Shape Optimization of Clutch Disc Using Differential Evolution Method

1 Copyright 2015 by ASME

Proceedings of the ASME 2015 International Mechanical Engineering Congress & Exposition IMECE2015

November 13-19, 2015, Houston, Texas

DRAFT

IMECE2015-51378

SHAPE OPTIMIZATION OF CLUTCH CUSHION DISC USING DIFFERENTIAL EVOLUTION METHOD

N. KAYA Department of Mechanical Engineering

Uludag University, Bursa, Turkey

[email protected]

S. KARTAL Valeo A..

Bursa, Turkey [email protected]

T. AKMAK Valeo A..


F. KARPAT Department of Mechanical Engineering

Uludag University, Bursa, Turkey

[email protected]

A.KARADUMAN Valeo A..


ABSTRACT The clutch is an element which makes a temporary connection

between gear box and vehicle engine. It transmits not only

engine torque, but also ensures comfort and drivability during

slippage. One of the main functions of clutch disc is to transmit

the engine torque with absorbing vibrations. It allows a soft

gradual reengagement of torque transmission. Cushion disc

which is located between two clutch facings has wavy surface,

thus it behaves like a spring during engagement and

disengagement. The axial elastic stiffness of the clutch disc is

obtained by a cushion disc. The load-deflection curve is obtained

by compressing clutch disc between two plates, representing

pressure plate and flywheel. The wavy shape of the cushion disc

provides progressive stiffness curve of the clutch disc.

The cushion disc participates in drivers comfort during

engagement of the clutch. The comfort depends on the limits of

the progressive stiffness curve. Outside the limits of this cushion

function, the clutch engagement would be harsh and

uncomfortable for the driver. Besides, engine torque may not be

transmitted during the later service lifetime and life of the clutch

might be decreased. In the case of, cushion disc has no

cushioning function, engine might be stopped. Additionally,

improper cushioning function cause to heat and deform of the

pressure plate and it also decreases the transmitted engine torque.

Therefore, cushion disc has to have certain cushioning

characteristics in order to overcome these problems.

In this study, optimum shape design of cushion disc was

performed using an evolutionary optimization algorithm.

Differential evolution algorithm was selected as optimization

method because it guarantees the global optimum. Design of

experiment method has been employed to construct the response

surface that approximates the behavior of the objective function

inside a certain design space. Three shape parameters of cushion

disc have been selected. The objective of the shape optimization

is to find the optimum shape parameters that provide the target

stiffness curve. After solving the optimization problem with

differential evolution method, optimum shape parameters of

cushion disc have been found for two case studies. A Pascal code

based differential evolution optimization code was developed for

shape optimization and Ansys finite element software was used

for calculating stiffness curve of cushion disc.

INTRODUCTION Vehicle clutch disc transmits the engine torque with absorbing

vibrations and allows a soft gradual reengagement of torque

transmission. The axial elastic stiffness of the clutch disc is

obtained by a cushion disc. The load-deflection curve is obtained

by compressing clutch disc between two plates. The wavy shape


of the cushion disc provides progressive stiffness curve of the

clutch disc.

Design of the clutch starts with the pedal force designated by the

vehicle manufacturer, then target cushion disc stiffness curve is

calculated. This process is mainly based on designers experience

or trial and error method, therefore it is time consuming and

costly process. In this study, a methodology is proposed in order

to overcome these problems.

Studies on cushion disc in the literature are quite limited. Only a

few studies have been found about finite element modeling of

cushion disc. Parameter studies were performed to determine the

effective parameters on the stiffness curve, but no study has been

found about shape optimization of cushion disc to have target

stiffness curve.

It is aimed to investigate the temperature influence on the

cushion spring characteristic modification and the consequent

torque transmissibility curve in [1]. It is highlighted that an

increment of the temperature level result in a decrease of the

material stiffness and this is underlined by a curve slope

modification. In our study, influence of the temperature on the

cushion spring load-deflection characteristic has not taken into

account. Sfarni et al. [2] studied the influence of geometrical

parameters of cushion disc by design of experiments (DOE).

They identified the most influent geometrical parameters. Sfarni

et al. [3,4] proposed a finite element riveted clutch disc model in

order to predict the cushion curve. In order to avoid the wear

facing which degrades the cushion curve stability and the

drivers comfort, they verified that the knowledge of the contact pressure distribution enables its prediction for different designs

of riveted clutch discs. Sfarni et al. [5] proposed a comparison

between the values of contact pressures obtained with a FE

model and a test. Their work leading to a framework for the

drawing up of design rules for riveted clutch disc in term of

stability. The functions and design requirement return and

cushion spring are reviewed in [6]. A comparison of results in

terms of space, weight, costs and transmission performance is

also provided. It is intended to optimize the performance of the

automotive clutch system in [7]. The modeling and simulation of

an automotive clutch system is carried out and a control strategy

is derived for optimizing its performance. A mathematical model

of a simplified clutch system is built for the analysis of its

dynamic behavior. A sensitivity analysis was carried out to

evaluate the effective structure of the model. Kaya [8] performed

shape optimisation of an automobile clutch diaphragm spring

using a genetic algorithm. A design proposal is determined with

the topology optimisation approach, and then design

optimisation by response surface methodology was effectively

used to improve the new clutch fork design in [9].

