Σ n ij , Σ the geometry of the considered surface element S i can now be functionally approximated in the local coordinate system ( x , y , z ) . By adopting the same functional representation as considered in Section 2.4, we have: Σ

f i ( x , y , z ) = z − (a1i + a2i x + a3i y + + a4i x y + a5i x 2 + a6i y 2 + + a10i x 3 + a11i y 3 + a12i x 3 y 3 ) = 0. (13)

As in Eq. (12), but expressed in terms of the local coordinate system, the evaluation of the polynomial coefficients ami (m = 1, 2,...,12) is based on given topological requirements fulfilment.

Finally, with the interpolation function Σ f i ( x , y , z ) = 0 fully defined, the deviation dik of the considered edge point actΓ Pi k of the actual product geometry Gactk can be readily determined. Let us first remember that the coordinate z axis has been chosen in such a way as to coincide with the edge normal 244

dik = a1i , (14)

and the respective deviation vector dik is:

dik = dik ⋅ actΓ nik . (15)

Fig. 6. Edge geometry deviation determination

The deviation of the actual product edge geometry from the reference one, which is established by the deviation vector set {dik } , determines the appropriateness of the blank shape geometry Gblk used in the forming process simulation in the given iteration. The maximal shape deviation has been chosen as a measure of this appropriateness:

 2 + a9i x 2 y 2 + + a7i x 2 y + a8i xy

vector actΓ nik with the origin set at point Γ Pi k , i.e. Γ k act Pi = (0, 0, 0) . Thus, since the deviation is quantified by the distance measured from point actΓ Pi k to the auxiliary surface Σ G in the direction normal to the edge, it can be retrieved directly from the functional representation Σ f i ( x , y , z ) = 0 , when taking x = y = 0 in Eq. (13). The established deviation is evidently:

{ }

k d max = max dik , i = 1, 2,..., N act . (16)

The deviation of the actual product edge geometry is computed in each, say, kth iteration. Its determination requires computation of edge normal vectors actΓ nik at points lying on the edge Γ kact of the actual product geometry Gactk and finding a Σ corresponding surface element S on the auxiliary i Σ surface G . The respective surface normal vectors Σ Σ j ni ( j = 1, 2,3, 4) at points of the surface element S i , which are also required in computation of the actual product edge geometry deviation, are computed only once, i.e. after the auxiliary surface Σ G determination (see Section 3.1).

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3.3 Blank Shape Geometry Correction In the case where the convergence criterion is not k > dtol , a correction of the blank shape met, i.e. d max geometry Gblk is required. In our approach, this is carried out on two levels separately, first, by carrying out the repositioning of the edge points blΓ Pi k , which is then followed by readjustment of the interior points bl Pi k appertaining to the FE mesh. Correction of the edge blank shape geometry ΓGblk is carried out based on the established deviation vector set {dik } , Eq. (15), with repositioning of points blΓ Pi k applied in the direction of the edge normal vectors blΓ nik , Eq. (10), considering deviation magnitudes dik . The positions of the points lying on the edge Γblk+1 of the newly defined blank shape are thus computed by the equation:

Γ bl

a)

Pik +1 = blΓ Pik + dik ⋅ blΓ nik . (17)

In order to preserve the quality of the simulation in the subsequent iteration, the aspect ratio of the existing FE mesh, used in the description of the blank shape geometry Gblk , should not be significantly changed by repositioning of the edge points blΓ Pi k .

However, this threat can be efficiently neutralised by adequate readjustment of the interior points bl Pi k considering the applied edge repositioning, Eq. (17). Numerically, this can be achieved by exposing the blank having shape geometry Gblk to given displacement boundary conditions (Fig. 7a) and solving the corresponding boundary value problem, while assuming isotropic linear elastic behaviour. The blank edge nodes are displaced in accordance with the prescribed correction of the blank shape, Eq. (17), while nodes lying on the symmetry planes are subject to symmetric boundary conditions. The result of such computer analysis is the adjusted FE mesh specifying k +1 the blank shape geometry Gbl , which will be used in the subsequent iteration. The adjustment is carried out in accordance with the computed displacement field (Fig. 7b).

4 CASE STUDY - BLANK SHAPE OPTIMISATION In this section, the developed blank shape optimisation method is validated by considering a particular product with the prescribed edge geometry, manufactured by sheet metal forming (Fig. 8). The product’s geometry, which is originally double planar symmetric by design specification, is characterised by two deep cut outs extending downwards to the cup’s bottom.

b)

k

k +1

Fig. 7. Blank shape geometry correction Gbl → Gbl ; a) applied displacement boundary conditions, and b) adjustment of the FE’s mesh

Computer simulations supporting the blank shape geometry optimisation in the considered case study have been treated as three-dimensional and carried out using the FEM computer program ABAQUS. The numerical simulation of the complete production cycle of the formed product, which consists of a loading and unloading stage, associated respectively with irreversible plastic deformation and elastic strain relaxation, has been carried out considering particularities in the numerical treatment of the problem. Accordingly, the forming process simulation is carried out using an explicit time integration scheme in ABAQUS Explicit [30], whereas the springback simulation is carried out by ABAQUS Standard [31] using an implicit time integration scheme.

For computationally efficient integration of the constitutive equations, the NICE scheme, an explicit integration scheme developed in our laboratory,

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see Halilovič et al. [3] and Vrh et al. [4], has been implemented into ABAQUS/Explicit via the VUMAT subroutine [32]. The optimisation procedure is carried out using a closed loop Fortran program that has been upgraded by an Abaqus’ Python script to enable display of the optimisation results in each iteration.

After the optimisation blank shape geometry procedure is completed, which is treated in Section 4.2, both the conceived forming process simulation model and the proposed blank shape geometry optimisation procedure are validated experimentally, which is presented in Section 4.3.

the hardening plastic behaviour of the given material is plotted in Fig. 9. Since no significant reversed plastic bending occurs during the considered forming process, possible kinematic hardening is neglected and isotropic hardening is assumed.

4.1 Initial Blank Shape, Reference Product Geometry and Tool Geometry The geometry of the considered product (referred as the “reference geometry” in this paper) to be obtained by a sheet metal forming process, where a further sheet metal cutting phase is excluded, is shown in Fig. 8a. An aluminium sheet blank of 1.0 mm thickness is used. The cross-section of the corresponding forming tool assembly is shown in Fig. 8b. The forming tool is defined in such a way that its die geometry is determined by the reference product geometry, with a constant clearance of 1.2 mm ensured between the die and the punch, when the tool is in a closed position.

In accordance with the assumptions given in Section 2.5, the tool geometry is considered fixed and not subject to possible variation. The design parameters specifying the initial blank shape geometry, used to start the described numerical optimisation blank shape geometry procedure, are displayed in Fig. 8c.

a)

b)

4.2 Determination of Optimal Blank Shape Determination of the blank shape geometry appropriate for the production of the product, displayed in Fig. 8a, is carried out iteratively following the optimisation procedure, described in Section 3.

Thus, a corresponding forming process computer simulation is performed in each iteration, considering the actualised blank shape geometry.

In simulation, the following material and technology process data have been taken into consideration. It is assumed that the product is made from a 1mm thick aluminium sheet exhibiting orthotropic material behaviour, with a Young’s modulus of 72500 MPa, a Poisson’s ratio of 0.3, an initial yield limit of 105 MPa, and coefficients describing orthotropic material properties in relation to the potential function Hill 48 being: Rxx = 1.0 , Ryy = 0.95 and Rxy = 0.97 . The yield curve specifying 246

c)

Fig. 8. Product and forming process geometry design specification; a) product reference geometry, b) forming tool assembly, and c) initial blank shape geometry

To avoid wrinkling a blank-holder force of 5.2 kN in total is applied. The design prescribed

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clearance between the die and the punch in a closed position, which is set to 1.2 mm, allows for a potential increase in the sheet thickness during progressing of the punch. For the forming process simulation to be realistic the tribological conditions between the surfaces in contact with one another have to be adequately described. In this investigation the Coulomb friction law is adopted with the friction coefficient between the sheet metal and the tool parts assumed to equal 0.12.

Fig. 9. Yield curve of aluminium sheet

quadrilateral shell FEs with a characteristic size of 1.2 mm and with 11 integration points evenly distributed through the shell thickness.

Once the numerical simulation model is completely defined, the steps in the optimisation procedure, given by the flow chart in Fig. 3, can be carried out and the optimal blank shape geometry is achievable iteratively. In Fig. 11, the deviation of the product edge geometry from its reference geometry along the edge contour, as determined by the computer simulation in the considered iteration, is plotted. From these plots it is evident that the developed optimisation procedure is computationally very efficient. Namely, the maximal product edge deviation computed when the initial blank shape is considered in the forming simulation is equal to 3.1 mm. After the first iteration this deviation is reduced to less than 0.3 mm and after two additional iterations to less than 0.001 mm. The shape variation, to which the blank of initial shape geometry is subject when it is optimised using the presented approach, is displayed in Fig. 12.

4.3 Experimental Validation of the Blank Shape Optimisation

Fig. 10. Numerical model of the forming process

The forming process simulations are based on a FE model, which is conceived in such a way as to cope efficiently with all peculiarities encountered when numerically modelling such a complex problem.

Since both the tool and the blank shape geometries exhibit planar symmetry with respect to the xz and yz planes and orthotropic material properties are taken into account, only a quarter of the product can be considered in the model (Fig. 10), with symmetry boundary conditions applied and a given blankholder force of magnitude 1.3 kN. In the FE model, the tool surfaces are modelled by rigid FEs with a characteristic size of 1.0 mm, while the sheet metal is modelled by deformable reduced integration

In order to validate the developed approach to blank shape geometry optimisation in the investigated case study experimentally, the forming tool parts have been manufactured and assembled in accordance with the design scheme (Fig. 8b). In fact, the forming tool is designed in as simple a way as possible. The punch movement is guided by four cylindrical guides, which also have the function of limiting the punch bottom position. The blank-holder is attached to the tool die by eight sets of bolts and springs under tension so that required blank-holder force is ensured. The blanks used in the experiment are cut out of an aluminium sheet by a water jet cutter.

The experiment was carried out on a HI-KON (model HK250S2) forming press with corresponding measurements of the product edge geometry done by the 3D measuring machine Brown & Sharpe, Mistral 7-10-7, its accuracy classified as being Âą0.004 mm.

The product edge geometry was measured at 6 control points along the edge contour (see Fig. 13 for their position).

To validate the accuracy of the determined blank shape geometry and computed product shape, two experiments were performed, considering different blank shapes. In the first experiment, the product is formed from the blank of initially defined shape

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geometry (Fig. 8c) in order to confirm the physical and numerical appropriateness of the forming process simulation model definition used in the blank shape optimisation procedure. The difference between experimentally (Fig. 13, Exp. initial) and numerically (Fig. 13, Num. initial) obtained product edge geometry deviation is less than 0.20 mm.

the adequacy of the numerically optimised blank shape geometry. As it can be seen from the plots in Fig. 13, the difference between experimentally (Fig. 13, Exp. optimised) and numerically (Fig. 13, Num. optimised) determined product edge geometry is of the same order of magnitude as in the first experiment and is less than 0.22 mm. Therefore, we can confirm that the applied optimisation procedure is appropriate.

Photographs of the formed products, produced by the blanks having initial and optimised shape geometry, are displayed in Fig. 14.

Fig. 13. Product edge deviation

Fig. 11. Product edge deviations during iteration process

Fig. 14. Formed product with initial and optimised blank shape

5 CONCLUSION

Fig. 12. Optimised vs. initial blank shape geometry

The difference between the computer simulation and experiment being in the same range as the accuracy of the cutting machine (±0.1 mm), the adequacy of the simulation model can be confirmed.

The second experiment was carried out to confirm 248

In this paper, a blank shape optimisation method that enables determination of appropriate blank shape geometry, by which a formed product of prescribed geometry can be produced, has been presented. The developed method is based on an iterative procedure where in each iteration the adequacy of the actual blank shape geometry is checked by identifying the edge geometry deviation of the resulting product geometry, which is obtained by means of the

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corresponding forming process computer simulation.

As demonstrated numerically and subsequently validated experimentally, the blank shape optimisation method used here for a specific case study with rather complex product geometry is characterised by computational efficiency and robustness.

6 ACKNOWLEDGMENT The second author appreciates the support provided in the operation part financed by the European Union, European Social Fund. The operation was implemented within the framework of the Operational Programme for Human Resources Development for the Period 2007 to 2013.

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[3] Fan, J.P., Tang, C.Y., Tsui, C.P., Chan, L.C., Lee, T.C. (2006). 3d finite element simulation of deep drawing with damage development. International Journal of Machine Tools and Manufacture, vol. 46, no. 9, p.

1035-1044, DOI:10.1016/j.ijmachtools.2005.07.044.

[4] Vrh, M., Halilovič, M., Starman, B., Štok, B., Comsa, D., Banabic, D. (2011). Earing prediction in cup drawing using the bbc2008 yield criterion. AIP Conference Proceedings, vol. 1383 no.1, p. 142-149, DOI:10.1063/1.3623604.

[5] Halilovič, M., Vrh, M., Štok, B. (2009). Nice-an explicit numerical scheme for efficient integration of nonlinear constitutive equations. Mathematics and Computers in Simulation, vol. 80, no. 2, p. 294-313, DOI:10.1016/j. matcom.2009.06.030.

[6] Vrh, M., Halilovič, M., Štok, B. (2010). Improved explicit integration in plasticity. Inernational Journal of Numerical Methods in Engineering, vol. 81, no. 7, p. 910-938.

[7] Vrh, M., Halilovič, M., Štok, B. (2011). The evolution of effective elastic properties of a cold formed stainless steel sheet. Experimental Mechanics, vol. 51, p. 677695, DOI:10.1007/s11340-010-9371-1.

[8] Vrh, M., Halilovič, M., Starman, B., Štok, B. (2011). A new anisotropic elasto-plastic model with degradation of elastic modulus for accurate springback simulations.

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2, p. 217-225, DOI:10.1007/s12289-011-1029-8.

[9] Koc, P., Štok, B. (2008). Usage of the yield curve in numerical simulations. Strojniški vestnik - Journal of Mechanical Engineering, vol. 54, no. 12, p. 821-829.

[10] Batoz, J.L., Duroux, P., Guo, Y.Q., Detraux, J. M. (1989). An efficient algorithm to estimate the large strains in deep drawing. Proceedings of the NUMIFORM, p. 383-388.

[11] Batoz, J.L., Guo, Y.Q. (1997). Analysis and design of sheet forming parts using a simplified inverse approach.

COMPLAS V, Barcelona, p. 178-185.

[12] Batoz, J.L., Guo, Y.Q., Mercier, F. (1998). The inverse approach with simple triangular shell elements for large strain predictions of sheet metal forming parts.

Engineering Computations, vol. 15, no. 7, p. 864-892, DOI:10.1108/02644409810236894.

[13] Chung, K., Richmond, O. (1992). Ideal Forming-I. Homogeneous deformation with minimum plastic work. International Journal of Mechanical Sciences, vol. 34, no. 7, p. 575-591, DOI:10.1016/00207403(92)90032-C. [14] Chung, K., Richmond, O. (1992). Ideal Forming-II. Sheet forming with optimum deformation. International Journal of Mechanical Sciences, vol. 34, no. 8, p. 617633, DOI:10.1016/0020-7403(92)90059-P. [15] Azaouzi, M., Belouettar, S., Rauchs, G. (2010). A numerical method for the optimal blank shape design.

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[16] Azaouzi, M., Naceur, H., Delamoziere, A., Batoz, J.L., Belouettar, S. (2008). An heuristic optimization algorithm for the blank shape design of high precision metallic parts obtained by a particular stamping process.

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[17] Naceur, H., Guo, Y.Q., Batoz, J.L. (2004) Blank optimization in sheet metal forming using an evolutionary approach. Journal of Materials Processing Technology, vol. 151, no. 1-3, p. 183-191, DOI:10.1016/j.jmatprotec.2004.04.036.

[18] Park, S.H., Yoon, J.W., Yang, D.Y., Kim, Y.H. (1999) Optimum blank design in sheet metal forming by the deformation path iteration method. International Journal of Mechanical Sciences, vol. 41 no. 10, p.

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[19] Yeh, F.H., Wu, M.T., Li, C.L. (2007). Accurate optimization of blank design in stretch flange based on a forward inverse prediction scheme. International Journal of Machine Tools and Manufacture, vol.

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[20] Lin, C.T., Kwan, C.T. (2009). Application of abductive network and fem to predict the optimal blank contour of an elliptic cylindrical cup from deep drawing. Journal of Materials Processing Technology, vol. 209, no. 3, p.

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[21] Parsa, M.H., Pournia, P. (2007). Optimization of initial blank shape predicted based on inverse finite element

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[24] Hamammi, W., Padmanbhan, R., Oliveira, M.C., BelHadjSalah, H., Alves, J.L., Menzes, L.F. (2009), A deformation based blank design method for formed parts. International Journal of Mechanics and Materials in Design, vol. 5, no. 4, p. 303-314, DOI:10.1007/s10999-009-9103-9.

[25] Fazli, A., Arezoo, B. (2012). A comparison of numerical iteration based algorithms in blank optimization. Finite Elements in Analysis and Design, vol. 50, p. 207-216, DOI:10.1016/j.finel.2011.09.011.

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[27] Padmanabhana, R., Oliveiraa, M.C., Baptistaa, A.J., Alvesb, J.L., Menezesa, L.F. (2009). Blank design for deep drawn parts using parametric nurbs surfaces. Journal of Materials Processing Technology, vol. 209, no. 5, p. 2402-2411, DOI:10.1016/j. jmatprotec.2008.05.035.

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[30] SIMULIA. (2008). ABAQUS/Explicit, 6.8-1, Simulia, Providence.

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DOI:10.5545/sv-jme.2012.315 (1.01) Original Scientific Paper

Received for review: 2012-01-18 Received revised form: 2012-12-17 Accepted for publication: 2013-02-12

Exergy Based Performance Analysis of a Shower Cooling Tower Qi, X. – Liu, Y. – Liu, Z. Xiaoni Qi1,* – Yongqi Liu1 – Zhenyan Liu2

1 Shandong 2 School

University of Technology, College of Traffic and Vehicle Engineering, China of Mechanical and Power Engineering, Shanghai JiaoTong University, China

The present study provides a descriptive mathematical model of energy and exergy for a shower cooling tower (SCT). The model is used to predict the variation in temperature and exergy along the tower length. The validity of the model for predicting variations in gas and liquid characteristics along the tower length was examined against some operating data measured in a cooling tower company. The results show that the exergy of water decreases as tower height increases. The distribution of the exergy loss is high at the bottom and gradually decreases moving up to the top of the tower. Moreover, 1.50 m/s air velocity results in less exergy destruction. With a decrease in the size of the water droplets, the fluids carrying energy have more opportunities for mass and energy transfers.

Keywords: shower cooling tower, exergy, heat and mass transfer

0 INTRODUCTION Cooling towers remain widely used in industry for cooling circulating water [1] to [3]. In a conventional cooling tower, fill material acts as the media of heat and mass transfer, by means of which the waste heat is rejected from the system. The effect of the fill is to distribute the water stream and provide a larger surface area for contact between the air and the water. However, fouling of cooling tower fill is one of the most important factors affecting its thermal performance as it reduces cooling tower’s efficiency and capability. In a conventional cooling tower, due to salt deposition on the packing and subsequent airflow block, cooling performance of the tower declines after a period of operation time [4] to [5]. On the other hand, the existence of the fill strongly increases the draught drag, which is the main constituent of power consumption. Eliminating the fill leaves the tower completely empty which is of far-reaching significance in cooling turbid water with high temperature [6]. The application of packed cooling towers to industry is not practical due to salt deposition on the packing and subsequent blockage.

