Journal of Mechanical Engineering 2014 2

http://www.sv-jme.eu

60 (2014) 2

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

77

Marko Šimic, Mihael Debevec, Niko Herakovič: Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges

84

Bojan Starman, Marko Vrh, Mirko Halilovič, Boris Štok: Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution

93

Jixin Wang, Long Kong, Bangcai Liu, Xinpeng Hu, Xiangjun Yu, Weikang Kong: The Mathematical Model of Spiral Bevel Gears - A Review

106

David Vegelj, Boštjan Zajec, Peter Gregorčič, Janez Možina: Adaptive Pulsed-Laser Welding of Electrical Laminations

115

Erik Čuk, Matjaž Gams, Matej Možek, Franc Strle, Vera Maraspin Čarman, Jurij F. Tasič: Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis

124

Dragica Jošt, Aljaž Škerlavaj, Andrej Lipej: Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models

135

Fuhai Cai, Xin Wang, Jiquan Liu, Fuling Zhao: Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory

Journal of Mechanical Engineering - Strojniški vestnik

Contents

2 year 2014 volume 60 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.

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

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SV-JME, Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.eu Print: Littera Picta, printed in 420 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: Four high response digital hydraulic piezo poppet valves, presented as digital fluid control unit, could be used instead of conventional hydraulic servo valves.

Development of new poppet valve by using CFD (Computer Fluid Dynamics) and FEM (Finite Element Methods) analyses, new digital electronics and new control algorithms results in a better dynamic characteristics and lower power consumption compared to the conventional servo hydraulic valves.

Image Courtesy: University of Ljubljana, Faculty of Mechanical Engineering, Department of Manufacturing Technologies and Systems, Laboratory for Handling, Assembly and Pneumatics, Slovenia

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 Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical 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 Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical 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 Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical 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 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 60(2014)2 Contents

Contents Strojniški vestnik - Journal of Mechanical Engineering volume 60, (2014), number 2 Ljubljana, February 2014 ISSN 0039-2480 Published monthly

Papers Marko Šimic, Mihael Debevec, Niko Herakovič: Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges Bojan Starman, Marko Vrh, Mirko Halilovič, Boris Štok: Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution Jixin Wang, Long Kong, Bangcai Liu, Xinpeng Hu, Xiangjun Yu, Weikang Kong: The Mathematical Model of Spiral Bevel Gears - A Review David Vegelj, Boštjan Zajec, Peter Gregorčič, Janez Možina: Adaptive Pulsed-Laser Welding of Electrical Laminations Erik Čuk, Matjaž Gams, Matej Možek, Franc Strle, Vera Maraspin Čarman, Jurij F. Tasič: Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis Dragica Jošt, Aljaž Škerlavaj, Andrej Lipej: Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models Fuhai Cai, Xin Wang, Jiquan Liu, Fuling Zhao: Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory

77 84 93 106 115 124 135

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 77-83 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1104

Received for review: 2013-03-26 Received revised form: 2013-08-08 Accepted for publication: 2013-09-16

Original Scientific Paper

Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges Šimic, M – Debevec, M. – Herakovič, N. Marko Šimic* – Mihael Debevec – Niko Herakovič

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia This paper presents a new approach for modelling and simulation of hydraulic spool valves by using the already known simple mathematical expressions for describing the sliding spool geometry. The main objective of the research is to divide the hydraulic sliding spool into functional elements which can be described analytically. Such models can be implemented as micro components into any hydraulic simulation tool’s library. By using such an approach, hydraulic valves can be designed very flexibly, with different shapes of spool metering edges in combination with other functional elements. The user can quickly find the most appropriate spool construction for the desired hydraulic system performance. This can be done in advance, before designing the real system. The paper deals with and presents only one of the several different geometrical shapes of spool metering edges and the corresponding mathematical models of the volume flow characteristics that we have developed. Simulation results are verified and confirmed with experimental tests using a real valve geometry.

Keywords: hydraulic spool valve, spool notch geometry, mathematical modelling, simulation model

0 INTRODUCTION The development of new advanced hydraulic systems and their components consists mainly of two approaches. The first approach, which is based on an adequate tracking and intelligent closed loop control using adaptive feedback systems, satisfies most of the practical technical requirements for the quality of the dynamic behaviour of the hydraulic system [1] and [2]. The other approaches, such as direct numerical simulation using an immersed-boundary method [3] or three dimensional CFD analysis [4] and [5], provide more flexibility and reveal an opportunity for the development of new mechanical design solutions.

Many different types of spool valves assist in controlling the hydraulic fluid flow in hydraulic systems. The main function of the spool valve is to provide a desired balance in the system between the flow control and the pressure control. The improved dynamic behaviour of the system and its desired functionality are possible because of the use of hydraulic valves with specially designed metering edges of the spool which allow precise pressure and flow control [6] and [7]. The new design is usually based on the use of dedicated computer simulation tools operated by experts. A single valve is normally described with a very small number of parameters which often do not meet all the simulation criteria.

The missing parameters must be defined by using analytical or CFD simulation packages. Otherwise, experimental tests are necessary [8].

The main idea of our research is to divide the sliding spool of the valve into different modules which can be optionally put together by the user.

This approach allows the user to design a valve with

different metering edges of the spool and to quickly check through simulation how the valve influences the characteristics of a practical hydraulic system. When the results of the system characteristics satisfy the user’s expectations, the real valve can be ordered at the valve producer by giving them the exact demands regarding the geometry of the valve, especially the sliding spool. Thus, the user can get the best possible design of the valve from the producer and achieve the best possible performance of the hydraulic system.

One of the very well-known commercial simulation tools appropriate especially to perform the simulation and the analysis of the static and dynamic behaviour of hydraulic and pneumatic systems and components is the DSHplus software [9].

The functional elements of the sliding spool valve dealt with in our research are intended to be implemented into MCL (Micro Component Library) of the DSHplus [9].

1 MODELLING OF HYDRAULIC SPOOL VALVES The existing simulation valve models consist of classic sharp metering edges with mathematical models of linear volume flow rate characteristics without the possibility of modifications. The metering edge geometry is the key parameter for the determination of the volume flow characteristics. Sophisticated hydraulic valves require a new, advanced notch geometry of the spool, the characteristics of which are nonlinear [7].

The basic formula for the volume flow calculation is the orifice Eq. (1) [9]:

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

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 77-83

Q = αD ⋅ A⋅

2 ⋅ ∆p , (1) ρ

where the parameters are defined as follows; Q is the volume flow [l/min], αD the contraction flow coefficient [-], A the cross-sectional area [m2], ρ the density of the fluid [kg/m3], and Δp the pressure difference across the metering edge [bar].

One of the main variables of Eq. (1) is the crosssectional area A which is a function of the spool displacement and of the metering edge geometry.

An example of a real hydraulic valve with a specially designed sliding spool, used for pressure and flow control, is shown in Fig. 1 [7].

Fig. 1. Hydraulic valve with specially designed metering edges

control edges with three different geometries which can be divided into three different functional elements or modules. Detail A shows the possible geometry of the metering edge. By describing geometrical dependencies separately for each metering edge of the spool analytically with simple mathematical expressions, it is possible to accurately describe the characteristics of the hydraulic valve as a whole through simulation. Thus, we do not need to perform an experimental test for each new valve geometry.

Fig. 2. Detail view of one specially designed metering edge of the hydraulic spool

The detailed view of the hydraulic sliding spool from Fig. 1 is presented in Fig. 2 [7]. The spool has six

Fig. 3. The valve spool of Fig. 2 as sub-model in DSHplus valve model [8]

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Šimic, M – Debevec, M. – Herakovič, N.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 77-83

Fig. 3 presents the incorporated spool into the simulation valve model of the DSHplus tool by using various functional elements from its MCL. Specific functional elements of the spool are: its mass, mechanical stops, springs, and the most important »Metering-Edge-Notched« elements with different geometries. Static and dynamic characteristics of the valve are mostly influenced by combining different functional elements together into the sliding spool.

The »Metering-Edge-Notched« element, marked with a rectangle in Fig. 3 (marked with thick doted red line), presents the element described in the background by the mathematical calculation of the volume flow across the notched metering edge area.

2 ANALYTICAL MODELLING AND SIMULATION As the first step for the analytical modelling of the cross-sectional area A, the shape and the geometry of the metering edge of the real spool have to be defined. For this purpose, a 3D model of a hydraulic spool notch was designed in order to reach a better understanding of the direction and all the influential areas of the fluid flow during the spool displacement (Fig. 4).

consideration to verify the possibility of geometry simplification.

A schematic presentation of the curved radial cross-sectional area is shown in Fig. 5a, while a simplified surface of the radial area is presented as a flat surface in Fig. 5b.

a) b) Fig. 5. Schematic presentation of the radial cross-sectional area; a) curved and b) simplified

The schematic curved and simplified axial crosssectional areas shown in Figs. 6a and b present more or less a rectangle with a negligible curved area at the curvature line of the spool.

a) b) Fig. 6. Schematic presentation of the axial cross-sectional area; a) curved and b) simplified

a) b) c) Fig. 4. a) flow direction of the metering edge, b) radial and c) axial cross-sectional areas

In general, the 3D model shows that there are two types of fluid flow: radial flow going through the radial areas on the curvature of the spool and axial flow going through the axial areas on the face side of the spool. The direction of the flow is indicated by the flow vectors Ar and Aa in Fig. 4a. Figs. 4b and c show the annotations for those two areas: the radial area segment is labelled as Ar and the axial area segment as Aa. The fluid normally flows through the radial area first and then passes through the axial area into the hydraulic chamber.

For the mathematical calculation of the radial and axial cross-sectional areas of the spool metering edge, the curvature of the surface has to be taken into

Assuming that the spool diameter is 16 mm (the nominal size 10 of a proportional valve) and the radii of the notch is less than 2 mm, the curved areas presented in Fig. 5a and Fig. 6a can be neglected because the deviation between the real curved surface (Figs. 5a and 6a) and the simplified surface (Figs. 5b and 6b) is less than 0.5%. The modelling is therefore simplified from 3D (three dimensional) surface models to 2D (two dimensional).

The real geometry of the flow control metering edge (notch) of the spool is marked as detail B in Fig. 7.

The metering edge consists of two different geometrical parts presented in Fig. 8: half circular pockets presented as areas A1 and A3 and a rectangular pocket presented as area A2.

The mathematical description of the metering edge is therefore separated into the modelling of the circular and the rectangular areas.

Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges

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 B A1r ( x) = R12 arccos   − B 2 ⋅ R1 ⋅ x − x 2 . (2)  R1 

Axial circular segment area A1a(x) for the same range of spool displacement x0 < x < x1 can be calculated with Eq. (3): A1a ( x) = C1 ( x) ⋅ h1 , (3)

Fig. 7. The real metering edge of the sliding spool marked as detail B

where h1 represents the depth of the circular pocket A1, and C1(x) is given by Eq. (4): C1 ( x) = 2 ⋅ 2 ⋅ R1 ⋅ x − x 2 . (4)

The area A2 from Fig. 8 is shown in detail in Fig. 10. It represents the rectangular area for spool displacements from x1 to x2 where the mathematical formulation of the radial geometry is simple and therefore the mathematical model does not require a detailed analysis.

Fig. 8. The real metering edge of the sliding spool marked as detail B

Area A1 as presented in Fig. 8 can be calculated by using simple, well-known trigonometric equations of the circular segment [10]. The detailed view and all the influential parameters are shown in Fig. 9.

Fig. 10. Influential parameters of rectangular area A2

By taking into account the geometrical parameters (R1, R2, h1 and h2) presented in Fig. 10 and the spool displacement range x1 < x < x2, the mathematical model for radial cross-sectional area A2r(x) can be described with Eq. (5).

A2 r ( x) = 2 ⋅ R1 ⋅ x. (5)

Spool displacement x2 is given by Eq. (6).

Fig. 9. Influential parameters of the circular area A1

Radial circular area A1r(x) depends on radii R1 and on spool displacement x0 < x < x1. It can be calculated by using Eq. (2) which is given simply by the area of the triangular portion subtracted from the circular sector of the entire wedge-shaped portion.

Parameter B in Eq. (2) stands for R1–x.

80

2

x2 = R2 − R2 2 − ( R2 − x ) . (6)

Total axial area A2a(x) (Fig. 11) can be divided into two areas. The first one represents the constant rectangular area which depends on R1 and on depth of the pocket h1 while the second area depends on C2(x) and on depth h2 – h1.

Šimic, M – Debevec, M. – Herakovič, N.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 77-83

Axial circular segment area A3a(x) for spool displacement x2 < x < x3 is given by Eq. (10):

A3a ( x) = C2 ( x) ⋅ h2 , (10)

where h2 represents the depth of the A3 pocket, and C2(x) is already given with Eq. (8).

The final mathematical model of the total radial and axial cross-sectional areas (Eqs. (11) and (12)) is taking into account the spool displacement from the beginning of the spool metering edge x0 to the end of the metering edge x3 as shown in detail B of Fig. 8.

Total radial area Ar is determined as the sum of all radial areas and increases incrementally as described with the formulae of the individual segments.

Fig. 11. Geometry of axial area A2a(x)

Total axial area A2a(x) is represented with Eq. (7) where C2(x) can be calculated with Eq. (8).

A2 a ( x) = 2 ⋅ R1 ⋅ h1 + C2 ( x) ⋅ ( h2 − h1 ) , (7)

C2 ( x) = 2 ⋅ 2 ⋅ R2 ⋅ x − x 2 . (8)

The area A3 from Fig. 8 is defined in a similar way as area A1. The radial and axial cross-sectional areas consist of the parameters presented in Fig. 12.

Ar = n ⋅ ( A1r + A2 r + A3r ) . (11)

Total axial area Aa is determined as the minimal axial cross-sectional area at a given spool position xi (i = x0, …, x3, Fig. 6). Parameter n, considered in Eqs. (11) and (12), represents the total number of notches.

Aa = n ⋅  min ( A1a , A2 a , A3a )  . (12)

The volume flow characteristics Q(Ar) and Q(Aa) presented in Fig. 14 are calculated by using the final mathematical models of radial and axial crosssectional areas Ar and Aa in Eq. (1). The volume flow rate is calculated for 24 discrete points of the spool displacement from x = 0 to x = 1.5 mm. All other influential parameters considered in Eq. (1) are shown in Fig. 13.

Fig. 12. Geometry of the circular area A3

A final equation for the radial cross-sectional area A3r(x) for spool displacement x2 < x < x3 can be written as:

 D 2 A3r ( x) = R2 2 arccos   − D 2 ⋅ R2 ⋅ x − x , (9) R 2  

where D stands for R2 – x.

Fig. 13. Parameters for volume flow rate calculation and its corresponding geometry of the notch Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges

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From the results of the simulation, it is obvious (Fig. 14) that there is a great difference between the values of the volume flow rates Q(Ar) and Q(Aa), calculated with Ar and Aa respectively. Minimal volume flow rate is represented by the curve Q(Ar) which takes into account the cross-sectional area Ar which is smaller than Aa in each calculated point of the spool displacement.

Fig. 14. Volume flow rate curves as a result of simulation models

Table 1. Simulated volume flow rates at discrete spool displacements N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

82

x [mm] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.63 0.65 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50

Q(Ar) [l/min] 0.000 0.353 0.984 1.778 2.800 3.696 4.770 5.897 7.062 8.251 9.453 10.000 11.860 12.678 12.800 14.433 17.610 21.264 25.281 29.578 34.085 38.744 43.502 48.308

Q(Aa) [l/min] 0.000 10.492 14.443 17.190 19.257 20.846 22.062 22.963 23.585 23.951 24.071 32.000 42.000 50.182 53.000 60.660 72.200 80.880 87.555 92.659 96.443 99.057 100.593 101.100

Q min [l/min] 0.000 0.353 0.984 1.778 2.800 3.696 4.770 5.897 7.062 8.251 9.453 10.000 11.860 12.678 12.800 14.433 17.610 21.264 25.281 29.578 34.085 38.744 43.502 48.308

At the maximal spool displacement x = 1.5 mm, the value of the volume flow rate Q(Ar) is 48.3 l/min.

The exact value of the volume flow rate at a discrete spool displacement can be seen from Table 1.

3 EXPERIMENTAL VERIFICATION OF THE ANALYTICAL MODEL An experimental analysis of the valve has been performed with a real hydraulic spool valve of the KV4/3-10 type with the modified sliding spool geometry as shown in Figure 13. The HLP 46 hydraulic oil with the density of 840 kg/m3 was used during the experiment. The volume flow across the metering edge was measured at the same 24 discrete spool displacement points from x = 0 to x = 1.5 mm as in the simulation (Table 2). The pressure difference was set to 100 bar for every discrete spool position where the volume flow was measured.

The results of the simulation and the experiment are given in Table 2 and shown in Fig. 15.

Table 2. Simulated volume flow rates Qsim and experimentally determined volume flow rates Qexp N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

x [mm] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.63 0.65 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50

Qsim [l/min] 0.000 0.353 0.984 1.778 2.800 3.696 4.770 5.897 7.062 8.251 9.453 10.000 11.860 12.678 12.800 14.433 17.610 21.264 25.281 29.578 34.085 38.744 43.502 48.308

Qexp [l/min] 0.000 0.357 0.995 1.800 2.830 3.740 4.820 5.960 7.130 8.330 9.600 10.160 11.940 12.730 12.920 14.550 17.750 21.400 25.450 29.780 34.400 39.120 43.950 48.800

∆Q [%] 0.00 1.01 1.11 1.21 1.06 1.17 1.03 1.06 0.96 0.95 1.53 1.57 0.67 0.41 0.93 0.80 0.79 0.63 0.66 0.68 0.92 0.96 1.02 1.01

Experimental (Qexp) and simulated (Qsim) results presented in Fig. 15 fit very well together and therefore confirm the hypothesis of the minimal

Šimic, M – Debevec, M. – Herakovič, N.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 77-83

cross-sectional area. The measurement uncertainty of the experimental points is calculated as a standard deviation σ = 0.094 l/min. The maximal difference between simulated and experimental values (1.57%) appears at the spool displacement x = 0.55 mm which can be considered as a transition area where the large circular pocket begins. In general, the deviation between the simulated and the experimental results of the volume flow is no greater than 1.2%.

With the use of our approach, hydraulic valves can be designed in advance, very simply and flexibly, with different shapes of spool metering edges in combination with other functional elements. Such an approach enables the user to quickly find the most appropriate spool construction for the desired hydraulic system performance during the development phase.

5 REFERENCES

Fig. 15. Simulated and experimentally defined characteristic points of volume flow rate

4 CONCLUSION This research presents the new approach to modelling and simulation of hydraulic spool valves by using simple mathematical expressions for describing the sliding spool metering edge geometry. Developed models can be implemented into any hydraulic simulation tool’s library, such as MCL library of DSHplus.

The paper focuses on only one of the several different mathematical models of the specially designed metering edge volume flow characteristics that we have developed. The cross-sectional area of the metering edge has been simplified to a 2D geometrical area instead of using a 3D shape because the cross-sectional difference is less than 0.5%.

The results of the volume flow rate shown in Table 2 are calculated by using the simulation model which took into account the minimal cross-sectional area. The data sheet table is prepared as the input parameters for the valve simulation model. The simulation results of the volume flow rate model are verified and confirmed with experimental tests.

[1] Lazić, D.V. (2010). Practical tracking control of the electropneumatic piston drive. Strojniški vestnik Journal of Mechanical Engineering, vol. 56, no. 3, p.

1-6.

[2] Detiček, E. (2011). An intelligent electro-hydraulic servo drive positioning. Strojniški vestnik - Journal of Mechanical Engineering, vol. 57, no. 5, p. 394-404, DOI:10.5545/sv-jme.2010.081.

[3] Posa, A., Oresta, P., Lippolis, A. (2013). Analysis of a directional hydraulic valve by a direct numerical simulation using an immersed-boundary method.

Energy conversion and Management, vol. 65, p. 497506, DOI:10.1016/j.enconman.2012.07.012.

[4] Lisowski, E., Czyzycki, W., Rajda, J. (2013). Three dimensional CFD analysis and experimental test of flow force acting on the spool of solenoid operated directional control valve. Energy Conversion and Management, vol. 70, p. 220-229, DOI:10.1016/j. enconman.2013.02.016.

[5] Tic, V., Lovrec, D. (2012). Design of modern hydraulic tank using fluid flow simulation. International Journal of Simulation Modelling, vol. 11, no. 2, p. 77-88, DOI:10.2507/IJSIMM11(2)2.202.

[6] Herakovič, N. (2009). Flow-force analysis in a hydraulic sliding-spool valve. Strojarstvo, vol. 51, no.

6, p. 555-564.

[7] US-Patent 6.450.194 B1 (2002). Spool Notch Geometry for Hydraulic Spool Valve. Wasson, J.B., Swaim, D.W., Fiala, G.T. The United States Patent and Trademark Office, Alexandria.

[8] Herakovič, N., Noe, D. (2006). Analysis of the operation of pilot-stage piezo-actuator valves. Strojniški vestnik - Journal of Mechanical Engineering, vol. 52, no. 12, p. 835-851.

[9] FUIDON GmbH (2010). DSHplus. Micro Component Library, Aachen.

[10] Murrenhoff, H. (2001). Grundlagen der Fluidtechnik.

Teil 1: Hydraulik. Shaker Verlag. Aachen.

[11] Mathworld, Circular segment area (2012). from http:// mathworld.wolfram.com, accessed on [2012-03-08].

Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges

83

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1349

Original Scientific Paper

Received for review: 2013-07-31 Received revised form: 2013-10-02 Accepted for publication: 2013-10-21

Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution Starman, B. – Vrh, M. – Halilovič, M. – Štok, B. Bojan Starman – Marko Vrh – Mirko Halilovič – Boris Štok*

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia The paper focuses on the modelling of springback within a formed stainless steel sheet. The main subject of this work is the construction of a constitutive model which simultaneously considers sheet anisotropy, damage evolution, and stiffness degradation in material during forming.

The developed model is based on the Gurson–Tvergaard–Needleman damage model, which is adequately extended by the implementation of the anisotropic Hill48 plasticity and Mori–Tanaka’s approach to stiffness degradation. Considering the established relationships, some material parameters that are included in the model are characterised by the corresponding measurements. The experimental validation of the developed constitutive model is performed on a springback test, which consists of bending and releasing rectangular stainless steel specimens that were previously plastically prestrained to a different degree, either in the rolling or transverse direction. A comparison of the proposed modelling approach to the classical approach by using the Hill48 model clearly indicates that the simultaneous modelling of material phenomena, especially the coupling of stiffness degradation with anisotropic plasticity, can be the true key to obtaining a more accurate prediction of the springback in sheet-metal-forming applications.

Keywords: springback, damage, elastic properties, stiffness degradation, anisotropy, plastic prestrain

0 INTRODUCTION In recent years, stainless steels have been increasingly utilised in the automotive industry and in the production of domestic appliances because of its excellent strength, good formability and resistance to corrosion. While the use of stainless steel increases mechanical properties of a formed part, there is another merely technological disadvantage. We refer to the so–called springback effect, a phenomenon that is associated with the reversible elastic strain recovery which happens after the removal of the tools and complete unloading of the formed part. Because of the high stress state that is achieved in stainless steels during forming, and because of smaller sheet thickness that is usually required in order to reduce the weight of the produced component, the springback of stainless steels in regard to forming mild steels is even more intense and has been long recognised as a major problem for companies’ development departments.

From the point of view of tool design, there is no doubt that a reliable springback prediction, which is based on the corresponding numerical simulation of the forming process, is the key to resolving this problem. In this regard, a considerable amount of work that is specifically related to the proper mathematical modelling of the springback has been already done. Above all, it turns out that the precise experimental observation of the material’s behaviour and the inclusion of its revealed physical relations into a corresponding constitutive model are the most promising way. Models that have been presented up until now mostly deal with the precise modelling of 84

anisotropy [1], the Bauschinger effect [2], and damage [3]. Recently, we have proposed that, for reliable springback prediction, elastic modulus degradation must also be included in the constitutive modelling; in [4] we studied the effect of coupling the Mori–Tanaka model with isotropic plasticity. However, sheet metals usually exhibit significant plastic anisotropy. This paper is an attempt to build a corresponding physically consistent constitutive model that is capable of taking into account the resulting stiffness degradation during plastic deforming and plastic anisotropy in material.

The topic presented here, which is focused on the anisotropy modelling and experimental verification of the basic assumptions, is in this regard a continuation and an upgrade of our previous research, as reported in [5] and [4]. The work is based on a combined experimental–analytical–numerical approach; with the proven experimental evidence being analytically modelled, the physical adequacy of the built constitutive equations is established by means of a numerical simulation of given experiments. The conceived constitutive model is implemented into a FEM-based program; in our case, ABAQUS/Explicit [6]. In the development stage of the constitutive modelling, the FEM simulations are used purely for the purpose of constructing a constitutive model that is as consistent as possible with the given experimental evidence. In the end, with the developed constitutive model being confirmed in regard to its original experimental framework, it is further numerically validated on a simulation of a series of experimentally performed springback tests.

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

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

1 CONSTITUTIVE MODELLING AND NUMERICAL IMPLEMENTATION In [5] we showed that a considerable amount of voids tends to evolve during plastic straining. This evidence was determined by observing the microstructure of stainless sheet metal which had been plastically prestrained to different degrees. According to damage mechanics [7], the arisen voids are the main cause for the stiffness degradation in ductile materials, which means that in order to more realistically capture the material behaviour, this evidence must be considered in the constitutive modelling. In particular, this is important with respect to springback simulation analyses.

In the attempt to conceive a constitutive model that would incorporate the above experimental evidence to the greatest degree, we start in this work with the isotropic Gurson–Tvergaard–Needleman (GTN) model [8] to [10] and continue with its corresponding upgrading. In the GTN model, which establishes the respective constitutive laws for the evolution of ductile damage in porous materials, void nucleation and void growth (two essential elements of damage evolution besides void coalescence) are considered. In this regard, in order to consider physically consistent stiffness degradation in the material, we have coupled the GTN model with the Mori-Tanaka model [11], which considers stiffness degradation due to inclusion of spherical voids. Further, since the sheet metal is also highly anisotropic due to rolling, the anisotropic Hill48 model has been consistently compounded into the model as well. Therefore, we propose the model which is physically consistently compounded from the GTN, Hill48, and Mori–Tanaka model. Considering the initials of all three models, it can be designated as the ‘GHM model’.

σ eq2 = F (σ 22 − σ 33 ) + G (σ 33 − σ 11 ) + 2

(σ ) Φ=

2

eq

(σ M )

2

 3q σ + 2 f q1 cosh  2 H  2σ M

 2  − (1 + q3 f ) , (1) 

where the anisotropy is modelled following the Hill48 definition of the equivalent stress σeq as:

(

)

2

2 + H (σ 11 − σ 22 ) + 2 Lσ 23 + M σ 132 + Nσ 122 . (2) 2

In the above equation, the stress tensor σij is given in the coordinate system (1, 2, 3), which is defined by the material principal axes, whereas the coefficients F, G, H, L, M, N are the corresponding material parameters that characterise the material anisotropy. For the simplicity of further description, σeq can be equivalently introduced by means of fourthorder Hill tensor Hijkl and the stress deviator tensor sij = σij – 1 / 3σkk δij as:

σ eq2 = sij H ijkl skl . (3)

In Hijkl only the following components are nonzero:

H1111 = G + H, H3333 = F + G, H1133 = H3311 = – G, H1212 = H2121 = N, H2323 = H3232 = L.

H2222 = F + H, H1122 = H2211 = – H, H2233 = H3322 = – F, H1313 = H3131 = M,

(4)

The material anisotropy parameters F, G, H, L, ij / σ ref , M, N are related to yield stress ratios Rij = σ yield ij σ where σref and yield are respectively the adopted reference yield stress and actual yield stresses from the uniaxial (i = j = 1, 2, 3) and shear (i ≠ j) experiments, in the following way: 2F = −

1.1 Plastic Potential In order to simultaneously consider the anisotropy and damage, we have adopted a yield criterion that is based on the upgrade of the Gurson type (GTN) potential with the anisotropy [12]:

2

1 1 1 1 + + , 2N = 2 , R112 R222 R332 R12

2G =

1 1 1 − 2 + 2 , 2 R11 R22 R33

2M =

2H =

1 1 1 + 2 − 2 , 2 R11 R22 R33

2L =

1 , (5) R123 1 .

R232

The parameters of the introduced model can be grouped with regard to their action in two subsets.

The parameters F, G, H, L, M, N or, equivalently, Rij, describe the anisotropy, while parameters q1, q2 and q3, describe the influence of damage on yielding. In Eq. (1), σM is the yield stress of the matrix material, which is defined as a function of the equivalent strain of matrix material ε Mp . In this work, the hardening curve is constructed from 10 cubic splines up to ε Mp = 0.46, and the equivalent plastic strain of the matrix material ε Mp is obtained from the following equivalent plastic work expression:

(1− f )σ M dε Mp = σ ij dε ijp . (6)

Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution

85

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

The remaining state variables in Eq. (1), σH and f, are, respectively, the hydrostatic stress σH = σkk / 3 and void volume fraction or porosity in the material.

1.2 Evolution of Porosity The law governing the evolution of porosity considers two mechanisms, void growth and void nucleation, respectively: df = dfgrowth + dfnucleation .

df growth = (1 − f ) dε kkp , (8)

whereas the nucleation of voids due to microcracking and decohesion of the particle–matrix interface is related to the plastic deformation of the matrix material: df nucleation = An dε Mp . (9)

as:

In the GTN model, the parameter An is computed

An =

fn sn 2π

 1  εM − εn      , (10)  2  sn   p

2

exp  −

following a normal distribution about the mean nucleation strain εn with a standard deviation sn.