Some researches include the finite element model of clutch

elements such as cushion disc, diaphragm spring etc. Parametric

studies were performed using design of experiment method and

most effective parameter was determined [2,3,4,5]. Less

attention has been paid to optimization of the clutch elements

and no study has been found about shape optimization of cushion

disc to have target stiffness curve.

Recently, the use of non-deterministic algorithms have attracted

the researchers to find global optimum. Among the non-

deterministic methods, the Differential Evolution (DE)

algorithm produced good results in the literature for different

applications in science and engineering. DE and Particle Swarm

Optimization methods have been applied to the design of

minimum weight toroidal shells subject to internal pressure. The

optimization process is performed by Fortran routines coupled

with finite element analysis code Abaqus [10]. An investigation

into structural topology optimization using a modified binary DE

with a newly proposed binary mutation operator is performed

[11]. Carrigan et al. [12] introduced and demonstrated a fully

automated process for optimizing the airfoil cross-section of a

vertical-axis wind turbine using a parallel DE algorithm. A

framework for the shape optimization of aerodynamics profiles

using computational fluid dynamics and genetic algorithms

proposed by Lopez et al. [13]. A DE optimization based

technique is proposed to find the optimum value of a modified

Bezier curve. The proposed equation contains shaping

parameters to adjust the shape of the fitted curve [14].

In this study, shape optimization of cushion disc to have a desired

stiffness curve has been performed. Stiffness curves of cushion

disc were obtained by finite element method. Design of

experiment study was conducted and curve fitting was applied to

determine the objective function. A Pascal code based

differential evolution algorithm was developed for shape

optimization. Developed optimization software were tested with

two test functions, then optimization was performed for two case

studies.

DIFFERENTIAL EVOLUTION ALGORITHM

One of the main shortcoming of classical optimization methods

is sticking into local optimum instead of global optimum.

Genetic Algorithm and Differential Evolution algorithms are

evolutionary optimization algorithms, they were developed for

finding the global optimum of the optimization problems. DE is

a relatively new evolutionary optimization algorithm. It is a

population-based optimization method introduced by Storn et al.

[15]. They developed a new robust, versatile and easy-to-use

global optimization algorithm and published it under the name

differential evolution algorithm in 1995. This algorithm, like

other evolutionary algorithms, has a population-based structure,

and it attacks the starting point problem using a real-coded

system and a new differential mutation operator. The DE

algorithms main strategy is to generate new individuals by calculating vector differences between other individuals of the

population. The DE algorithm includes three important

operators: mutation, crossover and selection. In the DE, a

population vectors are randomly created at the start of iteration.

This population is successfully improved by applying mutation,

crossover and selection operators, respectively. Mutation and


crossover are used to generate new vectors (trial vectors), and

selection then are used to determine whether or not the new

generated vectors can survive the next iteration. Among the

strategies in DE algorithm, DE/rand/1/bin DE strategy was used.

The details of DE algorithm are given below.

DE consists of two fundamental phases: initialization and

evolution [16]. In the initialization phase, just like in other

evolutionary algorithms, an initial population (P0) is generated.

After that, the P0 population evolves to P1, P1 evolves to P2 and

so on. In this way, evolution of new populations is continued

until the termination conditions are fulfilled. While evolving

from the Pn to Pn+1, three evolutionary operations are executed

on the individuals in the current population. These operations are

differential mutation, crossover and selection [16].

Initialization

In this stage, the initial population P0 is randomly created from

Np number of individuals:

0, =

+ (

), 1 (1)

where 0 means the initial population, i is the sequence of the

population, j is the number of individuals in the population,

is the real random number generator in the ith population and jth

individual, is the lower value of the jth individual and

is

the upper value of the jth individual.

Differential Mutation

In mutation, a mutant (vn+1,i) and a mutant vector (xn+1,v,i) are

created for each pn,i individual, called a mother, in the Pn

population. It should not be forgotten that x is a vector that

represents all individuals in the current population (x = x1, x2, , xN).

Mutant vector xn+1,v,i is created as follows:

x+1,, = x,, + (x,1 x,2)

1

,

1 1 2 (2)

where xn,b,i is the base vector (b) selected for the new individual

that will be created for the ith old individual in the nth population,

xn,P1,i is the P1yth individual selected randomly from between

[1,NP] integers. Similarly, xn,P2,i is the P2yth individual selected

randomly from between [1,NP] integers, and Fy is the scale factor

for the yth vector difference in the range of [0,1].

The xn,b,i base vector can be selected in different ways:

from the current vector: x,, = x,, , ( = ), from the best vector: x,, = x,,, (b = the best), from the better vector: x,, = x,, , (b = the better), from a random vector: x,, = x,, , (b = random).

After the mutation process, the new individual can be created

outside the range of [,

]. Various methods have been

proposed for infeasible individuals.