Studies of shower cooling tower (SCT) have been reported sporadically over the years. Givoni developed a kind of SCT used for cooling buildings in 1995 [7] to [8], which consists of an open shaft with showers at the top and a collecting pond at the bottom. A pond at the bottom of the shaft collects the sprayed water for recirculation by a small pump. Givoni introduced the system and performed a test to analyze thermal performance. His students tested and compared the performances of this system in three very different climates in another paper. However, they didn’t give any theoretical analysis to assist the system. In China, SCT attracts many investigators’ attention and is becoming a research hotspot. However, there are no *Corr. Author’s Address: Shandong University of Technology, Zibo, China, qixiaoni@sjtu.org

papers presenting a detailed analysis of the cooling characteristics of an SCT. Given the present state of research, no systematic theory for studying SCT is available. Most studies are focused on experiments, which deal with many parameters. So perpetual modifications in a great deal of experiments are needed to finalize the results [9]. Furthermore, the investigators come up against variour problems of technology and theory in the experimental investigation. The first part includes the movement mechanism of the droplets and the matching of various parameters. In order to solve this problem, the researchers need to carry out largescale system experiments, which consume a lot of manpower and material resources. Moreover, the flexibility is worse when different experimental results are used to analyze the same problem, because the experimental results cannot conform to the diversity of the environment. The other part includes the creation of a mathematical model for SCT. The first part can be solved by solving the second part.

Therefore it is important to establish a reasonable mathematical model of heat and mass transfer for the process of cooling hot water in the SCT, because the results of the numerical simulation can then guide the experimental research and theoretical analysis.

However, the process of heat and mass transfer between the air and the water in the tower is very complicated. The influence factors include the interior structure, the direction and amount of spraying water, environmental factors, the air mass flow rate, wind velocity, the mass flow rate and temperature of inlet water and so on. The cooling performance has a complicated non-linear relationship with these factors and it is difficult to establish perfect mathematical model for a SCT. The standard method for establishing a model can identify some quantitative relationships to show the heat and mass performance, but there are 251

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too many assumptions, the process is complicated, and the precision is lower.

Exergy is the maximum work potential that can be obtained from a form of energy [10] and [11].

Exergy analysis is a useful method to complement, but not to replace, energy analysis. Exergy analysis yields useful results because it deals with irreversibility minimization or maximum exergy delivery. Exergy analysis can indicate the possibilities for thermodynamic improvement of the process under consideration. Exergy analysis has proven to be a powerful tool in thermodynamic analyses of energy systems. Recently, the concept of exergy has received great attention from scientists, researchers, and engineers, and the exergy concept has been applied to various utility sectors and thermal processes. In general, more meaningful efficiency can be evaluated by using exergy analysis rather than energy analysis, since exergy efficiency is always a measure of the approach to the ideal.

One important feature of exergy analysis for a system that undergoes a psychrometric process, such as in cooling tower operations, is that the total exergy can be split into thermomechanical and chemical components, thereby enabling one to quantify the contribution of each term to the total exergy through the tower. Shukuya and Hammache [12] stated that thermomechanical and chemical exergy play important roles in assessing the actual thermodynamic merit of psychrometric process application. Thirapong [13] used exergy analysis to explain the performance of a conventional cooling tower.

Currently, little is known about the applicability of exergy analysis to shower cooling tower studies.

In this paper, shower cooling tower performance is predicted by using heat and mass transfer between water and air to drive the solution to steady-state conditions. The second law is used to take account of exergy distributions of water and air in the cooling tower. An investigation of the calculated results can be used to further understand the details of exergy in shower cooling towers.

1.1 Mathematical Model at the Water Droplet Level To better investigate the cooling performance of the SCT, it is helpful to first study the heat and mass transfer process at the single water droplet level.

This situation is presented schematically in Fig. 1. The motion acceleration, turbulence intensity, internal gyration, and evaporation of the water droplet influence the motion characteristics in SCT. However, in order to describe the motion characteristics of the water droplet in mathematical language, some assumptions concerning the water droplet are made.

As shown in Fig.2, the water droplet is assumed to be spherical and its diameter is small enough so that the temperature of the water droplet can be assumed to be uniform; all water droplets have a uniform diameter and uniform motion tracks; the lesser impact factors such as the possibility of collision or scatteration of the water droplet during the motion process, libration and the nonuniformity of internal flow and temperature distribution and so on are ignored in this investigation.

Qe Ta

Tw Water droplet Air inlet

In a SCT, for the sake of computational simplification, when deriving governing equations, the following assumptions were made, which have no evident influence on the cooling result:

Air outlet

Qc

dd

1 EXPERIMENTAL SETUP HEAT AND MASS TRANSFER OF AN SCT

252

1) Heat and mass transfer coefficients are constant within the tower. Both the cool air and hot water have constant physical properties.

2) The vapor pressure in the tower is so low that it has little influence on the pressure throughout the entire tower. Therefore the average value of atmospheric pressure is used in the calculation.

3) The water droplets moving in the tower are in the shape of ball, and the Soter average diameter is assumed to be equivalent to the diameter of the drop.

4) The exterior and interior temperature of the drop is uniform, and thermal resistance for the water drop is negligible.

5) The Lewis factor is equal to 1 [14].

6) Because the whole motion direction of the water droplet is vertical, it is assumed that the water droplet rises or falls vertically in one dimension.

ud

ua

Hot water inlet

Air inlet

Cold water outlet

Fig. 1. Schematic representation of the energy exchange at the water droplet level

Qi, X. â&#x20AC;&#x201C; Liu, Y. â&#x20AC;&#x201C; Liu, Z.

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The heat rejected from the water droplets includes convective heat and evaporative heat. The water droplet loses heat to the air at the expense of its internal energy, and an energy balance on a control surface surrounding the water droplet yields:

where:

dU d = − ( Qdc + Qde ) , (1) dt

U d = md c pw Tw , (2)

Qdc = hc Ad (Tw − Ta ), (3)

Qde = hd Ad ( wsw − wa ) iv , (4)

forces exerted on the droplet moving with certain velocity include gravity, Gd, air buoyancy, Fd, and air resistance, Rd. They are expressed as gravity:

Gd = md g = π d d3 ρ w g / 6, (12)

buoyancy:

Fd = π d d3 ρ a g / 6, (13)

drag:

Rd = π Cd ρ a U 2 d d2 / 8. (14)

According to the motion condition, the drag coefficient, Cd, is expressed as [9]:

Combining the above equations gives:

Cd = 18.5 / Re3/ 5 ,

dTw [h (T − T ) + hd ( wsw − w) iv ] Ad =− c w a = ρ w md dt

=−

6hd [le f (imasw − ima ) + (1− le f )( wsw − w) iv ] c pw ρ w d d

. (5)

The water evaporation rate, associated with mass transfer, is equal to:

dmd = hd ( wsw − wa ) Ad , (6) dt

where

hc = N ud λ / d d , (7)

Red = (ud + ua )d d / ν , (8)

Pr = µ c pa / λ , (9)

Nu = 2 + 0.6 Re1/ 2 Pr1/3 , (10)

dmd dm dwa = w = N d = ma dt ma m h ( w − wa ) Ad = ( w ) d sw dz. (11) ma md ud

1.9169 ≤ Re < 508.3917, ( transition flow ) (15)

Re ≥ 508.3917, ( turbulence flow ) . (16)

Cd = 0.44,

The physical definition and Newton’s second law of motion is given as: ud = dz / dt , (17)

= ad

Combining the two equations given above, we obtain the kinetic equation for water droplets in the SCT as:

ρ wu d

dud = ( ρ w − ρ a ) g − 3Cd ρ a (ud + ua ) 2 / 4d d . (19) dz

1.3 Heat Transfer Equation in the Tower

mw+dmw iw+diw

ma(1+w+dw) ima+dima

mw iw

ma(1+w) ima

dz

Please refer to Bosnjakovic [8] for a discussion on the derivation and development of the Lewis factor.

1.2 Force Analysis of Water Droplets In our previous publication, the force exerted on a single water droplet was analyzed in detail. Only the results are used in this work. In the motion process of a water droplet sprayed from the nozzle, the

dud du = ud d . (18) dt dz

Fig. 2. Control volume of SCT

As shown in Fig. 2, energy should be kept balanced between the droplet and the air, so the equation can be expressed as:

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dQ = dQc + dQe , (20)

where dQc is the sensible heat between the air and the water caused by the temperature difference, which can be expressed as:

dQc = hc (Tw − Ta )dA. (21)

dQe is the evaporative heat caused by the concentration difference between the saturated vapor around the water droplets and the ambient air which is expressed as:

high (Re < 10), it can be regarded as an evaporation process for a static drop in the air; when the relative velocity of the liquid droplet to the airstream is higher (10 < Re < 1800), heat and mass transfer is promoted.

Under this latter condition we can use a some of the heat and mass transfer equations given in previous references. After comparing the application range, the Frossling equations, whose application range is 10 < Re < 1800, were adopted.

As Ta is between 0 and 100 °C, let Sc = 0.63 and substitute this value into Eq. (11) to obtain the following:

dQe = iv hd ( wsw − wa )dA. (22)

hd =

As the water temperature is Tw, the enthalpy of the water vapor is iv = ifgw0 + cpv Tw , and the enthalpy of the saturated water vapor is:

Dc =

imasw = c paTw + wiv + ( wsw − w)iv . (23)

Combining the above two equations yields the following equation:

imasw = c paTw + wiv + ( wsw − w)iv . (24)

Because the enthalpy of wet air per unit mass of wet air can be expressed as:

ima = c paTa + w(i fgw0 + c pvTa ). (25) Subtracting Eq. (25) from Eq. (24) yields:

Tw − Ta =

(imasw − ima ) − ( wsw − w)iv . (26) c pma

Supposing lef as 1 yields: dQ hd = (imasw − ima )dA. (28) ma ma

In the cooling tower without fill:

dima dQ m 6hd = =( w) (imasw − ima ). (29) dz ma ma ρ wud d d

According to reference [15], when the relative velocity of the liquid droplet to the air stream is not 254

(1 + 0.276 Re d 1/ 2 Scd 1/ 3 ), (30)

0.0805 Ta + 273 1.8 4 ( ) × 9.8 × 10 . (31) pa 273

In order to study the temperature and heat transfer of the water droplet, we divided the whole motion process z into N finite sections with thickness dz.

Section n is filled with water droplets that enter the section at a temperature of Tw,n and exit at Tw,n+1.

Consequently, the velocity, retention time, height at the outlet of the section n: uw,n+1, tn+1 and zn+1, respectively differ from the inlet values- uw,n, tn and zn, respectively. The Runge-Kutta method is applied to solve Eqs. (5), (11), (19), (30) and (31), thus allowing us to obtain the temperature of the droplets in each differential section and the outlet water temperature.

2 EXERGY CALCULATION

 h  h (imasw − ima ) + (hd − )iv ( wsw − w)  dA. (27) dQ =  c pma  c pma 

dima =

dd

1.4 Numerical Solution

Substituting Eq. (26) into Eq. (20) yields:

ρ a Dc

According to recent definitions, the exergy of a system may be classified as having thermomechanical and chemical exergies. The former may be divided into three types: physical, kinetic, and potential exergies.

Physical exergy is the maximum amount of obtainable work when the stream is brought, by a reversible process, from its initial state at T and p to a state at T0 and p0 that is in thermal and mechanical equilibrium with the environment. The equilibrium state is referred to as the restricted dead state [10]. Most of the time, variations in potential and kinetic exergies can be neglected and hence they are not considered in exergy analysis.

Let each species j reach its partial pressure in the mixture to get to chemical equilibrium. The system will then reach its dead state. Therefore, the

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chemical exergy of a material stream is the maximum achievable work needed to go from one restricted dead state to another dead state. It is said that the system is at dead state when its pressure, temperature, composition, velocity, or elevation are equal to the corresponding environment parameters.

The total exergy content of a material stream is calculated by summing up these above-mentioned exergies. The specific exergy in the psychrometric process – such as in the cooling tower operating mechanism, without the effect of kinetic and potential energy, in a steady state – can thus be generally represented as:

ex = exth + exme + exch . (32)

The total exergy consists of three parts: thermalmechanical exergy, exth, mechanical exergy, exme, and chemical exergy, exch. The three items are represented as:

exth = (c pa + ω c pv )T0 (

T T − 1 − ln ), (33) T0 T0

exme = (1 + 1.608ω ) RaT0 ln

exch = RaT0 [(1 + 1.608ω )ln

p , (34) p0

1 + 1.608ω0 s . (35) 1 + 1.608ω

So the total exergy can be written as: X air = ma [(c pa + ω c pv )(T − T0 − T0 ln

p + p0

+ RaT0 (1 + 1.608ω ) ln

1 + 1.608ω0 s + 1 + 1.608ω

ω ]. ω0 s

Total exergy entering

= [ X w, z ( j ) + X air , z ( j +1) ] + . (37) {I 14442444 3 Destroyed exergy Total exergy leaving

After rearrangement, the exergy destruction for the discrete height dH will be:

I = [ X w, z ( j +1) − X w, z ( j ) ] + [ X air , z ( j ) − X air , z ( j +1) ]. (38) 3 EXPERIMENTAL PROCEDURE

A schematic diagram of an experimental apparatus carrying out this study is shown in Fig. 1. The tested cooling tower is a mechanical draft-counterflow type where the height of the nozzle can be adjusted from 5 to 8 m, the cross sectional area is 10.87 m2. Hot water is pumped from the storage tank, which is heated by heaters and controlled by temperature controller, and enters the cooling tower at the top. The water flow rate is adjusted by the flow control valve installed next to the pump where its value is read by flow meter. Cooled water exits the tower at the bottom and recirculates to the storage tank throughout the duration of the test. Moist air is brought in by a fan and enters from the bottom, flowing past the tower, and is finally discharged into the atmosphere at the top of the tower.

T )+ T0

+(1 + 1.608ω ) RaT0 ln

+ RaT01.608ω ln

[ X w, z ( j +1) + X air , z ( j ) ] = 14442444 3

(36)

The rate of exergy destruction I, which is the loss in potential of air to recover the exergy supplied by the water, can be determined from the control-volume exergy balance equation. This equation is applied under steady state conditions and undergoes an adiabatic process with no work delivered. Assuming that the air-water thermodynamics properties are known at discrete points along the tower height, the exergy destruction for each incremental tower height dz is:

Fig. 3. Schematic diagram of experimental unit

The flow rate can also be measured by anemometer for air velocity with a known tower cross sectional area. The humidity ratio and dry bulb temperature distribution inside the tower are measured via hygrometer (psychrometer) in which the measuring tip is protected from contact with the water. The distribution of water temperatures inside the tower

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are measured by thermocouples. Before doing the experiment, water that was collected in the storage tank was heated to the desired temperature. The flow control valve was adjusted and the water flow rate was measured. Air velocity was measured at the air inlet. During the experiment, water was uniformly distributed from nozzles, while air was blown upwards. Water temperatures and air conditions were measured after steady state operating conditions were obtained. The diameter of the droplets were measured using an intensified high-speed camera.

the water mass flowrate is 55.21 m3/h, the air mass flowrate is 44206 m3/h.

4 RESULTS AND DISCUSSIONS 4.1 Model Verification The presented model was validated by using plant data obtained from Jiangsu Seagull Cooling Tower Co., ltd. in China. Table 1 represents the operating conditions and the characteristics of working fluids through the shower cooling tower. The comparative results are the outlet water temperature. It should be noted that the error between the predicted and experimental values are within 10%. Thus, this model can be used for predicting the conditions of water and air in shower cooling towers.

Table 1. Comparison of results of the heat and mass transfer model and the experiment data Nozzle height [m] Droplet diameter [mm] Initial velocity [m/s] Air velocity [m/s] Air to water ratio Dry-bulb temperature of inlet air [°C] Humidity ratio of inlet air Inlet water temperature [°C] Experimental outlet water temperature[°C] Computed water temperature based on heat and mass transfer model [°C] Reletive error [%]

8 1.1 4 2 0.8 34 0.78 50

6 0.8 6 2.5 0.9 33 0.7 44

7 0.8 5 2.3 0.9 34 0.68 42

43.7

37.8

36.2

44.3

38.1

36.7

9.52

3.85

8.67

4.2 Exergy Analysis Following the validation of the model, it seems appropriate to evaluate some exergic interactions in the SCT. Atmospheric conditions are considered to be at dead state with a dry-bulb temperature of 29 °C, wet-bulb temperature of 21.11 °C, and a pressure of pt = 100.4 kPa. The cross-sectional area of the tower is 10.87 m2, and the height of the nozzle is 5 m, 256

Fig. 4. Temperature profiles of water and air, and humidity ratio profile through the shower cooling tower

Fig. 4 is a plot of the water temperature, the drybulb temperature, and the humidity ratio of the air versus height of the tower. The water falls from the top and its temperature, Tw, decreases continuously as it approaches the bottom of the tower. This is generally expected in a shower cooling tower because the water loses heat both by convection and evaporation. It is interesting to see that the air, which enters from the bottom of the tower with an initial dry bulb temperature, Tdb, decreases in temperature and then increases before leaving from the top of the tower.

This can be explained by the fact that the water, which enters from the top of the tower, when it reaches the lower part, is cooled because of a predominantly evaporative mechanism. In this region, the water temperature, Tw, is much lower than the entering air dry bulb temperature, Tdb, however, as we note from Fig. 3, when the tower height from the top reaches above 2.9 m, the water temperature is less than Tdb. This results in heat transfer from the air to the water (i.e. negative convection). The intersection point of the Tdb and Tw curves indicates no temperature difference.

At this point, there is no convection heat exchange between the water and the air. Furthermore, below this point Tdb is less than Tw, which results in heat transfer from the water to the air (i.e. positive convection).

Fig. 5 is a plot of water exergy, Xw, and water temperature, Tw, versus tower height. As in a conventional cooling tower [13], water exergy, defined as the available energy carried by supplying water, decreases continuously from top to bottom. This can be explained by the fact that water temperature decreases from top to bottom as a result of supplying its exergy to air.

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 251-259

Fig. 5. Exergy of water and water temperature profiles through the shower cooling tower

when the tower bottom height is changed from 0.00 to 0.56 m. Furthermore, another 0.876 kW of exergy destruction is also destroyed when the tower height is changed from 0.56 to 1.12 m, and so on. These distributions of exergy destruction indicate that these are high at bottom and gradually a low at the top. The minimum value is found at the top. Fig. 8 also shows the effect of water droplet diameter on overall exergy destruction. Obviously, with a decrease in the water droplet diameter, the fluids carrying energy have more opportunities for mass and energy transfers. Highly intensive mass and energy interactions always occur with thermodynamic irreversibilities and entropy production, which result in increasing in exergy destruction.

Fig. 6. Effect of tower height on water exergy through the shower cooling tower

Fig. 7. Effect of air velocity on exergy destruction through the SCT (diameter is 15 mm)

The effect of the tower height on the exergy of water through the SCT is shown in Fig. 6. As the tower height increases, the exergy of water decreases because the heat and mass transfer rate from the water to the air is greater. The exergy of water supplied to the air decreases with decreasing height.

Exergy destruction is represented by the difference between the exergy change of water and the exergy change of air. Fig. 7. shows the exergy destruction along the SCT for three air velocities. It shows that the distribution of the exergy loss is higher at the bottom and gradually decreases towards the top of the tower. Hence, minimum exergy destruction is seen at the top of the tower for each air velocity.

Moreover, it shows that 1.50 m/s air velocity resulted in less exergy destruction. This is due to a large transfer from water to air.