Parameter fn represents the maximum possible nucleated void volume fraction. In this study, the possible decrease of the strength of material due to extensive void coalescence is omitted.

1.3 Stiffness Degradation

E=

ν =

2 E0 (1 − f ) ( 7 − 5ν 0 )

(

2 ( 7 − 5ν 0 ) + f 13 − 2ν 0 − 15ν 02

(

2ν 0 ( 7 − 5ν 0 ) + f 3 − 2ν 0 − 5ν 02

(

2 ( 7 − 5ν 0 ) + f 13 − 2ν 0 − 15ν 02

, (11)

) ) , (12) )

where E0 and ν0 are Young’s modulus and Poisson’s ratio of the virgin material (i.e., undeformed material).

Note that the Hill48 model can be derived from the GHM model when specific values of the material model parameters are considered. The Hill48 model can be simply obtained when the porosity is set to p zero. In this case, σ M (ε M ) becomes the classical yield stress as a function of equivalent plastic strain p σ y (ε eq ) .

1.4 Numerical Implementation The above conceived constitutive model has been implemented in a general purpose finite element code ABAQUS via VUMAT subroutine. For the integration of the constitutive equations, a new highly efficient explicit integration scheme is used, which was recently developed by the authors. More about the application of the new scheme and its implementation within FEM can be found in [20], whereas the reader is invited to study [21] for the theoretical background.

2 EXPERIMENTAL OBSERVATIONS AND MEASUREMENTS

Stiffness degradation due to inclusion of spherical voids in the elastic continuum was studied in many papers, such as [13] to [15]. To characterise it, Eshelby’s equivalence principle and his solution of the elastic field of an ellipsoidal inclusion in an infinite elastic medium [16] can be used. Eshelby’s principle is best combined with Mori–Tanaka’s concept of average stress in the matrix [11] and [13], which was verified several times for different materials (e.g., in [17] for aluminium, in [18] for graphite, and in [19] for metal composites). Combining the GTN model with the Eshelby and Mori–Tanaka approach gives a firm basis for building a proper constitutive model that is capable of simulating the measured material response.

86

(7)

The first term on the right-hand side can be formulated by considering mass conservation:

e

Here, linear and isotropic elastic law σ ij = Cijkl ε kl , i.e., Hooke’s law, will be used with degradation of stiffness taken into account. The effective values of Young’s modulus and Poisson’s ratio, E and ν are related, according to the Mori-Tanaka approach, to the porosity of the material that contains spherical voids in the following way:

For the investigated stainless steel EN 1.4031, the standard tensile tests in three specific directions, namely in direction 0, 90 and 45° regarding the rolling direction of the sheet, have been performed in order to obtain respective yield curves. In addition, Young’s modulus degradation has been measured on the tensile sheet specimens that were plastically prestrained in the rolling direction.

2.1 Plastic Anisotropy and Hardening The overall hardening and plastic anisotropy of the observed steel was measured by means of the standard tensile test, performed in the Tira 2300 tensile test

Starman, B. – Vrh, M. – Halilovič, M. – Štok, B.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

machine. The initial thickness and width of the tensile sheet specimens were 0.67 and 20.2 mm, respectively.

In the experiment, the tensile force F and elongation ΔL of the gauge length L0 = 80 mm were measured.

The established F – ΔL relationships for the three considered directions, which include the rolling direction (0°), transverse direction (90°) and diagonal direction (45°), are graphically displayed in Fig. 1.

From three hardening curves which are displayed in Fig. 1, one can observe that cold rolled metal sheet EN 1.4301 exhibits a significant degree of plastic anisotropy, which certainly cannot be neglected in numerical simulations.

unloading of the plastically prestrained specimen, the effective elastic modulus can be calculated considering Hooke’s law by using the interpolation of the measurement data of length, cross-sectional area, and force. In order to retrieve the effective Young’s modulus degradation as a function of the longitudinal plastic strain, the described procedure is repeated for different degrees of the applied plastic prestrain. As can be clearly seen from the plotted graph in Fig. 3, it is beyond all question that the evidenced degradation of the effective Young’s modulus is directly correlated to the degree of the applied plastic prestrain. The fact that, in our experiment, plastic prestraining was achieved under the condition of a uniaxial stress state certainly does not affect the general statement. The phenomenon described here can be found also in [22] and [23].

Fig. 1. Tensile test measurements

2.2 Effective Young’s Modulus Degradation The degradation of the effective Young’s modulus was measured as a function of the longitudinal plastic strain, which can be defined for the uniaxial stress case as εp = ln ((L0 + ΔL) / L0). The corresponding experimental procedure is a two-stage loading procedure in which standard specimens are first plastically prestrained in the tensile test machine to a certain degree of the equivalent plastic strain and then released. The thickness and width of each prestrained specimen is precisely measured in order to evaluate the respective cross-sectional area which will be needed in subsequent stiffness analysis. In the second stage, each plastically prestrained specimen is clamped again in the tensile test machine and loaded only elastically.

In order to accurately follow the elastic response, the machine is equipped with a precise dynamometer, which can measure force in a range up to ±10 kN with accuracy class being 1 (ISO 376, EN 10002–3), whereas the strain transducer, which is mounted on the specimen as shown in Fig. 2, is of accuracy class 0.1 while its nominal displacement range is ±2.5 mm.

From the measured force-displacement relationship registered by elastic loading and

Fig. 2. Measurement of elastic elongation

Fig. 3. Young’s modulus degradation due to plastic straining

3 IDENTIFICATION OF THE PARAMETERS The parameters of the herein deduced GHM model have been identified using the least square method and the Levenberg-Marquardt optimization algorithm

Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution

87

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

Fig. 4. Inverse identification results: experiment vs. calculation

[24] and [25]. The goal of the optimization is to fit with the respective numerical simulation, when performed by considering the GHM model, the performed experiments to a greatest extent. Thus, in the optimization, the model parameters that are under consideration take the role of design variables, while the cost function to be minimised is defined as the discrepancy between the numerical and experimental responses. With a corresponding adjustment of the model parameters’ values, we have thus tried to numerically obtain both the measured force-displacement curve (F – ΔL) in three observed directions (0, 45, and 90°) and the measured effective Young’s modulus degradation at the same time.

Parameters q1, q2 and q3 are essentially an improvement of the basic Gurson’s constitutive model, upgrading thus the original plastic potential [9]. Since the values of those parameters are similar for all metals [6], we adopt them by taking q1 = 1.5, 2 q2 = 1 and q= q= 2.25 . Further, we have observed 3 1 that the model fits more adequately the measured effective Young’s modulus degradation when An in Eq. (9) does not follow the normal distribution. Thus, An has been adopted as the independent parameter of the model; in that case only one parameter needs to be identified instead of three. Also, considering specific of the planar anisotropy in the rolled sheet only two yield stress ratios, R22 and R12, have been chosen as relevant for the characterization of the anisotropic behaviour (parameters of Hill48). Accordingly, only R22 and R12 have been identified, whereas the remaining yield stress ratios were set to 1. ε p Further, the control points σ M = σ M (ε M ) of the cubic spline, which is used for the definition of the 88

hardening behaviour, have been the subject of the identification as well. Values of all the GHM model parameters, those that are identified and those that are assumed, are tabulated in Table 1. The values, which are denoted in the table by an asterisk, are assumed to be fixed and are not subject to identification.

As can be seen from Fig. 4, a very good agreement between the calculated and measured F – ΔL curves and the effective Young’s modulus degradation as a function of the elongation of gauge length is obtained for the optimised GHM model.

To provide elements for a comparative analysis between different constitutive models, the same procedure for the parameters’ identification has been used for Hill48 model, but considering a reduced experimental data set. In the Hill48 model parameters identification, only the deviation in F – ΔL curves is considered as the cost function. The corresponding parameter values for the Hill48 constitutive models are p tabulated in Table 1, where σ yε = σ y (ε eq ) represents the control points of the cubic spline for the definition of the hardening behaviour.

4 VALIDATION OF THE MODEL ON SPRINGBACK TEST The developed constitutive model is experimentally validated by numerically simulating a springback test. The test consists of the bending and releasing of rectangular stainless steel sheet specimens that were previously plastically prestrained to different degrees in the rolling direction. In the experiment, the rectangular specimens are first uniaxially stretched to a certain level of plastic deformation and then released. As a measure of plastic prestrain, let us

Starman, B. – Vrh, M. – Halilovič, M. – Štok, B.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

introduce geometrically defined longitudinal plastic strain ε p =ln((L0 +∆L)/L0 ) , which is defined with the elongation of the respective gauge length. Such plastically prestrained specimens are then clamped, in the second part of the experiment, into a special bending tool (Fig. 5a) and subsequently bent in an angle of γ = 90° (Fig. 5b). As a result, the initially homogeneous plastic strain state, which is due to the applied plastic prestrain in the first part of the experiment, is now subject to change into a nonhomogeneous one because of the developed plastic strains by bending.

Nevertheless, the applied plastic prestrain and its amount still has a decisive role in the resulting nonhomogeneous plastic strain and stress distribution, which will be clearly seen from the exhibited springback behaviour (Fig. 5c).

Lastly, in the considered experiment, after the removal of the bending load and re–established equilibrium of the bent specimen under residual stresses, its deviation from perpendicularity (γ = 90°), denoted in Fig. 5a by angle α, is measured. In fact, in this experiment, the angle α is a clearly visible measure of the exhibited springback behaviour of the bent steel sheet. From a photograph of one set of the bent specimens, shown in Fig. 5c, one can clearly see how the degree of plastic prestrain is directly related to the intensity of the exhibited springback. More extensive springback, which is evidenced with a larger amount of plastic prestrain, may be attributed to the following: greater actual yield stress due to the occurred hardening, thinner specimens, and lower effective Young's modulus. All of these effects are a direct consequence of the previous plastic prestraining, with their variation being proportional in a non-linear way to the degree of the applied plastic prestrain.

Fig. 5. Springback test; bending of prestrained specimens; a) experiment, b) simulation, c) bent specimens Table 2. Comparison of the springback angle, experiment vs. calculations (rolling direction) prestrain εp [-] in rolling direction 0 0.053 0.097 0.144 0.203 0.244 0.300 0.353 0.402

angle α [°] bending in rolling direction numerical experimental GHM Hill48 10.3 10.2 10.2 13.9 14.2 14.0 17.4 17.4 16.9 20.8 20.5 19.6 24.8 24.1 22.7 26.8 26.1 25.3 30.4 30.3 27.9 33.1 33.5 30.7 36.8 36.6 33.5

The measured springback angle α is then compared in the validation test with the results obtained by a numerical simulation of the considered springback test. The computer simulation is based

Table 1. Values of the identified and assumed fixed parameters GHM [ -, N/mm2] * R11

1

Hill48 [ -, N/mm2]

R22

* R33

* R23

* R11

R12

R13

R22

* R33

R12

R13

0.944

1

0.921

1

1

1

0.940

1

0.914

1

0.03 σM

* R23

σ y0

σ y0.045

σ y0.09

σ y0.135

q1*

q2*

q3*

An

0 σM

1.5

1

2.25

0.151

301.8

405.1

1

300.9

442.8

559.5

661.8

0.05 σM

0.1 σM

0.15 σM

0.2 σM

0.25 σM

0.3 σM

σ y0.18

σ y0.225

σ y0.28

σ y0.32

σ y0.37

465.3

611.7

730.3

844.5

947.9

1048.3

751.7

833.1

926.2

991.1

1070.9

0.34 σM

0.38 σM

0.46 σM

E0

ν 0*

-

σ y0.41

σ y0.46

E0

ν 0*

-

1129.5

1208.1

1367.3

208800

0.3

-

1130.4

1190.9

208000

0.3

-

Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

on the FEM code ABAQUS/Explicit by using the VUMAT subroutine for the implementation of the GHM model. In the simulation, 100 shell elements with a reduced integration (S4R) and 11 through thickness section points were used to model the specimen, while the tools were assumed to be rigid.

The results of the calculated and measured springback angle a for the ‘rolling direction specimens’ are presented in Table 2, whereas in Table 3, the respective springback angle α values for the ‘transverse direction specimens’ are displayed.

Table 3. Comparison of the springback angle, experiment vs. calculations (transverse direction) prestrain εp [-] in transverse direction 0 0.056 0.090 0.147 0.200 0.234 0.300 0.340 0.368

angle α [°] bending in transverse direction numerical experimental GHM Hill48 9.8 9.6 9.6 15.2 13.6 13.3 17.7 15.9 15.3 20.7 19.5 18.5 23.9 22.5 21.0 26.8 25.3 23.6 30.4 28.3 26.1 33.0 31.7 29.2 35.8 34.0 31.3

From Tables 2 and 3, it can be seen that the approach presented here, which is formulated as the GHM model, gives much more accurate results. From the histograms, plotted in Figs. 6 and 7, showing for both considered constitutive models the respective absolute value of the established relative deviations between the simulated and the experimental springback, the advantage of the GHM model is even more distinct.

It is beyond all doubt that the overall departure of the calculated springback from the experimental one is smaller when the GHM model is used in comparison with the Hill48 model. Nevertheless, some further detailed perceptions can be extracted from the obtained results: • When plastic prestraining in the rolling direction is small, both of the constitutive models that are considered here give similar results because the stiffness degradation is negligible and anisotropy has a minor influence.

• At a higher degree of plastic prestraining in the rolling direction, however, the Hill48 model underestimates springback, as is seen in Table 2.

The main cause lies in the fact that it neglects the degradation of the effective Young’s modulus.

90

•

There, the advantage of the GHM model becomes visible.

The deviations from the experimental evidence are larger in the case of prestraining in the transverse direction for both models. The error may be attributed to the adoption of the isotropic evolution of the stiffness degradation in the GHM model and the adoption of the Hill48 anisotropy, the latter having limited possibility of anisotropy modelling due to its simplicity. Nevertheless, the springback prediction with the GHM model is, again, much more precise than the classical approach.

Fig. 6. Absolute value of relative deviations of springback (rolling direction)

Fig. 7. Absolute value of relative deviations of springback (transverse direction)

5 CONCLUDING REMARKS This paper presents a construction of the advanced constitutive model which simultaneously considers sheet anisotropy, damage evolution, and stiffness degradation in sheet metal during forming. From the comparative analysis of the numerical and experimental results for springback in the investigated validation test, it can be concluded that only

Starman, B. – Vrh, M. – Halilovič, M. – Štok, B.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 84-92

simultaneous modelling of the stiffness degradation and anisotropy can be a true key for resolving the problem of physically reliable simulation of springback. It is clearly seen from the obtained results that, in the calculations of springback, neither stiffness degradation nor anisotropy in sheet metal should be neglected. One way to take those effects into account is the approach elaborated and validated in this article, which has resulted in the so-called GHM model.

Although the stiffness degradation due to occurrence of damage could be tackled in a much more sophisticated way (which is, however, a matter of further research), the approach used with the effective Young’s modulus degradation considered covers the phenomenological background adequately enough to qualitatively demonstrate its impact on the numerical springback behaviour.

6 ACKNOWLEDGEMENT The authors would like to express their gratitude to the company Kovinoplastika Lož d.d., which enabled the experimental research that is presented in the paper, and to the Slovenian Research Agency for its financial support.

7 REFERENCES [1] Geng, L.M., Shen, Y., Wagoner, R.H. (2002).

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18, no. 5-6, p. 743-767.

[2] Chun, B.K., Kim, H.Y., Lee, J.K. (2002). Modeling the Bauschinger effect for sheet metals, part II: applications. International Journal of Plasticity, vol.

18, no. 5-6, p. 597-616.

[3] Bonora, N., Gentile, D., Pirondi, A., Newaz, G. (2005).

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[4] Vrh, M., Halilovic, M., Stok, B. (2008). Impact of Young’s modulus degradation on springback calculation in steel sheet drawing. Strojniski vestnik Journal of Mechanical Engineering, vol. 54, no. 4, p.

288-296.

[5] Halilovic, M., Vrh, M., Stok, B. (2009). Prediction of elastic strain recovery of a formed steel sheet considering stiffness degradation. Meccanica, vol. 44, no. 3, p. 321-338, DOI:10.1007/s11012-008-9169-8.

[6] ABAQUS. (2008). User’s Manual, Ver. 6.7. Simulia, Providence.

[7] Kachanov, L.M. (1958). Rupture time under creep conditions. Otdelenie techniceskih Nauk, Izvestiya Akademii Nauk SSSR, vol. 8, p. 26-31 (in Russian).

English translation (1999): Rupture time under creep conditions. International Journal of Fracture, vol. 97, no. 1-4, p. 11-18.

[8] Gurson, A.L. (1977). Continuum theory of ductile rupture by void nucleation and growth 1. Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology-Transactions of the ASME, vol. 99, no. 1, p. 2-15.

[9] Tvergaard, V. (1981). Influence of voids on shear band instabilities under plane-strain conditions. International Journal of Fracture, vol. 17, no. 4, p. 389-407, DOI:10.1007/BF00036191.

[10] Chu, C.C., Needleman, A. (1980). Void nucleation effects in biaxially stretched sheets. Journal of Engineering Materials and Technology-Transactions of the ASME, vol. 102, no. 3, p. 249-256, DOI:10.1115/1.3224807.

[11] Mori, T., Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, vol. 21, no. 5, p. 571574, DOI:10.1016/0001-6160(73)90064-3.

[12] Benzerga, A.A., Besson, J. (2001). Plastic potentials for anisotropic porous solids. European Journal of Mechanics - A/Solids, vol. 20, no. 3, p. 397-434, DOI:10.1016/S0997-7538(01)01147-0.

[13] Zhao, Y.H., Tandon, G.P., Weng, G.J. (1989).

Elastic-Moduli for a class of porous materials. Acta Mechanica, vol. 76, no. 1-2, p. 105-130, DOI:10.1007/ BF01175799.

[14] Hu, G.K., Weng, G.J. (2000). Some reflections on the Mori-Tanaka and Ponte Castaneda-Willis methods with randomly oriented ellipsoidal inclusions. Acta Mechanica, vol. 140, no. 1-2, p. 31-40.

[15] Riccardi, A., Montheillet, F. (1999). A generalized self-consistent method for solids containing randomly oriented spheroidal inclusions. Acta Mechanica, vol.

133, no. 1-4, p. 39-56, DOI:10.1007/BF01179009.

[16] Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems.

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376-396, DOI:10.1098/rspa.1957.0133.

[17] Carvalho, F.C.S., Labuz, J.F. (1996). Experiments on effective elastic modulus of two-dimensional solids with cracks and holes. International Journal of Solids and Structures, vol. 33, no. 28, p. 4119-4130, DOI:10.1016/0020-7683(95)00269-3.

[18] Pundale, S.H., Rogers, R.J., Nadkarni, G.R. (1998).

Finite element modeling of elastic modulus in ductile irons: Effect of graphite morphology. AFS Transactions, vol. 106, p. 99-105.

[19] Bohm, H.J., Eckschlager, A., Han, W. (2002). Multiinclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements.

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42-53, DOI:10.1016/S0927-0256(02)00248-3.

[20] Halilovic, M., Vrh, M., Stok, B. (2009). NICE-An explicit numerical scheme for efficient integration of nonlinear constitutive equations. Mathematics and

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[22] Morestin, F., Boivin, M. (1996). On the necessity of taking into account the variation in the Young modulus with plastic strain in elastic-plastic software. Nuclear Engineering and Design, vol. 162, no. 1, p. 107-116, DOI:10.1016/0029-5493(95)01123-4.

[23] Yang, M., Akiyama, Y., Sasaki, T. (2004). Evaluation of change in material properties due to plastic deformation.

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Starman, B. – Vrh, M. – Halilovič, M. – Štok, B.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 93-105 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1357

Review Scientific Paper

Received for review: 2013-08-05 Received revised form: 2013-10-14 Accepted for publication: 2013-12-03

The Mathematical Model of Spiral Bevel Gears - A Review Jixin

Wang1

Wang, J. – Kong, L. – Liu, B – Hu, X. – Yu, X. – Kong, W. – Long Kong1,* – Bangcai Liu2 – Xinpeng Hu1 – Xiangjun Yu3 – Weikang Kong1 1 Jilin

University, School of Mechanical Science and Engineering, China 2 XCMGH Hydraulic Component Co., Ltd, China 3 Kunming University, College of Automatic Control and Mechanical Engineering, China The spiral bevel gear (SBG) is a key component of the power transmission of intersection axes. Since the mathematical model of the SBG is a basis for stress and thermal analysis, the optimization of machine-tool settings, frictional contact analysis in lubricated condition, and advanced manufacturing technology, research on designing and manufacturing of SBGs based on mathematical models of SBG has long been a topic of considerable interest in the field of mechanical transmission. The significance of research on the mathematical model lies not only in analysing and building the tooth surface model, but also in investigating the design principles and manufacturing processes. This paper conducts a comprehensive literature review regarding the mathematical modelling of SBGs. The methods of building mathematical models, such as the matrix method, the vector method and the geometry method, are illustrated, compared and summarized in detail. Furthermore, the research history and applications of each method of building a mathematical model of SBGs are presented for better understanding.

Based on applications of the mathematical model of SBGs, it is also indicated that more manufacturing methods could be updated or explored with the future development of universal milling machine technologies and computer aided manufacturing methods.

Keywords: spiral bevel gear, mathematical model, matrix method, vector method, geometry method

0 INTRODUCTION The spiral bevel gear (SBG), with its high contact ratio, high strength and smooth driving, is widely used to transmit dynamic power in various mechanical products, including vehicles, mining machinery, aerospace engineering, and helicopters [1] to [4].

Typical SBGs are shown in Figs. 1a and b. The SBG has been a subject of research for almost a century, and there is a significant amount of literature on the mathematical model of SBGs. The tooth surface of an SBG is a complicated curved surface with a kinematic performance directly bonded to the special cutting process [5]. The mathematical model has significantly contributed to the Computer-Aided Design and Manufacturing (CAD/CAM) of SBGs, because the mathematical model of SBG can be constructed to determine the processing method [6] and [7], to calculate machine-tool settings [8] to [11], to optimize tooth surface topography [12] to [15], to build models of Finite Element Analysis (FEA) [16] to [18], Tooth Contact Analysis (TCA) and Loaded Tooth Contact Analysis (LTCA) [19] to [21], and to develop new SBG types, as shown in Fig. 2. Therefore, the study of the mathematical model construction significantly influences the technological development of the SBG. The most popular method of manufacturing SBGs is that used by Gleason, Oerlikon, and Klingeinberg.

The basic structural forms of special machines include the traditional cradle-type hypoid and computer numerical control (CNC) hypoid generators. The typical feature and manufacturing principle of these special machines are to cut the workpiece using

a rotating cutter head. To analyse the process of manufacturing SBGs, the mathematical model of the tooth surface can be considered to be a spatial trajectory of the cutter blade [22].

a) b) Fig. 1. 3D model of the typical SBG; a) SBG of drive axle of vehicle, and b) SBG of aerospace transmission

To eliminate the restriction of the applied range, reduce processing costs, and improve the universal properties of the special manufacturing system, new manufacturing technologies and design methods of SBGs in universal machines could be investigated.

The mathematical model based on geometry characteristics can guide the manufacture of SBGs in universal milling machines. The overview, analysis and comparison of mathematical models are valuable for improving the manufacturing process, machine tool technology, and design method.

This paper reviews almost all related literature on the mathematical model of SBGs and primarily summarizes three methods: the matrix method, the vector method and the geometry method. The matrix method and vector method are based on the special machining processes. The relationship between the manufacturing principle of a particular machine and the mathematical model is illustrated. As the geometry

*Corr. Author’s Address: Jilin University, 5988 Renmin Street, Changchun, China, goodkong.8810@163.com

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Fig. 2. Application of mathematical model of SBGs

method provides a theoretical model for the new manufacturing method, its development trend is also analysed and discussed.

1 MATRIX METHOD The matrix method is a mathematical tool used to induce spatial transformation in different coordinate systems. Litvin et al. [23] was the first to use the matrix method specifically to build a mathematical model of an SBG. The conjugation theory of space surfaces [24] and [25], which includes the global and local conjugation theories, is essential to establish a relationship between different surfaces in building the mathematical model of an SBG. The imaginary generating gear, formed by the motion of a rotating cutter blade, maintains line contact with the workpiece during the manufacturing process, in accordance with global conjugation theory and the meshing equation [25]. Based on the local synthesis method [26], the tooth surfaces of the gear and pinion conform with local conjugation theory at the mean contact point; therefore, this theory is used to determine the pinion machine-tool settings of the special machine. The universal conjugation theory of spatial surfaces is shown by Eq. (1) and the meshing equation is shown by Eq. (2).

r2 = m + r1 , (1)  n 2 = n1

f ( sg ,θ g ,φ1 ) = n ⋅ v12 = 0. (2)

In Eq. (1), r1, r2 are surface vectors of Surface 1 and surface 2, m is the relative position vector between coordinate systems of Surface 1 and Surface 2. In Eq. (2), sg, θg are surface parameters of the cutter 94

head, ϕ1 is workpiece spindle rotational angle, n is the normal vector of conjugate plane of gear surface, v12 is the relative velocity between Surfaces 1 and 2.

Litvin et al. [23] used the matrix method to synthesize and optimize the tooth surface of SBGs. Matrix transformation effectively represents the spatial transformation of the position vector of the cutter blade. However, the meshing equation is decided by the hypothetical conjugate surface instead of the approximate actual non-conjugate tooth surface. Therefore, a gap is created between the mathematical model and the actual SBG tooth surface.

Based on differential geometry and the theory of conjugate surfaces, Fong and Stay [27] investigated the mathematical model of SBGs generated by circular plane cutter. In [23], a matrix translation corresponding to the manufacturing process was illustrated, and tooth parameters were solved with the meshing equation. Moreover, a modular arrangement facilitates the conversion of the mathematical model into computer language. Fong and Stay [28] also investigated a mathematical model of the Gleason SBG and derived the undercutting equations through this model, which developed a method used to check the undercutting condition of SBGs. Rao et al. [29] compared a mathematical model created with the matrix method with a geometry model and computed the deviation, which confirmed the high uniformity of the two models. Handschuh [30] described a matrix method to build the SBG mathematical model proposed by Litvin and reviewed advances in applications for analysing SBGs. Clearly, building a mathematical model is the foundation for the thermal and structural analysis of SBGs. For example, Fuentes et al. [31] described how to build an accurate SBG mathematical model for finite element analysis in FORTRAN instead of CAD software.

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Flank modifications primarily include lengthwise crowning, profile modification of tools, and a flank twist from toe to heel. These modifications affect the result of the mathematical model, as shown in Table 1.

To determine the machine-tool settings of the tilted-head cutter, Litvin et al. [32] developed a series of matrix transformations corresponding to the manufacturing process, especially the cutter spindle angle and cutter swivel angle. To build a mathematical model of an SBG manufactured by modified roll, Lin et al. [33] illustrated the kinematic mechanism of a modified roll generation train for manufacturing an SBG and proposed a method to calculate the variable of roll ratio ηa, which is presented by Eq. (3). Fuentes et al. [31] proposed a design method for SBGs manufactured by modified roll. The variable of roll ratio was calculated with the parabolic function of transmission error and TCA output, and this function also provided the variable of roll ratio in terms of design. A CNC hypoid generator facilitates the implementation of nonlinear and higher-order kinematic correction motion to manufacture SBGs. Stadtfeld and Gaiser [34] proposed the theory of Ultimate Motion Graph and Ultimate Motion Concept (UMG/UMC), which is an effective theory for the flank modifications of a CNC hypoid generator.

Fourth-order kinematic correction motion was used to generate the gear geometry with low noise and high strength. However, the relationship between machine-tool settings and flank modifications was not described, because it is difficult to build a mathematical model of SBGs. To reduce transmission error, Simon [35] proposed a method to determine optimal polynomial functions. Fifth-order Polynomial functions were used to determine the relationship between the angle of the cradle rotation and the workpiece. Based on modified machine-tool settings, Fan [36] proposed a mathematical model expressed in terms of the sixth-order polynomial function of the cradle roll increment and angle. This model can be used to simulate the flank modifications manufactured

by CNC hypoid generators.

Table 1. Modified parameters of different flank modifications [34] Flank modifications Lengthwise crowning Profile modification of tool Flank twist from toe to heel

Modified parameters

Expression

Cutter radius

R0 SR = R(ϕc) rt = r(sg, θg) ϕ1 = s(ϕc) Em =E(ϕc) i, j

Modified radial motion Tooth profile Variable of roll Helical motion Cutter tilted

In Table 1, R0 is the cutter radius, SR is the radial setting of the cutter head, ϕc is the cradle rotational angle, rt is the vector of cutter tool, Em is the blank offset for gear or pinion, i is the cutter swivel angle and j is the cutter spindle angle.

dφ1   Ra = dφ = 1+C cos(φa − j ) / c   / {a0 cos(φe − φa + j ) +  . (3)   ru  + ru cos  C (φe − φa + j )  }     Ti η a = T Ra p 

In Eq. (3), C is the distance between the cradle centre and the rotational centre of the input shaft, ϕa is the angle of rotation of cradle, a0 is half the distance between two cam guide ways, ϕe is the angle of rotation of the input shaft, ru is the pitch radius of the generating cams, Ti and Tp are the tooth numbers of the index interval and pinion, and Ra is the instantaneous roll ratio of the cam-follower reciprocator.

Considering the different cutting methods and flank modifications, Fong [37] proposed a universal mathematical model that utilizes the matrix method and facilitates the compilation of object-oriented computer programming. In the future, more details about the simulation of universal face hobbing for SBG can be added to this model.

Face milling and face hobbing are two major cutting systems used to manufacture SBG. The differences between these systems are shown in Table 2.

Table 2. Features of two major cutting systems Cutting system Indexing motion Lengthwise tooth curve

Face milling Single indexing

Face hobbing Timed continuous

Circular arc

Extended epicycloid

Finish machining

Uniform or tapered tooth depth system Grinding and lapping

Machining feature

High tooth accuracy

Uniform tooth depth system Lapping High production efficiency

Tooth depth system

As face hobbing combines the timed continuous indexing and generating rolling, it is a more complicated process. The mathematical model is closely related to the generalized kinematic model of face hobbing. Fan [38] proposed a complete modelling of a face-hobbing SBG generated with a Phoenix®II hypoid generator. This method divided the creation of

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a mathematical model into four sections, disassembled the kinematic motion of the machine, and expressed the machine-tool settings as a function of the cradle increment angle. Shih et al. [39] proposed a universal mathematical model of face-hobbing generation with a wide application range. Vimercati [40] proposed a mathematical model of SBG that can be used to simulate the cutting process of face hobbing, and confirmed its high accuracy through an actual case.

Compared with that of face milling, the complexity of the mathematical model of a face-hobbing SBG is primarily reflected in the mathematical model of the cutter blade and the relative motion between the imaginary generating gear and workpiece.

The traditional cradle-type hypoid generator is being gradually replaced by the CNC hypoid generator. The machine-tool settings of the CNC hypoid generator are transformed from those of the virtual traditional cradle-type universal hypoid generator, and the mathematical model is also built by the traditional method. Shih and Fong [41] proposed a mathematical model of the Cartesian-type hypoid generator. The machine-tool settings of three rectilinear motions and three rotational motions in the Cartesian system were converted from a previously proposed universal hypoid generator [39]. Simon [42] developed an algorithm to ensure the relationship between the machine-tool settings of the CNC hypoid generator and those of the cradle-type generator. In the future, a method to directly build the mathematical model of SBG generated by CNC hypoid generator can be investigated.

As the tooth surface of an SBG is the motion trajectory of the cutter blade, the mathematical model of the cutter blade is of key importance. A straight cutting blade is often used to cut the workpiece during face milling [27], [28], [30] and [31]. To obtain high strength and low noise, however, a parabolic profile blade is used to generate the SBG. Litvin et al. [43] provided equations for three shapes of blade profile and confirmed the satisfactory transmission performance of SBG generated by a parabolic profile blade. The mathematical model of the face-hobbing cutter blade is more complex. Fan [38], Vimercati [40], and Shih and Fong [41] provided the matrix equation of the position vector of the cutter blade in face hobbing generation. Vimercati [40] also analysed an actual face-hobbing cutter head and presented a complex equation of a curved blade with Toprem.

The equation was obtained by analysing the complex cutter blade, which included the bottom, fillet, Toprem and curved blade. However, a more accurate model of the tooth surface is required to analyse genuine cutter 96

geometric models. Xie [44] described a genuine facemilling cutter geometric model with the parameters of blade angle, rake angle and relief angle. In simplified cutter geometry, the side and circular cutting edges of the blade are expressed on the normal plane. In [44], the blade rake plane was used to replace the normal plane, which matches real cutter geometry. Obviously, this research provided a method to improve the accuracy of the mathematical model of the facemilling cutter blade, and more studies are expected to extend to the face-hobbing cutter geometric model.

The mathematical model based on the matrix method has been developed into basic technology for computer-integrated methods to design, manufacture and analyse SBGs in special hypoid generation. As the mathematical model follows the manufacturing principle of special machine, it closely matches the actual gear. Although complicated, the matrix method is a clear spatial transformation process that yields a universal mathematical model adopted by most existing cutting systems. However, this method can be used only in special hypoid generation. Furthermore, the nonlinear meshing equation is difficult to solve, especially in the tooth root segment. The formation of the generated surface equations and their derivatives lead to inefficiency in solving the computer programming and contact algorithms.

2 VECTOR METHOD The vector method, proposed by Di Puccio et al.

[45], is an alternative formulation of gear theory; this formulation of the mathematical model of SBGs is clearer and more compact. The advantage of the vector method is that only vector formulation is used to express the surface model. The spatial transformation of the vector method conforms with the principle of rotating vectors. The vector method also avoids using the reference coordinate system in building mathematical models of SBGs.

Fig. 3. Vector rotation of position vector

The vector is rotated to translate the cutter blade spatially, and vector rotation around a mobile axis can simulate all translation processes in one expression.

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According to Fig. 3, the vector p is the rotation of the position vector p0 around the unit vector a by an  can be obtained by rotating angle β0, and the vector p position vector p around the unit vector b by an angle α0. The vector translation process can be expressed as Eq. (4) [45].

p = R (p 0 , a, β 0 ) . (4)  p = R (p, b,α 0 ) = R ( R(p 0 , a, β 0 ), b,α 0 )

The derivative of the vector method with respect to the surface parameters is a simplified expression for solving the meshing equation. Eq. (5) [45] shows that the derivatives of the tooth surface parameters are compact. ,s = R(p ( s ,θ ), a,α (φ )) P g , sg g g c   . (5) P ,θ g = R(p,θ g ( sg ,θ g ), a,α (φc ))  ,φ = R(p ( s ,θ ,φ ), a,α (φ )) + α'a × p ( s ,θ ,φ ) P c g g c c g g c ,φc

The meshing equation is represented by vector formation and requires no reference system. To avoid the application of a kinematic concept and relative differentiation, the meshing equation can be expressed as Eq. (6) [45]. f ( s g ,θ , φ ) = = [p e ,sg ( sg ,θ g )p e ,θ g ( sg ,θ g )h e ( sg ,θ g ,ηaφc )] =

= m e ( sg ,θ g ) × h e ( sg ,θ g ,η aφc ) = 0.

(6)

Di Puccio et al. [45] described the relative concept and formula derivation of the vector method in gear theory, analysed its application in building a mathematical model of an SBG, and described the characteristics of vector method through a numerical example of aerospace transmission application.

However, this application is involved in a simple facemilling process of the traditional cradle-type special generator. A complementary description of the vector method was proposed by Di Puccio et al. [46]. The principle of vector rotation around a mobile axis was proposed to express the complex spatial translation of the position vector of the cutter blade; this principle might even be used in supplemental spatial motions of the modern free-form cutting machine. A numerical example illustrates the convenience of constructing a mathematical model of SBGs. Further research could be conducted to build universal mathematical models by vector method.

The curvature of SBG tooth surfaces is analysed to evaluate their geometric features, mechanical properties, and physical characteristics. Di Puccio et al. [47] compared the different characteristics of Litvin’s approach [48], Chen’s approach [49], Wu and Luo’s approach [50], and the vector method for curvature analysis. In [47], vectors and tensors were introduced to analyse the curvature in vector method, and curvature tensors were used to simplify the analysis. Puccio et al. [51] used the proposed vector method to analyse curvature. The vector method can avoid using the reference system and provides explicit formulas to analyse curvature. Indeed, the vector method is a more compact and computationally efficient method for analysing curvature than other methods.

The vector method and matrix method are applied to simulate the similar machining processes and actual transformation path of the cutting blade, as shown in Fig. 4. However, they have different formulations in building mathematical models, as shown in Table 3 [37] and [46]. By avoiding the reference system, the vector method for expressing tooth surface is more compact and cleaner. Obviously, the vector method is an alternative formulation for the mathematical modelling of SBGs, and it facilitates the simplification of computer programming and the improvement of computational efficiency. Thus, the application range of the vector method could be extended.

In Fig. 4, q is the instalment angle for the cutter head, γm is the machine root angle, xb is the sliding base for gear or pinion, xp is the increment of machine centre to back.

In Table 3, Rg is the cutter head point radius, αg is the blade angle of the cutter head, ηa is the instantaneous roll ratio, [L1t (ϕc)], [Lq] is the 3×3 homogeneous transformation matrix, [Mij] is the 4×4 homogeneous transformation matrix from coordinate systems Sj to Si .

3 GEOMETRY METHOD Based on the principle of the geometry method, the geometry model of an SBG is determined by basic geometric parameters instead of machine-tool settings.

In fact, the geometry model is a theoretical model, and it emphasizes the guidance for manufacturing SBG. Geometric characteristics include tooth profile and centreline. The tooth profile primarily includes the spherical involute, approximate spherical involute, and circuit arc. Many spirals, such as the logarithmic spiral, circular cut spiral and involute spiral, can serve as the tooth centreline.

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Fig. 4. Transformation of cutting blade vector in machining process [37] Table 3. Comparison of matrix method and vector method [37] and [46] Mathematical model Position vector of cutter head Meshing equation Spatial translation Equation of tooth surface

98

Vector method

Matrix method

( Rg ± sg sin α g ) cos θ g    pe ( sg ,θ g ) =  ( Rg ± sg sin α g ) sin θ g    − sg cos α g  

( Rg ± sg sin α g ) cos θ g    rt ( sg ,θ g ) =  ( Rg ± sg sin α g ) sin θ g    − sg cos α g  

f ( sg ,θ g ,φc ) = m e ( sg ,θ g ) × h e ( sg ,θ g ,η aφc ) = 0

f ( sg ,θ g ,φc ) = n1 × v12 = 0

pg ( sg ,θ g ,φc ) = R( R( R( R(p e ( sg ,θ g ) +

[M1t ] = M1f  × M fe  × [M ed ] × ×[ M dc ] × [ M cb ] × [ M ba ] × [ M at ]

+e(φc ), ra (φc ), ε a (φc )), a(φc ),ψ (φc )) − −d(φc ), b(φc ), −ϕ (φc )), rb (φc ), −ε b (φc )) sg = R( R([L q ]p e ,[L q ]a,η aφc ) − [L q ]d a b ,[L q ]b, −φc )

Wang, J. – Kong, L. – Liu, B – Hu, X. – Yu, X. – Kong, W.

r ( sg= ,θ g ,φc )

[L1t (φc )] ⋅ rt ( sg ,θ g )

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3.1 Logarithmic Spiral Bevel Gear The logarithmic spiral bevel gear (LSBG) provides excellent transmission performance with a constant spiral angle. Huston and Coy [52] provided the geometric characteristics and model of LSBG and proposed the concept of the “ideal SBG”. Tsai and Chin [53] investigated the tooth surface geometry of LSBG and provided an equation of logarithmic spiral on pitch plane, which is presented in Eq. (7). The geometry model of LSBG can be obtained with Eq. (8).

r = Rm ⋅ ecotψ ⋅θ . (7)

generating angle, and θ is the polar angle on X–Y plane, as shown in Fig. 6.

 β2 2 sin cos ( sin α ) + X = r α θ + β 1 − ( )  t 2   β2 2 + r β sin α sin (θ + β ) 1 − sin α  4   β2 2 sin α ) − . (8)  Yt = r sin α sin (θ + β ) (1 − 2   β2 2 −r β sin α cos (θ + β ) 1 − sin α  4   β2 2  Z t = r cos α (1 − sin α ) 2 

In Eq. (7), r is the radial distance on the pitch plane, θ is the polar angle on the pitch plane, r = Rm when θ = 0, ψm is the mid-spiral angle, as shown in Fig. 5.

Fig. 5. Logarithmic spiral on pitch plane [53]

However, the consistency of the LSBG tooth centreline between the manufactured gear model and the geometry model is a key issue in the manufacturing process. Li et al. [54] investigated the meshing equation and the spatial relationship between the cutter blade and workpiece. This investigation is a theoretical exploration of the processing technology of LSBGs, and further research on its application in an actual manufacturing process can be conducted. Ju [55] developed a strategy to control the tool path of a finger-milling cutter of five-axis universal milling in manufacturing processes. A geometry model was built in Pro/e and translated into NC code in UG CAM. This method provides guidance for manufacturing LSBG. However, the accuracy of the generated LSBG still could be improved. Alves et al. [56] proposed a reliable design and manufacturing method of a universal five-axis milling machine, including a geometry analysis of the tooth surface, LTCA, and tooth modifications. The accuracy of the generated LSBG was validated via surface measurements.

In Eq. (8), Xt , Yt and Zt are the coordinate of the tooth surface, α is the root cone angle, β is the involute

Fig. 6. Spiral tooth centreline on X-Y plane [53]

Duan et al. [57] proposed a theory of loxodromic normal circular-arc spiral bevel gear (LCBG), which is a typical LSBG. The tooth profile is a circular arc; the tooth alignment curve is a logarithmic spiral, and the centre of the tooth profile is located on the tooth alignment curve. The mathematical expression of the tooth profile and tooth alignment curve was proposed.

The manufacturing process was investigated by Duan et al. [58], including the parameters of the form milling cutter, contact conditions, and relative motion relationship between the form milling cutter and the workpiece, which illustrated excellent machinability in the universal four-axis machining centre. Duan et al. [59] developed a complete process of designing and manufacturing LCBGs, introduced the basic idea of the mathematical model of LCBGs, and investigated the determination method of the tool path and tooth alignment curve. The manufacturing programming of LCBGs was used to achieve geometric characteristics, which provided a new generation theory of manufacturing LCBGs.

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In short, an LSBG is an excellent transmission component, in theory. However, the manufacturing techniques of LSBGs are still to be improved. The challenge in manufacturing LSBGs is to achieve its geometric characteristics, especially the tooth centreline.

exact spherical involute, which is obtained via a circuit tangent plane rolling over the base cone, as shown in Table 4. The ideal model is formed by the trace line on the tangent plane rolling over the base cone. Eq. (10) shows that the points of the tooth surface can be achieved by changing the value of the parameters α1 , β1 and υ.

3.2 Circuit Cut Spiral Bevel Gear Because the tooth centreline is a circuit spiral, a “circuit cut” SBG is conveniently manufactured by a rotary cutter blade and other special tools, which is the manufacturing principle adopted by Gleason. Tsai and Chin [53] proposed an equation for the tooth centreline on a pitch plane, as described in Eq. (9). The geometry model was built with a spherical involute and circuit spiral, and an actual Gleason SBG was used to verify its accuracy.

In Eq. (9), Rc is the cutter radius. Point O is the gear centre, and Point C is the cutter centre, as shown in Fig. 7.

Al-Daccak et al. [60] introduced a method to build a model of a circuit-cut SBG by using an exact spherical involute, which was defined as the curve on a sphere. The spherical involute was generated by rolling the circuit plane over the base cone, and a solid model of SBG can be created by twisting spherical involute along the tooth centreline.

Tsai and Chin [53] proposed the approximate spherical involute, which is obtained via a rectangle tangent plane rolling over the base cone. Shunmugam et al. [61] investigated a mathematical model with an

Fig. 7. Top view of circular-cut tooth centreline on pitch plane [53]

( X − Rm + Rc sinψ m ) 2 + (Y − Rc cosψ m ) 2 = Rc2 . (9)

 x = r0 cos( β sin α 0 − υ )sin α 0 cos β +  +r0 sin( β sin α 0 − υ )sin β    y = r0 cos( β sin α 0 − υ )sin α 0 sin β − (10)  −r0 sin( β sin α 0 − υ ) cos β   z = r0 cos( β sin α 0 − υ ) cos α 0

Table 4. Comparison of two spherical involutes [53] and [61] Approximate spherical involute

Exact spherical involute

Schematic diagram

Equation

Difference

100

( X p − r0 sin α 0 cos β ) 2 + (Yp − r0 sin α 0 sin β ) 2 +

( X p − r0 sin α 0 cos β ) 2 + (Yp − r0 sin α 0 sin β ) 2 +

+( Z p − r0 cos α 0 ) 2 = r02 β 2 sin 2 α 0

+( Z p − r0 cos α 0 ) 2 = 4r02 sin 2 (

Tangent plane is rectangle Wang, J. – Kong, L. – Liu, B – Hu, X. – Yu, X. – Kong, W.

β sin α 0 ) 2

Tangent plane is circuit

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In Table 4 and Eq. (10), Xp, Yp, Zp are the coordinates of typical point P of the spherical involute. r0 is the radius of the sphere in case of spherical involute (the cone distance), β is the rotation angle of the tangent plane over base cone, α0 is the base cone angle, and υ is the polar angle of a point on the circular arc.

Suh et al. [62] proposed a sculpted surface machining method for manufacturing SBGs with a three-axis CNC milling machine interfaced with a rotary table. The bi-parametric surface model can be derived via spatial translation with geometric characteristics. The CC point was sampled from a bi-parametric surface model, and a CC-parametric scheme was applied to control the tool path. Although the machining time is not ideal, the broad cutting range and generating a special type of gear can be implemented, which shows potential applications of this method. Suh et al. [63] investigated the method of manufacturing SBGs with a crown. The crown model was built via crown functions in longitudinal and involute curve directions. This geometry model was implemented in the GearCAM system with fouraxis CN milling, which verified the validity of this manufacturing method.

Tsai and Hsu [64] investigated a manufacturing and design method for the new point contact-type SBG. The mathematical model was built using tooth profiles and circular-arc contact paths.

Safavi et al. [65] invented the form milling method of manufacturing SBGs with an additional PLC module. Commercial software was used to build a CAD model, simulate the manufacturing process, and generate tool paths. This method provides a more automatic and simple process technology for manufacturing SBGs. However, the application of commercial software based on the specific manufacturing principle requires further research to verify the precision of generated gear.

Zhang et al. [66] developed a generating method for SBGs with spherical involute tooth curves. The tooth surface model was formed by the relative rolling motion of the tracing line on the tangent plane. The motion of the cutting edge of the cutter simulated the actual tracing line rolling on the base cone. The same tracing line was applied to cut a pair of gears, and the pinion ensures stable and proper meshing conditions.

The kinematic velocity and processing principle were used to illustrate the control theory of the CNC machine. The construction of the machine, as well as motion control, is simple and not subject to restriction of gear size. This research also analysed the straight tracing line. Additional research can focus on other

types of tracing lines, such as the logarithmic spiral, circuit arc spiral, and involute spiral. To confirm its excellent transmission performance, the research on contact characteristics of SBGs generated by this method should be conducted.

The mathematical model of the circuit arc SBG and manufacturing method based on its geometric characteristics are reviewed in this section. As the tooth centreline can easily be controlled, a circuit arc SBG has a wider application range. In particular, Gleason’s manufacturing principle is a typical application of the geometry model.

3.3 Involute Spiral Bevel Gear The involute spiral bevel gear is another SBG type, which is a theoretical model of Klingeinberg and Oerlikon’s method. Tsai and Chin [53] formulated the equation of an involute spiral on a tangent plane. Additional research focusing on the contact characteristics and transmission performance of this gear type can be conducted. The current application is in Klingeinberg and Oerlikon’s gear manufacturing method. However, the mathematical model of involute SBG could be applied to more situations generated by the universal milling machine. The tool path is also a challenge in implementing new manufacturing methods.

As analysed from sections 3.1 to 3.3, the process of building a mathematical model of an SBG can be simplified with the geometry method, which avoids the difficulties of solving the meshing equation and spatial transformation. The geometry model can be easily built using CAD software; based on the geometry model, several manufacturing methods have already been developed. The geometry method provides a theoretical model and can be used to explain the manufacturing principle.

4 CONCLUSIONS The development of CAD/CAM technology has made mathematical models indispensable in the design and manufacturing of SBGs. To analyse the process of building a mathematical model and the application of a mathematical model in design and manufacture, this paper reviews the methods of building mathematical models of SBGs, including the matrix method, vector method, and geometry method.

The matrix method and vector method are special methods based on the special machining principle of SBGs. The mathematical models are derived from actual machine-tool settings and are thus consistent

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with the actual manufacturing tooth surfaces. These two methods have a close relationship with the manufacturing process; this relationship benefits the application of the loop manufacturing system.

Coordinate transformation in matrix representation can produce a clear spatial translation process.

However, matrix expression is complicated, and converting it into computer language is difficult. The vector method presents a more compact and clearer formulation and does not need any coordinate system in vector rotation. Although the expression forms of spatial transformation are different, the theory and application range of these two methods are similar.

Solving the meshing equation of both methods is complicated, especially in the tooth root segment.

The geometry model places more emphasis on the theoretical model, which is used to develop new manufacturing principles. To research new types of SBGs, the geometry model could be built primarily for the presentation of a design idea.

Several research fields, which could be further targeted, include: (1) The matrix and vector methods are modelling methods based on manufacturing principles.

Further research may illustrate more details of new manufacturing methods, including the machine motion, cutter geometric model, and the relationship between the cutter blade and workpiece, which will benefit the application of building a mathematical model of SBGs by these two methods.

(2) The geometry method is proposed as a “theoretical model”, and it is a breakthrough in the study of new SBG theories. However, the rationality of current methods of controlling the tool path with commercial software will be further confirmed.

Thus, more research on building the relationship between geometric features and the machining process could be conducted. Future studies on the application of the geometry model are likely to consider the shapes of the milling cutter and the tool path. In practice, the disk milling cutter is an efficient tool, but the tool path is difficult to control; the finger milling cutter is easy to control, but its productivity can be improved. Further research on the application of the geometry model can focus on new process technology, such as the forging manufacturing technique, roll forming, and powder forming technology.

(3) The manufacture of large-scale SBGs is a challenge because of the high demand for control accuracy and large machining distortion. The application of the mathematical model may 102

provide more reliable and effective methods for manufacturing large-scale SBGs. (4) The transmission performance and contact characteristics are different in various types of SBGs. The evaluation criterion can be built by analysis methods, such as FEM, TCA, and LTCA. The evaluation results can be used to guide the application of different types of SBGs in power transmission.

5 ACKNOWLEDGEMENTS The authors acknowledge the financial support from National Natural Science Foundation of China (No. 51075179), the Chinese Government’s Executive Program for Exploring the Deep Interior Beneath the Chinese Continent - Instrumentation Development for Deep Continental Scientific Drilling (Sinoprobe-09-05) and Scientific Frontier and Interdisciplinary Merit Aid Projects of Jilin University, China (No. 2013ZY08).

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loxodromic-type normal circular-arc spiral bevel gear.

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Establishing gear tooth surface geometry and normal deviation Part II - bevel gears. Mechanism and Machine Theory, vol. 33, no. 5, p. 525-534, DOI:10.1016/S0094114X(97)00076-1.

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Sculptured surface machining of spiral bevel gears with CNC milling. International Journal of Machine Tools and Manufacture, vol. 41, no. 6, p. 833-850, DOI:10.1016/S0890-6955(00)00104-8.

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[64] Tsai, Y.C., Hsu, W.Y. (2008). The study on the design of spiral bevel gear sets with circular-arc contact paths and tooth profiles. Mechanism and Machine Theory, vol. 43, no. 9, p. 1158-1174, DOI:10.1016/j. mechmachtheory.2007.08.004.

[65] Safavi, S.M., Mirian, S.S., Abedinzadeh, R., Karimian, M. (2010). Use of PLC module to control a rotary table to cut spiral bevel gear with three-axis CNC milling.

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 106-114 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1407

Received for review: 2013-08-29 Received revised form: 2013-11-05 Accepted for publication: 2013-11-13

Original Scientific Paper

Adaptive Pulsed-Laser Welding of Electrical Laminations Vegelj, D. – Zajec, B. – Gregorčič, P. – Možina, J. David Vegelj1,* – Boštjan Zajec1 – Peter Gregorčič2 – Janez Možina2 2 University

1 Hidria Rotomatika, Slovenia of Ljubljana, Faculty of Mechanical Engineering, Slovenia

In this contribution we develop and describe a new, adaptive, pulsed-laser welding method for thin electrical laminations. By using a reflective laser sensor and a custom-made control unit with the appropriate software the system triggers the laser pulse only at the contacts between the laminations. In this way the total laser pulse energy and, consequently, the specific power losses of the electromotor, where the stack is used, are significantly reduced. This new method has great potential in the serial production of electromotors.

Keywords: laser welding, lamination welding, magnetic properties, stator, rotor

0 INTRODUCTION The laser welding of electrical laminations is a new technology that has been developed in the past decade and is of great interest in the field of manufacturing highly efficient electromotors for both hybrid and pure-electric vehicles, where the stator and rotor stacks are made of thin electrical steel laminations in order to reduce the eddy currents. In alreadyestablished procedures, the laminations are made using a technology of stamping or blanking [1] and then fastened together in stacks by riveting, interlocking, conventional and laser welding, and sticking technology [2] to [4]. The main problem with these classic methods is that they all have a significant effect on the magnetic properties of the stack.

In the past few years, interest in the technology of laser welding electrical laminations has rapidly increased; this is because the laser welds tend to be small and, consequently, their influence on the magnetic properties of the stacks is smaller than the influence of the already-established procedures [5]. Additionally, the classic interlocking system is becoming less reliable as the thickness of the laminations decreases [6]; this is especially important in high-end products. It is with such cases that laser technology shows a great deal of promise. Laserwelding technology also uses less energy, while ensuring that the mechanical properties of the welds are satisfactory for use in electrical motors.

In general, there are two techniques for laser welding that are currently in use: (i) continuous-laser welding and (ii) pulsed-laser welding, with a large degree of overlap [7] and [8]. Although other authors [9] and [10] have compared these two regimes and shown that, in general, there is a significant difference between them, there is no research that analyses the differences in the specific application of welding laminations. Here, in both regimes, the laser radiation 106

melts the whole material of the lamination surface.

However, this is unnecessary, since the joints are only needed at the contacts between the individual laminations. The main aim of our work is therefore to upgrade the existing methods by developing a pulsed-laser welding system that reduces the volume of molten material. In this contribution we present the developed system, which (i) significantly reduces the total-pulse energy per weld needed for the production of mechanically sufficient and acceptable welds, and (ii) enables smaller total magnetic losses and a higher relative permeability of the stator or rotor stack.

1 COMPARISON OF DIFFERENT LASER-WELDING APPROACHES The main ideas behind the different approaches to the laser welding of electrical laminations are schematically presented in Fig. 1. The electrical laminations with the marked regions of the welds are sketched in the middle of different schemes for pulsepower vs. distance.

The main idea behind the classic, continuous welding is presented in Fig. 1a. Here, a continuous laser with a constant power is used. The main advantages of such an approach are reliable welds, fast welding, no need for extra sensors, and less maintenance of the laser source.

On the other hand, since the laser power is all the time higher than the threshold for the material melting, this leads to strong and large welds, resulting in significant magnetic losses for the electromotor.

Continuous welding can be improved in the context of reducing the volume of molten material by employing adaptive, continuous welding, which is shown in Fig. 1b. Although this approach still uses a continuous laser, we modulate its power during the processing so that the maximum power is achieved at the gaps between the laminations, while the minimum

*Corr. Author’s Address: Hidria Rotomatika d.o.o., Spodnja Kanomlja 23, 5281 Spodnja Idrija, Slovenia, david.vegelj@hidria.com

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Fig. 1. The main ideas behind different approaches to the laser welding of electrical laminations; a) classic continuous welding, b) adaptive continuous welding, c) classic pulsed welding, and d) adaptive pulsed welding

power is maintained on the lamination surfaces, where the welds are unnecessary. During such a procedure, less laser energy is needed, since less material is being melted. However, if we keep the maximum power at the same level as in the case of continuous welding, this approach leads to unreliable welds with frequent cracks and blowholes. In contrast, increasing the total energy per a weld results in appropriate welds in terms of the mechanical properties, but in this case there is no advantage of using adaptive, continuous welding in comparison with the classic, continuous welding; the welds are still large and they have a significant influence on the magnetic properties of the electromotor.

Another possible regime is pulsed-laser welding, which is schematically presented in Fig. 1c. Here, the laser pulses start at the beginning of the stack and finish at the end of the stack, with a large overlapping of the pulses. This approach is usually slower than continuous welding, since we need a certain frequency of the pulses to achieve a sufficient degree of overlapping (usually around 70%) at the desired speed. The advantages of pulsed welding are smaller but deeper welds and less energy consumption in comparison with continuous welding.

The classic form of pulsed welding can be significantly improved by using an adaptive, pulsedwelding method, schematically presented in Fig. 1d.

In this case we only trigger the pulses at the gaps between the laminations. In this new approach, a

sensor for monitoring the gaps is necessary, since the laminations are not equally thick due to the manufacturing process of the electrical steel; their thicknesses can vary by up to 8% [11].

2 MATERIALS AND METHODS 2.1 Experimental Setup We have developed a new, adaptive, pulsed-laser welding system, which is schematically presented in Fig. 2. Our system first determines the position of the gap between two laminations and then triggers the laser pulse that welds two successive laminations.

In order to determine the gaps between the laminations we use a reflective laser sensor (Keyence LV-NH37, Japan) with a working distance of 70±5 mm. It consists of a probe laser (λ = 660 nm) that illuminates the laminations and the receiving sensor that observes the amount of reflected probe-laser light.

The spot diameter of the probe beam at the working distance is 50 µm. At the output the sensor gives an analog signal in the range 1 to 5 V. Its response time is shorter than 1 ms. The analog signal from the sensor is first amplified and filtered using a low-pass, 30-Hz filter (Stanford Research Systems SR560). The signal from the amplifier goes to the control unit.

The control unit uses two microcontrollers (Arduino Mini) for the signal processing and a relay for the laser triggering. Microcontroller 1 (see Fig. 2)

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makes an analog-to-digital (A/D) signal conversion.