The effect of the size of water droplets on exergy destruction through the SCT is shown in Fig. 8. For example, 1.19 kW of exergy destruction is destroyed

Fig. 8. Effect of size of water droplets on exergy destruction through the SCT

5 CONCLUSIONS In this study, energy and exergy analysis are carried out on a shower cooling tower based on mathematical

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modeling and simulation results. The method was validated using experimental data. In fact, the irreversibilities of any process destroy some inlet exergies. The results also show that the exergy of water is not completely absorbed by air and a notable portion of the exergy is always destroyed, much more in the bottom sections. The exergy analysis shows that exergy destruction increases with increasing water droplet diameter.

Water exergy, defined as the available energy carried by supplying water, decreases continuously from top to bottom. For the air side, exergy means the available energy of air to recover or utilize that supplied by water. There are two kinds of exergy in air that are due to the exergy of air via convective heat transfer and the exergy of air via evaporative heat transfer. Exergy destruction is high at the bottom and lower at the top. 1.19 kW of exergy destruction is destroyed when the tower bottom height is changed from 0.00 to 0.56 m. Furthermore, another 0.876 kW of exergy destruction is also destroyed when the tower height is changed from 0.56 to 1.12 m, and so on. The distributions of exergy destruction can be used as a guideline for finding the optimal potential for improving cooling tower performance. As water droplets drop to the bottom, the droplets' velocity increases rapidly, which means that the heat and mass transfer process between the water and fresh air is extremely short. Therefore, the useful energy delivered from water to air decreases, that is, the exergy destruction increases. For a shower cooling tower, this is an important constraining factor.

5 ACKNOWLEDGMENTS This work was supported by the Natural Science Foundation of China under contract No. 51249001 and the Jiangsu Seagull Cooling Tower Company Ltd.

imasw

iv I m i Lef Nu Pr Q Rd Re Sc Sh t T u U w z

Greek Letters λ thermal conductivity coefficient [W/m·K] μ dynamic viscosity coefficient [Pa/s] ρ density [kg/m3] Subscripts a air c convection d droplet db dry bulb e evaporation s saturated v vapor w water 1 inlet 2 outlet 7 REFERENCES

6 NOMENCLATURE A c Cd Dc Fd g Gd d ex hc hd ifgw0 258

specific enthalpy of saturated air evalutated at the local bulk water temperature [kJ/kg] specific vaporization heat of water [kJ/kg] exergy destruction [kW] mass [kg] specific enthalpy [kJ/kg] Lewis factor [-] Nusselt number Prantdl number heat transfer rate [W] drag [N] Reynold number Schmidt number Sherwood number time [s] temperature [°C] velocity [m/s] internal energy [J] humidity ratio of moist air evaluated at Ta [-] vertical coordinate [m]

area [m2] specific heat [kJ/(kg·°C)] drag coefficient on droplet [-] diffusion coefficient [m2/s] buoyancy [N] gravitational acceleration [m/s2] gravity [N] equivalent diameter [m] specific exergy [kJ/kg] heat transfer coefficient[W/m2·°C] mass transfer coefficient [kg/m2·s] specific latent heat of water at 0 °C [kJ/kg]

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[2] Kröger, D.G. (1998). Air-cooled Heat Exchangers and Cooling Towers, Begell House, New York, p. 50-56.

[3] Geoffrey, F.H. (2002). Heat Exchanger Design Handbook, Begell House, London, p. 137-151.

[4] Shi, L.Q. (2002). Application of the no-padding and atomizing cooling tower to industrial circulating cooling water. Metallurgical Power, vol. 3, p. 56-59.

[5] Li, P., Yang, L.Q. (2003). Application of the no-padding and atomizing cooling tower to circulating cooling

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water in steel industry. Shandong Metallurgy, vol. 4, p.

66-68.

[6] Li, H.L. (2003). Investigation and application of the nopadding and atomizing cooling tower. Industrial Water Treatment, vol. 8, p. 56-58.

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[9] Kotas, T.J. (2006). The Exergy Method of Thermal Plant Analysis. Butterworths, London, p. 223-237.

[10] Dovjak, M., Shukuya, M., Krainer, A. (2012). Exergy analysis of conventional and low exergy systems for heating and cooling of near zero energy buildings.

Strojniški vestnik - Journal of Mechanical Engineering, vol. 58, no. 7-8, p. 453-461, DOI:10.5545/svjme.2011.158.

[11] Moran, M.J. (1982). Availability Analysis: A Guide to Efficient Energy Use. Prentice-Hall, New Jersey, p. 91108.

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[13] Thirapong, M., Wanchai, A., Somchai, W. (2007). An exergy analysis on the performance of a counterflow wet cooling tower. Applied Thermal Engineering, vol. 27, no. 5-6, p. 910-917, DOI:10.1016/j. applthermaleng.2006.08.012.

[14] Johannes, C.K., Detlev, G.K. (2005). The Lewis factor and its influence on the performance prediction of wet-cooling towers. International Journal of Thermal Sciences, vol. 44, no. 9, p. 879-884, DOI:10.1016/j. ijthermalsci.2005.03.006.

[15] Guo, L.J. (2002). Two-phase and Multi-phase Hydrokinetics. Xian Jiaotong University, Xian, p. 251268.

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DOI:10.5545/sv-jme.2012.569 (1.01) Original Scientific Paper

Received for review: 2012-05-03 Received revised form: 2012-09-24 Accepted for publication: 2012-11-06

Experimental Identification of Overall Structural Damping of System Radoičić, G. – Jovanović, M. Goran Radoičić – Miomir Jovanović*

University of Niš, Faculty of Mechanical Engineering, Serbia The dynamic behaviour of large and complex structures largely depends on damping resistance in the structure. A portion of the structural energy is lost to deformations in material, friction between the contact surfaces, and relative motion within the structure. Often, in an analysis of numerical models, before the dynamic analysis of transient events (transient analysis), the damping resistance is adopted on the basis of recommendations, which implies an error of transient response (introduced by frequencies, logarithmic decrements and maximal amplitudes).

Decreasing amortized vibratory movement is dependent on the extent of the structural damping. This paper presents the importance of structural damping in structural analysis and shows the experimental and theoretical procedure for identifying G values ​​of the structural damping coefficient. A model for determining the G coefficient is shown in the example of a real tower crane structure. The experimentally obtained values were then used in the transient numerical FEM analysis, as the basis for adopting the conclusions about the dynamic behaviour of this class of structures (transportation machines). The effect of the external perturbation force of trapezoidal impulse form (lifting and quickly lowering of load) is introduced and the dynamic task, as an example of the use of the G coefficient G, is solved. The experimentally determined damping (theoretically isolated for tall truss structures) can be used in similar transient analyses.

Keywords: tower crane, truss structure, structural damping, transient response, large amplitude

0 INTRODUCTION Building sites with large construction machinery structures are very numerous and widespread today.

On a single building site there is often a significant number of tower cranes, indicating the importance of this class of construction machinery. Permanent exposure to natural forces (strong winds, earthquakes, etc.) and combinations of working conditions increase the risk of a possible incident. An understanding of the dynamic behaviour of the support structures for such machinery will minimize the negative impact caused by improper handling, random events and force majeure.

The support structures of cranes are tall and heavy, and they can be defined as complex truss structures due to different shapes and properties of the elements. The structure and behaviour of the representatives of this great class of construction machinery are the subjects of current studies, [1] and [2]. The connections between various elements, held together by various methods (riveting, bolted joint and welding), also affect the complexity of the structure.

Prior to dynamic analysis, it is very important to perform quality modelling of the complex structure [3]. The Finite Element Method (FEM) can be successfully used to provide a model of the complexity of the support structure. The FEM model allows us to define the geometrical and material properties of a structure very well using a large number of finite elements. The model includes all significant impacts, which define the loads, constraints and connections between elements.

260

The choice of damping model introduces the two damping coefficients, viscous and structural, and plays a key role in dynamic modelling. The viscous damping force is proportional to the velocity.

The coefficient of proportionality is named the viscous damping coefficient. This coefficient is determined by defining the material properties of the elements forming the model for numerical analysis.

The structural damping force is proportional to displacement. Structural damping includes: damping in a material of a structure, losses due to friction between contact areas, and dissipation in medium due to relative motion within the system [4]. The structural damping, thus, essentially covers the entire mechanical system (structure). The damping force is only shown through structural damping in [5]. The introduction of structural damping in the numerical analysis is estimated by the assumed coefficient G, [6] and [7].

The influence of damping resistance on the transient response of the mechanical structure is significant and difficult to determine accurately.

Problems of identification of damping resistance and evaluation of impacts of this resistance on the dynamic response of mechanical model structures are still very current, [8] to [11].

A number of researchers introduce the “hysteretic” damping which includes a delay of elastic structure in returning to the previous position after the end of the effect of excitation force [12]. The viscous damping matrix is then proportional to the “hysteretic” damping matrix, where the coefficient of proportionality is inverse to the absolute value of

*Corr. Author’s Address: University of Niš, Faculty of Mechanical Engineering, A.Medvedeva 14, Nis, Serbia, miomir@masfak.ni.ac.rs

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 260-268

circular frequency ω at which vibrating motion is carried.

According to the recent literature, aerodynamic damping is also used in dynamic analysis of the tall truss structures [13]. This type of damping of truss structure vibrations occurs with the motion of wind in the direction of the longitudinal axis of the tower.

The increment of the average value of wind velocity increases the aerodynamic component of damping.

Thus, the effect of the aerodynamic damping on vibratory movement of the structure becomes more expressed and closer to the impact of structural damping.

The impact of structural damping on tall and elastic structures with large dimensions and more degrees of freedom is particularly explored in the area of civil engineering, earthquake engineering, and structural dynamics.

Fig. 1. Tower crane POTAIN 744e

The overall structural damping can be determined experimentally for certain eigenvalues of vibrations.

The aim of this paper is to determine a single value that corresponds to the experiment on the basis of a group of several structural damping coefficients. In other words, a parallel observation of free vibrations of the structure for different values of the structural damping coefficient G in numerical models enables the determination of the real structural damping on basis of the experiment. For this purpose, we can introduce a set of deformation ratios Ri (experimentally determined) that realistically show the influence of the coefficient G on the damping of free vibrations. Parameter Ri represents the ratio of the maximal displacement amplitude (U0-initial)

and 2ith displacement amplitude (same phase) of an selected element of the structure for harmonic laws of vibrations, or:

Ri =

U0 ; U 2i

( i = 1, 2,...) .

(1)

This paper shows, in an original example, the determination and influence of the structural damping coefficient on the dynamic behaviour of the structure represented by the tower crane in Fig. 1.

1 THEORETICAL BASIS FOR THE IDENTIFICATION OF STRUCTURAL DAMPING COEFFICIENT G Complex structures of different materials and shapes represent vibratory systems where some portion of energy is lost in the damping of motion (dissipation).

Damping resistance occurs within the material (viscous damping), as well as within the mechanical structure of the vibratory system (structural damping).

The influence of the damping resistance on the dynamic behaviour of a complex system, such as the truss structure shown in Fig. 1, is expressed in the dampened vibrating motion. The search for the dynamic response of a structure can, theoretically, be performed numerically by using FEM with appropriate software support [6] and [7]. Defining conditions for the execution of the dynamic analysis by the direct transient method, [6] and [7], requires the introduction of a number of necessary values, one of which is the overall structural damping coefficient G. The dynamic response of the Multiple DegreeOf-Freedom (MDOF) model is required by direct numerical integration of the system of differential equations with the elementary matrix form [14]:

[ M ]{u( t )} + [ B ]{u ( t )} + [ KT ]{u ( t )} = { p ( t )} ,

(2)

where [M] is the inertial mass matrix, [B] is the viscous damping matrix, [KT] is the complex stiffness matrix which includes complex terms, {u ( t )} is the displacement vector, {u ( t )} is the velocity vector, {u( t )} is the acceleration vector, and { p ( t )} is the vector of external generalized forces.

The solution of the system equations (Eq. (2)) is the displacement vector {u ( t )} and it represents the basic response of the crane structure to the effect of external forces. By differentiating the values of the displacements in time, other transient responses are also obtained (velocities, accelerations).

Structural damping is included in the equations of motion as complex terms in the stiffness matrix,

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[4], [14] and [15]. Dynamic analysis, executed by application software [6] and [7], defines the use of material damping GE (MATi) or the coefficient G (PARAM, G) to get the overall structural damping.

After the addition of the abovementioned complex terms, the complex stiffness matrix takes on the following form:

[ KT ] = (1 + iG ) [ K ] + i ∑ GE [ K E ] ,

(3)

where G is the overall structural damping coefficient, GE is the element structural damping coefficient, [K] is the global stiffness matrix, and [KE] is the element stiffness matrix.

Since the transient response analysis, [6] and [7], does not include complex numbers in the calculation procedure, the conversion of structural damping G, as well as GE, into an equivalent viscous damping should be performed [14]. This conversion is done using Eq. (4), which is further used in the transient analysis:

G

1

3

4

[ BT ] = [ B ] + ω [ K ] + ω ∑ GE [ K E ] , (4)

where [BT] is the complex damping matrix, [B] is the viscous damping matrix, and ω3 and ω4 are the dominant circular frequencies in radians per unit time.

Matrix [B] and matrix [K] are known values, obtained in the modelling process when elementary materials and the geometry of the FEM model are defined (property).

The dominant circular frequencies ω3 and ω4 are user-provided parameters, which structural analysts take from the modal analysis. Usually, these are lower frequencies of dominant dynamic states.

Transient response uses various dominant modal frequencies, therefore, the software approximation (Eq. (4)) implies the use of the two modal frequencies closest to the dominant dynamic behaviour, according to [14]. The exclusion of these two parameters (ω3 = ω4 = 0) would lead to a complete neglect of the structural damping impact and thus the approximation (Eq. (4)) would represent only viscous damping, according to [6]. From the software point of view, the values of these frequencies (ω3, ω4) are not consecutive modal values, but they rather represent the frequencies that correspond to the type of transient analysis (dynamic behaviour in the propagation direction). In this case study, the vibratory states of the structure in the vertical global plane are dominant, and the mode shapes that these states describe by the lowest frequency ω3, as well as ω4, are selected.

262

The values of GE are obtained by introducing the materials into the analysis procedure (GE on the MATi entry, [6] and [7]), while G represents an unknown value of structural damping that needs to be defined prior to the transient analysis (PARAM, G).

In order to obtain an accurate transient response of the structure to harmonic excitations of the same eigenvalue, ω2, it is first necessary to experimentally determine the value of the structural damping coefficient G. The determination of parameter G is performed by calculating the logarithmic decrement of the vibrating motion. The logarithmic decrement ∆, or the decrease in declining vibratory movement [16], can be calculated as: ∆ = ln

un un +1

=δ

T , 2

(5)

where |un| and |un+1| are the absolute values of successive amplitudes, and d is the damping coefficient.

In addition, an alternative logarithmic decrement ∆′ can be calculated using the following equation: ∆' = ln

u (t ) u (t + T )

= δT ,

(6)

where the values |u(t)| and |u(t+T)| are related to the two same-phase amplitudes at the distance of one period of damped vibratory movement T. Overall structural damping depends on the coefficient δ (G = f (d)) which can be obtained experimentally, after determining the values of logarithmic decrement ∆ (or ∆′) of the declining vibrating motion and the period of damped vibratory movement T [16], or:

δ=

U (t ) 2∆ ∆ ' 1 . (7) = = ln T T T U (t + T )

2 THE EXPERIMENTAL-THEORETICAL MODEL FOR IDENTIFICATION OF COEFFICIENT G – AN EXAMPLE The dynamic response of a large truss structure on the effect of the external disturbance force (due to a sudden lowering of load to the ground) is required in the experiment. The tower crane P744e in Fig. 1 was selected as an object of examination to perform the experiment.

Radoičić, G. – Jovanović, M.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 260-268

The foundation of the crane pylon was composed of reinforced concrete and was located on ground with a compressibility class II. The base of the crane was supported only elastically and defined by the basic stiffness of the ground. The measurement of local movement of the base in relation to the surrounding ground under the working loads showed insignificantly minor values. The trial load was 5.3 or 2.65 t in the range of 20.88 or 45 m, and it was slightly larger than the nominal value for causing vibrations of larger amplitudes. Special attention was directed to preserving the stability of the crane and the stressstrain state in the field of elasticity.

The following values were measured experimentally: a) deformations at the root of the pylon on opposite sides using two strain gauges; b) change of force on the hook using a force transducer; c) deflection of the boom in the vertical plane for two characteristic ranges (20.88 and 45 m) using a pen on a vertical plotter (Fig. 2).

On this occasion, the following HBM measuring equipment was used [17]: an MGC Plus measuring amplifier, an LY 10/120W strain gauge, and a U2A 10t dynamometer. The measuring results were processed using HBM Catman software.The testing was performed according to the following scenario: 1)

the trial load was hoisted to the maximal height and stopped; 2) the next step was the calming of the load in order to dampen the vibrations of the crane caused by hoisting; 3) finally, by suddenly lowering the load until striking the ground, unloading was caused, along with the free damped vibrations of the entire structure.

Fig. 3 shows the results of vibrating motion for two opposite referent points of the root structure of the tower crane, measured by strain gauges. The abscissa shows the time of the experiment in seconds, and the ordinate shows the values of the measured deformations of considered structural elements in µm/m. Dynamically, the most interesting period occurs after sudden unloading of the structure, when maximal amplitudes are also observed. That is why the diagram in Fig. 3 does not contain parts of the vibratory movement that are related to the initial period of the experiment that lasted for 50 sec (hoisting and calming). The lighter curve MG1 is related to the measuring gauge at the front side – the side of the load, and the darker curve MG2 to the measuring gauge at the opposite side of the pylon.

The maximal value of the micro-deformations is a little over 100µm/m, which is within the boundaries of elasticity.

Fig. 2. The layout of the measuring system on the tower crane P744e Experimental Identification of Overall Structural Damping of System

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Fig. 3. Micro-dilatations (εMG1, εMG2) of the tower root where measuring gauges MG1 and MG2 were placed

The decreased vibrating motion was characterized by a changeable period of vibration (insignificant differences), thus the structural damping coefficient should be examined for an optimal number of logarithmic decrements and appropriate periods of vibrations. In the research measurements, the determination of the logarithmic decrement D (or D’) is more reliable if the number of measurements of the same mode of vibration (w) is larger. In the presented experiment, twenty-four deformation amplitude changes e were found before the structural calm, after which the deformation vibration level fell below 8%, and they were not observed further. A wider range of measured deformations introduces random influences that generate non-regular vibrations.

Areas with regular vibrations are necessary for the study (“proper” vibration without noise). Even the choice of measuring gauge and measuring technology was performed to ensure a better and purer signal in relation to other types of physical measuring values (accelerations). Therefore, the zones of relatively weak signals (small deformations) with a relatively greater presence of random effects were avoided.

Substituting the variables for the amplitudes shown in Fig. 3 into Eq. (7), the general expression used to calculate the particular values of the damping coefficient can be written as:

264

δi =

1 U 2( i −1) ln ; i = 1,2 ,… , (8) Ti U 2i

where U2i is the amplitude of displacement and U2(i-1) is the previous same-phase amplitude of displacement at the distance of one period of vibration Ti. Fig. 4 shows the results of the experimental determination of the damping coefficient according to Eq. (8). The discrete values of the logarithmic decrements and the periods of vibrations from the moment of unloading the structure to the end of the experimental test (tmax = 120 s) are introduced. The measured values of the damping coefficient d are shown with crosses. The third-degree polynomial approximation (solid line) shows the change in this coefficient in the discrete time calculations.