The main idea of our A/D conversion is to eliminate the noise in the signal. The processed signal is then converted into a digital signal, which is transferred to the second microcontroller (Microcontroller 2 in Fig. 2). This second microcontroller allows the setting of the delay between the input digital signal and the output triggering signal. The setting of this delay is necessary in order to synchronize the positions of the sensor and the processing laser beam. The triggering signal is then connected to the relay, which triggers the laser according to the detected gaps. For such control we wrote custom software in the open-source Arduino programming language.

Fig. 2. A schematic presentation of our adaptive pulsed-laser welding system with three main components: (i) reflective-laser sensor, (ii) control unit, and (iii) welding laser

As the welding laser we used a high-energy Nd:YAG laser (SpotLight Plus, Fotona, Slovenia, λ = 1064 nm). The laser works in a pulsed mode with pulse energy in the range 1 to 120 J. The laser beam was focused onto the surface of the laminations using a lens with a focal length of 95 mm. Due to the zoom beam-expander built into the welding laser, we were able to change the beam diameter at the focus from 0.3 to 2.0 mm. In our experiments we used the laserwelding parameters that are presented in Table 1.

During the experiments we chose the lowest possible pulse energies still ensuring the appropriate mechanical properties of the weld. During the welding we also used argon as a shielding gas with a flow of approximately 6 l/min. The samples were attached to a linear, motorized stage (Standa, model 8MT177-100) connected to a PC. The motorized stage moves with a constant velocity of 1 to 3 mm/s. The velocity depends on the thickness of the sample. The laser pulses were only irradiated when a gap between the laminations was detected by the reflective laser sensor.

108

At the end of each weld we obtained the totalpulse energy by multiplying the single-pulse energy by the measured number of pulses that corresponds to the number of laminations in a single stack.

Table 1. Laser welding parameters Sample Pulse energy [J] Pulse time [ms] Spot [mm] Comment A1 30.0 6.0 1.2 CPW A2 8.7 2.5 0.8 A3 31.3 4.0 1.2 A4 16.1 2.0 1.0 A5 12.5 2.0 1.0 APW A6 9.5 2.0 1.0 A7 7.0 2.0 0.8 B1 12.5 2.0 1.0 CPW B2 7.0 2.0 0.8 B3 7.8 2.0 0.6 B4 5.6 2.0 0.6 B5 4.7 2.0 0.5 APW B6 3.9 2.0 0.5 B7 2.6 2.0 0.3 C1 8.8 2.0 0.8 CPW C2 6.3 2.0 0.7 C3 5.3 2.0 0.7 APW C4 4.7 2.0 0.6 C5 3.9 2.0 0.3 CPW – classic pulsed welding, APW – adaptive pulsed welding

2.2 Electrical–Lamination Samples The experiments with our laser spot-welding system were performed on three different stator stacks (A, B and C). The laminations of each stack had the same dimensions, but different material grades, chemical composition and thickness. The characteristics of samples’ laminations are listed in Table 2.

Table 2. The materials of tested electrical laminations Sample Material C [%] Si [%] Thickness [mm] No. of laminations in the stack Hardness [HV 5]

A M700-100A 0.0020 2.40 1.0

B M800-50A 0.0015 1.05 0.5

C M330-35A 0.0020 2.41 0.35

16

33

47

164

128

155

All the laminations had a circular shape (e.g., see Fig. 3a) with an outside diameter of 50±0.05 mm and an inside diameter of 46±0.05 mm. The electrical laminations were built into a stator stack manually, i.e., they were fixed with a custom-made clamping tool,

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as shown in Fig. 3b. The black dotted line in Fig. 3b represents the position of the weld. All the manually made stacks had the same height of 16.3±0.2 mm.

Each sample was welded four times on the outside diameter where the welds were shifted by 90° around the symmetrical axes going through the center of the laminations. For each sample we performed 10 experiments and from these measurements we calculated the average and the standard deviation.

Fig. 3. a) typical shape of electrical laminations, and b) a clamping tool; here, the black dotted line shows the position of the weld

Fig. 4 is a magnified image of the lamination surfaces. Here, the gaps and the thicknesses of all three samples are clearly visible. For a better understanding, the gaps in Fig. 4 are marked with red circles. The laminations of sample A have a thickness of 1000±37 µm with a gap of 178±6 µm; the thickness of the laminations of sample B is 500±23 µm with gaps of 84±13 µm; while the laminations of sample C have a thickness of 350±17 µm with gaps of 47±7 µm.

2.3 A Real Stator Stack In addition to the experiments made on the sample that we described in Subsection 3.2, we also tested our new, adaptive, pulsed-laser welding system on an example of a real stator stack. As a real sample we used the stator core of a hybrid vehicle that is already being manufactured using classic, continuous-laser welding.

The laminations are made of the material M27035A, which is very similar to the material M33035A described in Table 2 (see sample C). The stator has an outside diameter of 180.0±0.1 mm, a height of 140.0+0.7 mm and a weight of approximately 9.9 kg. When fully assembled, the electromotor reaches around 30 kW in its normal operating mode. We compared the magnetic properties of a motor with a stator stack that was welded (i) classically, i.e., with continuous-laser welding, and (ii) with our new, adaptive, pulsed-laser welding method. In both cases we made 10 welds around the perimeter of the stack.

2.4 Magnetic-Properties Testing Magnetic measurements are very important for assessing the quality of the manufactured stator or the rotor stacks. Thus, after the welding we measured the relative permeability (µr) and specific core power losses (PS) of all the samples.

Here, the specific power losses are the losses that occur due to the rotation of the core in the magnetic field of the poles and they are a combination of the hysteresis and eddy-current losses [1], [12] and [13].

They are calculated on the basis of the mass of the tested stack. While the welding does not affect the hysteresis losses, it causes the laminations to come into electrical contact and thus increase the eddycurrent losses. The specific power losses appear as heat and thus raise the temperature of the electromotor.

On the other hand, the permeability is the degree of magnetization that a material achieves in response to an applied magnetic field. The relative permeability is the ratio between the permeability of the material and that of free space.

Fig. 4. Magnified image of the lamination’s surface of sample a) A, b) B, and c) C; the circles indicate the gaps between the laminations

For measuring purposes we used a Brockhaus Messtechnik MPG 100D AC/DC system that is widely used in the automotive industry. The measurements on this system were made according to the ISO standard IEC 60404. More details of the magnetic measurements can be found elsewhere, e.g., in [14] and [15]. To determine the magnetic properties of the samples A, B and C, they were wrapped with 257 primary and secondary coils, while the real stator was wrapped with 90 primary and secondary coils.

During the measurements we used flux densities of 1.5 T. Since high-end applications usually work at frequencies higher than 50 Hz, we used frequencies of 50 and 400 Hz in our testing.

2.5 Breaking-Force Measurement The tensile strength of a weld is only important until the winding of the copper wire. After the winding, the welds lose their functionality and only represent a

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distraction that affects the magnetic properties of the electromotor [15]. This does not mean that the welds can have cracks or other errors. It only means that the welds should be as small as possible, while still being mechanically acceptable. In this context we tested our samples with a modified breaking-force test.

For the testing of the welds we used an Instron 4206 tensile system. We attached the samples to the lower and upper plates of the tensile system with a two-component adhesive and tore them apart with a constant speed of 5 mm/min. Load was applied in the longitudinal direction of the samples. In this way we measured the breaking force at which the first of four welds on the sample was broken, i.e., it has not joined the laminations anymore. For all the small sample stators we decided that the welds were acceptable if they sustain a breaking force larger than 100 N. This force threshold was set on the basis of production demand.

3 RESULTS AND DISCUSSION The advantage of our new, adaptive method is clear from Fig. 5, which shows a comparison between the weld produced using classic, pulsed welding (Fig. 5a) and the weld produced using our new, adaptive, pulsed-laser welding system (Fig. 5b). In the case of the classic, pulsed method, a large overlap of the laser pulses is seen. On the other hand, our new method only irradiates the processing pulses at the gaps between the laminations. Thus, in this case a significantly smaller total-pulse energy is necessary.

For this particular example (presented in Fig. 5), our method uses only 23% of the total-pulse energy in comparison with the classic, pulsed-welding method.

Fig. 5. Magnified image of a typical weld on laminations produced by; a) classic pulsed-welding method, and b) our new, adaptive, pulsed-laser welding system

Typical cross-section in the transversal direction (e.g., see the broken line A-A in Fig. 5b) is presented in Fig. 6. In this particular case the pulse energy was 110

4.7 J and the thickness of the laminations was 0.5 mm (sample B). The shape of the weld suggests that our laser parameters were close to the threshold for the keyhole welding.

Fig. 6. Cross-section of a single-pulse weld in the transversal direction of sample B

3.1 Total-Pulse Energy Reduction The dependence of the breaking force as a function of the total-pulse energy for all three samples and for both laser-welding techniques, i.e., for our new, adaptive pulsed-laser welding and for the classic, pulsed-welding method, are presented in Figs. 7 (for samples A), 8 (for samples B) and 9 (for samples C).

The horizontal dotted lines in all the figures show the smallest, still acceptable, breaking force (i.e., larger than 100 N), while the dashed lines show a linear fit to the measured data. From these results we can conclude that for still-acceptable mechanical properties of the weld, our method uses significantly less energy in comparison with the classic, pulsedwelding technique. The ratio:

η=

E APW , (1) EPW

between the total-pulse energy needed per stillacceptable weld for our adaptive (EAPW) and for the classic, pulsed-welding (EPW) method equals around 21% for samples A, 29% for samples B and 52% for samples C. Fig. 7 shows the breaking force versus totalpulse energy per weld for samples A. In this case the thicker laminations (1 mm; see Table 2) were used and the stack consisted of 15 laminations. From these results it can be concluded that our new method needs a significantly lower total-pulse energy than the classic, pulsed-welding method for welds with similar mechanical properties. For example, for the welds to sustain a tensile force of approximately 200 N, five-times less total-pulse energy per weld is needed

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with our new welding method (105 J per weld) in comparison with the classic, pulsed technique (490 J per weld).

Fig. 7. The breaking force as a function of the total-pulse energy per weld for samples A welded by the adaptive, pulsed method (squares) and by the classic, pulsed method (triangles)

When using a lower total-pulse energy per weld than 105 J, the laminations in the stack welded with our method do not stand together. Similarly, the lowest total-pulse energy per weld with the classic method is 490 J; if a lower total-pulse energy is used with this method, the stack decays into laminations as soon as it is taken out of the clamping tool.

The dependence of the breaking force on the total-pulse energy for the samples B is shown in Fig. 8.

Here, the stacks were assembled from 33 laminations with a thickness of 0.5 mm (see Table 2). In this case the lowest total-pulse energy per still-acceptable weld is around 100 J for our method and 350 J for the classic, pulsed-welding method.

So, our new method in this case enables stillacceptable welds with only 29% of the total-pulse energy of the classic method. On the other hand, for the same breaking force (around 200 N), our method uses around 2 to 3 times less total-pulse energy than the classic welding method for the laminations of type B. Fig. 9 shows the dependence of the breaking force as a function of the total-pulse energy per weld for samples C. Here, thinner laminations with a thickness of 0.35 mm were used and the stack consisted of 47 laminations. In this case the lowest total-pulse energy per still-acceptable weld (i.e., a weld retaining a tensile force larger than 100 N and without cracks or other errors after the welding) equals 215 J for our adaptive method and 415 J for the classic, pulsedwelding method. Thus, the ratio η [see Eq. (1)] in this case equals 52%. At this point it should be noted that the samples C are made from the material M33035A that is less appropriate for welding due to its high Si content (see Table 2). For this reason, the welds obtained with the classic method do not hold the stack together for total-pulse energies lower than approximately 400 J. If a lower energy is used, the stack decays into individual laminations as soon as it is taken out of the clamping tool.

Fig. 9. The breaking force as a function of the total-pulse energy per weld for samples C welded by the adaptive, pulsed method (squares) and by the classic, pulsed method (triangles)

Fig. 8. The breaking force as a function of the total-pulse energy per weld for samples B welded by the adaptive, pulsed method (squares) and by the classic, pulsed method (triangles)

From the results presented in Figs. 7 to 9 the trend of η is clearly visible. When the lamination thickness goes to zero, the ratio η goes to 1. The reason for this lies in the fact that for the laminations thinner than the spot size, there is no difference between our new, adaptive method and the classic, pulsed-welding method, since in both cases an overlapping of the successive pulses is achieved. On the other hand, our method gives significantly better results in the context of the total-pulse energy per weld reduction in the

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Fig. 10. The total specific losses (PS) for lamination samples made of three different materials; a) material A, b) material B, and c) material C; the magnetic properties in all cases were measured at f = 400 Hz and B = 1.5 T

case of thick laminations (e.g., for samples A, where the thickness of the laminations is 1 mm).

3.2 Magnetic Properties of the Welds The influence of the welds produced by our adaptive, pulsed-laser welding and classic, laser pulsed welding on (i) the specific power losses and (ii) the relative permeability was examined by measuring the magnetic properties of the electrical-lamination samples as well as of a real stator stack. Different authors [14] and [16] compared the welded and non-welded stacks and showed that the specific power losses increase, while the relative permeability decreases, when the stack is welded.

The specific power losses at a frequency of 400 Hz and a flux density 1.5 T as a function of the total-pulse energy per weld for all three electricallamination samples are shown in Figs. 10a (for samples A), 10b (for samples B) and 10c (for samples C). From our results it can be concluded that there is no significant difference in the specific power losses between classic welded stacks (the full triangles in Fig. 10) and the stacks welded using our new, adaptive method (the full rectangles in Fig. 10).

The difference in this case is within ±1%, which is also the measurement uncertainty of the measuring system. We can explain this result with the size of the welds. Since the samples are small, their welds do not significantly influence the magnetic properties of the sample stacks.

However, we developed our method for welding the real stator stacks that are used in the electromotors of vehicles. Thus, we also tested the magnetic properties of a real stator stack welded with our new welding method. In contrast to the electricallamination samples, a real sample of a stator stack is much larger, as already described in Subsection 3.3.

All the tested real samples had fully acceptable mechanical properties. In the case of a real stator stack that means that the welds were strong enough to hold 112

the laminations together during the winding of the copper wire. The welds in all the tested real samples were also free of cracks and other failures.

The specific power losses and the relative permeability as a function of the total-pulse energy for the real stator sample are presented in Fig. 11.

Here, the full rectangles show the measurements obtained from the stacks that were welded with our new, adaptive, pulsed-laser welding method, while the full triangles correspond to the samples welded with a classic,continuous welding technique (already used in serial production). All the measurements were obtained at a frequency of 50 Hz and a flux density 1.5 T.

Fig. 11. a) the specific power losses (PS), and b) the relative permeability (µr) as a function of the total-pulse energy for a real stack welded with the classic, continuous-welding technique (triangles) and with our new, adaptive pulsed method (squares)

From the results in Fig. 11a it can be concluded that our new method results in around 10% lower specific power losses than the classic, continuous

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welding method and needs only 70 % of the totalpulse energy compared to the classic method.

Similarly, Fig. 11b reveals that our new method provides a 16% higher permeability of the stator stack than the classic technique. Here, we should also note that the classic, continuous welding technique is already used in serial production and therefore already optimized, while there is still space for further parameter optimization in our new, adaptive, pulsedwelding method.

The results in Fig. 11 confirm our hypothesis that the adaptive welding technique (i) results in significantly lower specific power losses than the already-established continuous welding technique, and (ii) it ensures acceptable mechanical properties with a significantly less total-pulse energy per weld.

4 CONCLUSIONS We have developed and presented an experimental system for the new technique of adaptive, pulsed-laser welding. The system is based on an on-line monitoring of the gap positions between the electrical laminations and thus enables the efficient welding of the laminated electrical cores for electromotors.

The results of visual, mechanical and magnetic testing on two different groups of samples, i.e., the small experimental stator samples and the real stator samples, revealed that (i) our system is capable of producing laser welds that are made with up to five times less total-pulse energy per a weld, while maintaining equal mechanical strength in comparison with the classic, pulsed-welding technique, and (ii) that adaptive, pulsed welds produce lower specific power losses and increase the relative permeability of the samples in comparison with conventional, continuous laser welding that is already used in industrial production.

Thus, our method brings advantages to the production as well as to the final product and can be easily implemented in serial manufacturing.

5 ACKNOWLEDGMENT This research has been partly supported and financed by the European Union, European Social Fund, 2009.

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Vegelj, D. – Zajec, B. – Gregorčič, P. – Možina, J.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 115-123 © 2014 Journal of Mechanical Engineering. All rights reserved.

DOI:10.5545/sv-jme.2013.1046

Original Scientific Paper

Received for review: 2013-02-11 Received revised form: 2013-10-18 Accepted for publication: 2013-11-13

Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis Čuk, E. – Gams, M. – Možek, M. – Strle, F. – Maraspin Čarman, V. – Tasič, J.F. Erik Čuk1,2,3,* – Matjaž Gams2 – Matej Možek3 – Franc Strle4 – Vera Maraspin Čarman4 – Jurij F. Tasič3 1LOTRIČ Metrology, Slovenia Jožef Stefan, Department of Intelligent Systems, Slovenia 3University of Ljubljana, Faculty of Electrical Engineering, Slovenia 4University Medical Centre Ljubljana, Department of Infectious Diseases, Slovenia 2Institute

Lyme borreliosis is the most common human tick-borne infectious disease in the northern hemisphere, occurring predominantly in temperate regions of North America, Europe and Asia. The disease’s most frequent manifestation is erythema migrans, a skin lesion that appears within days to weeks of a tick bite. Early recognition of the lesion is important since it enables proper management and thus prevention of later consequences of the disease which can hamper normal life. In this article, a novel visual system for recognition of erythema migrans is presented based on new multimedia interactive terminal technology available also on smartphones. For potential erythema migrans skin lesion edge detection, we compared three different methods: GrowCut, maximal similarity based region merging and random walker segmentation method. The results obtained with GrowCut method are better than those obtained with random walker method. The GrowCut method, improved with our new finger draw (FD1) marker yields comparable results to those obtained with maximal similarity based region merging method. Several classification algorithms including naive Bayes, support vector machine, AdaBoost, random forest, and neural network were compared and used for classification of skin lesions into ellipse, the most common shape of erythema migrans and erythema migrans class.

Keywords: Lyme borreliosis, erythema migrans, finger draw, segmentation, recognition, attributes

0 INTRODUCTION For machine-supported detection of segments of interest in medical images and understanding the content with learning algorithms [1] and [2], neural networks [3] and [4], naive Bayes, support vector machine (SVM) and others [5] are among key diagnostic tools of modern medicine. In several tasks, processing of visual content consists of image processing operations which include acquisition of image, pre-processing, segmentation procedure (performed with GrowCut (GW), random walker (RW) and other methods), attribute (feature) extraction and classification [2]. Improvement of existing analyses of medical images leads to better medical diagnosis [6]. This article represents a new effort to establish machine-supported analysis of medical images related to erythema migrans, an early manifestation of Lyme borreliosis.

Lyme borreliosis is a multisystem disease [7], caused by Borrelia burgdorferi sensu lato and transmitted by a tick bite. The early course of Lyme borreliosis is characterized by an expanding skin lesion named erythema migrans. The lesion typically appears about one week after a tick bite (range, 3 to 30 days; median 7 to 10 days) as a small redness at the site of the bite and expands over a period of days to weeks, often to an oval lesion with central clearing [8] and [9]. In Europe, about one third of patients with erythema migrans have a concomitant

viral-like illness (the “summer flu”) characterized by myalgias, arthralgias, headache, and fatigue. Fever is rarely present. Without treatment, erythema migrans skin lesion resolves itself within several weeks to months but the infection may progress and affect skin, nervous system and/or joints, and less frequently eyes and/or heart [7], [8] and [10]. Early recognition and proper antibiotic treatment can successfully prevent harmful effects of Lyme borreliosis and enable faster disappearance of erythema migrans.

Since visual recognition is part of the clinical diagnosis of erythema migrans, we decided to design a visual system that would serve as a complement in the diagnosis of erythema migrans either at the medical exam or at home through smartphones. However, execution of erythema migrans edge detection is a challenging task since the colours of erythema migrans typically vary from very pale to very intensive red, less frequently blue or brown shades or even a combination of two colours. Furthermore, the diameters of erythema migrans lesions in our database vary from 2 to 30 cm, less frequently more than 30 cm.

Most literature on visual detection and recognition of skin patterns relates to skin cancer, but there are no journal articles related to visual pattern recognition of erythema migrans. Therefore we analysed related domain articles for computer supported pattern recognition which is a two step procedure. In the first step, the object of interest is identified based

*Corr. Author’s Address: LOTRIČ Metrology d.o.o, Selca 163, 4227 Selca, Slovenia, erik.cuk@lotric.si

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on edge detection. As a result of rapid development in the image processing of skin lesions, a modified skin cancer segmentation of colour-texture regions in images and video (JSEG) [11] was proposed. The basic idea behind the algorithm is a division of the segmentation process into two independent stages: colour quantization and spatial distribution. Tang [12] propose a multi directional gradient vector flow algorithm. They modified the algorithm in a way that anisotropic diffusion (AD) filter is less sensitive to noise. The proposed AD filter uses adaptive threshold and a new way of calculating the gradient, which can effectively remove hair. Another very accurate multimodal segmentation technique suggested by Yuan et al. [13] is based on the region fusion and graph division of narrow energy band. The method is effective on complex characteristics of skin lesion changes such as topology, weak or false edges, and asymmetry. In segmentation of tumour, Schmid [14] proposed a colour based segmentation method Fuzzy c-means. Furthermore, Gomez et al. [15] developed an unsupervised algorithm called independent histogram pursuit, which can be easily combined with the majority of classification algorithms, while Zhou et al. [16] developed another segmentation method mean shift based on fuzzy c-means, which consumes less computing time with respect to other methods, while the detection of skin lesion edge remains precise and effective.

Edge detection is often obstructed by skin lines and hair. Xie et al. [17] similarly to Abbas et al. [18] removed hair using line detection in combination with exemplar-based inpainting.

After successful edge detection and extraction of skin lesion area, the second step is performed in order to recognize the lesion on the bases of attributes calculated from the first step. SVM [19], Neural network [20] and AdaBoost [21] classifiers were applied for recognition of a skin cancer lesion [19] to [21].

Another related visual detection and recognition system is classification of colour and texture features of human tongue that reflects the health status of a patient [22]. A segmentation method maximal similarity based region merging (MSRM) [22] and [23] contains a built-in initial flood segmentation method.

Results are reflected in the precise segmentation of the human tongue. Furthermore, Shi et al. [24] proposed a novel approach for tongue image segmentation called C2G2FSnake. They combine a geometrical snake model with a parameterized gradient vector flow (GVF) snake model. For the health status analysis of 116

a patient, a classification transductive SVM method is proposed by Zhang et al. [25].

Roullot et al. [26] proposed semiautomatic measurement which was used for detecting allergic rash. Semiautomatic measurements are useful for different kinds of allergic reactions that need to be visually detected and classified. Huttunen et al. [27] also proposed segmentation where user has to point the position of an allergic rash.

In this article, a supervised visual system for recognition of erythema migrans is presented. The article is divided into several sections. The first section describes the use and composition of the system, while the second section contains a description of image database, user marker inputs, segmentation and “finger draw 1” (FD1) user marker input, shape properties and attribute calculation and classification methods used in the experiment. This section also describes the methods for evaluation of segmentation and classification. In the third section experimental results and discussion for the segmentation and classification of skin lesions are presented. In the last, conclusion section, our findings are summarized.

1 SYSTEM FOR VISUAL RECOGNITION OF ERYTHEMA MIGRANS The primary targeted application of our system for visual recognition of erythema migrans is a mobilemarket system for smartphones with touchscreen display as a marker input.

1.1 User Input – Touchscreen Touchscreen interfaces are used in a wide range of technologies, from mobile phones to in-car systems [28]. A smartphone touchscreen is an electronic visual display that can recognize finger touch either with responding to a mechanical force applied to the material, infrared-based approaches or others [29]. In our experiments, touchscreen display was simulated with laptop touchpad that allows the user to draw a marker input with a finger.

1.2 Image Analysis The process of skin lesion image analysis [19] and [20] consists of two steps: edge detection and recognition of skin lesion.

Since edge detection affects the accuracy of subsequent step mentioned above, the process of edge detection is of vital importance [18]. Many features such as asymmetry, border irregularity and other

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shape properties are calculated from the edge of the skin lesion [19].

Most commonly, skin lesion edge detection is divided into three steps. The first step is called image scaling and is essential for execution of the following two steps. Image pre-processing is the second step in which data is processed in order to improve contrast [30] and remove noise. The third step in edge detection is segmentation.

Typically, segmentation methods need a user marker input that defines the position of seed points for extraction of skin lesion area. This is a step where skin lesion is extracted from the captured image: the image is divided into foreground that includes the skin lesion and the background that includes all other parts of the image such as skin without the lesion, different clothing and the surrounding environment. The background is then excluded from further analysis.

The most commonly used segmentation methods in medical imaging [31], and therefore the most suitable for segmentation of skin lesions are contour tracking methods, region growing method [26] and [32], GCmethod [33], RW method [34], active contours [35] and [36], level sets, geodesic active contours, Graph Cut method [37] and [38], watershed algorithms [39], scale multiplication algorithms [40] and [41], generic segmentation models and deformable segmentation models [42] and [43].

Similar to the edge detection, recognition of skin lesion is divided into steps. The first step in recognition of skin lesion is shape properties and attribute calculation where attributes are identified.

The second step is called classification of skin lesion.

Fig. 1 illustrates edge detection and recog- nition of erythema migrans. The structure of our system in Fig. 1 corresponds to a modern visual recognition system. The novelty lies in methods and input type.

The methods for segmentation and classification used in the experiment are described in the next section.

a)

Sca led a nd preprocessed ima ge

User ma rker input

Segmenta tion

b)

Sha pe properties a nd a ttribute ca lcula tion

Cla ssifica tion

Fig. 1. System for visual recognition of erythema migrans; a) edge detection, b) recognition

2 EXPERIMENTAL SET-UP AND TOOLS 2.1 Image Database The recognition was performed on a database of 143 images. An expert, a physician, experienced in the diagnosis of Lyme borreliosis, determined 91 positive cases of erythema migrans at different parts of human body, leaving out the intimate parts and head. The remaining 52 skin lesions represent negative close match cases of EM skin lesion. Images were taken at Department of Infectious Diseases of the University Medical Centre Ljubljana, Slovenia, using a commercial camera Canon EOS 600D with maximal output resolution 5184×3456. Standard flash and tripod were used to reduce vibration and prevent blurriness.

2.2 User Marker Inputs Four different user marker inputs were used in the experiment. In the user marker input (FD1), the user has to draw a curve marker around the skin lesion with a finger. In the case of “finger draw 2” (FD2), the inside curve marker is also needed. In the “point marker 6” (PM6) user marker input, the user clicks six evenly distributed points around skin lesion. In the case of “point marker 2” (PM2), the user has to click one inlier and one outlier marker point. These marker inputs were used as inputs to segmentation methods GC k > 6 (GC-FD1), GC k = 6 (GC-PM6), MSRM k > 6 (MSRM-FD2), RW k = 2 (RW-PM2) and GC k = 2 (GC-PM2). Variable k is the number of marker points.

2.3 Segmentation and FD1 User Marker Input Our approach is based on the GC method introduced

by Vezhnevets and Konouchine [44] that is also used for melanoma detection proposed by Ayoub et al. [33].

In order to successfully detect potential erythema migrans skin lesion, segmentation methods need several evenly distributed inlier and outlier points.

We improved the functioning of GC method with our FD1 marker input in a way that the user draws a curve around potential erythema migrans skin lesion.

The type of input was enabled with introduction of multimedia interactive terminals. From the outlier curve or the outlier points, we automatically calculate inlier points by uniformly decreasing outlier curve using Matlab function maketform [45] and affine transformation. Matrix for affine scaling [46] transformation is as follows:

Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis

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 vx  0 0 

0 vy 0

0  0  , (1) 1 

where experimentally obtained scaling factor values for our image database are vx = vy = 0.3.

During our experiments we compared the influence of segmentation methods: GC-FD1 with GC-PM6, MSRM-FD2 [22] based on initial watershed segmentation, Grady’s RW-PM2 [47] and GC-PM2.

In the third section besides segmentation results for all methods, we also present the results in combination with MSRM-FD2 and GC-FD1 segmentation methods related to erythema migrans computer- aided diagnosis.

2.4 Recognition 2.4.1 Shape Properties and Attribute Calculation Generally, second or third order splines [48] are used for shape description where the number of spline anchors has to be minimized. Due to the fact that the analysed skin lesions are of oval shape, we used basic shape properties. These shape properties of segmented potential erythema migrans skin lesion were calculated with Matlab regionprops [49] function. The properties were: minor axes b, major axes a, orientation o, eccentricity ε and circumference c properties. Attributes, calculated from the shape properties and used as an input for classification algorithms were: • eccentricity: Matlab regionprops function, • small and large axis ratio: b/a , (2) 2

• ellipse focus:

2

a b f =   −   , (3) 2 2

• ellipse angular ε: α = sin–1 (ε), (4) •

c c − , (5) rb ra

where c is circumference, ra = a / 2 and rb = b / 2, • orientation: Matlab regionprops function.

2.3.2 Classification We classified potential erythema migrans skin lesions with two types of experiments. In the first type the default class was ellipse, where the ground truth was determined by the expert. The second type includes 118

the determination of class erythema migrans skin lesion and the ground truth was also determined by the expert. Table 1 shows the number of true and negative cases for the two types of experiments.

Table 1. True and negative cases for ellipse and erythema migrans class Class Ellipse Erythema migrans

True 70 91

Negative 73 52

Attributes were calculated with equations in shape properties and attribute calculation section which were used as inputs to Matlab implementations of classification algorithms naive Bayes, SVM, AdaBoost, random forest and neural network.

Accuracy (Acc) [50] was calculated to estimate the performance of classifiers and is defined as:

Acc =

(TP + TN )

(TP + TN + FP + FN )′

, (6)

where numerator (true positive and true negative) are correctly classified skin lesions to class ellipse or erythema migrans and denominator (true positive, true negative, false positive and false negative) represents all skin lesions, i.e. 143.

2.5 Evaluation of Segmentation and Classification For comparison between segmentation of image lesions where the MSRM, RW and GC segmentation methods using different user marker inputs were used on one side, and manually labelled image lesions where these labels were used as ground truth defined by an expert on the other side, we used a known method for segmentation evaluation [23]. Afterwards, automatically segmented labels and ground truth were applied for computation of true positive rate (TPR) and false positive rate (FPR). The TPR is defined as:

TPR =

TP , (7) TP + FN

where the numerator is the number of correctly classified object pixels and the denominator the number of total object pixels in the ground truth.

Equation for FPR is defined as:

FPR =

FP , (8) FP + TN

where the numerator is the number of background pixels classified as object pixels and the denominator

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the number of background pixels in the ground truth.

Moreover, we calculated the mean value of true positive rate (TPRA) and the mean false positive rate (FPRA) for all 143 images. The higher the TPRA and the lower FPRA get, the better the method. We also calculated the corresponding standard deviation of TPRA (σTPRA) and FPRA (σFPRA).

Classification to class ellipse and erythema migrans was evaluated with K-fold cross-validation [50] with K = 10 for estimation of predictive model performance in practice. For every fold of both types of experiments, a paired student’s ‘t test’ [50] was performed between classifiers combined with GCFD1 and classifiers combined with MSRM-FD2. In addition, a standard deviation for the Acc results (σAcc) was calculated for the second type of the experiment.

3 EXPERIMENTAL RESULTS AND DISCUSSION Table 2 shows segmentation results for five different approaches with k marker points for all 143 skin lesions. We tested GC-PM2 and RW-PM2 in the way where a non-expert computer specialist clicked one inlier and outlier marker point placed on the same position for both methods. For GC-PM2, TPRA was 36.64% higher than for RW-PM2. GC-PM2 has a higher FPRA value but lower σFPRA. Our GCFD1 yielded comparable results to the state of the art MSRM-FD2. In case of GC-PM6 the goal was

to estimate how many marker points are needed to achieve results that are comparable to GC-FD1.

Table 2. Comparison of segmentation methods for 143 skin lesions and k marker points Method RW-PM2 GC-PM2 GC-PM6 GC-FD1 MSRM-FD2

k k=2 k=2 k=6 k>6 k>6

TPRA [%] 44.32 80.96 83.16 83.67 82.14

σTPRA [%] 43.34 13.64 11.35 10.63 14.82

FPRA [%] 29.49 41.49 1.38 0.69 4.78

σFPRA [%] 39.67 20.3 2.49 0.87 4.96

In addition to Table 2, Fig. 2 presents the performance of the first three best segmentation approaches for the three skin lesions. Manually drawn ground truth (first line on Fig. 2) was compared with segments computed by GC-PM6 (second line on Fig. 2) , GC-FD1 (third line on Fig. 2) and MSRM-FD2 (fourth line on Fig. 2). The marker points or lines for the three methods are presented in the first column.

Fig. 2 shows that our GC-FD1 approach satisfactorily detects the proper shape for all three segmented skin lesions, where results obtained with GC-FD1 in Table 5 additionally support this claim.

For better understanding of segmented lesions in Fig. 2, TPR in Table 3 and FPR in Table 4 are presented.

In the last part of this section, experiments for classification of skin lesions are presented. Tables 5

Fig. 2. Segmentation methods comparison for the three skin lesions Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis

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and 6 show Acc results for the ellipse and erythema migrans class.

Table 3. TPR segmentation evaluation for all three segmented skin lesions in Fig. 2 Method, TPR [%] GC-PM6 GC-FD1 MSRM-FD2

Image 1 77.79 90.4 88.94

Image 2 78.7 82.57 87.94

Image 3 82.85 83.53 71.84

Table 4. FPR segmentation evaluation for all three segmented skin lesions in Fig. 2 Method, FPR [%] GC-PM6 GC-FD1 MSRM-FD2

Image 1 1.01 1.68 3.27

Image 2 0.76 0.65 3.95

Image 3 1.31 1.05 0.68

The results in Table 5 show that best classification Acc results for ellipse recognition, which is the most common shape of erythema migrans, are the ones where classifiers were combined with GCFD1. We also performed paired student‘s ‘t test’ and confirmed that results for all classification algorithms in combination with GC-FD1 for the class ellipse are significantly better than the results obtained with the classification algorithms combined with MSRM-FD2.

Table 5. Classification to class ellipse for 143 skin lesions GC-FD1 Acc [%]

MSRM-FD2 Acc [%]

naive Bayes

80.42

60.84

SVM

76.22

58.04

AdaBoost

78.32

60.84

Random forest

76.22

55.94

Neural network

76.19

54.55

Method

Table 6 shows that GC-FD1 in combination with all classifiers achieves best Acc and its standard deviation (σAcc) results for class erythema migrans.

Calculation of paired student’s ‘t test’ for the Acc for every fold showed that results between GC-FD1 and MSRM-FD2 in combination with naive Bayes, SVM and AdaBoost are significantly better. However, the test did not prove significance for the random forest and neural network classifier.

For the automatic edge detection of a human tongue, the MSRM method needs a prior knowledge that the tongue body usually locates in the centre of the tongue image. In this case a circle marker is positioned in the centre of the tongue image. Such a scenario of marker positioning is not possible for potential erythema migrans edge detection, as 120

potential erythema migrans are of different sizes. In addition, potential erythema migrans have different shades of colours which make the task even more challenging and require precise placement of marker points. Although the results for the class erythema migrans are satisfactory, shape recognition does not give the final diagnosis of erythema migrans, since the diagnosis also demands answers from the patients.

Table 6. Classification to class erythema migrans for 143 skin lesions Method naive Bayes SVM AdaBoost Random forest Neural network

GC-FD1 GC-FD1 MSRM-FD2 MSRM-FD2 Acc [%] σAcc [%] Acc [%] σAcc [%] 74.13 9.15 60.14 11.34 75.52 7.37 59.44 13.02 76.22 9.18 61.54 9.23 73.43 11.16 62.42 14.45 69.23 6.11 65.73 18.77

Our image database was acquired with equipment with better technical capabilities than that available in smartphones since the quality of the lens in our Canon EOS 600D is better than the quality of the lens available in smartphones. However, it is worth noting that smartphone external lens can already substantially improve the quality of image. Regarding the fact that the development of smart devices, especially smartphones, is evolving quickly we expect that smartphone technology will reach a satisfactory image quality level in a few years. Currently, the best smartphone camera available on the market has a 41 megapixel resolution which is approximately 23 megapixel higher compared to Canon EOS 600D. 4 CONCLUSION The novelty of our approach stems from the use of pattern recognition methods in the field of infectious diseases including erythema migrans, development of new features that can be used for medical purposes, and from the mass introduction of multimedia interactive terminal, available also on smartphones.

This type of input enables simple finger drawing to improve visual detection. However, as experimentally shown, different inputs and different methods give different results for segmentation and classification.

From experimental results we can conclude that GCPM2 performed better than RW-PM2 segmentation method and that GC-FD1 was comparable to the MSRM-FD2 (see Table 2) segmentation method. For satisfactory segmentation results of potential erythema migrans six or more evenly distributed marker points around potential erythema migrans are needed. From a

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practical point of view it is easier and faster for a user to draw a curve around the potential erythema migrans than clicking six or more evenly distributed marker points using the multimedia interactive terminal technology. The results of our experiments indicate that this technical innovation improves the diagnostic results in a reasonable manner. The experiments showed that the GC-FD1 approach is robust to various shapes and colours of potential erythema migrans.

Moreover, our approach gives better classification Acc results for the ellipse class and better Acc and σAcc results for erythema migrans class (see Tables 5 and 6).

In near future we intend to improve the Acc of the existing visual system for recognition of erythema migrans with colour and Gabor based attributes.

However, for efficient diagnosis of erythema migrans, hybrid method will be needed. We plan to combine medical image analysis with patient textual data involving machine learning methods. The reason for combining image analysis with text description usage is based on current best practices in treatment of erythema migrans.

5 ACKNOWLEDGMENTS The authors would like to thank the rest of Department of Infectious Diseases of the University Medical Centre Ljubljana personnel for helping with their expertise on Lyme borreliosis diagnosis and helping create an image database for visual recognition of erythema migrans. Our work was partly financed

by European Union. The study was approved by the Medical Ethics Committee of the Ministry of Health of Republic of Slovenia, No 130/05/12, which assesses the compliance with the Helsinki declaration.

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DOI:10.5545/sv-jme.2013.1222

Original Scientific Paper

Received for review: 2013-05-17 Received revised form: 2013-09-20 Accepted for publication: 2013-12-10

Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models Jošt, D. – Škerlavaj, A. – Lipej, A. Dragica Jošt1,* – Aljaž Škerlavaj1 – Andrej Lipej1 1Turboinštitut,

Slovenia

A comparison between numerical simulations and measurements of a six-blade Kaplan turbine is presented in order to determine an appropriate numerical setup for accurate and reliable simulations of Kaplan turbines. Values of discharge, torque and losses obtained by different turbulence models are compared to each other and to the measurements. Steady state simulations with various turbulence models tend to predict large errors at full discharge rate, which are the result of underestimated torque on the shaft and overestimated flow energy losses in the draft tube. The results were slightly improved with the curvature correction (CC) and Kato-Launder (KL) limiter of turbulence production. Transient simulations were performed with shear-stress-transport (SST) turbulence model, the scale-adaptive-simulation (SAS) SST model, and with zonal large-eddy-simulation (ZLES). Details about turbulent structures in the draft tube are illustrated in order to explain the reasons for differences in flow energy losses obtained by different turbulence models. The effects of advection schemes and mesh refinement were tested. It was shown that all of the transient simulations considerably improved results at full discharge rate. The largest improvement was achieved with the SAS SST and the ZLES models in combination with the bounded central differential scheme. In addition, it was shown that the ZLES model produced accurate results at all operating points, with discrepancy lower than 1%.

Keywords: water turbine, Kaplan turbine, efficiency prediction, CFD, turbulence models, ZLES

0 INTRODUCTION For more than 20 years, computational fluid dynamics (CFD) has played an important role in the gradual improvement of the characteristics of axial water turbines. The relative effect of hydraulic shape modifications on the efficiency curve can be quite reliably predicted, yet the prediction of the absolute value of efficiency is still a challenge. The main problem is an inaccurate flow simulation in the draft tube. In this paper we present a case where extremely poor results of steady-state analysis were improved by transient simulations and advanced turbulence models.

A simulation of the Kaplan turbine is in general more time consuming than a simulation of the Francis turbine. The Kaplan turbine is double-regulated.

Therefore, the overall efficiency curve is an envelope of partial efficiency curves, which are determined at a fixed angle of runner blades. The specifics of the Kaplan turbine are hub and tip clearance. At large guide vane openings there is often a clearance between overhanging guide vanes and the bottom ring. For accurate simulations it is recommended to take all such details into account. However, this results in large computational grids.

Often engineers have to make a compromise in order to reduce computational time. An extreme case is to simulate only one periodic segment of runner blades and a draft tube. In [1] it is reported that such an analysis was not satisfactory, until the effect of the spiral casing with stay and guide vane cascades on conditions at the runner inlet, were taken into account.

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Another compromise is to perform steady-state simulations instead of transient ones. Flow in turbines is unsteady and usually a converged steady-state solution cannot be obtained [2]. Besides, two-equation Reynolds-averaged Navier Stokes (RANS) models, including the shear-stress-transport (SST) model that became a standard in turbomachinery, are not able to model all flow structures in the draft tube.

An important attempt to determine the applicability of state-of-the-art CFD simulations for a Kaplan draft tube was the Turbine 99 workshop held in Porjus, Sweden, in the years 1999, 2001 and 2005.

The axial and swirl velocity components at the inlet (with rms-values and one Reynolds’ stress component) and the pressure distribution around the outlet cross section were available before the workshop. For the third meeting, besides two inlet boundary conditions (the first one with a prescribed axi-symmetrical inlet velocity profile, and the second one with a measured phase-averaged resolved inlet velocity profile) also computational grids with 1 million and 2.5 million nodes with values for y+ equal to 1 and 50 were provided by the organiser. Numerical results obtained by different CFD codes were compared to the experimental data for pressure recovery factor, pressure distributions along the draft tube walls and a detailed velocity field in one downstream cross section. One of the conclusions was that much attention must be paid to grid quality and boundary conditions [3]. Too coarse a grid and too high values of y+ significantly reduced the accuracy of the pressure recovery factor. All tested RANS models predicted too weak a secondary flow [4]. Although the results

*Corr. Author’s Address: Turboinštitut, Rovšnikova 7, 1210 Ljubljana, Slovenia, dragica.jost@turboinstitut.si

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of SST were significantly better than the results of the k–ε model [5], they did not reach the same level of accuracy throughout the draft tube as the LES model.

The best results were obtained by Kurosawa and Nakamura [6] using the dynamic Smagorinsky LES model and axi-symmetrical inlet boundary profile.

Since 2005, several papers with the same draft tube using measured quantities from the Turbine 99 case, have been published [7] and [8]. In [7], the results obtained with the SST model, with time dependent angular resolved inlet velocity boundary condition on a mesh, with 6 million cells, were presented.

The conclusion was that by using time dependent angular resolved inlet boundary conditions and grid refinement, only limited improvement of numerical results can be achieved. A possible means of further improvement might be in turbulence modelling.

In 2011, a numerical research project about the modelling of axial turbines was carried out at Turboinštitut [9]. Its purpose was to determine the optimal setup of a numerical model, which would be capable of reproducing the measured overall efficiency curve. The study was focused on the effect of hub and tip clearance, on the mesh refinement effect, effect of two- or seven- equation RANS turbulence models, and type of simulation (steady-state or transient).

None of the results were satisfactory. Therefore, it was concluded that the only way to predict efficiency accurately at all operating regimes was to use advanced turbulence models, such as the scaleadaptive-simulation SST turbulence model (SAS SST), and zonal large eddy simulation (ZLES). The improvement of the results by these two turbulence models was presented in [2]. Such an analysis is too time-consuming to be used in the design process, but at least for the final geometry it is worth doing.

In this paper the results are presented for the same Kaplan turbine as in [2]. While the paper [2] was focused on comparison of predicted turbine efficiency to the measured values, in this paper also the values of discharge, torque and losses obtained by different turbulence models are compared to each other and to the measurements. The effects of different discretisation schemes for advection term and of grid density are also presented.

Numerical simulations were performed with the Ansys CFX solver [10].

1 TURBULENCE MODELS AND DISCRETISATION SCHEMES In this paper, several turbulence models were used.

Most of them are well known RANS turbulence models, such as the standard k–ε turbulence model,

the Wilcox k–ω model [11], the Baseline (BSL) k–ω model [12], the SST model [12] and [13] and the ε-based SSG RSM [14]. We have used two scaleresolving simulation (SRS) models, the SAS SST and the ZLES model.

The SAS SST turbulence model [15] is a socalled second generation URANS model, according to classification [16]. The model is essentially the SST turbulence model with an additional source term QSAS in the ω transport equation [17]:

 QSAS = max  f 

 L     , 0  . (1)  LvK  

The term QSAS can detect the unsteadiness of the solution through the comparison of the RANS length scale L to the von Karman length scale LvK. The result of the unsteadiness is an increased value of QSAS, which results in decreased turbulent viscosity.

Consequently, the SAS SST develops an LES-like solution in unsteady regions. At the same time, the model provides standard RANS capabilities in stable flow regions. If the time step size is too large the unsteady structures can’t be resolved and the model obtains an RANS or URANS solution [18].

The main idea of the ZLES model [17] is to resolve the flow inside a predefined zone with the LES model, and the rest of the domain with the RANS model. In CFX, the model source term in the k-equation forces the eddy viscosity to be equal to the LES subgrid-scale viscosity inside the user-specified zone. The synthetic turbulence at the RANS-LES boundary is based on harmonic flow generator [19] acting through a special source term in the momentum equation. In CFX, the ZLES model is available with all k–ω turbulence models (Wilcox, BSL, SST, BSL, explicit algebraic RSM, DES SST and SAS SST). In the presented simulations, the zone of the ZLES model was defined within the SAS SST simulation. The zone started just after the interface between the runner and the draft tube, and it is extended to the outlet of the computational domain.

Some turbulence models were used in combination with curvature correction (CC) [20] and with the Kato-Launder limiter (KL) [21] of turbulence production. The CC option captures the effects of streamline curvature and system rotation. When the CC option is selected, the production terms in k and ω transport equations are multiplied by the upwards and downwards limited curvature correction function [20].

The function is defined as:

Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 124-134

2r *  1 − cr3 tan −1 (cr2 r )  − cr1 , (2)  1 + r*  where the r* and r are functions of the strain rate tensor Sij and of the effective rotation rate tensor Wij, which includes the term of the system rotation Ωrot. The empirical constants cr1, cr2 and cr3 are set equal to 1, 2 and 1, respectively. The lower limit of the function is 0, which represents strong convex curvature, whereas the upper limit is 1.25, which represents strong concave curvature.

The KL limiter uses a different formulation to deal with the excessive production of turbulence energy at stagnation points, than it is the default in CFX. In CFX, the limiter is defined as a constant, multiplied by the density and the turbulence eddy dissipation. The KL limiter replaces one of the strain rates in the production term with the magnitude of vorticity [10] and [21].

In the simulations, we used either ‘high resolution’ or the bounded central difference scheme. The highresolution scheme (HRS) is a bounded second order upwind biased discretisation. The scheme calculates the bounded values based on the procedure of the Barth and Jespersen’s scheme [22] and [23]. The bounded central difference scheme (BCDS) is based on the normalised variable diagram and blends from the central difference scheme (CDS) to the first-order upwind scheme when the convection boundedness criterion [24] is violated.

For convenience, short labels are used for turbulence models and discretisation schemes: SST for the Shear Stress Transport model, SAS for the SAS SST model, ZLES for the SAS SST ZLES model. So, for example, ZLES BCDS means the SAS SST ZLES turbulence model with BCDS used for the advection term.

f rot = (1 + cr1 )

2 NUMERICAL PREDICTION OF FLOW AND ENERGETIC CHARACTERISTICS In this paper the results of a detailed numerical analysis of flow in a 6-blade Kaplan turbine which operates at middle head (ψ = 0.44) are presented.

The model of the turbine was tested on a test rig in accordance with international standard IEC 60193 [25], so we were able to compare the numerical results with the measured ones. Numerical simulations were done for three angles of runner blades at constant head.

The turbine consists of semi-spiral casing with two vertical piers, 11 stay vanes and a nose, 28 guide vanes, a 6-blade runner and an elbow draft tube with 126

two vertical piers. Tip clearance was modelled while hub clearance was neglected. The grid in the spiral casing with stay vanes was unstructured, while the grids in the other turbine parts were structured. All the grids satisfied the recommendations for orthogonality and the aspect ratio of elements. Near the walls the grids were refined to get recommended values of y+.

For the draft tube and the draft tube prolongation, basic and refined grids were used. For the other turbine parts, our previous studies [9] showed that with proper values of y+, the number of nodes used in this case (see Table 1) is sufficient, and that by additional grid refinement only a negligible improvement of results can be obtained. The computational grid for the complete computational domain can be seen in Fig. 1.

Fig. 1. Computational domain and basic grid Table 1. Number of nodes for all turbine parts (in basic grid – BG and in fine grid – FG) Turbine part Semi spiral casing with stay vanes Guide vane cascade Runner Draft tube (BG) Draft tube prolongation (BG) Draft tube (FG) Draft tube prolongation (FG) Total (BG) Total (FG)

Number of nodes 1,480,999 2,755,496 1,858,374 1,786,432 398,056 6,169,935 1,681,992 8,279,357 13,946,796

The numerical analysis was done in three stages.

Firstly, a steady-state analysis at the local best efficiency points for three angles of runner blades was performed with several turbulence models. Secondly, a transient analysis was done with three turbulence models using two discretisation schemes (SST HRS, SAS HRS, SAS BCDS and ZLES BCDS) at only one operating point. Finally, a transient analysis was performed at several operating points for three angles of runner blades, with the ZLES turbulence model and BCDS.

Jošt, D. – Škerlavaj, A. – Lipej, A.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 124-134

2.1 Steady-State Analysis with Different Turbulence Models Steady-state analysis was performed on the basic grid with k–ε, k–ω, BSL, SST and SSG-RSM turbulence models. For two-equation RANS models, simulations were repeated with CC and KL limiter of production term. For the k–ε and SSG RSM models, scalable wall functions were used. For k–ω based turbulence models, automatic near-wall treatment was used. The automatic treatment allows a gradual switch between wall functions and the low-Reynolds number method.

For discretisation of the advection term, the HRS implemented in ANSYS-CFX was used.

In the case of the SST CC KL and the SSG RSM turbulence models, three partial efficiency curves were simulated (see Section 2.3). In the other cases,

only analysis at the local best efficiency points for three angles of runner blades (see Table 2) was performed, with guide vane opening being determined from experimental data.

Table 2. Operating points for steady state analysis by different turbulence models Operating point OP1 OP2 OP3

β [°] 12 20 28

φ/φBEP 0.64 0.95 1.31

ψ/ψBEP 0.86 0.86 0.86

The simulations were done at constant head and rotational speed. From numerical results the values of discharge, torque on the shaft, flow energy losses in all turbine parts and turbine efficiency, were obtained.

Fig. 2. Comparison between steady-state results obtained with different turbulence models and the measured values Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models

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The results at local best efficiency points for each blade angle are presented in Fig. 2. The values of discharge, torque on the shaft and efficiency were divided by measured values at the same operating points. That means that all measured values in relative form are equal to 1 and are therefore not presented in the diagrams. The values of flow energy losses are presented in percentage of head.

For all operating points and all turbulence models, the calculated values of discharge were larger than the measured ones (relative values are larger than 1). The discrepancy was the largest when turbulence models k–ε, k–ε CC KL and SSG RSM were used.

Consequently, these three models gave the largest values of torque on the shaft. At OP1, calculated values of torque were too large when k–ε, k–ε CC KL and SSG RSM models were used, while for the other models the values were very close to the measured ones.

At OP2, turbulence models k–ε, k–ε CC KL and SSG RSM gave larger values of torque while all the other models gave smaller values than the measurements. At OP3 all models obtained too small values of torque.

For flow energy losses before the runner, in the runner and in the draft tube no measured values were available, so only the values obtained by different turbulence models can be compared with each other. For all turbulence models the values of losses before the runner, in the runner and in the draft tube decreased by using the KL limiter of production term and the CC. The KL reduces flow energy losses at stagnation points, therefore its effect was significant in stay and guide vane cascades, and to a smaller degree also in the runner. Curvature correction acted mostly in the draft tube, where it reduced flow energy losses, especially for large discharge. When SSG RSM was used, flow energy losses before the runner and in the runner were smaller than those obtained by two equation models. The difference in losses before the runner obtained by the SST CC KL and SSG RSM models was between 0.3 and 0.4% of head at all three operating points. The differences in losses in the runner obtained by SST CC KL and SSG RSM were at OP1, OP2 and OP3; equal to 0.5, 0.35 and 0.31% of head, respectively. The losses in the draft tube, predicted by the SSG RSM model, were at OP1 slightly larger, at OP2 approximately the same and at OP3 significantly larger than those obtained by two-equation models with KL and CC, the difference between SST CC KL and the SSG RSM at OP3 was about 0.95% of head.

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Calculated efficiency values at all three operating points and for all turbulence models were smaller than the measured ones (relative values are smaller than 1).

For all two-equation models the values of efficiency increased by using KL and CC, due to smaller flow losses. At OP1 and OP2 all turbulence models predicted the efficiency values rather well. At OP1 the differences between measured efficiency and those obtained by SSG RSM and by SST CC KL were 0.38 and 1%, respectively. The same differences at OP2 were 0.8 and 1.43%. At OP3, the discrepancy between calculated and measured efficiency values was very large, for some of the two-equation models without CC and KL even more than 5%. The agreement between measured and numerical results was the best when the SST CC KL or SSG RSM models were used, but the discrepancies were still 4 and 4.4%, respectively.

2.2 Transient Flow Simulation with Different Turbulence Models at One Operating Point We tried to improve the results by transient analysis.

Three turbulence models were used in this respect: SST, SAS and ZLES. In the case of the SST turbulence model, the HRS was used for the advection term.

The simulation by the SAS model was done firstly with BCDS and then also with a HRS, in order to see the influence of a discretisation scheme. In the case of the ZLES model, the BCDS was used. For time discretisation, a second order backward Euler scheme was used.

In order to reduce computational time, the domain for this analysis was without the spiral casing and stay vane cascade. For the draft tube and draft tube prolongation, refined grids were used (see Table 1).

For an appropriate comparison of results the steadystate simulation with the SST model was repeated on the fine grid. In order to see the effect of grid density on results, a simulation with ZLES BCDS was performed on both grids, the basic and the fine one.

The inlet boundary and initial conditions were prescribed from the steady-state solution of the SST model. The velocity components were prescribed at the inlet boundary (at guide vane inlet), which means that the value of discharge was prescribed while the value of head was a result of the simulation. Time step size corresponded to 0.5 degrees of runner revolution. The average value of Courant number was, in all turbine parts, less than 0.3, in the draft tube even less than 0.02. Total simulation time corresponded to 30 runner revolutions. Transient analysis is very time consuming, therefore it was done only at one operating point for the blade angle of 28 degrees. This

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operating point is not the local best efficiency point.

Its guide vane opening and discharge are smaller. The numerical analysis had started before the measured results were available, and the exact positions of the local best efficiency points had not been known yet.

In the case of transient simulations, a certain amount of time is required for the oscillating values of head, torque and efficiency to stabilise around average values. A detailed analysis of the results presented in [2] had showed that turbulent structures in the flow were after 10 runner revolutions not entirely developed, therefore the simulation time was extended to 30 runner revolutions. The results of the last ten runner revolutions were used to obtain averaged values of torque, head and efficiency.

In Fig. 3 a comparison of flow in the draft tube obtained by four simulations (steady-state SST, transient SST, SAS and ZLES) is presented. In the case of steady-state analysis by SST model there is a large swirl at the end of the cone, which with transient simulations nearly disappeared. Streamlines obtained with the SAS and the ZLES models are more curled than those obtained with SST. For the SAS model, the difference in flow due to the discretisation scheme was hardly visible, therefore only the results of SAS BCDS are illustrated. For ZLES, a coarser grid in the draft tube had a very small influence on streamlines, therefore this picture is also omitted. Turbulent structure in the flow can be better seen by the isosurface of velocity invariant Q (see Fig. 4), coloured by viscosity ratio (ratio between eddy and dynamic viscosity). With the steady-state SST and also by the transient SST model, only large structures in the flow were obtained, but with SAS and especially ZLES, also small turbulent structures in the flow were well resolved. Besides, it can be seen that values of viscosity ratio are large in the case of the SST model, and much smaller when the SAS and especially the ZLES models were used (see Table 3). While the influence of discretisation scheme in the case of SAS can hardly be seen, the grid density in the case of ZLES did have an effect on the size of turbulent structures and also on the value of viscosity ratio.

Small turbulent structures in the flow can be obtained only on a fine grid where also the values of viscosity ratio are smaller.

In Fig. 5, the results of steady-state analysis (SST HRS) and of transient simulations (SST HRS, SAS HRS, SAS BCDS, ZLES BCDS) are presented. The values of head, torque and efficiency were divided by the measured values. The flow energy losses were divided by the measured value of head. The value of

discharge was the input data and it was equal for all simulations.

Table 3. The values of viscosity ratio in the draft tube Numerical modeling and grid density Steady-state SST HRS, FG Transient SST HRS, FG SAS HRS, FG SAS BCDS, FG ZLES CDS, FG ZLES CDS, BG

Eddy viscosity / Dynamic viscosity maximal averaged 10152 2527 6005 1439 1146 169 1311 180 850 81 1020 95

The spiral casing with stay vanes was not included in the computational domain, therefore the flow energy losses before the runner were calculated as a sum of the losses in a guide vane cascade obtained by transient simulation, and the losses in the spiral casing obtained by the previous steady-state analysis. It can be seen that the losses before the runner are nearly the same in all cases.

In the runner, the highest losses were obtained with steady-state, and transient analysis with the SST HRS (3.81 and 3.93%, respectively) and the smallest with SAS BCDS (3.41%) and ZLES BCDS (3.51%).

Losses in the runner obtained by SAS HRS (3.74%) are closer to the losses of steady-state and transient SST HRS than to SAS BCDS. Flow energy losses in the draft tube differ significantly due to steady-state or transient analysis and due to the choice of a turbulence model and discretisation scheme. In the case of steady-state analysis by SST HRS, the losses in the draft tube exceed 6.6% of turbine head, while in case of transient analysis with the same model they reduce to 3.47%.