The diagram in Fig. 4 suggests the nonlinear nature of the structural damping in complex structures, i.e. the damping coefficient does not represent a constant default value in numerical analyses, which is caused by the change in the vibrating mass of the system (rejection of load). In this experiment, the damping coefficient takes two extreme values: δmin = 0.004 and δmax = 0.108, which are specific to the very end of the damped vibration measurement.

Greater dissipation of coefficient d (Eq. (8)) always occurs in experimental recordings at the end of the vibration process. If the observation area is narrowed, then the propagation damping area is also narrower (due to a smaller number of observed amplitudes). Therefore, the measured results were statistically processed and the mean value of damping δ = 0.050853 (Eq. (9)), as well as the empirical

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Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 260-268

Fig. 4. Change of the damping coefficient d, dependent on the logarithmic decrement ∆’

variance of the statistical set Var(d)˝= 0.00079131 (Eq. (10)), were obtained.

δ =

1 n 1 U 2( i −1) ∑ ln U , n = 24, (9) n i =1 Ti 2i

Var (δ ) =

1 n ∑ δi − δ n − 1 i =1

(

)

2

. (10)

If Eq. (8) is calculated for more same-phase amplitudes, then the dissipation is smaller. Regardless of the dissipation in Fig. 4, the calming process in Fig. 3 (obtained by measurement) is very orderly and asymptotically damped in accordance with the theory (the tops of the amplitudes are located on the envelopes).For the transient analysis that follows, the nonlinear nature of the structural damping should be represented as linear. The problem is simplified by introducing into the calculation the rounded average value of the measured results of the damping coefficient δexp= δ = 0.05 (Eq. (9)) at the selected measuring point of the tower crane P744e (Fig. 4, the long dash-dot-dot line).

characteristics: model mass MMOD = 72204 kg, number of finite elements NELEMENT = 1667, number of nodes NNODE = 1146, overall number of degrees of freedom NDOF = 6876.

The dynamic behaviour of the structure is caused by the external force that occurs during the sudden unloading of the structure due to an unexpected, highspeed lowering of the load. This can be observed as an incident (for example, the falling of a load). Structural calming, after hoisting the load, is achieved by choosing the optimal calming time. The new vibrating motion is excited by the force of the sudden unloading during the lowering of the load to the ground. The duration of unloading is taken from the experiment and it is ∆t = 0.83 s (the diagram in Fig. 5).

3 MODEL APPLICATION The application of the previously obtained experimental results in dynamic analyses and the impact of the overall structural damping on the transient response of the structure are shown in an example of direct transient analysis in the FEM model of the tower crane. The transient response of the structure of the tower crane P744e is calculated numerically by using FEM in the previously developed model with the following general

Fig. 5. The measured changes in force on the hook

The theoretical perturbation force has a trapezoid impulse form [18], as shown in Fig. 6. To numerically

Experimental Identification of Overall Structural Damping of System

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solve the dynamic problem of free damped vibrations of the system, the MDOF application software, [6] and [7], is used.

coefficient δ, and it has taken values ranged from G = 0.01 to 0.108. The abovementioned damping coefficients were functionally dependent G = f (d).

It is difficult to examine internal losses (mathematical dependencies) in the structural damping coefficient G at the level of this complex system, and the total presence of all internal influences, according to the Eq. (3), can only be experimentally determined.

The character of the connection between G and δ depends on the particular construction of structural elements and their mutual contacts.

Fig. 6. Impulse shape force

The significant circular frequencies of the numerical model were determined by modal analysis of the first one hundred eigenvalues. The basic circular frequency ω1,m = 0.1687 Hz is specific for the rotation around the main vertical axis of the crane pylon. This frequency is not significant for the transient analysis that is performed in this application.

The dominant vibratory movement occurs in the vertical plane during the lowering/falling of load.

The circular frequencies in the second and the fourth vibration mode (ω2,m = 0.4749 Hz, ω4,m = 1.5444 Hz) best represent the structural vibrating nature in the vertical plane, and they are further used in the transient analysis (Eq. (4)). Fig. 7 shows the second (Mode 2), the fourth (Mode 4), and the sixth (Mode 6) eigenvalue, i.e. the vibration mode of the developed FEM model.

Matrices of mass, stiffness, and viscous damping are determined on the basis of the selected materials and the geometry of the model in the modelling process. The best value of structural damping from the experiment was adopted and converted into the value of viscous damping in Eq. (4) by using two specific circular frequencies (ω3 = ω2,m; ω4 = ω4,m), which were obtained in the modal analysis. By numerical integration of the system equations in Eq. (2) in the direct transient analysis, node displacements u(t) were obtained for the model that freely vibrates after unloading.

Apart from determining the transient response using the structural damping coefficient G = 0.05 = Gexp, other transient responses of the model were also numerically determined. The behaviour of the numerical model under the influence of the same load set was observed, i.e. the same conditions of unloading only at different values of the structural damping of the system. The overall structural damping coefficient G, in numerical simulations, corresponded to the 266

Fig. 7. Eigenvalues of vibrations; a) Mode 2, forward-back (ω2,m = 0.4749 Hz), b) Mode 4, forward-back (w4,m = 1.5444 Hz), c) Mode 6, up-down(w6,m = 2.0274 Hz)

For the given values of the coefficient G, the ratios of numerically determined amplitudes Ri were obtained according to Eq. (1). The obtained ratios Ri were grouped in curves according to the value of the constant G. The curves of changes in the numerical amplitude ratios Ri are shown in Fig. 8, and they represent the appropriate mathematical approximations in exponential form, obtained numerically. The same figure shows the values of the ratio Ri, obtained experimentally, with crosses. The abscissa shows the shortened time t of the experiment/ numerical test, which covers the period from the occurrence of the initial and highest vibration U0

Radoičić, G. – Jovanović, M.

StrojniĹĄki vestnik - Journal of Mechanical Engineering 59(2013)4, 260-268

up to the time t = 91.95 s. This shortened overview has greater visibility and shows values of the ratio Ri for the first 13 numerically obtained same-phase amplitudes (i = 1, â&#x20AC;Ś, 12), with only a few values of the coefficient G. The change in the amplitude ratio Ri, shown in the diagram (Fig. 8), is performed during the calming of the structure, yet it is not a time function but a function of position i of the amplitude U2i with regard to the initial amplitude U0 (e.g. if i = 4 â&#x2021;&#x2019; U2i = U8 then Ri = R4 = U0 / U8) and it can be represented using the approximate exponential form: Ri = C1eC2i , (11)

where C1 and C2 are the constants which depend on the form of the exponential function. For the experimentally obtained curve of the change in the amplitude ratio Ri, these constants take values C1 = 1.1882 and C2 = 0.133, thus the approximate exponential equation for the observed example can be written as:

(

on the value of coefficient G. In order to accurately determine coefficient G, it is desirable to perform experimental dynamic tests, which are presented in this paper.

A transient analysis of the complex numerical model of the tower crane leads to conclusions as to the impact of structural damping on damped vibrations in large support structures. The results indicate that similar vibrating systems, with increasing values of the structural damping coefficient, have a shorter calming period (a faster fall in vibrations). Vibrations of such systems are dampened more quickly (Fig. 8; G = 0.07, G = 0.1). Furthermore, it can be concluded that in weakly damped vibrations (at lower coefficients G) simpler mathematical approximations for the identification of the structural damping coefficient can also be applied. Therefore, Fig. 8 shows that the exponential curve Ri (G = 0.02) can be replaced by a linear function with acceptable accuracy.

)

Ri Gapprox . = 1.1882e0.133i . (12)

By increasing the value of the structural damping coefficient G, the amplitude ratios Ri become more pronounced and the damping of the vibration becomes greater, making the fastest calming of the structure correspond to the largest coefficient G = 0.1 (Fig. 8). The specific value of the structural damping coefficient G was found by changing the different values of this coefficient in the theoretical model [6] and [7]. This characteristic value of the coefficient G corresponds to real behaviour of the structure based on the experiment. The numerical approximation of damped vibration of the real structure with the appropriate structural damping coefficient (G = 0.05) is:

Ri ( G = 0.05 ) = 1.0002e0.1568i . (13) 4 CONCLUSION

Fig. 8. The influence of the structural damping coefficient G on a high truss structure (numerical simulation)

In the development of numerical models for the needs of dynamic analyses of large structures, it is very difficult to precisely determine the dissipation energy in the structure. The damping matrix represents the dissipation energy, which contains a constant of the overall structural damping G. This matrix is used in the software to determine the dynamic response of the structure. The damping of vibrating motion, as well as the duration of calming of the structure, depends

Models that use experimentally determined overall structural damping G are more reliable, with wide boundaries within which this ratio is changed.

In the absence of experimental values, using the coefficient G of similar categories and similar sizes of structures is acceptable. Experiments can give good descriptions of the initial conditions of movement, the velocities of unloading and loading of

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structures for impulse effects. In extreme cases, the observed amplitudes may belong to a class of large displacements, since they can reach up to 10% of the object’s height.

The experimental-numerical model for the identification of coefficient G, as shown in this paper, can be applied to similar structures (larger tower cranes, portal-rotating cranes, mobile cranes, long/tall objects, pylons and towers) that vibrate with significant amplitudes under the influence of the external perturbing force. Reliable values for the structural damping coefficient for further use in similar analyses can be obtained by applying this model.

5 ACKNOWLEDGEMENT The paper is a part of research carried out under project TR35049. The authors would like to thank the Ministry of Education and Science of the Republic of Serbia for their support.

6 REFERENCES [1] Raubar, E., Vrančić, D. (2012). Anti-sway system for ship-to-shore cranes. Strojniški vestnik - Journal of Mechanical Engineering, vol. 58, no. 5, p. 338-344, DOI: 10.5545/sv-jme.2010.127, DOI:10.5545/svjme.2010.127.

[2] Gašić, M., Savković, M., Bulatović, R. (2011).

Optimization of trapezoidal cross section of the truck crane boom by Lagrange’s multipliers and by differential evolution algorithm (DE). Strojniški vestnik - Journal of Mechanical Engineering, vol. 57, no. 4, p. 304-312, DOI:10.5545/sv-jme.2008.029.

[3] Jovanović, M., Radoičić, G., Petrović, G., Marković, D. (2011). Dynamical models quality of truss supporting structures. Facta Universitatis, Series: Mechanical Engineering, vol. 9, no. 2, p. 137-148.

[4] Gaul, L. (1999). The influence of damping on waves and vibrations. Mechanical Systems and Signal Processing, vol. 13, no. 1, p. 1-30, DOI:10.1006/ mssp.1997.0185.

[5] Bošnjak, S., Zrnić, N., Dragović, B. (2009). Dynamic response of mobile elevating work platform under wind excitation. Strojniški vestnik - Journal of Mechanical Engineering, vol. 55, no. 2, p. 104-113.

[6] MSC/Nastran v68 - Basic Dynamic Analysis (2004).

MSC.Software Corporation, from: http://www. mscsoftware.com, accessed at 2004-08-26.

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[7] MSC/Nastran v70 - Advanced Dynamic Analysis (2004).MSC.Software Corporation, from http://www. mscsoftware.com, accessed at 2004-08-26.

[8] Li, Q.S., Fang, J.Q., Jeary, A.P. (2000). Free vibration analysis of cantilevered tall structures under various axial loads. Engineering Structures, vol. 22, no. 5, p.

525-534, DOI:10.1016/S0141-0296(98)00124-2.

[9] Takewaki, I., Nakamura, M. (2000). Stiffness damping simultaneous identification using limited earthquake records. Earthquake Engineering and Structural Dynamics, vol. 29, no. 8, p.1219-1238, DOI:10.1002/1096-9845(200008)29:8<1219::AIDEQE968>3.0.CO;2-X. [10] Lin, Y.Y., Miranda, E., Chang, K.C. (2005). Evaluation of damping reduction factors for estimating elastic response of structures with high damping. Earthquake Engineering and Structural Dynamics, vol. 34, no. 11, p.1427-1443, DOI:10.1002/eqe.499.

[11] Johnson, C.D., Kienholz, D.A. (1983). Prediction of damping in structures with viscoelastic materials using MSC/Nastran. MSC/Nastran Users’ Conference Proceedings, paper no.17.

[12] Feriani, A., Perotti, F. (1996). The formation of viscous damping matrices for the dynamic analysis of MDOF systems. Earthquake Engineering and Structural Dynamics, vol. 25, no. 7, p. 689-709, DOI:10.1002/ (SICI)1096-9845(199607)25:7<689::AIDEQE575>3.0.CO;2-L. [13] Holmes, J.D. (1996). Along-wind response of lattice towers - II. aerodynamic damping and deflections.

Engineering Structures, vol. 18, no. 7, p. 483-488, DOI:10.1016/0141-0296(95)00131-X. [14] Rose, T. (2002). An approach to properly account for structural damping, frequency-dependent stiffness/ damping, and to use complex matrices in transient response. Worldwide Aerospace Conference & Technology Showcase, presentation no. 6, from:www. mscsoftware.com/events/aero2002/partner/index.cfm, accessed at 2002-04-08.

[15] Marsh, E., Slocum, A. (1996). An integrated approach to structural damping. Precision Engineering, vol. 18, no.

2-3, p. 103-109, DOI:10.1016/0141-6359(96)00051-7.

[16] Rašković, D. (1965). Theory of Vibration. Scientific Book, Belgrade.

[17] Hottinger Baldwin Messtechnik (2011). HBM, from: www.hbm.com, accessed at 2011-01-01.

[18] Prolović, V., Bonić, Z., Živković, S. (2004).

Dependence of the basic dynamic parameters on the disturbing force of the impact impulse type with different forms. FactaUniversitatis, Series: Architecture and Civil Engineering, vol. 3, no. 1, p. 2331, DOI:10.2298/FUACE0401023P.

Radoičić, G. – Jovanović, M.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278 © 2013 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2012.478 (1.01) Original Scientific Paper

Received for review: 2012-03-23 Received revised form: 2013-01-18 Accepted for publication: 2013-02-12

Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F. Han Lianfu1,2,* – Tang Wenyan1 – Wang Jun1 – Fu Changfeng2

1 Harbin

Institute of Technology, Institute of Precision Instrument, China Petroleum University, School of Electronic Science, China

2 Northeast

This paper studies the influence of small clamping eccentricity on the measurement and evaluation of involute profile, analyzes the cause of clamping eccentric error, establishes a clamping eccentric error compensation model, and introduces a computing method of clamping eccentric parameters. Experimental testing of the computing method of clamping eccentric parameters proves that the computing method of clamping eccentric parameters is correct and feasible when the clamping eccentric value is within three quarters of the measurement range of the probe. The results of the clamping eccentricity error compensation experiment show that the clamping eccentric error compensation model can compensate measurement results to the results attained under operating conditions, and achieve precision measurement of involute profiles. The highest allowed clamping eccentric value can be extended to three quarters of the measurement range of the probe by the clamping eccentricity error compensation model proposed in this paper, thereby saving time otherwise spent adjusting for clamping eccentricity.

Keywords: gear measuring center, clamping eccentricity, involute profile, measurement, evaluation

0 INTRODUCTION Involute gears are widely used in mechanical manufacturing, and the quality of the involute profile affects gear vibration and noise [1] and [2].

Measurement of involute profile is difficult, and the involute profile is usually measured by a gear measuring center [3] and [4]. Their measuring principle is the Electronic-generating method, which demands that the rotation of the center axis of the gear measuring center should coincide with the geometric center line of the gear [5] to [7]. It is easy to meet the clamping demand for middle and small size external gears with a core shaft and special clamp. But internal gears cannot be clamped on the measurement platform with a core shaft and special clamp because there is no position to clamp internal gears. Large gears also cannot be clamped on the measurement platform because they are too large to find a proper core shaft [7]. Large and internal gears are clamped on the measurement platform of the gear measuring center using a number of magnets. In the clamping process of large gears and internal gears, a manual method with a dial indicator is involved to assist in the clamping.

Therefore, the geometric center line of the gear often parallel deviates from the rotation center axis of the gear measuring center due to the limitations of manual micro-positioning ability and the accuracy of the auxiliary clamping instrument, resulting in clamping eccentricity, where the error caused by this is called the clamping eccentric error.

The clamping eccentric error inevitably reduces the measurement accuracy of gear measuring center. To decrease the influence of the clamping

eccentricity on the measurement of the involute profile, the value of the clamping eccentricity is adjusted to be as small as possible, but this takes a lot of time and the clamping eccentricity still exists [8]. Therefore, accurate measurement of the involute profile with clamping eccentricity is a real problem.

A method of compensating for clamping eccentricity error was proposed by Yang, and the method achieves clamping eccentric error compensation by correcting the transformation of the workpiece coordinate system and the instrument coordinate system. However, this method needs specially designed measurement nets to obtain the proper clamping eccentricity parameters [9]. Shi proposes a method to compensate for clamping eccentricity [10], which measures the net points on the involute profile and uses the least square method to estimate the clamping eccentricity parameters, but the method needs to include an extra measurement trajectory for the computation of eccentricity parameters. At present, the gear measuring center made

by Klingelnberg allows the measurement of involute profile with clamping eccentricity. The allowed maximum eccentricity value is 2 mm and the eccentric parameters are obtained by measurement of the core shaft. However, its disadvantage is its inaccessibility to internal and large gears [11].

This paper proposes a measurement method for involute profiles with small clamping eccentricity for internal gears and large gears based on a gear measuring center. The method not only allows precision measurement of involute profiles without changing the measurement trajectory, but also reduces the time needed to adjust the clamping eccentricity.

*Corr. Author’s Address: Harbin Institute of Technology, 92 West Dazhi Street 150001, Harbin , China, lianfuhan@hit.edu.cn

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1 THE INFLUENCE OF CLAMPING ECCENTICITY ON MEASUREMENT AND EVALUATION OF INVOLUTE PROFILE To measure the involute profile with a small clamping eccentricity, the influence of the clamping eccentricity on the measurement and evaluation of the involute profile should be considered. When there is no clamping eccentricity, the measurement trajectory of the probe relative to the gear is an involute curve attained through the linkage of the X-axis and main shaft. If the X-axis and main shaft still maintain their linkage in the default way in the clamping eccentric state, the profiles deviate from their default measurement position by the existence of the clamping eccentricity. So, any point P0(x0, y0, z0) attained by generating the measured values of involute profiles deviates from its default measurement position, as shown in Fig. 1a. The measured points are in the same plane, because the Z-axis remains static when the involute profile is measured. Although there is clamping eccentricity, the measured curve is part of the involute curve to be measured and the point P0(x0, y0, z0) is still in the curve to be measured. The point P0(x0, y0, z0) may delay or advance its actual position, and it may appear at position P1-point or P2-point, both of which are on the measured involute profile, as shown in Fig. 1b.

It is important to study the usage of the above rule in measuring the involute profile with clamping eccentricity without changing the measurement trajectory. It is assumed that involute profiles are by default in their measurement position during the process of measurement of involute profiles in the clamping eccentric measurement state, so the probe may impact the gear if the clamping eccentricity is too large. The measurement range of the probe is L, so the maximum allowed deviation Δdmax between the rotation center of the instrument and the geometric center of the gear is expressed as:

The output and input of the probe have a strictly linear relationship within three quarters of the measurement range of the probe, so the appropriate maximum value of the clamping eccentric value Δe is given by Eq. (2).

b) Fig. 1. The influence of clamping eccentricity on measurement and evaluation of the involute profile; a) gear in the clamping eccentricity condition, and b) the schematic diagram of a section

270

Δe = 0.75 L. (2)

For all kinds of gears, except for micro gears, the relation between d1 (the distance from the starting point of evaluation to the starting point of measurement) and d2 (the distance from the end point of evaluation to the end point of measurement) can be expressed as:

a)

Δdmax = L. (1)

 d1 ≈ 0.5( LAE − Lα ) >> ∆e = 0.75 L , (3)  d 2 ≈ 0.5( LAE − Lα ) >> ∆e = 0.75 L

where, LAE is the abbreviation of active length and Lα is the abbreviation of the profile evaluation range [12].

According to Eq.(3), the length of the profile that is not evaluated is much larger than the clamping eccentric value. The clamping eccentricity only affects the lengths of d1 and d2, and the evaluation length is not affected when the clamping eccentricity is within 0.75 L. So, the part of the curve involved in the evaluation can be measured without adjusting the measurement trajectory, but the profiles deviate from their default measurement position due to the existence of the clamping eccentricity. As shown in Fig. 1b, the measurement values that stand for the P0point in the operation condition stand for the P1-point in the clamping eccentricity condition, an error of PM is caused during the total deviation computation 1 of the tooth profile. The clamping eccentricity error should be compensated for in the evaluation process of involute profiles, although the measurement trajectory of involute profile does not need to be changed when the clamping eccentric value is 0.75 L.

Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278

2 ˝THE CLAMPING ECCENTRIC ERROR COMPENSATION MODEL 2.1 Cause of the Clamping Eccentric Error To establish the clamping eccentric error compensation model, the cause of the clamping eccentric error should be studied. The coordinates of involute profiles in the workpiece coordinate system should be transformed to the instrument coordinate system.

While evaluating involute profiles, the measured values in the instrument coordinate system should be transformed into the workpiece coordinate system.

Therefore, it is necessary to study the transformational relationship between these two coordinate systems.

The workpiece coordinate system is established with the geometrical center of the gear as the origin, and the X, Y and Z directions of the instrument coordinate system as X0, Y0 and Z0 directions of the workpiece coordinate system, as shown in Fig. 2. The geometric center line of the gear parallel deviates from the rotation center axis of gear measuring center in the clamping eccentric state and the gear is clamped onto the platform of a gear measuring center, the origin of workpiece coordinate system in the instrument coordinate system keeps changing and it is denoted as O0i.

a)

 xi   x0ii   0   x0ii           yi  =  y0ii  +  0  =  y0ii  + α , (4)  zi   z   b2   z   0ii   0ii 

where b is the face width, and α is the translation vector of the Z direction between the workpiece coordinate system and the instrument coordinate system in the ideal state.

As shown in Fig. 2b, the transformational relationship of the coordinates in the clamping eccentric state can be expressed as:

 xi   x0ii   0   mi   x0ii             yi  =  y0ii  +  0  +  ni  =  y0ii  + α + β i , (5)  zi   z   b2   0   z   0ii   0ii 

where βi is the clamping eccentricity transformational vector between the workpiece coordinate system and the instrument coordinate system, and (mi, ni, b/2) is the origin of the workpiece coordinate system in the instrument coordinate system.

When the clamping eccentric value is small, βi ≈ 0 and Eq. (5) is approximately valid if βi is neglected. Under this condition, the influence of neglecting βi on the transformational accuracy of the coordinates is tiny and the clamping eccentric error can be neglected. When the clamping eccentric value is large, βi > 0 and Eq. (5) is not valid if βi is neglected. The influence of neglecting βi on the transformational accuracy of the coordinates is large, so it would result in a large transformational error of the coordinates. Under this condition, the clamping eccentric error should not be neglected, but in fact the clamping eccentricity transformational vector βi is still not considered, so the clamping eccentric error appears. From the preceding analysis it is clear that the clamping eccentric error is caused by the transformation error between the workpiece coordinate system and the instrument coordinate system due to ignoring the clamping eccentricity transformational vector βi.

2.2 Establishment of the Clamping Eccentric Error Compensation Model

b) Fig. 2. Construction of the workpiece coordinate system; a) in the ideal state, and b) in the clamping eccentric state

As shown in Fig. 2a, the transformational relationship of the coordinates in the ideal state is expressed as:

According to Eq. (5), βi is determined by the origin O0i(mi, ni) of the workpiece coordinate system in the instrument coordinate system. Because the gear is clamped on the measurement platform of the gear measuring center, the distance between the origin of the workpiece coordinate system and the origin of

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the instrument coordinate system is a constant. When the gear revolves 2π around the rotation center axis of gear measuring center, the geometric center of the gear moves in circle O, as shown in Fig. 3. It is assumed that the origin of workpiece coordinate system at a certain measurement time is at the position point O00, the clamping eccentric value that the initial position of the gear geometrical center axis deviates from the rotation center axis is e0 = e, and the polar angle is ϕ0 and the reading of main shaft is θ0. After main shaft rotates θi, the origin of the workpiece coordinate system revolves to point O0i. As shown in Fig. 3, the eccentric value ei and the polar angle ϕi at this measurement moment can be obtained as: ei = e , (6)  ϕi = ϕ0 + θi

where θi is the angle that the main shaft rotates to the initial angle θ0.

The origin of the workpiece coordinate system in the instrument coordinate system is point O0i(mi, ni), and the equations of the orgin are as follows: mi = e cos ϕi = e cos(ϕ0 + θi ) . (7)   ni = e sin ϕi = e sin(ϕ0 + θi )

the ith workpiece coordinate system the i-position polar angle

clamping eccentric Y value

between the instrument coordinate system and the i-position of workpiece coordinate system is expressed as Eq.(8):

The measured values of involute profiles should be transferred into the same workpiece coordinate system in the evaluation. In this paper, the measured values are transfered into the initial workpiece coordinate system. As shown in Fig. 3, the coordinate transformational relationship between the initial workpiece coordinate system and the i-position of workpiece coordinate system can be expressed as:

e O

main shaft rotating angle

 x0ii     y0ii  (9) z ,  0ii   1 

where, M is the coordinate transformational matrix between the initial workpiece coordinate system and the i-position of the workpiece coordinate system and it is expressed as:

1 0 M = 0  0

0 1 0 0

0 −e[cos φ0 − cos(φ0 + θi )] 0 e[sin(φ0 + θi ) − sin φ0 ]  , (10)  1 0  0 1 

The clamping eccentric error compensation model is established according to Eqs. (8) and (9), and the expression of the model is given by Eq. (11).

X

the initial workpiece coordinate system

initial polar angle Fig. 3. Running track of the gear’s geometrical center The No.i workpiece coordinate system

When the origin of the workpiece coordinate system is at the position point O0i, the workpiece coordinate system is denoted as the ith workpiece coordinate system. For any point Pi(xi, yi, zi, θi) in the instrument coordinate system, according to Eqs. (2) and (7), the coordinate transformational relationship 272

 x00i     y00i  z  = M  00i   1 

Y0i O0i Y00 X 0i φ θi i X 00 φ0 O00

 xi   x0ii  e cos(ϕ0 + θi )         yi  =  y0ii  + α +  e sinn(ϕ0 + θi )  . (8)  zi   z    0  0ii 

 x00i   xi − e cos(φ0 + θi )     y − e sin(φ + θ )  y  00i  0 i   i = . (11) M b z    − z i 2 00i     1  1   

Once the coordinate transformation without error between the initial workpiece coordinate system and the instrument coordinate system is realized by the clamping eccentric error compensation model, then the involute profiles can be evaluated according to the evaluation standard. According to Eq. (11), there are only two unknown parameters (clamping eccentric value e and polar angle ϕ0) in the model.

Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278

2.3 Calculation of Clamping Eccentric Parameters e and ϕ0 This part discusses the calculation of the clamping eccentric parameters according to the measuring points of the tooth profile. As shown in Fig. 4a, the probe is adjusted to the base circle in the X direction.

The proportional relationship between the rotation angle of the main shaft and the moving distance of the Y-guideway is determined by the parameters of the gear to be measured. The geometrical center O00 (m0, n0) of the gear at a certain measurement moment is defined as the origin of the initial workpiece coordinate system, as shown in Fig. 4b.

As shown in Fig. 4b, the involute curve P0Pm is the involute profile to be measured, and O00 (m0, n0) is the center of the base circle and rb is the radius.

The radius vector rkm at the point Pm (xm, ym) of the involute profile is expressed as: rkm = ( xm − m0 ) 2 + ( ym − n0 ) 2 . (13)

As shown in Fig. 4b, involute curve P0Pm is the involute profile to be measured, and O00 (m0, n0) is the center of the base circle and rb is the radius.

According to the characteristics of the involute, the expression for generating angle θ km at the point Pm (xm, ym) of the involute profile is as follows:

θ km = tan α km − α km , (14)

Y

Base circle Probe of gear measuring center X Geometric circle of gear

where, α km is the pressure angle at the point Pm (xm, ym) of the involute profile and can be expressed as: rb   α km = arccos r km  . (15)  2 rkm − rb2   tan α km = rb 

Rotation circle of gear measuring center

a)

As shown in Fig. 4b, βm is expressed as Eq. (16):

β m = ∠DO00 P0 − ∠DO00 Pm = y −n y −n = arctan 0 0 − arctan m 0 . (16) x0 − m 0 xm − m0

As shown in Fig. 4b, βm can also be expressed as Eq. (17).

β m = θ km − θ k0 . (17)

Let

b)

Fig. 4. Determination of circle center using the involute curve; a) measurement of involute profiles, and b) example of an involute curve

The clamping eccentric angle ϕ0 and the clamping eccentric value e can be expressed as:

φ0 = ∠O00OX , (12)   e = O00O

f m = θ km − θ k0 − ∠DO00 P0 − ∠DO00 Pm . (18)

Eq. (19) should hold in theory, but it does not hold for the manufacturing error and measurement error.

f m = 0. (19)

According to the principle of the least square method, the following model for the identification of clamping eccentric parameters is established:

N  2  F (m0 , n0 , rb ) = min ∑ f m . (20)  m =1  > 0 s . t r b 

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Eq. (20) is a nonlinear least square problem and it can be solved by recursive methods [13]. In theory, as long as the measurement values of one tooth profile are obtained, m0 and n0 can be estimated. But it may result in a large deviation between the obtained eccentricity parameters and the true value. To reduce such a large deviation, the measurement values of teeth evenly distributed in the gear are adopted. The circle center is obtained from each tooth is O0i (m0i, n0i). The optimum identification model of the circle center is established as follows:

is determined by the parameters of the gear to be measured. The involute profile with small clamping eccentricity is compensated such that it is in the ideal state using the clamping eccentric error compensation model. Involute profiles are evaluated according to ISO1328-1 [12].

z0

d (m0 , n0 ) = min ∑ [(m0i − m0 ) 2 + (n0i − n0 ) 2 ], (21) i =1

where, z0 is the measured number of teeth. The binary function d (m0, n0) has its minimum value at the inflexion, so we take the partial derivative of d (m0, n0) with respect to m0 and n0 to find the inflexion.

 ∂d (m0 , n0 ) =0  ∂m  0 . (22)   ∂d (m0 , n0 ) = 0  ∂n0

The inflexion (m0, n0),which is also the optimized circle center, can be obtained using Eq. (22). The clamping eccentric value e and the eccentric angle ϕ0 expressed as:

ϕ0 = arctan(n0 / m0 ) . (23)  2 2  e = m0 + n0

Fig. 5. The experimental equipment

3.1 Experiment on the Calculation Method Of Clamping Eccentric Parameters The dial indicator is used to monitor the geometrical center position of the internal gear and to control the clamping eccentric value within 0.75L. The points of the addendum circle are measured every 0.5° by the gear measuring center. As shown in Fig. 6, pi (xi, yi) is one measurement point in the addendum circle, the circle center o2 (aˆ , bˆ) is obtained using the measured values of the points in the addendum circle using the Limacon Model [14].

3 EXPERIMENTS To verify the correctness of the calculation method for clamping eccentricity parameters and the validity of the clamping eccentricity error compensation model, the following experiments were carried out with the internal gear on a L45 type gear measuring center. The measurement range of the probe in the experiment is 0.70 mm and the parameters of the internal gear are as follows: modulus is 3.25; tooth number is 60; modification coefficient is 0; face width is 30 mm; pressure angle is 20°; spiral angle is 0°.

Fig. 5 shows the measurement equipment. The gear is fixed on the platform with six magnets, the probe is adjusted to the base circle in the X direction. The proportional relationship between the rotation angle of main shaft and the moving distance of the Y-guideway 274

 a =   b = 

1 N −1 ∑ xi N i =0 1 N −1 ∑ yi N i =0

. (24)

The involute profiles of the gear are measured under the same clamping eccentric condition, and the center of base circle is obtained using the Involute Model proposed in this paper. To decrease contingency, a series of experiments were carried out under different clamping eccentric value conditions.

3.2 The Clamping Eccentric Error Compensation Experiment Using the Involute Model proposed in this paper, the clamping eccentric value is 164.3 mm and the clamping eccentric angle is 43.62°. The clamping eccentric error compensation model is applied in the evaluation of the involute profiles.

Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278

y

p

i

(

xi yi ,

)

measurement curve optimize fitted curve

ri

θi

O2(a,b) o1

x

r0

Fig. 6. Limacon Model

To decrease the influence of chance in the error compensation experiment, the position of the gear is adjusted and the involute profiles are measured 20 times for each clamping eccentric value. Then the preceding operation is repeated with a different clamping eccentric value.

3.3 The Experiment of Saving Time in Using Adjusted Clamping Eccentricity To compare the saving time in using adjusted clamping eccentricity, a number of experiments were carried out with a series of gears whose tip diameter is less than 3000 mm. To ensure the accuracy of the measurement, the clamping eccentric value is required to be as small as possible. For example, for a gear whose tip diameter is no more than 450 mm, the clamping eccentric value is usually required to be less than 2 μm. The highest allowed clamping eccentric value can extend to 0.75 L using the clamping eccentricity error compensation model proposed in this paper.

4 RESULTS AND DISCUSSION 4.1 Results and Discussion of the Calculation Method of Clamping Eccentric Parameters In theory, the circle center of the addendum circle should coincide with the circle center of the base circle, and the clamping eccentric value and the clamping eccentric angle can be attained by the

circle center according to Eq. (23). Table 1 shows a comparison of the results of the experiment on the calculation method of clamping eccentric parameters.

As shown in Table 1, both the eccentric values attained by Limacon Model and those attained by the Involute Model basically tally with those using the Indicator. The maximum deviation of clamping eccentric values is 1.9 mm, which can be ignored compared to the maximum allowed clamping eccentric value of 0.75 L. The maximal deviation of the eccentric angle between two models is 0.09°, which can also be ignored. This shows that the eccentricity parameter estimation method raised in this paper is correct and feasible.

4.2 Results and Discussion of the Clamping Eccentric Error Compensation Experiment The measurement results before compensation are shown in Fig. 7a, then the clamping eccentric error compensation model is applied to the evaluation of the involute profiles and the deviation diagrams are shown in Fig. 7b.

The measurement results are shown in Table 2.

There are eight deviation curves of involute profiles in every deviation diagram in Fig. 7, and the deviation distance from the vertical lines in every curve stands for the deviation of the involute profiles.

The deviation curves of involute profiles should be straight lines if there are no deviation in the profiles, and there is 1 mm per unit of profile deviation.. To display the deviation clearly, the deviations are enlarged 1000 times in Fig. 7. The numbers “1, 16, 31, 46” mean the numbers of the measured tooth, and the numbers on the left stand for the number of left tooth profiles and those on the right stand for the number of right tooth profiles. Fig. 7a and c are obtained under the same measurement condition for the same four involute profiles. However, for the clamping eccentricity, the inclined degree of the deviation curves is obviously different, which indicates that ±fHα (profile slope deviation) increases greatly. The curve shape of the deviation curves is also different, which means that ffα (profile form deviation) and Fα (total profile deviation) are affected by the change in

Table 1. Estimated clamping eccentric parameters of different models Limacon Model Involute Model Dial Indicator

e [μm] Φ0 [°] e [μm] Φ0 [°] e [μm]

29.3 3.30 29.1 3.28 30

69.1 355.27 68.7 355.34 70

100.4 42.45 98.5 42.36 100

164.4 43.71 164.3 43.62 160

228.5 9.29 227.2 9.33 230

299.8 186.62 298.7 186.71 300

Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center

357.2 190.78 356.1 190.74 360

491.7 6.78 490.3 6.73 490

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Table 2. Measurement results of involute profiles under different conditions Under operating conditions Before compensation After compensation

Fα [μm] 8.0 35.7 8.3

Grade 5 9 5

the evaluation position resulting from the clamping eccentricity. As shown in Table 2, due to clamping eccentricity, the 5th grade Fα is incorrectly judged as grade 9, the 6th grade Ffα is incorrectly judged as grade 10, and the 4th grade ±fHα as grade 9. Therefore, the measurement results are seriously affected. So, the measurement of tooth profile is seriously affected under conditiond of clamping eccentricity if the clamping eccentricity compensation is not taken into account. After compensation, the curves in Fig. 7b mostly accord with those in Fig. 7c in the evaluation range. According to Table 2, the measurement results after compensation are close to those under operating conditions and do not influence the evaluation grade.

ffα [μm] 7.9 37.2 7.6

Grade 6 10 6

± fHα [μm] -3.2 -17.8 -3.1

Grade 4 9 4

The average measurement results of the involute profiles before and after compensation with different clamping eccentric values are shown in Fig. 8. As shown in Fig. 8, as the clamping eccentric value grows, the measurement errors of Fα, Ffα and ±fHα all increase. The maximum measurement errors of Fα, Ffα and ±fHα are, respectively, 31.6, 29.5 and 16.6 μm, so the measurement results seriously deviate from the true results under operating conditions.

a)

a)

b) b)

c) Fig. 7. Comparison of measurement results; a) the measurement result before compensation, b) the measurement result after compensation, c) the measurement result under operating conditions

276

c)

Fig. 8. Comparison of profile deviation before and after compensation; a) comparison of total profile deviation, b) comparison of profile form deviation, and c) comparison of profile slope deviation

Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, 269-278

After compensation, the following measurement results are attained.

• The average value of Fα is 8.1 mm, which approximates to the true value 8.0 mm. Fα is the 5th grade of accuracy class, which is also the same as the true grade.

• The average value of Ffα is 7.9 mm, which approximates to the true value. Ffα is the 6th grade, which is the same as the true grade.

• The average value of ±fHα is –3.3 mm, which approximates to the true value –3.4 mm. ±fHα is the 4th grade of accuracy class, which is also the same as the true grade.

From the results of the preceding experiments, the measurement results after compensation basically tally with those obtained under operating conditions. Therefore, the clamping eccentricity error compensation model is correct.

4.3 Results and Discussion of the saving time in adjusting the clamping eccentricity The highest allowed clamping eccentric value can extend to 0.75 L using the clamping eccentricity error compensation model proposed in this paper. The time saved adjusting the clamping eccentricity is shown in Fig. 9.

5 CONCLUSIONS This paper draws the following conclusions.

• The measurement trajectory of involute profiles does not need to be changed if the clamping eccentric value is within 0.75 L, but the profiles should be compensated to the default measurement position in the evaluation.

• The clamping eccentric error is caused by a coordinate transformation error between the workpiece coordinate system and the instrument coordinate system when the clamping eccentricity transformational vector βi is ignored.

• The eccentricity parameters obtained using the involute Model raised in this paper are consistent with those obtained using the Limacon model.

Therefore, the eccentricity parameters estimation method proposed in this paper is correct and feasible.

• The clamping eccentric error compensation model compensates evaluation results to the state without eccentricity, allowing precision measurement of involute profile.

• The highest allowed clamping eccentric value can extend to 0.75 L, which saves a lot of time otherwise spent in adjusting clamping eccentricity.

6 ACKNOWLEDGEMENT This work was supported by the Technology Program of the Educational Office of Hei Longjiang Province, China (No.11553016).