The losses obtained with SAS HRS, SAS BCDS and ZLES BCDS are 2.93, 3.0 and 2.94%, respectively.

Steady-state simulation with SST HRS underestimated the value of torque on the shaft by 5.07%. With transient simulations the prediction of torque improved significantly. Transient SST HRS and SAS HRS underestimated values of torque by 0.78 and 0.71%, respectively. SAS BCDS and ZLES BCDS overrated torque by 0.039 and 0.036%, respectively. It seems that for torque prediction a choice of differential scheme for advection term was more important than the choice of turbulence model.

When HRS was used, SST and SAS predicted nearly the same values of torque. Similarly, when BCDS was used, SAS and ZLES predicted nearly the same value.

On the other hand, the difference between values obtained with SAS HRS and SAS BCDS is significant in spite of the same turbulence model. It is likely

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Fig. 3. Streamlines and velocity contours in the draft tube; a) steady-state SST, HRS, FG, b) transient SST, HRS, FG, c) SAS, BCDS, FG, and d) ZLES, BCDS, FG

Fig. 4. Isosurfaces of velocity invariant Q = 0; a) steady-state SST, HRS, FG, b) transient SST, HRS, FG, c) SAS, HRS, FG, d) SAS, BCDS, FG, e) ZLES, BCDS, FG, and f) ZLES, BCDS, BG

that in the runner the QSAS term was less important because close to the runner surface the SAS model acted as the SST model. On the contrary, the choice 130

of discretisation scheme had a direct effect on the predictions of flow close to the runner surface.

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a)

b)

c)

d) 0 = measurement; 1 = steady-state SST, HRS, FG; 2 = transient SST, HRS, FG; 3 = SAS, HRS, FG; 4 = SAS, BCDS, FG; 5 = ZLES, BCDS, FG; 6 = ZLES, BCDS, BG Fig. 5. Comparison of results obtained with different turbulence models to the measurements of a) head, b) flow energy losses, c) torque, d) efficiency

The values of calculated efficiency differ mostly due to different values of flow energy losses in the draft tube and different values of torque. The efficiency value calculated from steady-state solution with SST HRS was 4.42% smaller than the measured one. With transient analysis the results improved significantly. With SST HRS and SAS HRS the efficiency values were smaller than the measured ones by 1.01 and 0.24%. The agreement between measured and numerical values is excellent for SAS BCDS and ZLES BCDS, where the discrepancy is 0.09 and 0.05%, respectively. Based on the results it can be concluded that the SAS and ZLES models with BCDS are very suitable for flow simulation at operating points with large discharge. Results obtained by SAS HRS are less accurate mostly due to underestimated torque.

Comparing the results of ZLES BCDS on fine and basic grids it was seen that the effect of grid density on the calculated torque, head and efficiency was negligible. Flow energy losses in the draft tube were only higher by 0.18% of head, than on the fine grid. Therefore, for transient simulations for different

operating regimes (Section 2.3) the basic grid was used.

2.3 Transient Flow Simulation with the Zonal LES Model for Different Operating Regimes The purpose of this study is to find a turbulence model that would be capable of predicting efficiency accurately for all operating regimes (OP1, OP2 and OP3). In the previous section, the ZLES model (at operation regime close to OP3) showed such potential.

In order to see whether this model is suitable for all operating regimes it should be thoroughly tested at several operating points for three angles of runner blades.

In this section, simulations with ZLES were performed for the whole turbine. To reduce computational time, basic grids for the draft tube and draft tube prolongation were used. In all transient simulations CC and KL are included. The input data consisted of geometry, head and rotational speed. The results were the values of discharge, torque on the shaft, flow energy losses and efficiency.

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Fig. 6. Efficiency diagram – measurements and CFD Table 4. Calculated flow energy losses in the draft tube. ΔH/H×100 [%] Simulation type Steady-state SST CC KL Steady-state SSG RSM SAS SST CC KL ZLES

OP1 1.88 2.22 1.12

OP2 2.29 2.43 1.56

OP3 6.32 7.26 3.33

The efficiency prediction is presented in Fig. 6.

All values were divided by the measured efficiency at OP2. The diagram shows a clear distinction between the results of steady-state analysis and the results of the transient ones obtained with the ZLES model.

With ZLES, the efficiency prediction was improved at all operating points, but the improvement was most significant for runner blade angle 28 degrees, where the decrease of calculated flow energy losses in the draft tube was the largest (see Table 4). At the local best efficiency point for runner blade angle 12 degrees (OP1), the calculated and measured values were practically the same. At OP2 (blade angle 20 degrees), the efficiency value obtained with ZLES was about 0.5% smaller than the measured one. In the section 2.2 for blade angle 28 degrees, the discrepancy in efficiency was only 0.05%. That operating point corresponded to the third point from the left on the ZLES curve for 28 degrees In Fig. 6 the efficiency values at the first three points on the ZLES curve for blade angle 28 degrees agreed with the measured results well. At higher discharge, the discrepancy increased. Peak to peak difference in efficiency values was 0.44% but numerically obtained position of the local best efficiency point was shifted to the left. The tendency that numerical prediction is less accurate at 132

operating points with large discharge still remains, but the improvement with the ZLES model was significant.

3 COMPUTATIONAL EFFORT Transient numerical simulations are very time consuming. To get reliable results the grids have to be refined enough and time step must not be too large to get proper values of Courant number. Besides, it takes a long time before the values of efficiency stabilise.

Usually more than 20 runner revolutions are needed to get stable values.

The simulations were run on a supercomputer cluster with 512 Quad-Core Intel Xeon processors L5335. For the simulation on the basic grid (8.3 million nodes) with the ZLES model, 8 quad-core processors (64 cores) were used and the computational time was about 28 hours for one runner revolution.

Too long CPU time is the main disadvantage of transient simulations and the reason for their limited use in the design process. It can be expected that with future development of hardware and software the problem will be overcome.

4 CONCLUSIONS •

Steady-state analysis failed entirely to predict flow in the Kaplan turbine. Even for small and optimal runner blade angles, where efficiency was quite accurately predicted, a detailed analysis of results showed that the prediction of torque and discharge values (head as input data) was

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•

•

•

•

•

not accurate enough. At full rate the efficiency prediction was extremely poor. The discrepancy was more than 4%, mostly due to underestimated torque on the shaft and overestimated flow energy losses in the draft tube. The results were only slightly improved by using the KL and CC. Transient simulations with SST HRS, SAS HRS, SAS BCDS and ZLES BCDS were performed at one operating point for a maximal runner blade angle. The results were significantly improved.

The largest improvement was achieved with SAS BCDS and ZLES BCDS, where not only the efficiency values but also the values of torque and head (discharge as input data) were almost equal to the measured values.

Comparing transient results of SST HRS, SAS HRS and SAS BCDS, it can be concluded that the improvement due to the use of BCDS instead of HRS, was even larger than the improvement due to the use of SAS instead of SST. With BCDS, the agreement with measurements was improved mostly because of smaller losses in the runner and better prediction of torque on the shaft.

The effect of grid refinement in the draft tube, on efficiency prediction with ZLES BCDS, was negligible in spite of clearly seen differences in vortex structures. A grid with about 2 million nodes was in this case fine enough for reliable results.

It is important that transient simulations are not stopped too soon, otherwise swirls from the steady-state simulations used as initial conditions still remain, and also vortex structures are not entirely developed.

Simulations with ZLES BCDS were performed at several operating points for three runner blade angles. The discrepancy in efficiency values was smaller than 1% at all operating points. It can be concluded that ZLES model with BCDS is suitable for all operating regimes.

5 ACKNOWLEDGEMENTS

The authors are grateful to their colleagues and especially to the head of the Turbine R&D Department, Dr. Vesko Djelić, for geometrical data and measured results. The research was partially funded by the Slovenian Research Agency ARRS – Contracts No. P2-0196 and 1000-09-160263.

6 NOMENCLATURE cr1, cr2, cr3 [-] [-] frot [m·s–2] g [m] H [m] HExp [m] ΔH k L LvK M MExp Ptot Q QSAS r r , r*

Empirical constants for frot Curvature correction function Gravitation Head Head, experimental value Flow energy losses ΔH = ΔPtot/(ρg) [m2·s–2] Turbulence kinetic energy per unit mass [m] RANS length scale [m] Von Karman length scale [Nm] Torque on the shaft [Nm] Torque on the shaft, experimental value [Nm–2] Total pressure [m3 s–1] Discharge [kg·m–3·s–2] Source term in ω-equation for the SAS SST model [m] Runner radius [-] Non-dimensional arguments in frot

β ε η ηExp ηrel

[degrees] [m2 s–3] [-] [-] [-]

φ

[-]

ρ ψ

[kg m–3] [-]

ω ω

[s–1] [s–1]

BCDS BEP BG CC FG HRS KL

Bounded central differential scheme Best efficiency point Basic grid Curvature correction Fine grid High resolution scheme Kato-Launder limiter of production term in equation for turbulent kinetic energy Scale adaptive simulation Shear stress transport Zonal large eddy simulation

SAS SST ZLES

Runner blade angle Turbulence dissipation rate Efficiency Efficiency, experimental value Efficiency divided by measured efficiency at OP2 Discharge coefficient φ = Q / (πωr3) Density Energy coefficient ψ = 2gH/(ωr)2 Runner speed Turbulence frequency

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[5] Marjavaara, D., Kamakoti, R., Lundstöm, T.S., Siddharth, T., Wright, J., Schy, W. (2005). Steady and unsteady CFD simulations of the Turbine-99 draft tube CFX-5 and STREAM. Cervantes, M.J., Engström, T.F., Gustavsson, L.H. (eds.), Turbine-99 III. Luleå University of Technology, Luleå, p. 83-100.

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[8] Nilsson, H., Cervantes, M.J. (2012). Effects of inlet boundary conditions on the computed flow in the Turbine-99 draft tube, using OpenFOAM and CFX. IOP Conference Series: Materials Science and Engineering, vol. 15, no. 3, p. 032002, DOI:10.1088/17551315/15/3/032002.

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Ansys, Canonsburg.

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[18] Menter, F., Egorov, Y. (2009). Formulation of the Scale-Adaptive Simulation (SAS) Model during the DESIDER Project. Haase, W., Braza, M., Revell, A. (eds.), DESider - A European Effort on Hybrid RANS-LES Modelling. Springer, Berlin, p. 51-62, DOI:10.1007/978-3-540-92773-0.

[19] Adamian, D., Travin, A. (2011). An efficient generator of synthetic turbulence at RANS–LES interface in embedded LES of wall-bounded and free shear flows.

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DOI:10.5545/sv-jme.2013.1368

Original Scientific Paper

Received for review: 2013-08-13 Received revised form: 2013-11-15 Accepted for publication: 2013-12-11

Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory Cai, F. – Wang, X. – Liu, J. – Zhao, F. Fuhai Cai1,* – Xin Wang1 – Jiquan Liu2 – Fuling Zhao1

1Dalian

University of Technology, Department of Mechanical Engineering, China 2Jiangsu Bada Heavy Industry Machinery, China

There are large numbers of welds on cranes’ lattice booms. Due to notably complicated force conditions, the fatigue life analysis of the lattice boom is difficult when designing the limit life. The fatigue cycle of the utilization level of a lattice boom crane is in the overlap zone of low-cycle fatigue and high-cycle fatigue. The poor accuracy of measurement of the fatigue life of lattice booms indicates that the calculation dispersion of the linear Palmgren-Miner (PM) rule was high. To obtain real crane stress spectra inexpensively and conveniently, a new stress spectra acquisition method based on the ‘measured & simulated & compared & statistics’ integrated strategy of crane K-type welded joints is proposed. The errors of the maximum stress amplitudes between the measured stress spectra and the simulated stress spectra were less than 10%. The fatigue test results also indicated that errors of the test fatigue life were less than 10% under both the simulated and the measured stress spectra. A new, simplified Huffman non-linear cumulative damage theory is proposed to calculate the fatigue life of crane K-type welded joints based on the notch stress and strain approaches. The calculation results indicated that the accuracy of the non-linear damage accumulation was higher than that of the PM rule, although the calculation result based on the non-linear damage accumulation method was slightly un-conservative when the initial damage was not considered in the calculation. By setting different initial damage conditions, various results were analysed, which revealed that the calculation errors of fatigue life based on the non-linear theory were less than 10% when the initial damage levels were set from 0.02 to 0.04. Such results are appropriate for engineering applications. When the fatigue life calculation needs to be more conservative, the initial damage levels may be set from 0.04 to 0.07; the resulting calculation errors could be less than 25%. As the Huffman non-linear cumulative damage theory requires data from only a few material properties, such as the cyclic stress-strain curve and the constant amplitude strain-life curve, it could therefore be more suitable for engineering applications with higher calculation accuracy and fewer costs.

Keywords: crane, lattice boom, K-type welded joints, fatigue life analysis, non-linear cumulative damage theory

0 INTRODUCTION Engineering fatigue fracture failure is one of the most common phenomena of mechanical and structural failure, statistically representing approximately 50 to 90% of total mechanical damage [1]. Lattice structures have been widely used in crane booms, steel plants, parking garages, offshore platforms and other fields [2] to [4]. In a crane, the lattice boom has better mechanical properties compared to a box boom of the same weight. However, the welding process of a lattice structure is complex. The defects on the structures located at high levels are occasionally difficult to detect, especially in the crane boom. If the boom welds are not examined in a timely manner, there will be a massive security risk. Therefore, lattice boom fatigue life assessment of a crane has considerable significance in ensuring safety during use.

Fatigue life assessment methods applied in the crane field applications began nearly two decades ago.

The primary objects of the study were bridge cranes, gantry cranes, harbour cranes and other cranes with high levels of utilization [5] to [7]. The mainly welded structures of bridge cranes and gantry cranes are composed of flat butt weld or fillet weld structures, for

which the welding process and the stress conditions are simpler than those of the lattice boom.

Most scholars focused on the overall life estimation of the entire metal structure and the life assessment of the flat welded areas of the structure.

There are many difficulties in performing fatigue life analysis of the lattice boom compared to other structures: (1) Because a lattice boom is primarily welded by a large number of K-type and T-type pipe welded joints, a high probability of cracks or failure occurs in the poor working conditions in the latter part of the working time than for other types of cranes.

(2) The fatigue cycle of the utilization level of a lattice boom crane is in the overlap zone of lowcycle fatigue and high-cycle fatigue [1]. Strength analysis parameters, such as stress or strain, are difficult to choose because the structure stress state is complex.

(3) Because the structure of a lattice boom is more complex than that of a box boom, the fatigue experiments must design a special tooling equipment installation for the lattice boom connector. Therefore, few studies have been performed involving appropriate tooling design for the experiments.

*Corr. Author’s Address: Dalian University of Technology, Department of Mechanical Engineering, Dalian, China, cfhdlut@163.com

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(4) It is costly and time-consuming to obtain effective stress-time spectra. The traditional method is to place strain gauges on the structure for an extended time to obtain information on structure deformation and strain. However, as the crane boom is quite large, the measurement points cannot cover the structures as much as desired.

(5) The Palmgren-Miner (PM) rule is a linear accumulation damage approach that is widely used for predicting the part’s lifespan under variable amplitude loading [8]. However, there is a considerable dispersion rate in the damage calculation, which results in a lower life assessment accuracy. In recent years, models have been developed based on a variety of techniques that join theories of fracture mechanics and empirical observations [9] to [12]. Other methods take into account the residual stresses caused by the plasticity of the material at the crack tip and the crack tip closing phenomena. Although these modelling methods are more accurate than the PM rule, they require substantially more experimental data to fit the necessary parameters.

To solve such difficulties, a new method of obtaining the stress-time spectra is proposed in this paper, based on the simulation software with the advantages of being economical and convenient. A simplified non-linear cumulative damage theory based on the strain parameter is introduced; it is suitable for engineering applications within elastic-plastic deformation structures, such as a crane lattice boom, because it requires only a small amount of material properties data, such as the cyclic stress-strain curve and the constant amplitude strain-life curve.

1 STRUCTURE LOAD CHARACTERISTICS There are many K-type welded joints on a lattice boom. The flowchart of the fatigue life assessment is

shown in Fig. 1. Each step of the flowchart will be described in this paper.

Fig. 1. Flowchart of fatigue life assessment

1.1 Load Characteristics Analysis A notable feature of a jib crane is that it has an overhang with a rotating boom as the main working component. A jib crane can work in a round or oblong space. The load characteristics of a lattice boom crane are shown as follows: (1) The load condition is complicated. The boom is mainly subjected to a plane axial load force and a bending load force in the luffing plane, as shown in Fig. 2a. The boom is subjected to a lateral bending plane load force in the turning plane, as shown in Fig. 2b. The boom can withstand a compressive load, and the K-type weld joints are subjected to tension and compression composite fatigue loads, as is shown in Fig. 2c.

(2) The fatigue design parameter does not clearly correspond to the crane’s fatigue cycles. The utilization level of all cranes can be divided into

a) b) c) F1-working load; F2-cable tension T-lateral load N1, N2 in plane axial load; M out of plane bending Fig. 2. Loading diagrams a) in the luffing plane; b) in the turning plane; and c) of K-type welded joints

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10 classes, defined as U0 to U9, which represent the frequency of the use of the crane [13]. The lattice boom crane utilization level is generally U4 to U6. U4 represents the total number of working cycles 1.25×105＜N≤2.5×105. U6 represents the total number of working cycles 2.5×105＜N≤1.0×106. These levels indicate that the crane’s total working cycles is in the range of 1.25×105~1×106. When the number of crane working cycles exceeds the range, the crane may be in the state of fatigue failure, which is extremely dangerous. Regarding fatigue research, we believe that a low-cycle fatigue failure of cycles is N＜104~105 cycles, which the strain parameter often chosen as the design parameter. The high-cycle fatigue failure of cycles is N＞104~105, which indicates the stress parameter is often chosen as the design parameter [1]. However, the utilization level of the lattice boom crane with a range of 1.25×105 to 1×106 cycles is in the overlap zone of lowcycle fatigue and high-cycle fatigue. Therefore, a more suitable parameter for cranes requires more considerations.

(3) The load history is random with a large amount of scatter. The load state level of all cranes can be divided into four classes, defined as Q1 to Q4, which indicate whether the load is heavy or light in the daily operation of the crane [13].

Q1 indicates that the crane rarely lifts the rated load but always lifts decidedly lighter loads. Q4 indicates the crane currently lifts the rated load. In transportation construction, the small and middle tonnage of lattice boom crane is usually used, e.g.

25 to 100 ton cranes. For this kind of crane, the load state is generally Q2 to Q3, i.e. occasionally lifting rated loads. Part of the structure may suffer plastic deformation because of the high stress concentration or poor weld quality under exceptional poor load conditions. As a result, cracks initiate in the boom weaknesses, followed by crack propagation. In large buildings and engineering lifting, the large tonnage of the lattice boom crane is usually used. For this kind of crane, the load state is generally Q3 to Q4, i.e. lifting rated load frequently. Furthermore, cracks may initiate in the high stress concentration.

As the crane experiences a variety of working conditions, e.g. different boom lengths, different working ranges and different weights of the work according to various work demand, the loads will be changed randomly.

Therefore, lattice boom and welded joints have complex stress characteristics. K-type pipe joints of the booms are welded together with circular hollow pipes. The chords and braces are intersected.

Damage will occur under fatigue load, especially in the seam weld, which is subjected to high stress and variable amplitude multi-axial loading. Some plastic deformation will occur in the weak areas because of the stress concentration and poor weld quality. The single stress parameter can no longer reflect the actual force conditions. Therefore, the strain parameter can be chosen as a reasonable design parameter for the fatigue life calculation [14] and [15].

1.2 Determination of the Critical Points of the Boom In this paper, a 25 t and 18 m long lattice boom crane, as shown in Fig. 3, was investigated in depth. The crane has been in service for eight years. Two typical operating conditions were selected for analysing the crane, as described in Table 1.

Fig. 3. A 25 t lattice boom crane

According to the finite element method (FEM) analysis, the crack growth was mainly due to the stress perpendicular to the weld at which large cracks were found in the area of high structural stress and high stress concentration [16]. The stresses in the chord at the welded joints on the bottom section near the variable cross-section were found to be relatively higher compared to those at the other sections.

Although the critical points determined by FEM calculation are not at the same locations under each working condition. The locations of some of these points will be changed. However, several typical critical points can be determined by calculation under each condition. Seven critical points were determined

by ANSYS finite element calculation software, as described in Table 2 and Fig. 4.

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b) c) Fig. 4. The critical points: a) on the bottom section; b) on the middle section; c) on the top section

a)

Table 1. Calculation conditions Conditions 1 2

Load 85% of the rated load

Boom amplitude [m] Actions 8 Lifting-turningunloading 12

Table 2. The critical points determined by calculated stress amplitudes Points index 1 2 3 4 5 6 7

Locations Connection of the chord and the stiffener plate on the bottom section Connection of the chord and the side brace on the bottom section Connection of the chord and the space brace on the bottom section The chord hinge ear root on the bottom section The chord hinge ear root on the middle section Connection of the chord and the space brace on the middle section Connection of the chord and the space brace on the top section

Condition 1 Condition 2 [MPa] [MPa] 129.23

149.12

150.11

170.23

125.34

138.45

117.22

134.32

104.32

122.51

113.56

134.34

136.34

150.45

To ensure whether the critical points that were determined by the software matched the real locations on the boom, the crane boom was inspected. Most of the calculated points were found to match the real locations.

Because the crane has been in service for eight years, several cracks were detected. The working cycle is approximately 2.8×105, which is beyond the U4 level. There were different degrees of cracks, which were found in the welding area of the jib structure, as shown in Figs. 5 and 6. At point 2, the mean crack length is approximately 3.0 to 5.5 mm. At point 7, the mean crack length is approximately 3.5 to 6.0 mm. The max length of the crack on this kind of boom joints is typically approximately 10 to 15 mm.

138

As can be seen in the pictures, the stress concentration is high because of the poor weld quality. Therefore, the fact that cracks were found there means that the boom had been very seriously damaged. Most of the cracks were in a state of stable crack growth or that of rapid crack growth. This was sufficiently dangerous that the boom of the crane needed to be repaired.

After performing an in-depth examination, the majority of the cracks were found in the critical area or near the area accounting for approximately 80% of the total number of cracks. After determining the critical points, the fatigue analysis range was narrowed to a small number of priority hazardous areas. Point 2 on the K-type welded joint is considered as the focused analysis object in the following analysis.

Fig. 5. Cracks at critical point 2; crack length = 3.0 to 5.5 mm

Fig. 6. Cracks at critical point 7; crack length = 3.5 to 6.0 mm

Cai, F. – Wang, X. – Liu, J. – Zhao, F.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 135-144

1.3 Simulated Stress Spectra Acquisition Method A statistical method that is based on a large sample specimen is not suitable for a crane’s complex metal components, because the number of crane experimental samples is relatively small. As a result, a new method of obtaining the stress-time spectra based on simulation is proposed here. The details are described as follows.

(1) When the finite element model was built in Ansys, the beam188 element type was chosen for the structures of the pipes and the shell63 element type was chosen for the structures of the plates on the lattice boom [16]. After the static analysis, the most critical areas with high stress were determined. This next step was to narrow the study area that was shown in Fig. 4.

(2) To obtain the stress and stress distribution on the circumferential direction of the K-type welded joints on the chords and braces, the beam188 element type was modified to the shell63 element type. In the dynamic analysis, stress information cannot be extracted from the beam188 element type. To avoid the coupling effects between the two elements, the shell elements chords and braces area are built as large as possible to ensure that the coupling region was far away from the critical areas [17]. After modification, all the critical areas were built using shell63 element type.

(3) In Ansys, the model was transferred to the MSC. ADAMS software. Dynamic load was applied on the model in ADAMS [18]. Next, the simulated stress-time spectra could be obtained from the finite element model nodes after the dynamic simulation, as shown in Fig. 7. The axial forces and the bending moment of each chord and brace could be obtained from the dynamic simulation.

They will be used to load the chord in the fatigue test.

(4) Comparative analysis of the measured and simulated stress-time spectra is discussed here.

A strain gage was used to measure the actual strain information at point 2, as shown in Fig. 8.

Because the FEM model could only be simulated under determined conditions in the software, the measured conditions should be set the same as the simulated conditions. First, the measured points should be polished clearly to obtain the true strain. Second, the load of the crane should be the standard weight. Third, the actions of the crane working cycles should be the same between the measurement and the simulation. It means that, in each step of ‘lifting-turning-unloading’, the measured time and the simulated time should be the same. However, the time of different cycles can be different. From six working cycle simulations, a comparative analysis of the simulated and measured stress spectra of Condition 1 are described in Fig. 7 and Table 3. The error of the maximum stress amplitudes were less than 10%, with the majority being within the 5% error range of the allowable project application. The results indicated that the stresstime spectra obtained from dynamic software was accurate.

Fig. 8. Strain gauge at critical point 2

Fig. 7. Measured and simulated stress-time spectra of point 2; L-lifting; T-turning; U-unloading Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory

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Table 3. Maximum stress amplitude comparison Cycle number 1 2 3 4 5 6

Stress Amplitudes measured [MPa] simulated [MPa] -138.32 -140.13 -145.07 -138.22 -142.07 -138.32 -144.61 -139.34 -150.11 -139.41 -146.23 -139.42

Error [%] 1.31 -4.72 -2.64 -3.64 -7.13 -4.51

(5) The stress spectra shown in Fig. 7 were treated as a basic stress spectra block. The rain-flow counting method was used for statistical analysis.

The counting cycles corresponding to the stress amplitude and the mean stress were obtained as shown in Figs. 9 and 10 [19]. The basic stress spectra block could be extended to a long time stress spectra as the classic stress spectra of the crane. The same method could be used to obtain the stress spectra of Condition 2 and of other critical points. Although this is not the real stress spectra, the stress spectra of specified conditions can be obtained based on the virtual simulations.

Significant amounts of money and time can be saved via this method.

Fig. 9. Histograms of the rain-flow stress amplitudes cycles of Condition 1

Based on the above introductions, the new analysis method, including the ‘measured & simulated & compared & statistics’ combination strategy can be applied in the crane stress spectra acquisition. This method has been used in studies of bridge cranes and gantry cranes [20].

2 CALCULATION OF THE NON-LINEAR FATIGUE DAMAGE ACCUMULATION The PM rule can be expressed by Eq. (1): n D = ∑ i = ∑ Di , (1)

Ni where D is a constant that denotes the damage, and ni and Ni are the applied cycles and the total cycles to failure under ith constant amplitude loading level Si , respectively. Based on the assumptions made by the model, D should be 1; experimentally, it is found to range between 0.7 and 2.2. This variability is evidence of the failure of the PM rule to accurately predict fatigue lifespans. Because of its conceptual simplicity and the minimal amount of data necessary for its implementation, PM is a popular method for estimating the fatigue life.

A non-linear fatigue cumulative damage calculations approach from Huffman was used in this paper [21]. Only the cyclic stress-strain curve and the curve of the constant amplitude strain-life material properties are required in this method, which could be found easily from the research results of [1] and [22] to [24]. The model was first applied to calculate the K-type welded structure fatigue life with the intrinsically non-linear cumulative damage theory.

The model is calculated as follows.

DT is the sum of the damage of all reversals and Di is the normalized damage caused by the ith reversal ranging from i = 1 to i = 2NT; failure occurs when DT = 1.

2 NT

DT = ∑ Di . (2) i =1

The damage accrued in each step is calculated using a relatively simple algorithm and a constant amplitude strain–life data. The Basquin-MansonCoffin (BMC) equation is the most widely used constant amplitude strain-life approach, which is described in Eq. (3):

Fig. 10. Histograms of the rain-flow mean stress cycles of Condition 1

140

Cai, F. – Wang, X. – Liu, J. – Zhao, F.

∆ε ∆ε e ∆ε p = + = 2 2 2 σ' = f (2 N fail )b + ε 'f (2 N fail )c , (3) E

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 135-144

where Δε / 2 is half of the strain range, Δεe is the elastic strain, Δεp is the plastic strain, and 2Nfail is the number of reversals until failure. In this sample experiment, the material of K-type welded joints is Q345B, the elastic modulus E = 200741 MPa, the fatigue strength coefficient σf' = 947.1 MPa, the fatigue strength exponent b = –0.0943, the fatigue ductility coefficient εf' = 0.4644, the fatigue ductility exponent c = –0.5395.

A good description of the crack growth per reversal, for a constant amplitude strain, is given by the hyperbolic sine function. For a crack size a, the rate of crack tip advance, after 2Nk reversals, can be written as Eq. (4): 2Nk da ∝ sinh( ρ ), (4) dN 2 N fail

where 2Nfail is the total number of reversals to failure for the given strain amplitude, and ρ is a scaling factor, that is used to adjust the incremental damage. ρ = log(2Nfailγ), where 2Nfail was from the BMC equation and γ was taken as –c / 2εf'. In this way, da / DN has the correct functional relationship to the strain amplitude.

Eq. (4) was used for the damage calculation. The normalized damage due to the ith reversal is expressed as Eq. (5): NT +1

Di =

2Nk

∑ sinh( 2 N k =1

fail

NT

ρ ) − ∑ sinh( k =1

2N j

Nf

∑ sinh( 2 N j =1

NT +1

NT

≈

ρ)

2Nk

(5)

2Nk ρ )dN k 2 N fail k =1 fail , (6) DT = N f ≈ Nf 2N j 2N j sinh( ρ ) dN sinh( ρ ) ∫1 j ∑ 2 N fail 2 N fail j =1

ρ)

∫

NT

1

sinh(

2 N fail

× cosh −1 ( DT (cosh( ρ ) − (7)

ρ

−(cosh(

ρ ρ )) + cosh( )), 2 N fail 2 N fail

From Eqs. (1) to (7), the strain life calculation process based on non-linear cumulative damage is shown in Fig. 11.