7 REFERENCES

Fig. 9. Time saved adjusting clamping eccentricity

As shown in Fig.9, the time saved adjusting the clamping eccentricity increases with the increase of the addendum circle diameter. When the addendum circle diameter of a gear is within 600 mm, the method can save five minutes, on average. When the addendum circle diameter of the gear is beyond 600 mm, the time saved increases rapidly and the maximum time saved is 187 minutes. All in all, the method of involute profiles saves a lot of time that would otherwise be spent adjusting clamping eccentricity.

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[3] Lou, Z.F, Wang, L.D. (2010). Adjustment of the measurement point’s position in a double-disc

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Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F.

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4 Vsebina

Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 59, (2013), številka 4 Ljubljana, april 2013 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki člankov David Heath, Brane Širok, Marko Hočevar, Boštjan Pečnik: Uporaba kavitacije za preprečevanje nalaganja oblog CaCO3 Ani Luo, Heping Liu, Cheng Li, Yongfan Wang: Raziskava cvetu podobne zložljive konstrukcije Tomaž Katrašnik, Henrik Aleš Schuemie, Johann C. Wurzenberger: Konvergenca, stabilnost in računska zahtevnost numeričnih shem za 0-D modeliranje procesov v valju motorjev z notranjim zgorevanjem Nikolaj Mole, Gašper Cafuta, Boris Štok: Metoda za določitev optimalne geometrije platine za izdelavo izdelka s preoblikovanjem na osnovi rezultatov računalniške simulacije Xiaoni Qi, Yongqi Liu, Zhenyan Liu: Eksergijska analiza učinkovitosti hladilnega stolpa Goran Radoičić, Miomir Jovanović: Eksperimentalna identifikacija celotnega strukturnega dušenja sistema Han Lianfu, Tang Wenyan, Wang Jun, Fu Changfeng: Merjenje evolventnega profila z majhno ekscentričnostjo vpetja na preizkuševališču zobnikov Osebne vesti Doktorske disertacije, znanstvena magistrska dela, diplomske naloge

SI 41 SI 42 SI 43 SI 44 SI 45 SI 46 SI 47 SI 48

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 41 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-08-13 Prejeto popravljeno: 2013-01-11 Odobreno za objavo: 2013-02-12

Uporaba kavitacije za preprečevanje nalaganja oblog CaCO3 Heath, D. – Širok, B. – Hočevar, M. – Pečnik, B. David Heath1 – Brane Širok2 – Marko Hočevar2,* – Boštjan Pečnik1,3 1 Mednarodna 2 Univerza

podiplomska šola Jožefa Stefana, Slovenija v Ljubljani, Fakulteta za strojništvo, Slovenija 3 Gorenje d.d., Slovenija

V preglednem članku je podan pregled literature o kavitacijski eroziji krhkih kristaliničnih materialov z namenom boljšega razumevanja materialnih lastnosti oblog CaCO3 ter možnosti za odstranjevanje teh oblog s kavitacijo.

Obloge vodnega kamna CaCO3 so velika nadloga, s katero se soočamo v najrazličnejših industrijskih procesih. Vplivi vodnega kamna v prenosnikih toplote so dvojni: tanjše obloge vodnega kamna lahko bistveno zmanjšajo učinkovitost prenosa toplote, ker delujejo kot izolator pa lahko povzročijo tudi notranje pregrevanje grelnega elementa in posledično okvaro. Vodni kamen lahko tudi močno zmanjša pretok skozi cevi. Vodni kamen se na območjih s trdo vodo nabira tako v gospodinjskih aparatih, kot so bojlerji, pralni stroji in solarni kolektorji, kakor tudi v industrijskih napravah, kjer zmanjšujejo zmogljivost in življenjsko dobo. Neželene posledice vključujejo tudi povečanje vzdrževalnih stroškov, pogosto menjavanje opreme in višje stroške za energijo. Stroški za preprečevanje in odstranjevanje vodnega kamna v industrijskem in gospodinjskem okolju niso neznatni.

Oblikovanje oblog vodnega kamna in kavitacijski fenomeni, vključno z mehanizmi kavitacijske erozije, so v literaturi prikazani. Ob začetnem pregledu literature se izkaže, da je kavitacijska erozija tesno povezana tako z makroskopskimi lastnostmi kot z mikrostrukturo materiala, zato je pri raziskovanju tega problema najbolje uporabiti pristop znanosti o gradivih. Pregled objavljenih študij omogoča boljše razumevanje makro in mikrostrukturnih površinskih lastnosti vodnega kamna ter kavitacijske erozije krhkih kristaliničnih materialov.

Pregled literature je pokazal, da sta oblikovanje vodnega kamna in kavitacijska erozija kompleksna fenomena, ki sta odvisna od mnogih dejavnikov. Gostota, trdota in trpežnost oblog je odvisna od okolja, v katerem se nalagajo.

Okolje v primeru CaCO3 vpliva na materialne in površinske lastnosti nanosa vodnega kamna. Med različnimi mehanizmi kavitacije sta najpomembnejša kolaps oblaka mehurčkov ter učinek pljuskanja po oblikovanju curka.

Pri lomljenju krhkega materiala zaradi kavitacijske erozije prihaja do plastičnih deformacij le v omejenem obsegu, najpomembnejši mehanizem erozije pa je oblikovanje mikrorazpok, ki ga še krepi prisotnost strukturnih nehomogenosti. Boljše razumevanje mehanizmov poškodb zaradi kavitacije ni uporabno le za tiste, ki iščejo načine za izboljšanje učinkovitosti ultrazvočnega odstranjevanja trdovratnih nanosov mineralnih oblog, temveč tudi na področju sonokemije, medicine ter preučevanja naravnih procesov kavitacije pri karbonatnih materialih.

Mehanizmi za ustvarjanje kavitacijskih udarnih valov z dovolj veliko energijo, da lahko poškodujejo material, so trajni predmet preučevanja. Primerjava kavitacijskih tlačnih obremenitev je težavna, saj nekateri avtorji uporabljajo ocene, drugi pa kavitacijo merijo po različnih postopkih. Vrednost ekstremnega tlaka, ki nastane v središču mehurčka, se tudi močno zmanjša na kratki razdalji.

Boljše razumevanje mehanizmov poškodb zaradi kavitacije ni uporabno le za tiste, ki iščejo načine za izboljšanje učinkovitosti ultrazvočnega odstranjevanja trdovratnih nanosov mineralnih oblog, ampak tudi na področju sonokemije, medicine in preučevanja naravnih procesov kavitacije pri karbonatnih materialih.

Ključne besede: kavitacija, kavitacijska erozija, ultrazvok, vodni kamen, kalcijev karbonat, prenosnik toplote

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, marko.hocevar@fs.uni-lj.si

SI 41

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 42 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-03-36 Prejeto popravljeno: 2013-01-28 Odobreno za objavo: 2013-02-18

Raziskava cvetu podobne zložljive konstrukcije Luo, A. – Liu, H. – Li, C. – Wang, Y. Ani Luo – Heping Liu – Cheng Li – Yongfan Wang

Tehniška univerza v Harbinu, Fakulteta za strojništvo in elektrotehniko, Kitajska

Dežniki, pahljače in šotori so primeri zložljivih predmetov, ki se uporabljajo v vsakodnevnem življenju. Zloženi zasedejo malo prostora, da jih je mogoče enostavno shranjevati in prenašati, ko jih preuredimo v delovni položaj, pa dobimo velike površine ali prostornine. Takšne konstrukcije imenujemo zložljive konstrukcije. Z njimi se rešujejo zahtevni problemi pri snovanju vesoljske opreme, kot so npr. satelitske antene in polja sončnih celic.

Takšna vesoljska oprema v delovnem položaju zaseda veliko površino, ko potuje v vesolje, pa mora biti njena prostornina kar se da majhna. Zložljive konstrukcije so zato dobro raziskane in so se uveljavile na področju vesoljskih poletov, ki je zaslužno za njihov hiter razvoj. Članek obravnava raziskavo zložljive konstrukcije, ki je podobna cvetu. Zložljiva konstrukcija je mehanizem, ki se razpre v krožno ploskev in njegova oblika je podobna cvetu, kjer cvetni listi obkrožajo cvetišče. Mehanizem se premika tako, da se cvetni listi z vrtenjem odmikajo stran od fiksnega cvetišča, dokler ni mehanizem popolnoma razprt. Gibanje vseh cvetnih listov je enako in sinhronizirano. V članku je podan pregled cvetu podobne zložljive konstrukcije, ki izpolnjuje omenjene zahteve. Z raziskavo zložljive konstrukcije so bile ovrednotene tudi metode snovanja in analize takšne konstrukcije. Članek obravnava predvsem dimenzije in položaj vhodov na konstrukciji. Po analizi oblike mehanizma v zloženem in razprtem stanju je izračunana prostornina zloženega mehanizma in določeno število listov. Po izračunu prostostnih stopenj je izbrana povezovalna konstrukcija listov in število togih ravnin. Položaj vhodov je določen z metodo analize mehanske topologije in s teorijo vijaka. Končno je s simulacijo preverjena tudi uporabnost zložljive konstrukcije.

Osnova za večino ugotovitev v tem članku je analiza prostostnih stopenj mehanizma. Mehanizem je sestavljen iz več bližnjih verig, zato ugotavljanje prostostnih stopenj ni enostavna naloga, za dokončanje dela pa je treba uporabiti teorijo vijaka in metodo analize topologije.

Iz analize izhajajo naslednji zaključki: 1. Prostornina zložene vesoljske konstrukcije mora biti kar se da majhna, zato je smiselna uporaba metode optimizacije dimenzij konstrukcij mehanizma z namenom doseganja minimalne prostornine. To je preverjeno v članku.

2. Za lažje upravljanje mehanizma je treba zmanjšati število vhodov oz. število prostostnih stopenj. Število elementov in vrsto kinematičnih parov v mehanizmu je torej mogoče ugotoviti z analizo njegovih prostostnih stopenj.

3. Kombinirani teorija vijaka in metoda analize mehanske topologije omogočata določanje pravilnega položaja vhodov. Tudi to je v članku dokazano.

Opisane metode so bile uporabljene v članku in zasnovana je bila uporabna zložljiva konstrukcija. Metode so torej primerne za snovanje tovrstnih zložljivih konstrukcij. Čeprav je bila dokazana uporabnost zasnovanega mehanizma, pa je nujna sinhronizacija šestih vhodov. Na gibanje mehanizma ima velik vpliv sinhronizacija in njegovo upravljanje je zato težavno. Da bi se zmanjšala težavnost upravljanja mehanizma, je treba konstrukcijo še dodatno izpopolniti in zmanjšati število vhodov, kar bo tudi cilj prihodnjih raziskav.

V članku je opisana nova vrsta zložljive konstrukcije. Predstavljena analiza dokazuje, da je mogoče združiti teorijo vijaka in metodo analize topologije za reševanje prostostnih stopenj in iskanje mest za vhode na mehanizmu.

Teorija vijaka in metoda analize topologije omogočata tudi optimizacijo konstrukcije mehanizma z več bližnjimi verigami.

Ključne besede: zložljive konstrukcije, prostostne stopnje, topologija, teorija vijaka, mesta vhodov

SI 42

*Naslov avtorja za dopisovanje: Tehniška univerza v Harbinu, Fakulteta za strojništvo in elektrotehniko, 145 Nantong, Harbin, Kitajska, luoani@hrbeu.edu.cn

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 43 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-06-19 Prejeto popravljeno: 2012-12-28 Odobreno za objavo: 2013-01-18

Konvergenca, stabilnost in računska zahtevnost numeričnih shem za 0-D modeliranje procesov v valju motorjev z notranjim zgorevanjem Tomaž Katrašnik1,* – Henrik Aleš Schuemie2 – Johann C. Wurzenberger2 1 Univerza

2 AVL

v Ljubljani, Fakulteta za strojništvo, Slovenija List GmbH, Napredne simulacijske tehnologije, Avstrija

Razvoj 0-D modelov motorjev z notranjim zgorevanjem, ki so namenjeni simulacijam procesov v realnem času, zahteva izbiro ustreznih numeričnih metod z ozirom na konvergenco, stabilnost in računsko zahtevnost. Model valja predstavlja računsko najzahtevnejši element, saj so za valj značilni največji gradienti termodinamskih parametrov. Zato so lastnosti dveh eksplicitnih in štirih večkoračnih numeričnih shem za reševanje sistemov navadnih diferencialnih enačb analizirane na primeru enovaljnega motorja povezanega s polnilnim in izpušnim zbiralnikom.

Motivacija za izvedbo takšne analize izhaja iz dejstva, da takšna analiza še ni bila opravljena, in iz dejstva, da se za validacijo natančnosti simulacijskih modelov najpogosteje uporablja zgolj podatek o ujemanju z izmerjenimi parametri. Slednje ujemanje pa nikakor na zagotavlja, da je rešitev simulacijskega modela tudi numerično konvergentna ali stabilna. Nastavitve kalibracijskih parametrov, ki se v največji meri nanašajo na parametre modelov zgorevanja in parametre modelov prestopa toplote, so lahko zato uporabljene ne zgolj za povečevanje konsistentnosti med virtualnim modelom in simuliranim objektom, ampak tudi za odpravljanje numeričnih nekonsistentnosti. Težave takšnih modelov se odražajo v odstopanju parametrov med simulacijo motorja v širokem razponu delovnih pogojev in med delovanjem v prehodnem režimu.

V članku so podane vodilne enačbe modela valja in modela prenosa snovi skozi ventile. Numerični eksperiment je izveden za en valj realnega dizelskega motorja in več delovnih točk motorja. Doprinos članka izhaja tudi iz identifikacije delovnih pogojev motorja, delov cikla in parametrov, ki so primerni za analizo stabilnosti in konvergence rešitev. Analiza rezultatov sloni na analizah potekov krivulj parametrov v valju v odvisnosti od zasuka ročične gredi, ki so osnovni pokazatelj stabilnosti in konvergence rešitev. Te osnovne analize primerov so nato nadgrajene z analizo brezdimenzijske vsote kvadratov odstopkov posameznih parametrov čez celoten cikel in računskega časa. Takšna interpretacija rezultatov namreč omogoča enostavno primerjavo natančnosti rezultatov v odvisnosti od računske zahtevnosti posamezne sheme.

Rezultati analize kažejo, da se natančnost posameznih numeričnih shem znatno razlikuje ob enaki računski zahtevnosti, kar nakazuje potrebo po izbiri ustreznih shem. V vseh analiziranih primerih je bila v območju časovnih zahtevnosti, ki so primerne za simulacije v realnem času, natančnost rezultatov največja v primeru uporabe eksplicitne Runge-Kutta sheme četrtega reda, ki je ob tej računski zahtevnosti omogočala izračun konvergentnih in stabilnih rezultatov. V članku je tudi pokazano, da integralni motorski parametri, kot sta srednji indiciran in efektivni tlak procesa, niso primerni kot merilo za določitev konvergence in stabilnosti modela, saj dobro ujemanje integralnih parametrov ni zadosten pogoj za dobro ujemanje potekov parametrov v valju skozi celoten cikel. Članek tako poda jasne napotke za izbiro ustreznih metod za 0-D modeliranje procesov v valju motorja z notranjim zgorevanjem. V članku je prikazana tudi primerjava simulacijskih in izmerjenih podatkov tlakov v valju sodobnega tlačno polnjenega dizelskega motorja, ki potrjuje primernost in natančnost predstavljenega modela za modeliranje procesov v valju.

Ključne besede: modeliranje motorjev z notranjim zgorevanjem, integracijske sheme, konvergenca, stabilnost, računska zahtevnost

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, tomaz.katrasnik@fs.uni-lj.si

43

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 44 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-12-04 Prejeto popravljeno: 2013-01-25 Odobreno za objavo: 2013-02-12

Metoda za določitev optimalne geometrije platine za izdelavo izdelka s preoblikovanjem na osnovi rezultatov računalniške simulacije Nikolaj Mole1 – Gašper Cafuta2 – Boris Štok1,* 1 Univerza

v Ljubljani, Fakulteta za strojništvo, Slovenija d.d., Automobilska industrija, Slovenija

2 Cimos

V današnjem času se pri izdelavi serijskih izdelkov največ pozornosti posveča optimiranju izdelovalnih procesov z vidika ekonomske učinkovitosti in z vidika geometrijske ustreznosti izdelka. Pri preoblikovalnih procesih se, predvsem zaradi večje ekonomske učinkovitosti procesa, pogosto poskuša minimizirati število faz preoblikovalnega procesa, pri kateremu kot zadnja pogosto nastopa faza obreza izdelka. Če uspemo določiti tako geometrijo platine, da geometrija robu preoblikovanega izdelka dovolj dobro ustreza zahtevani geometriji le-tega, se je možno tej fazi v celoti izogniti. Poleg manjših proizvodnih stroškov za izdelavo takega izdelka, je ob ustreznem načrtovanju geometrije platine možno zmanjšati količino porabljenega materiala. Manj materiala v območju pridrževanja pločevine zmanjša tudi možnost pojava gubanja ali trganja pločevine. Ker se izognemo obrezovanju pločevine po preoblikovanju, uporaba platine optimalne oblike omogoča doseganje večje učinkovitosti preoblikovalnega procesa in posledično zmanjšanje proizvodnih stroškov.

V prispevku predstavljena metoda za določevanje optimalne oblike platine omogoča določitev geometrije platine za izdelavo izdelkov poljubnih prostorskih geometrij. Cilj določevanja optimalne oblike platine je čim boljše ujemanje robu geometrije preoblikovanega izdelka z njegovo zahtevano geometrijo. Iterativni proces določevanja optimalne oblike platine se začne z le približno določeno obliko platine, ki pa se z izvajanjem korakov predstavljene metode določi tako, da rob izdelka izdelanega iz te platine natančno ustreza njegovi zahtevani geometriji. V vsaki iteraciji je oblika platine določena tako, da se na podlagi izvedene računalniške simulacije preoblikovanja izdelka in elastične povrnitve izdelka izračuna odstopanje robu izdelka od njegove zahtevane geometrije ter se nato na podlagi tega odstopanja izvede korekcijo oblike platine glede na njeno predhodno obliko.

Računalniške simulacije, ki so osnova za optimizacijo oblike platine, so bile izvedene z računalniškim programom ABAQUS, ki je zasnovan na metodi končnih elementov.

V prvem delu se izvede računalniška simulacija procesa preoblikovanja in sicer, zaradi lažje obravnave kontaktnih razmer med pločevino in deli orodja, z uporabo eksplicitne sheme časovne integracije numeričnega problema. Sledi računalniška simulacija elastične povrnitve izdelka, pri kateri uporabimo implicitno shemo s čemer se ta problem obravnava kot statični problem.

Proces optimizacije oblike platine je popolnoma avtomatiziran, avtomatizacija pa je izvedena z izdelavo računalniškega programa v programskem jeziku Fortran. Ta program je nadgrajen z uporabo programskega jezika Python, ki omogoča branje in zapisovanje rezultatov posamezne iteracije optimizacijskega procesa. Ker je učinkovitost metode določevanja optimalne geometrije platine močno odvisna od fizikalne objektivnosti računalniške simulacije, je bil za potrditev učinkovitosti metode izveden tudi eksperiment.

Eksperiment je bil izveden v dveh korakih. S prvim smo potrdili ustreznost numeričnega modela za računalniško simulacijo preoblikovanja in elastične povrnitve, ki je osnova za izračun geometrije izdelka. V drugem koraku smo potrdili učinkovitost metode optimizacije oblike platine. Na podlagi rezultatov numeričnih izračunov in opravljenih eksperimentov lahko trdimo, da je predstavljena metoda numerično robustna in učinkovita.