Initial total damage DT Number of reversals until failure calculation N f

a

i

Total cycle calculation

N

Eq.(3)

l

Eq.(7)

T

Cycle damage calculation

Di

Eq.(5)

Total damage calculation

D= DT + ∑ Di T N

Eq.(2)

DT > 1

Fig. 11. Non-linear cumulative damage based on the strain life calculation process

sinh(

∑ sinh( 2 N

2 NT =

Y

where NT is the number of reversals required to achieve the accumulated damage, and DT is the damage due to a constant amplitude strain range Δεi / 2. The number of reversals to failure at this constant strain range is 2Nfail when the total damage DT = 1.

DT is the sum of Di , which can be expressed as Eq. (6): NT

Failture cylces calculation i

fail

2Nk ρ )dN k 2 N fail ≈ .

Nf 2N j ∫1 sinh( 2 N fail ρ )dN j

∫

2Nk ρ) 2 N fail

Eq. (6) can be solved to find:

3 FATIGUE LIFE CALCULATIONS 3.1 The Preparation of the Fatigue Test A specially designed test rig for K-type welded joints was applied, on which axial force and bending moment can be loaded. The rig is shown in Fig. 12. The specimen outline is shown in Fig. 13. The dimension was the same size as the K-type welded joint area, where the critical Point 2 located.

There are two kinds of test. They are described as Condition 1 and Condition 2 under simulated stress spectra. The total number of specimens is ten. The number of specimens for each fatigue test is five.

The axial forces are applied from the simulation by ADAMS. When all the tests were finished, the mean value of the specimens’ fatigue life was used for comparison.

Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory

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Fig. 12. Test rig for K-type welded joints

Fig. 13. Outline of the K-type connector specimen

3.2 The Purposes of the Fatigue Test There are three purposes for the fatigue test: 1. Influence analysis on the calculated fatigue life based on the measured and simulated stress-time spectra; 2. Calculated fatigue life error analysis based on the Huffman non-linear cumulative damage theory and the linear PM rule compared to the test data; 3. Influence analysis on calculated fatigue life of different initial damage.

The model assumed that failure occurs when a single critical crack propagates across the width of the chord. The damage calculation was performed with the Huffman non-linear damage model and the PM damage model.

3.3 Test Results Analysis In Table 4, all of the calculated cycles and the test cycles are shown and compared as 0 when the initial damage was considered. It can be inferred that: 142

1. The measured stress spectra can be replaced by the simulated stress spectra. The errors from the two calculated fatigue cycles do not exceed 10%, which satisfies the engineering requirements.

2. The calculation accuracy of the non-linear damage model is higher than that of the linear PM model. The non-linear calculation precision is approximately 1.14 to 1.19 under Conditions 1 and 2. The linear PM calculation precision is approximately 0.29 to 0.32 under Conditions 1 and 2. The linear calculation model is too conservative.

3. The calculated fatigue life based on the simulated spectra is longer than that of the measured spectra. The maximum error is less than 10%.

In the measured stress spectra, there are many stress mutations and stress fluctuations, leading to structural stress mutation prone microstructure grain boundary sliding and cracks, so the fatigue life is shorter. The simulated stress spectra is relatively stable, indicating that less stress mutation causes less damage to the structure; this may explain the longer life.

4. Although the accuracy of the non-linear damage model is higher than that of the linear PM damage model, there is still a gap between the calculation life and the test life. In addition, the calculation result tends to be slightly un-conservative. The error may come from the initial damage.

3.4 The Design Recommendations Based on the Initial Damage Analysis Using the initial damage in the calculations in the range from 0.01 to 0.08, the results are described in Table 5 and Fig. 14. The test fatigue life data from the simulated spectra were used as the comparison reference. It can be inferred that when the initial damage is set to 0.03 to 0.04, the calculation errors are less than 10% for the two conditions based on the non-linear model. When the initial damage is set to a value of over 0.07, the calculation error will be larger than 20%.

For the engineering application, there should be a certain amount of surplus for the design requirements.

The proposed initial damage of K-type welded joints may be set to 0.04 to 0.07; this choice will ensure accuracy while being moderately conservative at the same time.

Cai, F. – Wang, X. – Liu, J. – Zhao, F.

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, 135-144

Table 4. Calculation and test results (Initial damage DT = 0) Conditions

1

2

Approaches local stress strain method (Non-linear approach) local stress strain method (Linear approach) experimental fatigue life (Mean value of five specimens) calculation accuracy of the Non-linear approach calculation accuracy of the Linear approach local stress strain method (Non-linear approach) local stress strain method (Linear approach) experimental fatigue life (Mean value of five specimens) calculation accuracy of Non-linear approach calculation accuracy of Linear approach

Results from the simulated spectra 4.80×105 1.30×105 4.04×105 1.19 0.32 2.58×105 8.70×104 2.11×105 1.22 0.41

Results from the measured spectra 4.60×105 1.20×105 / 1.14 0.29 2.41×105 7.98×104 / 1.14 0.38

Error [%] –4.35 –8.33 / / / –7.05 –9.02 / / /

Table 5. Calculation errors based on the non-linear cumulative damage (fatigue test life chosen from simulated stress spectra) Conditions 1 2

Initial damage DT Calculation fatigue life (×105) Calculation error [%] Calculation fatigue life (×105) Calculation error [%]

0 4.80 18.81 2.58 22.27

0.01 4.60 13.91 2.38 12.89

Fig. 14. Calculation error comparison

4 CONCLUSIONS In this paper, a new non-linear cumulative damage calculation method based on strain parameters was introduced in detail; it is suitable for engineering applications because it only requires the constant amplitude fatigue strain-life data. This method is relatively simple compared to the other non-linear approaches.

1. The stress spectra acquisition method based on the ‘measured & simulated & compared & statistics’ integrated strategy of crane K-type welded joints was found to be feasible. The maximum error of the stress amplitude in every working cycle does not exceed 10% between the measured stress spectra and the simulated stress spectra. When the two spectra were used in the calculations,

0.02 4.40 8.86 2.29 8.67

0.03 4.15 2.72 2.15 1.90

0.04 3.88 –4.03 1.91 –9.71

0.05 3.68 –8.99 1.84 –12.78

0.06 3.57 –11.63 1.77 –16.16

0.07 3.22 –20.30 1.59 –24.53

0.08 3.02 –25.25 1.55 –26.51

the results indicated that the errors were no more than 10% under two conditions. This method can substantially reduce the workload and costs. The method may be applied to a variety of large and complex structures for stress spectra collection work.

2. The non-linear cumulative damage theory was found to have a higher accuracy than that of the PM rule. However, when the initial damage is set to 0, there is still a gap between the calculated data and the test data, and the calculated data tends to be non-conservative.

3. For welded structures, such as K-type welded joints, the initial damage can be set to 0.04 to 0.07. This choice of initial damage will ensure accuracy while being moderately conservative at the same time.

5 ACKNOWLEDGEMENTS This work was financially supported by The National Scientific and Technological Support Plan (grant no.

2011BAF04B01), Jiangsu Province Scientific and Technological Innovation and Technology Transfer Special Support Plan (grant no. BY2012089), and The National Scientific and Technological Support Plan of General Administration Quality Supervision, Inspection and Quarantine of China (grant no.

2013QK168).

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Cai, F. – Wang, X. – Liu, J. – Zhao, F.

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Vsebina Strojniški vestnik - Journal of Mechanical Engineering letnik 60, (2014), številka 2 Ljubljana, februar 2014 ISSN 0039-2480 Izhaja mesečno

Razširjeni povzetki člankov Marko Šimic, Mihael Debevec, Niko Herakovič: Modeliranje hidravličnih batnih ventilov s posebnimi oblikami krmilnih robov Bojan Starman, Marko Vrh, Mirko Halilovič, Boris Štok: Modeliranje procesa preoblikovanja pločevine z upoštevanjem plastične anizotropije in evolucije modula elastičnosti Jixin Wang, Long Kong, Bangcai Liu, Xinpeng Hu, Xiangjun Yu, Weikang Kong: Matematični model stožčastih zobnikov z ukrivljenim ozobjem – pregledni članek David Vegelj, Boštjan Zajec, Peter Gregorčič, Janez Možina: Adaptivno lasersko bliskovno varjenje električnih lamel Erik Čuk, Matjaž Gams, Matej Možek, Franc Strle, Vera Maraspin Čarman, Jurij F. Tasič: Nadzorovani vizualni sistem za razpoznavanje migrirajočega eritema, zgodnje kožne spremembe klopne borelioze Dragica Jošt, Aljaž Škerlavaj, Andrej Lipej: Izboljšanje napovedi izkoristka Kaplanove turbine z naprednimi turbulentnimi modeli Fuhai Cai, Xin Wang, Jiquan Liu, Fuling Zhao: Analiza utrujenostne trajnostne dobe K-zvarov pri žerjavih s pomočjo nelinearne teorije kumulativnih poškodb Osebne vesti Doktorske disertacije, diplomske naloge

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Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 20 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-03-26 Prejeto popravljeno: 2013-08-08 Odobreno za objavo: 2013-09-16

Modeliranje hidravličnih batnih ventilov s posebnimi oblikami krmilnih robov Šimic, M – Debevec, M. – Herakovič, N. Marko Šimic* – Mihael Debevec – Niko Herakovič Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija

Glavni namen prispevka je predstaviti novo metodo simulacije hidravličnih drsniških batnih ventilov in posredno celotnih hidravličnih sistemov. Zamisel se uresničuje v razdelitvi hidravličnega krmilnega bata ventila na posamezne oblikovne funkcijske elemente (krmilne robove), ki jih je možno popisati z znanimi in enostavnimi matematičnimi izrazi. Tako oblikovane matematične modele oblikovnih funkcijskih elementov nato vnesemo v knjižnice poljubnih simulacijskih orodij, namenjenih simulaciji hidravličnih komponent in sistemov. Na ta način lahko hidravlični krmilni bat zelo fleksibilno sestavimo iz različnih funkcijskih elementov krmilnih robov v kombinaciji z drugimi elementi krmilnega bata. Želene dinamične lastnosti hidravličnega sistema so v večji meri odvisne od dinamičnih lastnosti ventila, ki so pogojene z ustrezno obliko in velikostjo pomika krmilnega bata. Oblika in velikost pomika bata pa vplivata na obliko in velikost presečne ravnine, ki neposredno vpliva na karakteristiko volumskega toka ventila. Do sedaj znani postopki za določanje ustrezne oblike krmilnega bata so predvsem dolgotrajni in se opirajo na napredna CFD-orodja, za katera potrebujemo ekspertna znanja. Slabost takega postopka se kaže v tem, da je treba celoten postopek simulacije ponoviti že ob najmanjši spremembi oblike krmilnega bata. Poleg tega pa je postopek omejen le na optimizacijo geometrijskih parametrov posameznih hidravličnih komponent (hidravličnega bata ventila). Vpliva oblike krmilnega bata oz. vpliva karakteristike ventila na celoten hidravličen sistem z obstoječimi metodami ni mogoče analizirati.

Nova metoda geometrijske optimizacije krmilnih batov drsniških ventilov temelji na analitično-simulacijskem pristopu. Vsak funkcijski element, ki predstavlja specifično obliko krmilnega roba oz. zareze na robu, popišemo analitično z matematičnim modelom. Zahtevnost analitičnega modela je odvisna od zahtevnosti geometrije zareze na krmilnem robu bata, vendar pa želimo za popis uporabiti kar se da enostavne in že znane modele. V prispevku je predstavljen postopek razvoja matematičnega modela karakteristike volumskega toka le za eno izmed mnogih oblik zarez na krmilnem robu bata. Omenjene matematične modele implementiramo v funkcijske elemente knjižnice simulacijskih programov. Krmilni bat lahko nato sestavimo modularno iz poljubnih funkcijskih elementov, ki imajo lahko enako ali pa različno nelinearno karakteristiko volumskega toka. Celotni simulacijski model krmilnega bata upošteva tudi način povezave in medsebojni vpliv elementov.

Pri analitičnem modeliranju karakteristike volumskega toka ventila je upoštevana poenostavitev 3D-geometrije zareze na 2D, pri čemer napaka v velikosti presečne ravnine oz. teoretičnega volumskega toka ne presega 0,5 %.

Prav tako je med aksialno in radialno upoštevana minimalna presečna ravnina, skozi katero prehaja fluid. Analitični rezultati volumskega toka v odvisnosti od pomika krmilnega bata so primerjani z eksperimentalnimi rezultati.

Odstopanje volumskega toka v povprečju ne presega 1,2 %, kar potrjuje možnost poenostavitve analitičnega modela in pravilnost upoštevanja minimalne presečne ravnine za določitev prave karakteristike volumskega toka ventila.

Glavni prispevek članka je zamisel o razdelitvi krmilnega bata drsniškega ventila na funkcijske elemente, katerim je možno analitično določiti karakteristiko volumskega toka. Nova simulacijska metoda omogoča, da s pomočjo različnih kombinacij funkcijskih elementov krmilnega bata ventila hitro in učinkovito poiščemo optimalno obliko krmilnega bata za doseganje želenih dinamičnih lastnosti hidravličnega sistema, še preden načrtujemo končni realni hidravlični sistem.

Ključne besede: hidravlični batni ventili, krmilni robovi, zareze posebnih oblik, matematično modeliranje, modularna gradnja, simulacijski model ventila

SI 20

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, marko.simic@fs.uni-lj.si

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Prejeto v recenzijo: 2013-07-31 Prejeto popravljeno: 2013-10-02 Odobreno za objavo: 2013-10-21

Modeliranje procesa preoblikovanja pločevine z upoštevanjem plastične anizotropije in evolucije modula elastičnosti Starman, B. – Vrh, M. – Halilovič, M. – Štok, B. Bojan Starman – Marko Vrh – Mirko Halilovič – Boris Štok* Univerza v Ljubljani, Fakulteta za strojništvo, Slovenija

Članek obravnava konstitutivno in numerično modeliranje procesa preoblikovanja nerjavne pločevine s poudarkom na napovedi elastične povračljivosti. Znano je, da se med obremenjevanjem v pločevini poleg trajne deformacije razvija tudi poškodovanost v obliki mikropraznin, votlinic in razpok. Pojav je razpoznaven z opazovanjem mikrostrukture pločevine, ki je bila predhodno obremenjena do določene stopnje trajne deformacije. Glede na teorijo mehanike poškodb so tako nastale praznine in mikrorazpoke glavni vzrok za degradacijo togosti v duktilnih materialih, kar ima nadalje pomembno vlogo pri napovedovanju elastične povračljivosti. Ker je pločevina zaradi procesa izdelave anizotropna, ima tudi to pri napovedovanju pomemben vpliv.

Glavni predmet obravnave je izgradnja konstitutivnega modela, ki iz fenomenološkega vidika sočasno upošteva anizotropijo pločevine, evolucijo poškodovanosti ter degradacijo togosti v materialu med preoblikovanjem. Predstavljeni elastoplastični model tako temelji na Gurson–Tvergaard–Needlemanovem (GTN) poškodbenem modelu, ki je ustrezno razširjen z vgrajenim modelom plastične anizotropije Hill48 ter Mori-Tanaka pristopom degradacije togosti na nivoju materialne točke. Konstitutivni model, poimenovan GHM, je bil nadalje implementiran v program končnih elementov ABAQUS/Explicit prek uporabniškega podprograma VUMAT, parametri modela pa so bili usklajeni z eksperimentalnimi vrednostmi.

Za namen uskladitve parametrov konstitutivnega modela so bili izvedeni standardni natezni preizkusi v treh smereh (0°, 90° in 45°) glede na smer valjanja pločevine, ter preizkusi, pri katerih so standardni preizkušanci najprej predhodno deformirani do določene stopnje plastične deformacije in nato razbremenjeni. Mehanski odziv za karakterizacijo mehanskih lastnosti je nadalje določen na osnovi ponovnega obremenjevanja in razbremenjevanja teh preizkušancev v elastičnem področju v vzdolžni in pravokotni smeri glede na smer valjanja. Uskladitev parametrov konstitutivnega modela je bila izvedena z Levenberg-Marquardtovim optimizacijskim algoritmom na podlagi simulacije izvedenih preizkusov.

Zasnovani konstitutivni model je bil v zadnji fazi eksperimentalno verificiran v preizkusu elastične povračljivosti. Preizkus je sestavljen iz upogibnega obremenjevanja in relaksacije pravokotnega pločevinastega vzorca, ki je bil predhodno plastično deformiran do določene stopnje tako v pravokotni kot tudi v vzdolžni smeri glede na smer valjanja pločevine. Primerjava predlaganega pristopa k modeliranju in klasičnega pristopa samo z upoštevanjem utrjevanja in anizotropije pločevine v modelu Hill48 jasno pokaže, da sočasno modeliranje materialnih pojavov, še posebej z združevanjem anizotropije in degradacije materialne togosti, vodi v natančnejšo napoved elastične povračljivosti v procesih preoblikovanja pločevine.

Ključne besede: elastična povračljivost, poškodovanost, elastične lastnosti, degradacija togosti, plastična anizotropija

*Naslov avtorja za dopisovanje: Univerza v Ljubljani, Fakulteta za strojništvo, Aškerčeva 6, 1000 Ljubljana, Slovenija, boris.stok@fs.uni-lj.si

SI 21

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Prejeto v recenzijo: 2013-08-05 Prejeto popravljeno: 2013-10-14 Odobreno za objavo: 2013-12-03

Matematični model stožčastih zobnikov z ukrivljenim ozobjem Jixin

Wang1

Wang, J. – Kong, L. – Liu, B – Hu, X. – Yu, X. – Kong, W. – Long Kong1,* – Bangcai Liu2 – Xinpeng Hu1 – Xiangjun Yu3 – Weikang Kong1 1 Univerza

Jilin, Fakulteta za tehniške vede in strojništvo, Kitajska hidravličnekomponente, Kitajska 3 Univerza Kunming, Visoka šola za avtomatizacijo in strojništvo, Kitajska 2 XCMGH

Članek podaja smernice za tehnološki razvoj stožčastih zobnikov z ukrivljenim ozobjem, zlasti novih tipov zobnikov. Stožčasti zobniki z ukrivljenim ozobjem imajo zahtevno ukrivljeno površino s kinematičnimi lastnostmi, ki so neposredno povezane s posebnim procesom odrezavanja. Raziskave matematičnih modelov stožčastih zobnikov z ukrivljenim ozobjem so zato že dolgo vroča tema pri razvoju mehanskih prenosnikov. Matematični model je osnova za konstruiranje, izdelavo in analizo stožčastih zobnikov z ukrivljenim ozobjem. Pomen raziskav matematičnega modela ni le v analizi in izdelavi modela površine ozobja, temveč tudi v preučevanju načel konstruiranja in proizvodnih postopkov.

V članku je podan celovit pregled literature o matematičnem modeliranju stožčastih zobnikov z ukrivljenim ozobjem. Ilustrirane, primerjane in povzete so metode za gradnjo matematičnih modelov kot so matrična, vektorska in geometrijska metoda. Članek obravnava raziskave matematičnih modelov stožčastih zobnikov z ukrivljenim ozobjem za konstruiranje in izdelavo mehanskih prenosnikov, zgodovino raziskav ter uporabo posameznih metod za izdelavo matematičnih modelov stožčastih zobnikov z ukrivljenim ozobjem.

Osnova za matrično in vektorsko metodo modeliranja je proizvodni postopek, matematični modeli pa so izpeljani iz dejanskih nastavitev obdelovalnih strojev in torej ustrezajo dejanski izdelavi površine zob. Z nadaljnjimi raziskavami bo mogoče podrobneje opisati nove proizvodne postopke, vključno z gibanji stroja, geometrijskim modelom rezkarja ter zvezo med rezkarjem in obdelovancem, ti rezultati pa bodo prispevek k gradnji matematičnih modelov stožčastih zobnikov z ukrivljenim ozobjem po teh dveh metodah. Geometrijska metoda je predlagana kot teoretični model in predstavlja preboj na področju raziskav novih teorij stožčastih zobnikov z ukrivljenim ozobjem. Geometrijski modeli bodo odigrali vlogo predvsem pri predstavitvah novih konstrukcijskih zasnov stožčastih zobnikov z ukrivljenim ozobjem. Geometrijski model stožčastih zobnikov z ukrivljenim ozobjem namesto nastavitev obdelovalnega stroja opredeljujejo osnovni geometrijski parametri, pri čemer sta glavni geometrijski značilnosti profil zoba in srednjica. Za implementacijo geometrijskega modela so potrebne dodatne raziskave za določitev zveze med geometrijskimi značilnostmi in procesom obdelave. Nadaljnje študije uporabe geometrijskega modela bodo verjetno usmerjene v obliko rezkarjev in poti orodja.

V članku je prvič podan celovit pregled metod gradnje matematičnih modelov stožčastih zobnikov z ukrivljenim ozobjem. Z ozirom na teorijo in matematične pristope primerja in klasificira metode iz objavljenih člankov, vključno z matrično, vektorsko in geometrijsko metodo. Matrična in vektorska metoda sta implementirani z analizo procesa izdelave stožčastih zobnikov z ukrivljenim ozobjem na posebnem obdelovalnem stroju.

Geometrijska metoda je implementirana z analizo procesa konstrukcije novega tipa stožčastih zobnikov z ukrivljenim ozobjem za izdelavo na univerzalnem stroju. Matematični model stožčastih zobnikov z ukrivljenim ozobjem omogoča posodobitve in iskanje novih proizvodnih postopkov, skladno z razvojem tehnologije univerzalnih rezkalnih strojev in metod računalniško podprte proizvodnje.

Ključne besede: stožčasti zobnik z ukrivljenim ozobjem, matematični model, matrična metoda, vektorska metoda, geometrijska metoda, mehanski prenosnik

SI 22

*Naslov avtorja za dopisovanje: Univerza Jilin, 5988 Renmin ulica, Changchun, Kitajska, goodkong.8810@163.com

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 23 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-08-29 Prejeto popravljeno: 2013-11-05 Odobreno za objavo: 2013-11-13

Adaptivno lasersko bliskovno varjenje električnih lamel Vegelj, D. – Zajec, B. – Gregorčič, P. – Možina, J. David Vegelj1,* – Boštjan Zajec1 – Peter Gregorčič2 – Janez Možina2 2 Univerza

1 Hidria

Rotomatika, Slovenija

v Ljubljani, Fakulteta za strojništvo, Slovenija

Hibridna in električna vozila se vse bolj uveljavljajo na svetovnem trgu vozil. S tem se povečuje tudi potreba po učinkovitih elektromotorjih. Za njihovo izdelavo potrebujemo boljše materiale, primernejšo izbiro postopka izdelave ali uporabo frekvenčnih pretvornikov. Bliskovno in kontinuirno lasersko varjenje lamel v jedrih elektromotorjev se uveljavlja kot eden izmed glavnih postopkov izdelave. Oba trenutna postopka pa imata veliko pomanjkljivost, saj pretalita celotno stransko ploskev lamel. Poleg pretirane porabe optične energije za taljenje materiala, ki v zvaru ne sodeluje, pa trenutni postopki povzročajo tudi velike zvare. Velikost zvara posredno vpliva na magnetne lastnosti elektromotorja. V članku smo zato predstavili sistem adaptivnega bliskovnega laserskega varjenja in rezultate raziskav na področju adaptivnega varjenja. Dokažemo, da je razviti in predstavljeni postopek varjenja boljši od trenutno poznanih postopkov, in sicer tako z vidika porabljene optične energije kot tudi z vidika kakovosti izdelave posameznih zvarov.

V okviru raziskovalnega dela smo razvili sistem adaptivnega bliskovnega laserskega varjenja električnih lamel, ki avtomatsko zazna reže med lamelami. Za zaznavanje rež uporablja na trgu dostopen laserski odbojnostni senzor. Na podlagi signala, ki ga iz senzorja dobimo na režah, s krmilno enoto ustrezno krmilimo laserski izvor, ki deluje v načinu posameznih bliskov. Pri tem uporabljamo dva mikrokrmilnika, v katerih delujeta posebej za ta namen razvita programa. Varili smo s klasičnim bliskovnim Nd:YAG laserjem. Laserske parametre (čas in vršno moč bliska) smo nastavljali ročno na laserskem izvoru. Sistem posamezne bliske proži zgolj na tistih mestih, kjer potrebujemo zvare, t.j. na stiku med dvema lamelama.

Razviti adaptivni sistem smo preizkusili na dveh tipih vzorcev. Kot prvi tip vzorec smo uporabili manjše statorske pakete, izdelane iz različnih tipov in debelin elektropločevine, kot drugi tip vzorcev pa smo uporabili standardne statorske pakete, ki jih v proizvodnji trenutno varijo v kontinuirnem načinu. Zavarjene pakete smo vizualno, mehansko in magnetno preskusili. Mehanski test smo izvedli z modificiranim nateznim preskusom, pri čemer smo vzorce v vzdolžni smeri obremenili do točke porušitve zvara. Magnetno testiranje je vključevalo merjenje specifičnih skupnih izgub in relativne permeabilnosti paketa. Na manjših vzorcih smo izvedli vse opisane analize, medtem ko smo na standardnih vzorcih izvedli le magnetno testiranje, saj je bila mehanska trdnost zvarov povsem sprejemljiva.

Rezultati analiz so pokazali, da adaptivno bliskovno lasersko varjenje porabi do približno 80 % manj optične energije na zvar, pri tem pa je bila mehanska trdnost adaptivno varjenih vzorcev povsem primerljiva s standardnim bliskovnim načinom varjenja. Magnetno testiranje je pokazalo, da uporaba adaptivnega načina varjenja pri malih vzorcih ne izboljša bistveno skupnih specifičnih izgub. Po drugi strani pa pri standardnih lamelnih paketih z adaptivnim načinom varjenja dosežemo bistveno izboljšanje magnetnih lastnosti paketa v primerjavi s kontinuirnim načinom varjenja.

Glavni prispevek tega članka je v razvoju metode in sistema za adaptivno bliskovno lasersko varjenje električnih lamel, ki porabi znatno manj optične energije za izdelavo boljših zvarov. Manjši zvari tudi omogočajo izdelavo učinkovitejših elektromotorjev. Predlagani sistem bi lahko še nadgradili v smeri izboljšanja zanesljivosti in hitrosti varjenja.

Ključne besede: lasersko varjenje, varjenje lamel, magnetne lastnosti, stator, rotor

*Naslov avtorja za dopisovanje: Hidria Rotomatika d.o.o., Spodnja Kanomlja 23, 5281 Spodnja Idrija, Slovenija, david.vegelj@hidria.com

SI 23

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Prejeto v recenzijo: 2013-02-11 Prejeto popravljeno: 2013-10-18 Odobreno za objavo: 2013-11-13

Nadzorovani vizualni sistem za razpoznavanje migrirajočega eritema, zgodnje kožne spremembe klopne borelioze

Čuk, E. – Gams, M. – Možek, M. – Strle, F. – Maraspin Čarman, V. – Tasič, J.F. Erik Čuk1,2,3,* – Matjaž Gams2 – Matej Možek3 – Franc Strle4 – Vera Maraspin Čarman4 – Jurij F. Tasič3 1 LOTRIČ meroslovje, Slovenija Jožef Stefan, Odsek za inteligentne sisteme, Slovenija 3 Univerza v Ljubljani, Fakulteta za elektrotehniko, Slovenija 4 Univerzitetni klinični center Ljubljana, Klinika za infekcijske bolezni in vročinska stanja, Slovenija 2 Inštitut

Klopna borelioza je nalezljiva bolezen, ki se na človeka prenaša preko klopnega vboda. Razširjena je na severni polobli, natančneje v Severni Ameriki, Evropi in Aziji. Najbolj pogost znak prisotnosti okužbe z boleznijo je pojav migrirajočega eritema, kožne spremembe, ki se pojavi v nekaj dneh oziroma tednih po vbodu klopa. Zgodnje razpoznavanje kožne spremembe je pomembno, saj omogoča ustrezno ukrepanje in preprečevanje kasnejših posledic bolezni, ki lahko ovirajo normalno življenje.

V članku so najprej predstavljeni splošni termini s področja obdelave slik, sledi opis poteka okužbe s klopno boreliozo, hkrati pa so obravnavani sorodni sistemi iz področij razpoznavanja kožnega raka, razpoznavanja zdravstvenega stanja pacienta na osnovi barve in teksture človeškega jezika ter detekcije kožnih alergij. Glavni cilj raziskave je predstavitev novega vizualnega sistema za razpoznavanje migrirajočega eritema. Sistem je osnovan na tehnologiji multimedijskega interaktivnega terminala, ki se uporablja tudi v pametnih telefonih. Načrtovanje takega sistema zahteva poznavanje medicinskega področja, zatem sledi izbira ustreznih metod s področja obdelave slik in strojnega učenja. Za detektiranje robu potencialnega migrirajočega eritema je bila izvedena primerjava treh metod segmentacije: “GrowCut”, “maximal similarity based region merging” in “random walker”. Rezultati, pridobljeni z metodo “GrowCut”, so boljši od rezultatov, pridobljenih z metodo “random walker”. “GrowCut” metoda je bila izboljšana z novim načinom vnosa točk za označevanje ozadja in kožne spremembe z algoritmom imenovanim “označevalec s prstom (FD1)”. Ta pristop je omogočil pridobitev rezultatov, primerljivih tistim, ki so bili pridobljeni z metodo “maximal similarity based region merging”. Za zadovoljive rezultate segmentacije je potrebno vnesti šest ali več označevalnih točk. S praktičnega vidika je enostavneje in hitreje narisati krivuljo okoli kožne spremembe, kot vnesti naprimer šest med seboj enakomerno porazdeljenih točk. Izboljšana “GrowCut” metoda je robustna na različne oblike in barve potencialnih migrirajočih eritemov. Primerjava klasifikacijskih algoritmov “naive Bayes”, “support vector machine”, “AdaBoost”, “random forest” in “neural network” je pokazala, da pristop z izboljšano “GrowCut” metodo da boljše rezultate pri klasifikaciji kožnih sprememb v razreda elipsa in migrirajoči eritem.