Ključne besede: preoblikovanje pločevine, optimizacija, kompenzacija elastične povrnitve, eksperimentalna potrditev, računalniška simulacija, iterativni proces

SI 44

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, boris.stok@fs.uni-lj.si

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI -45 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-01-18 Prejeto popravljeno: 2012-12-17 Odobreno za objavo: 2013-02-12

Eksergijska analiza učinkovitosti hladilnega stolpa Xiaoni Qi1, * – Yongqi Liu1 – Zhenyan Liu2 1 Tehniška

2 Fakulteta

univerza v Shandongu, Fakulteta za promet in vozila, Kitajska za strojništvo in elektrotehniko, Univerza Šanghaj Jiao Tong, Kitajska

V konvencionalnih hladilnih stolpih se hladilna zmogljivost sčasoma zmanjša zaradi nalaganja oblog na polnilu, ki ovirajo pretok zraka. Prisotnost polnjenja tudi občutno poveča upor, ki je glavni dejavnik porabe moči. Stolp z odstranjenim polnilom je prazen, to pa vpliva na hlajenje kalne vode z visoko temperaturo.

Pregled stanja raziskav kaže, da ne obstaja sistematična teorija za preučevanje hladilnih stolpov. Dejavniki vpliva so notranja zgradba, način pršenja vode, okoljski dejavniki, masni pretok zraka, hitrost vetra, masni pretok vode, vstopna temperatura vode itn. Odvisnost med temi dejavniki in učinkovitostjo hlajenja je zahtevna in nelinearna. Raziskovalci morajo za reševanje tega problema izvajati eksperimente na velikih sistemih, ki zahtevajo velik vložek dela in materialnih sredstev. Eksperimentalne metode prav tako ne morejo pokriti celotnega obsega možnih okoljskih pogojev. Zato je nujna vzpostavitev primernega matematičnega modela prenosa toplote in snovi za proces hlajenja vroče vode v hladilnem stolpu.

Eksergijska analiza lahko nakaže možnosti za termodinamične izboljšave obravnavanega procesa. Muangnoi je z eksergijsko analizo pojasnil zmogljivost običajnega hladilnega stolpa. Danes še ni veliko znanega o uporabnosti eksergijske analize za preučevanje hladilnih stolpov. V tem članku je učinkovitost hladilnega stolpa napovedana na osnovi prenosa toplote in snovi med vodo in zrakom, ki pripelje do rešitve v pogojih stacionarnega stanja. Za reševanje parcialnih diferencialnih enačb, ki dajejo temperaturo kapljic v posameznih sekcijah in izhodno temperaturo vode, je bila uporabljena metoda Runge-Kutta. Porazdelitev eksergije v vodi in zraku hladilnega stolpa je določena z uporabo drugega zakona termodinamike. Celotna eksergija sestoji iz treh delov: termomehanske energije exch, mehanske eksergije exme in kemične energije exch. Hitrost izgubljanja eksergije I ali izgubljeni potencial zraka, da prevzame eksergijo od vode, je mogoče določiti na podlagi enačbe ravnotežja eksergije za kontrolne volumne. Enačbe popisujejo stacionarno stanje in adiabaten proces brez vloženega dela.

Model je bil validiran s podatki, ki jih je priskrbelo kitajsko podjetje Jiangsu Seagull Cooling Tower Co., LTD. Primerjana je bila izhodna temperatura vode, pri čemer je bilo ugotovljeno, da je napaka med napovedanimi in eksperimentalnimi vrednostmi znotraj 10 %. Model je zato primeren za napovedovanje stanj vode in zraka v hladilnih stolpih.

Po validaciji modela so bile preučena nekatera eksergijska dogajanja v hladilnem stolpu. Nepovračljivost vsakega procesa uniči nekaj vstopne eksergije. Rezultati tudi kažejo, da zrak ne absorbira vse eksergije vode in da se vedno izgubi znaten del eksergije, predvsem v spodnjih delih stolpa. Eksergijska analiza kaže, da se izguba eksergije povečuje z rastjo premera vodnih kapljic. Eksergija vode, opredeljena kot razpoložljiva energija, ki jo je mogoče pridobiti iz vode, se zvezno zmanjšuje od vrha do dna. Eksergija na zračni strani je razpoložljiva energija zraka, ki jo ta prejme od vode. Na dnu se izgubi veliko eksergije, manj na vrhu. Na spremembi višine od 0,00 do 0,56 m se izgubi 1,19 kW eksergije. 0,876 kW eksergije se izgubi pri prehodu iz višine 0,56 na 1,12 m, in tako naprej. Porazdelitev izgube eksergije je mogoče uporabiti za pomoč pri iskanju optimalne učinkovitosti hladilnega stopa. Uporabna energija, ki se iz vode prenaša na zrak, se zmanjšuje, to pa pomeni izgubo eksergije.

To je pomemben omejitveni dejavnik pri hladilnem stolpu.

Ključne besede: hladilni stolp, prenos toplote in snovi, metoda Runge-Kutta, termomehanska eksergija, mehanska eksergija, kemična eksergija

*Naslov avtorja za dopisovanje: Tehniška univerza v Shandongu, Fakulteta za promet in vozila, Kitajska, qixiaoni@sjtu.org

SI 45

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 46 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-05-03 Prejeto popravljeno: 2012-09-24 Odobreno za objavo: 2012-11-06

Eksperimentalna identifikacija celotnega strukturnega dušenja sistema Goran Radoičić – Miomir Jovanović Goran Radoičić – Miomir Jovanović*

Univerza v Nišu, Fakulteta za strojništvo, Srbija

Vrednosti parametrov dušenja se v analizi strukturne dinamike najpogosteje povzamejo po empiričnih priporočilih, pri tem pa se uvede neznana napaka, ki povzroči nerealističen dinamični odziv konstrukcije. Zaradi pomena strukturnega dušenja pri strukturni analizi je nujna opredelitev preverjenega eksperimentalno-teoretičnega postopka za identifikacijo koeficienta celotnega strukturnega dušenja G. Model za določanje koeficienta G mora biti preverjen na kompleksni realni strukturi.

Realen dinamični odziv velike jeklene palične konstrukcije na zunanje motilne sile zaradi hitrega spuščanja bremena do udarca ob tla je bil določen eksperimentalno. Eksperimentalna metoda vključuje določanje periode vibracij, amplitud premikov na referenčnih točkah konstrukcije in logaritemskega dekrementa dušenega vibracijskega gibanja. Cilj raziskave je poiskati vrednost koeficienta G, ki ustreza eksperimentom, na osnovi več simulacij z različnim strukturnim dušenjem. Z drugimi besedami, vzporedno opazovanje prostih dušenih vibracij konstrukcije pri različnih koeficientih G z numerično simulacijo omogoča opredelitev vrednosti koeficienta G, ki daje enako dinamično vedenje konstrukcije kot pri eksperimentu.

Eksperimentalna raziskava strukturnega dušenja je bila zasnovana na meritvah napetosti. Uporabljena je bila sodobna merilna oprema HBM. Rezultati eksperimentov so bili obdelani s programsko opremo HBM Catman AP. Eksperiment je bil opravljen na stolpnem žerjavu Potain 744e. Parametri predhodno razvitega numeričnega modela so: masa 72204 kg, število končnih elementov 1667, število vozlišč 1146, število prostostnih stopenj 6876.

Za analizo dinamičnega odziva je bila uporabljena Newmarkova metoda časovne integracije. Privzeti so bili veliki premiki.

Najnižji lastni frekvenci v drugem in četrtem vibracijskem načinu ω2,m = 0,4749 Hz in ω4,m = 1,5444 Hz najbolje predstavljata dominantni značaj strukturnih vibracij v vertikalni ravnini pri vzbujanju. Lastne frekvence so bile ugotovljene z modalno analizo. Ker uporabljena analiza dinamičnega odziva ne vključuje dušenja v obliki kompleksnih števil, je bilo strukturno dušenje za najdene dominantne frekvence pretvorjeno v ekvivalentno viskozno dušenje. Pretvorba je bila opravljena z enačbo: [BT] = [B] + (1/ω3)·[K] + (1/ω4)·∑GE[KE].

Koeficient dušenja (amortizacije) karakterističnih točk pri meritvah je obsegal vrednosti v območju d = 0,01 do 0,108. Na podlagi ugotovljenih posameznih vrednosti dušenja vibracij je bila z numeričnimi simulacijami opredeljena vrednost celotnega koeficienta strukturnega dušenja G = 0,05, pri kateri se rezultati najbolje ujemajo z eksperimentom. Na osnovi raziskave so bile pridobljene matematične aproksimacije dušenja.

Natančno ugotavljanje disipacije energije v velikih konstrukcijah je kompleksen problem, zato je pri dinamični analizi bolj praktična uporaba celotnega koeficienta strukturnega dušenja G, ki vključuje različne oblike dušenja povezav v konstrukciji. Za natančno opredelitev vrednosti koeficienta G je smiselno izvesti eksperimentalne dinamične preizkuse.

Prispevek te raziskave je v praktični uporabi koeficienta strukturnega dušenja v simulacijah realnih konstrukcij z velikim številom elementov. Dušenje je vključeno v modelu z bolj praktično kompleksno enačbo [KT] = (1 + iG) ·[K] + ∑GE[KE]. Eksperimentalno določanje koeficienta G ima prednost, ker omogoča boljše ocenjevanje parametrov: opisa začetnih pogojev premikov ter hitrosti razbremenjevanja in obremenjevanja konstrukcije za impulzni učinek. Eksperimentalno ugotovljene amplitude v izjemnih primerih sodijo med velike premike, saj dosegajo celo 10 % višine objekta.

Znanstvena vrednost raziskave je tudi v eksperimentalni verifikaciji zelo zahtevnega numeričnega modela.

Z eksperimentalno vrednostjo koeficienta G je bila potrjena zadostna natančnost modela MKE in preverjen numerični postopek za analizo dinamičnega odziva na dano vzbujanje. Eksperimentalno-teoretični model za identifikacijo koeficienta G, predstavljen v članku, je mogoče uporabiti tudi za podobne konstrukcije kot so večji stolpni žerjavi, viličarji za kontejnerje, pristaniški in ladjedelniški žerjavi, stolpi.

Ključne besede: strukturno dušenje, dinamični odgovor, palična konstrukcija, koeficient dušenja, stolpni žerjav, velika amplituda

SI 46

*Naslov avtorja za dopisovanje: Univerza v Nišu, Fakulteta za strojništvo, A.Medvedeva 14, NIš, Srbija, miomir@masfak.ni.ac.rs

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 47 © 2013 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2012-03-23 Prejeto popravljeno: 2013-01-18 Odobreno za objavo: 2013-02-12

Merjenje evolventnega profila z majhno ekscentričnostjo vpetja na preizkuševališču zobnikov Lianfu, H. – Wenyan, T. – Jun, W. – Changfeng, F. Han Lianfu1,2,* – Tang Wenyan1 – Wang Jun1 – Fu Changfeng2

1 Tehnološki

institut v Harbinu, Institut za natančne instrumente, Kitajska univerza za petrokemijo, Fakulteta za elektroniko, Kitajska

2 Severovzhodna

Napaka ekscentričnosti vpetja neizogibno zmanjšuje natančnost merjenja na preizkuševališču zobnikov. V članku je opisana metoda merjenja evolventnega profila z majhno ekscentričnostjo vpetja na preizkuševališču zobnikov, s ciljem zmanjšanja vpliva ekscentričnosti vpetja pri merjenju evolventnega profila in prihranka časa za nastavljanje ekscentričnosti vpetja.

Evolventni profili se običajno merijo na preizkuševališču zobnikov. Veliki in notranji zobniki se na merilno platformo preizkuševališča pritrdijo z več magneti. Pri vpenjanju takšnih velikih in notranjih zobnikov si običajno pomagamo z ročnimi merilnimi urami. Geometrijska srednjica zobnika pogosto ni natančno vzporedna z rotacijsko osjo preizkuševališča zobnikov zaradi omejenih zmožnosti ročnega mikropozicioniranja in natančnosti pomožnih vpenjalnih pripomočkov. Posledica tega je napaka ekscentričnega vpetja.

Analiza načina merjenja evolventnega profila ekscentrično vpetega ozobja je pokazala, da je izmerjena krivulja del merjene evolventne krivulje in da se izmerjene točke kljub ekscentričnosti vpetja nahajajo na merjeni krivulji. Izmerjena točka je lahko zamaknjena naprej ali nazaj glede na dejanski položaj, še vedno pa se nahaja na merjenem evolventnem profilu. Merilne trajektorije evolventnih profilov zato ni treba spreminjati, če je vrednost ekscentričnosti vpetja znotraj 0,75 L. Pri analizi pa je nujen premik profilov v privzeti merjeni položaj.

Pri analizi razlik pri različnih pogojih se je izkazalo, da je napaka ekscentričnosti vpetja posledica napake transformacije med koordinatnim sistemom merjenca in koordinatnim sistemom instrumenta oz. neupoštevanja transformacijskega vektorja ekscentričnosti vpetja βi.

Po analizi merilne trajektorije evolventnih profilov v pogojih ekscentričnega vpetja je bil določen model kompenzacije napake ekscentričnosti vpetja, ki vsebuje samo dve neznanki (vrednost ekscentričnosti vpetja e in polarni kot ϕ0). Središče kroga za vsak zob se pridobi po enačbi evolventnega profila.

Metoda privzema majhno ekscentričnost vpetja pri merjenju evolventnega profila na preizkuševališču zobnikov.

Rezultati: (1) Merilne trajektorije evolventnih profilov ni treba spreminjati, če je ekscentričnost vpetja znotraj 0,75 L, profile z majhno ekscentričnostjo vpetja pa je treba pri analizi premakniti v privzeti merilni položaj.

(2) Napaka ekscentričnosti vpetja je posledica napake transformacije med koordinatnim sistemom merjenca in koordinatnim sistemom instrumenta oz. neupoštevanja transformacijskega vektorja ekscentričnosti vpetja βi.

(3) Parametri ekscentričnosti, pridobljeni po evolventnem modelu iz tega članka, se ujemajo s tistimi po modelu Limacon. Metoda ocenjevanja ekscentričnosti parametrov v tem članku je zato pravilna in praktično uporabna.

(4) Model kompenzacije napake ekscentričnega vpetja omogoča kompenzacijo rezultatov analize v stanje brez ekscentričnosti in s tem natančno merjenje evolventnega profila. (5) Največja dovoljena ekscentričnost vpetja je tri četrtine merilnega območja merilne glave in omogoča pomemben prihranek časa pri nastavljanju ekscentričnosti vpetja.

V članku je opisan postopek merjenja evolventnega profila z majhno ekscentričnostjo vpetja na preizkuševališču zobnikov. Metoda dopušča manjšo ekscentričnost vpetja pri merjenju evolventnega profila ter je uporabna za merjenje zelo velikih zobnikov. Ti so veliki in težki, zato jih je težko vpeti brez ekcentričnosti.

Metoda evolventnih profilov lahko zato prihrani veliko časa pri nastavljanju ekscentričnosti vpetja.

Ključne besede: preizkuševališče zobnikov, ekscentričnost vpetja, evolventni profil, meritev, analiza

*Naslov avtorja za dopisovanje: Tehnološki institut v Harbinu, Institut za natančne instrumente, 92 West Dazhi Street 150001, Harbin, Kitajska, lianfuhan@hit.edu.cn

SI 47

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 48-51 Osebne objave

Doktorske disertacije, znanstvena magistrska dela, diplomske naloge

DOKTORSKE DISERTACIJE Na Fakulteti za strojništvo Univerze v Mariboru so obranili svojo doktorsko disertacijo: ● dne 15. marca 2013 Shpetim LAJQI z naslovom: »Razvoj obes in krmilnega sistema terenskega vozila s štirimi gnanimi in štirimi krmiljenimi kolesi« (mentor: izr. prof. dr. Stanislav Pehan); Cilj doktorske disertacije je zasnovati, oblikovati, optimizirati in analizirati sistem obes in krmilja za štirikolesno vozilo, ki ima pogon in krmiljenje na vsa kolesa. Značilnost terenskih vozil so veliki pomiki koles, da se vozilo ne prevrne, ko se giblje po neravnem terenu. Znani sistemi obes in krmiljenja imajo slabosti. Za zagotavljanje višje udobnosti, varnosti vozila, in odzivnosti je treba razviti nove koncepte obes in nov krmilni sistem.

Predlagan sistem obes je učinkovita modifikacija dvojnega A-mehanizma z enakima, a dolgima rokama, kar zagotavlja velike vertikalne pomike koles, brez poslabšanja voznih karakteristik. Optimalna sila dušenja, ki je potrebna za izboljšanje udobja in varnosti, se ustvarja z navideznim oziroma resničnim aktivnim sistemom, ki je dodan sitemu obes. Obširne analize dokazujejo upravičenost uporabe aktivnih sistemov. Predlagani sistem obes učinkovito absorbira vibracije zaradi neravnin in zagotavlja relativno majhno bočno pomikanje koles, brez nepotrebnega nagibanja.

Cilj je razviti takšen krmilni sistem terenskega vozila, ki bi zagotavljal učinkovito upravljanje vozila pri nizkih hitrostih in zagotavljal stabilnost pri večjih hitrostih. To je izpolnjeno s krmiljem, ki deluje na dva različna načina. Pri enem načinu sta so aktivna vsa štiri kolesa, pri drugem pa samo sprednja. Predlagani krmilni mehanizem z gonili in gredmi deluje tako, da v vseh situacijah povsem sledi zakonitosti Ackermanna. Prikazan je koncept povsem novega principa učinkovitega krmilnega sistema, ki temelji na posebnih gonilih s specialnimi pari zobnikov; ● dne 15. marca 2013 Nadja ROMIH z naslovom: »Vpliv povečane koncentracije kovin v rastlinah na njihovo snovno in energijsko izrabo« (mentor: doc. dr.

Cvetka Ribarič Lasnik); Onesnažena tla s kovinami so posledica več stoletnega nekontroliranega antropogenega onesnaževanja v preteklosti. Kovine zaradi svoje nezmožnosti razgradnje, za razliko od večino organskih snovi, predstavljajo resno grožnjo za okolje, ljudi in živali in jih je potrebno primerno odstraniti.

Eden od ukrepov revitalizacije onesnaženega ozemlja SI 48

s kovinami je pozelenitev z energetskimi rastlinami in situ.

V raziskavi smo z metodo fitoekstrakcije opazovali distribucijo kovin v rastlinskih delih juvenilnih in odraslih hibridov PR45 D01 in PR46 W31 oljne ogrščice (Brassica napus L. var. napus) in situ na različno onesnaženih tleh (Ponikva, Šentjur, Medlog, Proseniško, Škofja vas, Črna na Koroškem in Mežica). Rezultati raziskave kažejo, da je privzem Cd, Pb in Zn v nad-zemne dele odraslih rastlin do 56, 35 in 52 % v zaporedju večji v primerjavi z nadzemnimi deli juvenilnih rastlin. Vsebnost kovin v rastlinah je predvsem odvisna od vsebnosti kovin v tleh in pH vrednosti tal. Hibrida oljne ogrščice privzemata kovine in As v svoje nadzemne dele in, predvsem pomembno, tudi v semena. Za vsako kovino je bil izračunan TF, to je razmerje med rastlinskimi deli, in BAF, to je razmerje med vsebnostjo tarčnega elementa v nadzemnem delu in vsebnostjo elementa v tleh. Rezultata TF(Cd) in BAF(Cd) > 1 kažeta, da lahko hibrid PR45 W31 uporabimo za fitoremediacijo na srednje onesnaženih tleh. Primerjava dveh metod pridobivanja olja iz semen je pokazala, da je superkritična ekstrakcija s CO2 bolj primerna metoda.

Pogača po hladnem mehanskem stiskanju in po SC CO2 semen ni vsebovala povečanih vsebnosti kovin in je primerna v proizvodih za prehrano živali. Termična obdelava stebel in pogače pri 700 °C je pokazala, da so vsebnosti Cd, Pb in Zn v ostanku najmanjše in vsebnosti Cu, Ni, Co, Mo, As in Ba v ostanku pri 700 °C največje. Pri termični obdelavi na 850 °C so bile vsebnosti Cd, Pb in Zn v ostankih značilno večje od vsebnosti Cu Ni, Co, Mo, As in Ba. Kurilne vrednosti stebel in pogače so primerljive bitumenskemu premogu, lignitu in lesu.

Vpliv povečane vsebnosti kovin v tleh ni vplival na snovno in energijsko izrabo oljne ogrščice. Uporaba enega od ukrepov remediacije, pozelenitev, ne samo, je ne le ekonomsko najbolj donosna sanacija, ampak bi tudi polepšala izgled krajine in izboljšala kvaliteto življenja ljudi, ki so izpostavljeni kontaminiranim področjem. ● dne 15. marca 2013 Suzana PRAJNC z naslovom: »Razvoj poenostavljene metodologije za nadzor emisij cestnega prometa v urbanem okolju« (mentor: doc. dr. Branka Trček); Za urbana območja so zaradi velikega števila prebivalcev in ekonomskega razvoja značilni veliki pritiski na okolje. V preteklosti so bili ti posledica delovanja industrijskih obratov in individualnih kurišč, zdaj pa so v veliki meri rezultat povečevanja

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 48-51

cestnega prometa. Dobro poznavanje razmer na področju onesnaženosti zraka, ki ga povzroča promet v mestu, je temeljni pogoj za izdelavo učinkovitih programov varstva okolja in načrtovanje ukrepov za zmanjšanje prekomernih koncentracij onesnaževal v zraku. Emisije onesnaževal iz cestnega prometa je težko izmeriti, zato jih računamo na podlagi prometnih podatkov in specifičnih emisijskih faktorjev za posamezne kategorije vozil. Metodologije, ki se uporabljajo za nadzor emisij iz prometa, se med seboj razlikujejo glede na vhodne podatke, emisijske faktorje in način izračuna emisij. Z uporabo metodologije COPERT IV, ki je predpisana za poročanje o nacionalnih emisijah v okviru Evropske okoljske agencije, je pa zaradi teže dostopnih vhodnih podatkov in zahtevnosti uporabe manj primerna za poročanje na ravni občin, smo določili emisije za mesti Maribor in Celje. Na podlagi rezultatov metodologije COPERT in podatkov iz literature smo razvili poenostavljeno metodologijo, ki zahteva manj vhodnih podatkov, je preprosta za uporabo in hkrati učinkovita. Primerjava rezultatov poenostavljene metodologije z izračuni metodologije COPERT IV je pokazala, da je poenostavljena metodologija primerna za oceno skupnih emisij onesnaževal CO, NOx in PM10 iz cestnega prometa za celotno mesto Maribor in manjša mestna območja, primerna je tudi za oceno emisij CO, NOx in PM10 iz cestnega prometa po kategorijah vozil za mestna območja s številom registriranih vozil nad 20.000 ter za oceno skupnih emisij CO, NOx in PM10 iz cestnega prometa za manjša slovenska mesta. Zaradi pričakovanih sprememb strukture vozil v mestih bi bilo treba poenostavljeno metodologijo preveriti in nadgraditi vsaka tri leta.

ZNANSTVENA MAGISTRSKA DELA Na Fakulteti za strojništvo Univerze v Ljubljani je z uspehom zagovarjal svoje magistrsko delo: ● dne 19. marca 2013 Jaka OGRIČ z naslovom: »Določitev modela toka materiala pri iztiskanju bakrovih legur« (mentor: prof. dr. Franc Kosel, somentor: izr. prof. dr. Janez Kušar); ● dne 19. marca 2013 Andrej KOBAL z naslovom: »Projektno vodenje sočasnega osvajanja serijskih izdelkov« (mentor: prof. dr. Marko Starbek).

DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv univerzitetni diplomirani inženir strojništva: dne 4. marca 2013:

Matej LIBENŠEK z naslovom: »Vpliv procesnih parametrov visokotlačnega struženja na obdelovalnost trdih jekel« (mentor: doc. dr. Davorin Kramar, somentor: prof. dr. Janez Kopač); Rok MAVRIČ z naslovom: »Dinamični odziv vzbujenega hidravličnega cevnega sistema« (mentor: prof. dr. Branko Širok, somentor: izr. prof. dr. Anton Bergant); Žan SMODE z naslovom: »Dimenzioniranje enoosnega veznega elementa iz gradiv z oblikovnim spominom« (mentor: prof. dr. Franc Kosel, somentor: doc. dr. Tomaž Videnič); Žiga TURK z naslovom: »Kinematika toka na izstopnih šobah prezračevalnega sistema hladilnika HZA« (mentor: prof. dr. Branko Širok, somentor: izr. prof. dr. Marko Hočevar); dne 26. marca 2013: Žiga JELEN z naslovom: »Razplinjanje lignitnega sloja v Premogovniku Velenje« (mentor: izr. prof. dr. Andrej Senegačnik); Kristijan JURKO z naslovom: »Gorivne celice v napredni energijski oskrbi« (mentor: izr. prof. dr.

Mihael Sekavčnik); Nejc JURKOVIČ z naslovom: »Motor na komprimiran zrak« (mentor: izr. prof. dr. Mihael Sekavčnik); Anže LONČAR z naslovom: »Obtočni ventilator kondenzacijskega sušilnega stroja« (mentor: prof. dr.

Branko Širok); Jaša POLUTNIK z naslovom: »Značilnosti aktivnega elektroenergetskega omrežja v primorski regiji« (mentor: izr. prof. dr. Andrej Senegačnik).

Jure POTRPIN z naslovom: »Pojav laminarnega separacijskega toka ob aerodinamskem profilu« (mentor: izr. prof. dr. Marko Hočevar, somentor: prof. dr. Branko Širok); dne 27. marca 2013: Dušan JAUŠOVEC z naslovom: »Vpliv dinamičnih obremenitev na poškodbe ležajev pri montaži AZ motorja« (mentor: prof. dr. Miha Boltežar, somentor: doc. dr. Janko Slavič); Eva JORDAN z naslovom: »Optimizacija toka materiala in vrednosti« (mentor: prof. dr. Marko Starbek, somentor: izr. prof. dr. Janez Kušar); Klemen POGAČNIK z naslovom: »Optimiranje končne montaže« (mentor: prof. dr. Marko Starbek Somentor: izr. prof. dr. Janez Kušar); Tomaž RIFL z naslovom: »Delovanje kondenzatorja v parnem krožnem procesu z organsko delovno snovjo« (mentor: izr. prof. dr. Andrej Senegačnik); Adam VRAN z naslovom: »Souporaba lesnih sekancev v obročni jaškasti peči« (mentor: izr. prof. dr. Andrej Senegačnik); dne 28. marca 2013: SI 49

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 48-51

Tjaša DUH z naslovom: »Tehnično-ekonomska analiza sistema daljinskega hlajenja « (mentor: prof. dr. Alojz Poredoš); Hedvika KORBAR z naslovom: »Izguba legirnih elementov med reparaturnim varjenjem orodnih jekel« (mentor: prof. dr. Janez Tušek); Jure PENCA z naslovom: »Eksperimentalni sistem za določanje frekvenčnih karakteristik komponent izpušnega sistema« (mentor: doc. dr.

Primož Podržaj, somentor: doc. dr. Jurij Prezelj); Dejan BROZOVIČ z naslovom: »Razvoj vmesnika za prikaz energetskih kazalnikov zgradb« (mentor: prof. dr. Sašo Medved); Matej PAHOR z naslovom: »Parametrični model rabe toplote v stavbah« (mentor: prof. dr. Sašo Medved); Luca PETAN z naslovom: »Integriteta površine po hladni deformaciji površinskega sloja aluminijeve zlitine« (prof. dr. Janez Grum); Miha PLESTENJAK z naslovom: »Lasersko oplastenje aluminijeve zlitine« (mentor: prof. dr.

Janez Grum Somentor: izr. prof. dr. Roman Šturm); dne 29. marca 2013: Martin BIZJAK z naslovom: »Vpliv porazdelitve velikosti delcev reciklirane granulirane gume na akustične lastnosti zvočno-izolacijskih plošč« (mentor: prof. dr. Igor Emri, somentor: prof. dr. Mirko Čudina); Primož KUTNAR z naslovom: »Postavitev CAM tehnologije za izdelavo orodij - modelov za formanje« (mentor: prof. dr. Janez Kopač, somentor: doc. dr.

Franci Pušavec); Mark MIŠIČ z naslovom: »Razvoj hidravlične tehtnice« (mentor: doc. dr. Franc Majdič); Andraž SNEDIC z naslovom: »Obdelovalnost nodularne (GGG70) in jeklene litine (1.2333) pri procesu obodnega frezanja« (mentor: doc. dr. Franci Pušavec, somentor: prof. dr. Janez Kopač); * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 22. marca 2013: Matej VERK z naslovom: »Konstruiranje nosilca viseče mreže« (mentor: prof. dr. Zoran Ren, somentor: asist. dr. Matej Borovinšek); dne 29. marca 2013: Denis KOHEK z naslovom: »Posodobitev tehnologije ovijanja paketov v podjetju Codex d.o.o.« (mentor: doc. dr. Iztok Palčič, somentor: prof. dr. Riko Šafarič); SI 50

Aleš MERČNIK z naslovom: »Konstruiranje moment ključa« (mentor: doc. dr. Aleš Belšak, somentor: izr. prof. dr. Miran Ulbin); Denis OMLADIČ z naslovom: »Znižanje stroškov in modifikacija obstoječega paličnega mešalnika v podjetju BSH Hišni aparati, d.o.o.« (mentor: doc. dr. Marjan Leber, somentor: izr. prof. dr.

Borut Buchmeister); Matej SKORNŠEK z naslovom: »Montaža in zagon glavnega jamskega črpališča v Premogovniku Velenje« (mentor: prof. dr. Miran Brezočnik, somentor: doc. dr. Darko Lovrec). * Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 7. marca 2013: Klemen DENŠA z naslovom: »Optimizacija odstranjevanja snega in ledu s površin letal na Letališču Jožeta Pučnika Ljubljana« (mentor: izr. prof. dr. Tadej Kosel, somentor: doc. dr. Robert Kunc); Sandi GALIJOT z naslovom: »Eksperimentalna proga za analizo sreženja na superhidrofobnih površinah« (mentor: prof. dr. Iztok Golobič); Anton KOVAČIČ z naslovom: »Vizualna kontrola avtomobila pred prevzemom in distribucijo« (mentor: prof. dr. Mirko Soković); Erik SOBAN z naslovom: »Tračni transporter za pekovsko peč« (mentor: doc. dr. Boris Jerman); dne 8. marca 2013: Gašper BURJAK z naslovom: »Uporovno varjenje žic premera od 4 do 6 mm« (mentor: prof. dr.

Janez Tušek); Admir JAŠARAGIĆ z naslovom: »Razvoj senčnika za meglenko s kotno funkcijo« (mentor: izr. prof. dr. Jernej Klemenc); Dejan ŠTEFANIČ z naslovom: »Razvoj strojnega vršičkarja za prirezovanje vršičkov vinskih trt« (mentor: izr. prof. dr. Jernej Klemenc). * Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva: dne 29. marca 2013: Darjan HANŽELIČ z naslovom: »Racionalizacija stroja za ovijanje palet« (mentorica: doc. dr. Marjan Leber, somentor: izr. prof. dr. Borut Buchmeister); Uroš RAUTER z naslovom: »Konstruiranje varilne priprave za varjenje avtomobilskega dela v robotski celici« (mentor: doc. dr. Janez Kramberger); Janez THALER z naslovom: »Tehnološka priprava za izdelavo gredi električne motorne žage« (mentor: prof. dr. Miran Brezočnik).

Strojniški vestnik - Journal of Mechanical Engineering 59(2013)4, SI 48-51

* Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani inženir strojništva (UN): dne 29. marca 2013: Tadej LOVREC z naslovom: »Načrtovanje tehnološkega procesa laserskega razreza delov za avtomatski ovijalni stroj« (mentor: izr. prof. dr. Borut Buchmeister, somentor: prof. dr. Franci Čuš). * Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv diplomirani gospodarski inženir (UN): dne 29. marca 2013:

Žiga VERONEK z naslovom: »Računalniško podprto merjenje komponent malih gospodinjskih aparatov z optičnim digitalizatorjem« (mentor: prof. dr. Bojan Ačko, somentor: doc. dr. Barbara Bradač Hojnik). * Na Fakulteti za strojništvo Univerze v Ljubljani je pridobil naziv diplomirani inženir strojništva (VS): dne 8. marca 2013: Tine BERGINC z naslovom: »Energetsko samostojno ogrevanje stavbe s toplotno črpalko podprto s fotonapetostnim sistemom« (mentor: prof. dr. Alojz Poredoš).

SI 51

Strojniški vestnik – Journal of Mechanical Engineering (SV-JME) Aim and Scope The international journal publishes original and (mini)review articles covering the concepts of materials science, mechanics, kinematics, thermodynamics, energy and environment, mechatronics and robotics, fluid mechanics, tribology, cybernetics, industrial engineering and structural analysis.

The journal follows new trends and progress proven practice in the mechanical engineering and also in the closely related sciences as are electrical, civil and process engineering, medicine, microbiology, ecology, agriculture, transport systems, aviation, and others, thus creating a unique forum for interdisciplinary or multidisciplinary dialogue.

The international conferences selected papers are welcome for publishing as a special issue of SV-JME with invited co-editor(s).

Editor in Chief Vincenc Butala University of Ljubljana Faculty of Mechanical Engineering, Slovenia Technical Editor Pika Škraba University of Ljubljana Faculty of Mechanical Engineering, Slovenia Editorial Office University of Ljubljana (UL) Faculty of Mechanical Engineering SV-JME Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386-(0)1-4771 137 Fax: 386-(0)1-2518 567 E-mail: info@sv-jme.eu, http://www.sv-jme.eu Print DZS, printed in 450 copies Founders and Publishers University of Ljubljana (UL) Faculty of Mechanical Engineering, Slovenia University of Maribor (UM) Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia

http://www.sv-jme.eu

59 (2013) 4

Chamber of Commerce and Industry of Slovenia Metal Processing Industry Association

Since 1955

Strojniški vestnik Journal of Mechanical Engineering

Journal of Mechanical Engineering - Strojniški vestnik

s

year

no. 4 2013 59

volume

Cover: Cavitation near the solid surface causes local pressure shock as a result of cavitation cloud collapse, which causes material destruction in the form of cavitation erosion.The figure shows the effect of ultrasonic cavitation on a lime stone coated surface. The figure shows vectors of velocity of fluid flow in the vicinity of the test sample.

Image Courtesy: Laboratory for Water and Turbine Machines, Faculty of Mechanical Engineering, University of Ljubljana.

International Editorial Board Koshi Adachi, Graduate School of Engineering,Tohoku University, Japan Bikramjit Basu, Indian Institute of Technology, Kanpur, India Anton Bergant, Litostroj Power, Slovenia Franci Čuš, UM, Faculty of Mech. Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mech. Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mech. Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mech. Engineering, Slovenia Imre Horvath, Delft University of Technology, Netherlands Julius Kaplunov, Brunel University, West London, UK Milan Kljajin, J.J. Strossmayer University of Osijek, Croatia Janez Kopač, UL, Faculty of Mech. Engineering, Slovenia Franc Kosel, UL, Faculty of Mech. Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mech. Engineering, Slovenia Miroslav Plančak, University of Novi Sad, Serbia Brian Prasad, California Institute of Technology, Pasadena, USA Bernd Sauer, University of Kaiserlautern, Germany Brane Širok, UL, Faculty of Mech. Engineering, Slovenia Leopold Škerget, UM, Faculty of Mech. Engineering, Slovenia George E. Totten, Portland State University, USA Nikos C. Tsourveloudis, Technical University of Crete, Greece Toma Udiljak, University of Zagreb, Croatia Arkady Voloshin, Lehigh University, Bethlehem, USA President of Publishing Council Jože Duhovnik UL, Faculty of Mechanical Engineering, Slovenia General information Strojniški vestnik – Journal of Mechanical Engineering is published in 11 issues per year (July and August is a double issue).

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REFERENCES A reference list must be included using the following information as a guide. Only cited text references are included. Each reference is referred to in the text by a number enclosed in a square bracket (i.e., [3] or [2] to [6] for more references). No reference to the author is necessary.

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Journal Papers: Surname 1, Initials, Surname 2, Initials (year). Title. Journal, volume, number, pages, DOI code.

[1] Hackenschmidt, R., Alber-Laukant, B., Rieg, F. (2010). Simulating nonlinear materials under centrifugal forces by using intelligent crosslinked simulations. Strojniški vestnik - Journal of Mechanical Engineering, vol. 57, no. 7-8, p. 531-538, DOI:10.5545/sv-jme.2011.013.

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[2] Groover, M.P. (2007). Fundamentals of Modern Manufacturing. John Wiley & Sons, Hoboken.

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[3] Carbone, G., Ceccarelli, M. (2005). Legged robotic systems. Kordić, V., Lazinica, A., Merdan, M. (Eds.), Cutting Edge Robotics. Pro literatur Verlag, Mammendorf, p. 553-576.

Proceedings Papers: Surname 1, Initials, Surname 2, Initials (year). Paper title. Proceedings title, pages.

[4] Štefanić, N., Martinčević-Mikić, S., Tošanović, N. (2009). Applied Lean System in Process Industry. MOTSP 2009 Conference Proceedings, p.

422-427.

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[5] ISO/DIS 16000-6.2:2002. Indoor Air – Part 6: Determination of Volatile Organic Compounds in Indoor and Chamber Air by Active Sampling on TENAX TA Sorbent, Thermal Desorption and Gas Chromatography using MSD/FID. International Organization for Standardization. Geneva. www pages: Surname, Initials or Company name. Title, from http://address, date of access.

[6] Rockwell Automation. Arena, from http://www.arenasimulation.com, accessed on 2009-09-07.

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Additional costs for a color page is 90.00 EUR.

http://www.sv-jme.eu

59 (2013) 4

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

203

David Heath, Brane Širok, Marko Hočevar, Boštjan Pečnik: The Use of the Cavitation Effect in the Mitigation of CaCO3 Deposits

216

Ani Luo, Heping Liu, Cheng Li, Yongfan Wang: Study of a Flowerlike Deployable Structure

223

Tomaž Katrašnik, Henrik Aleš Schuemie, Johann C. Wurzenberger: On the Convergence, Stability, and Computational Speed of Numerical Schemes for 0-D IC Engine Cylinder Modelling

237

Nikolaj Mole, Gašper Cafuta, Boris Štok: A Method for Optimal Blank Shape Determination in Sheet Metal Forming Based on Numerical Simulations

251

Xiaoni Qi, Yongqi Liu, Zhenyan Liu: Exergy Based Performance Analysis of a Shower Cooling Tower

260

Goran Radoičić, Miomir Jovanović: Experimental Identification of Overall Structural Damping of System

269

Han Lianfu, Tang Wenyan, Wang Jun, Fu Changfeng: Measurement of Involute Profile with Small Clamping Eccentricity in a Gear Measuring Center

Journal of Mechanical Engineering - Strojniški vestnik

Contents

4 year 2013 volume 59 no.