Za ustrezno delovanje sistema je potrebna interakcija s sistemom s strani uporabnika, ki mora fotografirati kožno spremembo ter z uporabo multimedijskega interaktivnega terminala narisati krivuljo okrog kožne spremembe. Za uporabnika je ključen odločitveni del sistema, ki s klasifikacijo kožne spremembe v razred migrirajoči eritem služi uporabniku kot pripomoček pri diagnozi migrirajočega eritema. Sistem je namenjen uporabi tako pri zdravniškem pregledu kot za domačo uporabo s pripomočki, kot so pametni telefoni. V bližnji prihodnosti nameravamo delovanje sistema za razpoznavanje migrirajočega eritema izboljšati z razvojem barvnih in Gaborjevih atributov. Poleg tega bodo za bolj učinkovito diagnozo uporabljeni tudi tekstovni podatki o simptomih pacientov. Razlog za kombiniranje slikovne analize s tekstovnimi podatki je sedanja uspešna praksa zdravljenja migrirajočega eritema. Izpostaviti je treba, da je bila baza slik potencialnih migrirajočih eritemov pridobljena z opremo z boljšimi tehničnimi zmogljivostmi, kot je tista, ki je trenutno na voljo v pametnih telefonih.

Vendar pa razvoj pametnih telefonov napreduje zelo hitro, zaradi česar lahko pričakujemo, da bo tehnologija pametnih telefonov v nekaj letih dosegla zadovoljivo kvaliteto fotografije.

Ključne besede: klopna borelioza, migrirajoči eritem, označevalec s prstom, segmentacija, razpoznavanje, atributi

SI 24

*Naslov avtorja za dopisovanje: LOTRIČ meroslovje, Selca 163, 4227 Selca, Slovenija, erik.cuk@lotric.si

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 25 © 2014Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-05-17 Prejeto popravljeno: 2013-09-20 Odobreno za objavo: 2013-12-10

Izboljšanje napovedi izkoristka Kaplanove turbine z naprednimi turbulentnimi modeli Jošt, D. – Škerlavaj, A. – Lipej, A. Dragica Jošt1,* – Aljaž Škerlavaj1 – Andrej Lipej1 1Turboinštitut,

Slovenija

Numerična analiza toka je v minulih dvajsetih letih pomembno prispevala k razumevanju pojavov v aksialnih vodnih turbinah in k izboljšanju njihovih karakteristik. Dokaj natančno lahko napovemo vpliv sprememb hidravličnih oblik na izkoristek turbine, manj točna pa je napoved same vrednosti izkoristka. Namen tega članka je pokazati, zakaj lahko s stacionarnimi izračuni dobimo napačne rezultate in kako jih s časovno odvisnimi izračuni in naprednimi turbulentnimi modeli bistveno izboljšamo.

V ta namen smo numerično analizirali tok v 6-lopatični Kaplanovi turbini za srednje padce. Model turbine je bil preizkušen na merilni postaji v Turboinštitutu v skladu z mednarodnimi standardi IEC 60193. Obravnavali smo tri kote gonilnih lopatic pri konstantnem padcu. Rezultati preračuna so navor na gredi turbine, izgube v posameznih delih turbine in izkoristek ter pretok ali padec. Kadar je podan padec, je rezultat preračuna pretok, in obratno. Numerične rezultate, razen izgub, smo primerjali z izmerjenimi vrednostmi.

Stacionarni izračuni z dvoenačbenimi turbulentnimi modeli (k–ε, k–ω, BSL, SST) in z modelom Reynoldsovih napetosti SSG RSM niso dali zadovoljivih rezultatov. Pri dveh manjših kotih gonilnika razlike med izračunanim in izmerjenim izkoristkom niso bile velike, vendar je podrobnejša analiza rezultatov zlasti pri najmanjšem kotu gonilnika pokazala znatna odstopanja pretoka in navora. Izredno slabi pa so bili rezultati pri največjem kotu gonilnika, kjer je bil izračunani izkoristek turbine zaradi precenjenih izgub v sesalni cevi in premajhnega navora za več kot 4% manjši od izmerjene vrednosti. Časovno odvisni izračuni s tremi turbulentnimi modeli (SST, SAS, ZLES) in dvema shemama za diskretizacijo advektivnega člena (HRS – angl. High Resolution Scheme in BCDS – angl. Bounded Central Differential Scheme) so bili izvedeni v eni obratovalni točki pri največjem kotu gonilnih lopatic. Primerjava izračunanih tokovnih razmer je pokazala, zakaj smo s stacionarnimi izračuni dobili precenjene izgube v sesalni cevi. Zaradi pogoja zamrznjenega gonilnika (angl.: frozen rotor condition) so pri stacionarnih izračunih sence za gonilnimi lopaticami ves čas na istem mestu. Iz teh senc nastanejo vrtinci, ki so vzrok za velike izgube v sesalni cevi. Pri časovno odvisnih izračunih zaradi vrtenja gonilnih lopatic prihaja do mešanja toka, sence za gonilnimi lopaticami so manj izrazite, njihov položaj pa se ves čas spreminja. Pri časovno odvisnih izračunih dobimo popolnoma drugačne in veliko manjše vrtinčne strukture v toku. Z modeloma SAS in ZLES je vrednost turbulentne viskoznosti veliko manjša kot pri modelu SST. Zaradi naštetega so izračunane izgube v sesalni cevi manjše, ujemanje z meritvami pa boljše. Z modeloma SAS in ZLES v kombinaciji s shemo BCDS smo pri podanem pretoku dobili odlično ujemanje izkoristka, navora in padca, napaka je bila pri vseh treh veličinah manjša od 0,1%. Primerjava rezultatov časovno odvisnih izračunov s SST HRS, SAS HRS in SAS BCDS je pokazala, da je vpliv sheme diskretizacije na izgube v gonilniku in na navor celo večji kot vpliv uporabe modela SAS namesto SST. Da je model ZLES primeren tudi za druge obratovalne režime, smo potrdili z izračunom toka v več obratovalnih točkah za vse tri kote gonilnih lopatic.

Povsod je bilo odstopanje izračunanega izkoristka od izmerjenih vrednosti manjše od 1%.

V članku je pojasnjena povezava med turbulentnimi modeli, vrtinčnimi strukturami in velikostjo turbulentne viskoznosti na eni strani, ter izračunanimi izgubami v sesalni cevi na drugi strani. Pri drugih avtorjih nismo zasledili uporabe modela ZLES v celotni turbini. Prvič je prikazan tudi vpliv sheme diskretizacije na tok v gonilniku.

Ključne besede: vodna turbina, aksialna turbina, napoved izkoristka, računalniška dinamika tekočin, turbulentni modeli, ZLES

*Naslov avtorja za dopisovanje: Turboinštitut, Rovšnikova 7, 1210 Ljubljana, Slovenija, dragica.jost@turboinstitut.si

SI 25

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 26 © 2014 Strojniški vestnik. Vse pravice pridržane.

Prejeto v recenzijo: 2013-08-13 Prejeto popravljeno: 2013-11-15 Odobreno za objavo: 2013-12-11

Analiza utrujenostne trajnostne dobe K-zvarov pri žerjavih s pomočjo nelinearne teorije kumulativnih poškodb Cai Fuhai1,* – Wang Xin1 - Liu Jiquan2 - Zhao Fuling1 univerza v Dalianu, Oddelek za strojništvo, Kitajska 2 Jiangsu Bada, Kitajska

1 Tehniška

Na paličnih ročicah žerjavov je veliko zvarov. Analiza utrujenostne trajnostne dobe palične ročice je težavna naloga predvsem zaradi zahtevnih obremenitvenih razmer. Režim obremenitev palične ročice žerjava je v območju prekrivanja nizkocikličnega in visokocikličnega utrujanja. Pomanjkljiva natančnost meritev utrujenostne trajnostne dobe je posledica velikega raztrosa pri izračunih po linearnem Minerjevem pravilu.

V zadnjih letih je bilo razvitih več modelov na osnovi pristopa združevanja teorij lomne mehanike in empiričnih ugotovitev. Čeprav so ti postopki modeliranja natančnejši od Palmgren-Minerjevega pravila, pa za določanje potrebnih parametrov zahtevajo bistveno več eksperimentalnih podatkov.

V članku je za razrešitev teh težav predlagana nova metoda za pridobivanje napetostno-časovnih spektrov s pomočjo simulacijske programske opreme, ki zmanjšuje stroške in poenostavlja delo. Uvedena je poenostavljena nelinearna teorija kumulativnih poškodb na osnovi parametra deformacij, ki je primerna za inženirske aplikacije konstrukcij z elastoplastičnimi značilnostmi kot je palična ročica žerjava. Metoda zahteva le manjšo količino podatkov o materialnih lastnostih, kot sta ciklična napetostno-deformacijska krivulja in krivulja utrujenostne dobe pri konstantni amplitudi deformacij.

Predlagana je integrirana strategija za ekonomično in prikladno ugotavljanje realnega spektra obremenitev žerjava s K-zvari, ki združuje meritve, simulacijo, primerjave in statistiko. Najprej je bila opravljena statična mehanska analiza s programsko opremo Ansys z elementi tipa beam188 in shell63 Temu je sledila dinamična analiza s programsko opremo MSC.ADAMS s povezovalnimi elementi. Po dinamični simulaciji je bilo mogoče iz vozlišč modela s končnimi elementi pridobiti simulirane napetostno-časovne spektre.

Primerjalna analiza izmerjenih in simuliranih napetostno-časovnih spektrov je pokazala, da je napaka maksimalne amplitude napetosti manjša od 10 %, v večini primerov pa je znotraj dovoljenega območja napake 5%. Za izračun utrujenostne trajnostne dobe K-zvarnih spojev žerjava je predlagana Huffmanova nelinearna teorija kumulativnih poškodb na podlagi napetosti in deformacij ob zarezi. V članku je uporabljena BasquinManson-Coffinova (BMC) enačba. Rast razpoke na obrat je za deformacije konstantne amplitude dobro popisana s hiperboličnim sinusom. Normalizirana poškodba Di, ki jo povzroči i-ti obrat, je izražena kot aproksimirana integralska funkcija. Vsota Di predstavlja nelinearno kumulativno poškodbo oz. Dt. Opravljena je bila vrsta preskusov utrujanja K-zvarov na ročici žerjava z izmerjenim in simuliranim spektrom napetosti.

Rezultati utrujenostnih preskusov kažejo, da je napaka utrujenostne dobe pri simuliranih in izmerjenih napetostnih spektrih manjša od 10%. Natančnost nelinearne akumulacije poškodb je večja kot pri Minerjevem pravilu, čeprav je rezultat izračunov po metodi nelinearne akumulacije poškodb nekoliko nekonzervativen, če v izračunu ni upoštevana začetna poškodba. Z izbiro različnih pogojev začetne poškodbe so bili pridobljeni različni rezultati, ki razkrivajo, da je napaka izračuna utrujenostne trajnostne dobe na osnovi nelinearne teorije pri začetni poškodbi od 0,02 do 0,04 manjša od 10 %. Če so potrebni konzervativnejši izračuni utrujenostne trajnostne dobe, je začetno poškodbo mogoče nastaviti v območju od 0,04 do 0,07 in napaka izračuna je tedaj manjša od 25 %.

Huffmanova nelinearna teorija akumulacije poškodb zahteva le malo podatkov o materialnih lastnostih, kot sta ciklična napetostno-deformacijska krivulja in krivulja utrujenostne dobe pri konstantni amplitudi deformacij, zato je s svojimi natančnejšimi izračuni in manjšimi stroški primernejša za inženirske aplikacije.

Pristop je bil prvič uporabljen pri K-zvarih palične ročice žerjava. Spekter obremenitev žerjava je kompleksen, zato bi bilo treba računski model v prihodnjih raziskavah preveriti z več spektri iz realnih delovnih pogojev žerjava. Metoda je uporabna tudi pri drugih zvarnih spojih, ki se pogosto uporabljajo pri gradbenih strojih kot so žerjavi, bagri itn.

Ključne besede: žerjav, palična ročica, K-zvari, analiza utrujenostne trajnostne dobe, parameter deformacij, nelinearna teorija kumulativnih poškodb, utrujenostni preskus

SI 26

*Naslov avtorja za dopisovanje: Tehnološka univerza Dalian, Oddelek za strojništvo, Dalian, Kitajska, cfhdlut@163.com

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 27-28 Osebne objave

Doktorske disertacije, diplomske naloge DOKTORSKE DISERTACIJE Na Fakulteti za strojništvo Univerze v Ljubljani je obranil svojo doktorsko disertacijo: ● dne 6. januarja 2014 Bojan GJEREK z naslovom: »Geometrijska optimizacija stabilnosti vitkega aeroprofila v fluidnem toku« (mentor: prof. dr. Franc Kosel); Obravnavan je stabilnostni problem aeroelastičnega sistema, ki je predstavljen kot elastično vpet vitek aeroprofil v toku fluida. Zasnova vitkega aeroprofila temelji na togem simetričnem aeroprofilu z deformabilno ploščo, pritrjeno na njegov zadnji rob. Tako mejo stabilnosti obravnavanega aeroelastičnega sistema določata flutter aeroprofila in flutter plošče. Problem flutter-ja je predstavljen s sistemom diferencialno-integralnih enačb v kompleksni obliki, ki je rešen z razvojem po ortogonalnem sistemu lastnih funkcij pripadajočega problema. Na osnovi teoretičnega modela je podana optimizacija deformabilne kompozitne plošče v smislu števila plasti laminata plošče za različne konfiguracije parametrov aeroelastičnega sistema z namenom povišanja meje stabilnosti. Izkaže se, da v območju bimodalnega odziva aeroelastičnega sistema, kjer se pojavita flutter aeroprofila in flutter plošče istočasno, dosežemo na račun deformabilnosti plošče znatno povišanje meje stabilnosti v primerjavi s togo ploščo. Za namen validacije teoretičnega modela in rezultatov optimizacije je predstavljen celovit eksperimentalni pristop k problemu določanja meje stabilnosti danega več parameterskega aeroelastičnega sistema.

* Na Fakulteti za strojništvo Univerze v Mariboru so obranili svojo doktorsko disertacijo: ● dne 20. januarja 2014 Tijana RISIĆ z naslovom: »Protimikrobne medicinske tekstilije na osnovi hitozanskih nanodelcev za ginekološko zdravljenje« (mentorica: izr. prof. dr. Lidija Fras Zemljič); Cilj predstavljene doktorske disertacije je razvoj medicinskega tampona za alternativno ginekološko zdravljenje z uporabo hitozanskih nanodelcev, ki bodo delovali kot protimikroben agent ali kot dostavni sistemi za zdravilne učinkovine. V ta namen smo uporabili viskozni tamponski trak, ki smo ga funkcionalizirali s hitozanskimi in trimetil hitozanskimi nanodelci. Sledila je poglobljena karakterizacija pripravljenih materialov. Na začetku smo analizirali raztopine hitozana (CS) in trimetil hitozana (TMC), kakor tudi nanodelce sintetizirane s postopkom ionskega geliranja. Njihova karakterizacija je bila usmerjena predvsem v določitev naboja in protimikrobne aktivnosti na najpogostejše patogene mikroorganizme.

Vpliv prisotnosti pozitivnega naboja na inhibicijo rasti mikrobov je bila potrjena. Ker so CS in TMC raztopine ter

disperzije nanodelcev pokazale protimikrobno delovanje na laktobacile, je bila izvedena podrobna študija mehanizma protimikrobnega delovanja hitozana z uporabo nove, difuzijske nuklearne magnetne resonance (D-NMR). S tem smo lahko spremljali intra- in ekstracelularno izmenjavo vode v celicah, ki kaže na poškodbo celične membrane in izhajanje celularnih komponent. Nadalje smo z namenom, da bi raziskali pojav adsorpcije in molekulskih interakcij med CS/TMC in celulozo, uporabili celulozne modelne filme, na katerih smo z uporabo kremenove mikrotehnice spremljali adsorpcijo. CS in TMC sta se na celulozne filme prednostno odlagala pri višjih ionskih jakostih in višjih pH vrednostih, t.j. karakteristikah, ki vodijo v manjšajo topnost polimera, kjer je prisotnost elektrostatskih interakcij zanemarljiva in prevladujejo ne-elektrostatske interakcije.

Ugotovitve pridobljene na modelnih površinah so bile izjemno koristne za prenos na realne sisteme, kakor tudi pri karakterizaciji funkcionaliziranih vlaken in dobljenih rezultatov. Imobilizacija CS in TMC (v obliki raztopine in/ali nanodelcev) na celulozna vlakna je bila potrjena s številnimi analiznimi metodami. Vezava hitozana na vlakna se je izkazala za reverzibilno, kar smo ugotovili s spremljanjem desorpcije v simuliranih pogojih vaginalne uporabe. Evaluacija protimikrobnih lastnosti je bila izvedena z dvema mikrobiološkima tehnikama, ki sta obe pokazali učinkovito inhibicijo testiranih mikroorganizmov.

Testiranje občutljivosti laktobacilov na vlakna obdelana s hitozanom ni pokazalo negativnega učinka na normalno vaginalno mikrobioto. Obdelani materiali prav tako niso pokazali citotoksičnega efekta pri testiranju citotoksičnosti v direktnem kontaktu. Hitozanske nanodelce, v katere smo pripeli modelno zdravilo, smo naknadno nanesli na celulozna vlakna z namenom kreiranja modernih, vaginalnih dostavnih sistemov. Protimikrobne medicinske tekstilije raziskane v sklopu te disertacije kažejo potencialno uporabo na ginekološkem področju, kot preventiva ali kurativa, brez neželenih stranskih učinkov za uporabnika; ● dne 27. januarja 2014 Mitja KRAJNC z naslovom: »Odločitveni model najustreznejših parametrov izdelave lesnih sekancev« (mentor: izr. prof. dr. Bojan Dolšak); Z uporabo naprednih tehnologij pri delovanju kotlov na lesno biomaso pridobiva lesno gorivo vse večji pomen.

Izdelava kvalitetnega lesnega goriva zahteva veliko znanja in izkušenj, povezana pa je z večjim številom vplivnih parametrov. Uvrstimo jih lahko v tri kategorije, in sicer materialne, konstrukcijske in tehnološke. Materialne se nanašajo na lesno biomaso, tehnološke in konstrukcijske pa na sekalnik. Na materialne parametre imamo omejen vpliv, saj v sekance običajno predelamo surovino, ki je trenutno na voljo. Lahko pa izboljšujemo proces proizvodnje z izbiro ustreznih vrednosti konstrukcijskih in tehnoloških parametrov.

SI 27

Strojniški vestnik - Journal of Mechanical Engineering 60(2014)2, SI 27-28

Na osnovi monitoringa dimnih plinov in analize pepela smo določili najprimernejše lastnosti lesnih sekancev, ki zagotavljajo najvišji termični izkoristek. Izvajali smo ga na kotlu, moči 3.500 kW, v kotlovnici Lenart. Ugotovili smo, da so za večje sisteme primernejši sekanci večjega velikostnega razreda pri mehkem lesu, ter manjšega velikostnega razreda pri trdem lesu. Lesni sekanci naj vsebujejo čim manj prahu ter imajo čim bolj enakomerno velikostno strukturo.

V okviru raziskovalnega dela smo izdelali računalniški model velikega bobnastega sekalnika, ki smo ga uporabljali pri praktični izdelavi lesnih sekancev. S pomočjo kinematične analize smo simulirali možne spremembe vplivnih parametrov in ocenili teoretično dolžino lesnih sekancev. Izbrali smo vplivne parametre, ki v največji meri vplivajo na kvaliteto sekancev, določili njihove vrednosti in raziskali njihov medsebojni vpliv.

V okviru eksperimentalnega dela doktorske disertacije smo proizvajali lesne sekance želenih lastnosti in jih primerjali z rezultati računalniške simulacije. Ugotovili smo odstopanja in pojasnili vzroke za njihov nastanek.

S pomočjo rezultatov praktičnih testiranj smo izdelali odločitveni model, ki na podlagi danih lastnosti vhodne surovine in želenih lastnosti lesnih sekancev predlaga najprimernejše vrednosti konstrukcijskih in tehnoloških parametrov. Odločitveni model smo vgradili v spletno aplikacijo, ki smo jo poimenovali Bober.

DIPLOMSKE NALOGE Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva: dne 22. januarja 2013: Domen ŽAGAR z naslovom: »Jeklena konstrukcija za solarne panele« (mentor: prof. dr. Jožef Duhovnik); dne 23. januarja 2013: Dejan OTONIČAR z naslovom: »Brezdotična kontrola premera kovičnih luknjic« (mentor: prof. dr. Janez Diaci); Janez STRLE z naslovom: »Vizualizacija uparjanja v ploščnem prenosniku toplote« (mentor: prof. dr. Iztok Golobič).

* Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv magister inženir strojništva: dne 22. januarja 2014: Tomaž ČAMPA z naslovom: »Analiza togosti in toplotne prevodnosti pultriranega kompozitnega nosilca« (mentor: prof. dr. Boris Štok, somentor: doc. dr. Miroslav Halilovič); dne 23. januarja 2013: Andrej OŠLAK z naslovom: »Laserska izdelava merilnih črtic po celotnem obodu okrogle merilne letve« (mentor: prof. dr. Janez Možina, somentor: doc. dr. Matija Jezeršek).

SI 28

* Na Fakulteti za strojništvo Univerze v Mariboru je pridobil naziv magister inženir strojništva: dne 29. januarja 2014: Matej ZAJC z naslovom: »Razvoj preizkusne naprave za kontaktne probleme« (mentor: prof. dr. Srečko Glodež).

* Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva: dne 9. januarja 2014: David FERLAT z naslovom: »Tlačne razmere znotraj hidravličnih sistemov s potnimi, tokovnimi in tlačnimi ventili« (mentor: doc. dr. Franc Majdič); Aleš MAVSER z naslovom: »Vstopniki zraka za hlajenje potisnega električnega motorja« (mentor: izr. prof. dr. Tadej Kosel, somentor: izr. prof. dr. Tomaž Katrašnik); Renato LEBAN z naslovom: »Posodobitev proizvodnje pločevinastih komponent za peskalne stroje« (mentor: prof. dr. Peter Butala); Aleš SEDMAK z naslovom: »Optimizacija procesa montaže sesalnika« (mentor: izr. prof. dr. Niko Herakovič); Janko ŠKARABOT z naslovom: »Zasnova hidravličnega krmilja naprave za praznjenje smetarskih sodov« (mentor: izr. prof. dr. Niko Herakovič); Matej ŽABAR z naslovom: »Vpliv visokotlačnega dovoda rezalnega olja na odrezovalnost« (mentor: doc. dr.

Davorin Kramar, somentor: prof. dr. Janez Kopač).

* Na Fakulteti za strojništvo Univerze v Ljubljani sta pridobila naziv diplomirani inženir strojništva (VS): dne 9. januarja 2014: Jure POCRNJIČ z naslovom: »Termodinamsko in emisijsko vrednotenje laboratorijskega turbinskega motorja« (mentor: izr. prof. dr. Tomaž Katrašnik); Jani KONDA z naslovom: »Izboljšanje tehnologije obdelave ohišja reduktorja« (mentor: prof. dr. Janez Kopač).

* Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva:

dne 29. januarja 2014: Anton ZORIČ z naslovom: »Planiranje proizvodnje v podjetju Vivapen d.o.o.« (mentor: izr. prof. dr. Borut Buchmeister, somentor: doc. dr. Marjan Leber); Blaž HROVATIČ z naslovom: »Konstrukcija traktorskega pluga za drenažo« (mentor: doc. dr. Janez Kramberger).

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

Founding Editor Bojan Kraut

University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Editorial Office University of Ljubljana, Faculty of Mechanical Engineering SV-JME, Aškerčeva 6, SI-1000 Ljubljana, Slovenia Phone: 386 (0)1 4771 137 Fax: 386 (0)1 2518 567 info@sv-jme.eu, http://www.sv-jme.eu Print: Littera Picta, printed in 420 copies Founders and Publishers University of Ljubljana, Faculty of Mechanical Engineering, Slovenia University of Maribor, Faculty of Mechanical Engineering, Slovenia Association of Mechanical Engineers of Slovenia Chamber of Commerce and Industry of Slovenia, Metal Processing Industry Association President of Publishing Council Branko Širok University of Ljubljana, Faculty of Mechanical Engineering, Slovenia

Vice-President of Publishing Council Jože Balič

University of Maribor, Faculty of Mechanical Engineering, Slovenia Cover: Four high response digital hydraulic piezo poppet valves, presented as digital fluid control unit, could be used instead of conventional hydraulic servo valves.

Development of new poppet valve by using CFD (Computer Fluid Dynamics) and FEM (Finite Element Methods) analyses, new digital electronics and new control algorithms results in a better dynamic characteristics and lower power consumption compared to the conventional servo hydraulic valves.

Image Courtesy: University of Ljubljana, Faculty of Mechanical Engineering, Department of Manufacturing Technologies and Systems, Laboratory for Handling, Assembly and Pneumatics, Slovenia

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 Mechanical Engineering, Slovenia Narendra B. Dahotre, University of Tennessee, Knoxville, USA Matija Fajdiga, UL, Faculty of Mechanical Engineering, Slovenia Imre Felde, Obuda University, Faculty of Informatics, Hungary Jože Flašker, UM, Faculty of Mechanical Engineering, Slovenia Bernard Franković, Faculty of Engineering Rijeka, Croatia Janez Grum, UL, Faculty of Mechanical 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 Mechanical Engineering, Slovenia Franc Kosel, UL, Faculty of Mechanical Engineering, Slovenia Thomas Lübben, University of Bremen, Germany Janez Možina, UL, Faculty of Mechanical 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 Mechanical Engineering, Slovenia Leopold Škerget, UM, Faculty of Mechanical 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 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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ISSN 0039-2480 © 2014 Strojniški vestnik - Journal of Mechanical Engineering.

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Acknowledgement of collaboration or preparation assistance may be included before References. Please note the source of funding for the research.

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.

References must be numbered and ordered according to where they are first mentioned in the paper, not alphabetically. All references must be complete and accurate. All non-English or. non-German titles must be translated into English with the added note (in language) at the end of reference. Examples follow.

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.

Note that the title of the book is italicized.

Chapters in Books: Surname 1, Initials, Surname 2, Initials (year). Chapter title. Editor(s) of book, book title. Publisher, place of publication, pages.

[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.

Standards: Standard-Code (year). Title. Organisation. Place.

[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.

EXTENDED ABSTRACT By the time the paper is accepted for publishing, the authors are requested to send the extended abstract (approx. one A4 page or 3.500 to 4.000 characters). The instructions for writing the extended abstract are published on the web page http://www.sv-jme.eu/ information-for-authors/.

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PUBLICATION FEE For all articles authors will be asked to pay a publication fee prior to the article appearing in the journal. However, this fee only needs to be paid after the article has been accepted for publishing. The fee is 300.00 EUR (for articles with maximum of 10 pages), 20.00 EUR for each addition page.

Additional costs for a color page is 90.00 EUR.

http://www.sv-jme.eu

60 (2014) 2

Strojniški vestnik Journal of Mechanical Engineering

Since 1955

Papers

77

Marko Šimic, Mihael Debevec, Niko Herakovič: Modelling of Hydraulic Spool-Valves with Specially Designed Metering Edges

84

Bojan Starman, Marko Vrh, Mirko Halilovič, Boris Štok: Advanced Modelling of Sheet Metal Forming Considering Anisotropy and Young’s Modulus Evolution

93

Jixin Wang, Long Kong, Bangcai Liu, Xinpeng Hu, Xiangjun Yu, Weikang Kong: The Mathematical Model of Spiral Bevel Gears - A Review

106

David Vegelj, Boštjan Zajec, Peter Gregorčič, Janez Možina: Adaptive Pulsed-Laser Welding of Electrical Laminations

115

Erik Čuk, Matjaž Gams, Matej Možek, Franc Strle, Vera Maraspin Čarman, Jurij F. Tasič: Supervised Visual System for Recognition of Erythema Migrans, an Early Skin Manifestation of Lyme Borreliosis

124

Dragica Jošt, Aljaž Škerlavaj, Andrej Lipej: Improvement of Efficiency Prediction for a Kaplan Turbine with Advanced Turbulence Models

135

Fuhai Cai, Xin Wang, Jiquan Liu, Fuling Zhao: Fatigue Life Analysis of Crane K-Type Welded Joints Based on Non-Linear Cumulative Damage Theory

Journal of Mechanical Engineering - Strojniški vestnik

Contents

2 year 2014 volume 60 no.