Crossover

In this process, a new child individual (cn+1,i) is created by mating

the new individual (xn+1,i) that is created in the mutation process

with the current individual (pn,i) in the population according to

the crossover probability Cr. Here, pn,i is referred to as the

mother, and xn+1,i is referred to as the father.

Selection

There is a competition between mother and child in the selection

operation. They compete with each other according to objective

function values to survive in the next generation. This

competition is formulated mathematically as follows:

p+1, = {c+1, , (c+1, > p,)

p, , (3)

The key parameters of control in DE are:

NP: the population size (number of individual),

Cr: the crossover constant (probability) (0.0 -1.0),

Fy: scaling factor that controls the amplification of

differential variations (0.0 2.0).

During the iterations of DE algorithm, various feasible and

unfeasible individuals may appear. Regular DE operators can

produce unfeasible individuals. It means that some individuals

may violate the constraints. For example at some stage of the

evolution process, a population may contain some feasible and

unfeasible individuals. Therefore, several trends for handling

unfeasible solutions have emerged in the area of evolutionary

computation. In this study, any individual which do not comply

with the constraints is eliminated and a new individual is created.

This insures that the size of the population remains constant even

when eliminating those individuals which violate the constraints.

Therefore, every individual in the population satisfies the

constraints.

In this study, DE algorithm was selected for shape optimization

due to following reasons [17];

- It finds the lowest fitness value for most of the problems,

- DE is robust; it is able to reproduce the same result consistently over many trials,

- It is simple, robust, converges fast, and finds the optimum in almost every run.

DE algorithm is slower than the other evolutionary algorithms

especially for noisy problems. This is the disadvantage of the DE

algorithm.


In this study, a Pascal programming language based DE

optimization software was developed and validated using two

test functions [18]. After validation of the developed DE

optimization software, optimum shape parameters of cushion

disc were determined.

FINITE ELEMENT MODELING OF CUSHION DISC

Cushion disc which is located between two clutch facings has

wavy surface, thus it behaves like a spring during engagement

and disengagement. Cushion disc is fixed by rivets between two

facings as shown in Figure 1.

Fig. 1: Vehicle clutch mechanism [2]

Stiffness curve is obtained by compressing a cushion disc

between two flat pressure plates. It has non-linear characteristics

as shown in Figure 2. As a design request, this curve is desirable

in between two limit curves. Corrugated type surface is the main

reason for obtaining this progressive cushion curve.

Fig. 2: A typical stiffness curve of cushion disc and its limits

In order to determine the stiffness curve of the cushion disc,

nonlinear finite model were defined using Ansys software.

Because of the thickness is constant, mid-surface is obtained

from the solid model and the surface is transferred to finite

element software for modeling. Surface geometry of cushion

disc is shown in Figure 3.

Fig. 3: Cushion disc mid-surface geometry

Cushion disc geometry is cyclic symmetric, therefore 1/4 model

was used in finite element model. The cushion disc is meshed

with shell elements. Disc thickness is 0.7 mm. Linear elastic

material law is employed. Finite element model is given as

shown in Figure 4.

Fig. 4: 1/4 cyclic symmetry finite element model

The pressure plates are modelled by two rigid plates (Figure 5).

Plates compress the disc axially to simulate its behavior during

the re-engagement. Two frictional contacts were defined

between cushion disc and rigid plates. In this study, augmented

lagrangian method was used to solve the contact problem.

Bottom plate is fixed and upper plate is also fixed except for axial

translation.


Fig. 5: Top and bottom plates

The top plate compresses the cushion disc by forces in axial

direction. Internal ring of cushion disc is restrained in order to

constrain rigid body motion. Cyclic symmetry boundary

conditions were applied on both sides of the cushion disc. During

the analysis, force and displacement values were stored to obtain

stiffness curve.

DESIGN OF EXPERIMENT AND RESPONSE SURFACE METHOD

The computation time required for structural analysis is a major

obstacle in structural optimization studies. Representative

metamodels empirically capture the inputoutput relationship of structural analysis for evaluating the objective functions and

constraints. They are utilized for two reasons, the first of which

is to obtain the global behaviour of the original functions. The

second is to shorten the optimization calculation time by using

surrogate functions that can quickly return approximate values

instead of relying on time-consuming functions [19].

In this study, the response surface method (RSM) is used for

obtaining objective function. The objective of the shape

optimization of cushion disc is to find the shape parameters that

provide the desired stiffness curve.

The difference between the calculated displacement point (ucalc)

and the target displacement point (utarget) data for each point in a

stiffness curve is measured by a statistical term called chi-square

which is given as follows:

chi square = (ucalc_iutarget_i)

2

utarget_i

ni=1 (4)

Here, n is the measurement points shown in Figure 6.

If the chi-square is large, then the calculated and target curves

are not close to each other. If the two curves are exactly the same,

chi-square will be zero. The large value means that two curves

are not identical and very close from each other. In this study,

chi-square must be as small as possible to have the desired

stiffness curve of cushion disc.

ucalc refers to the stiffness curve point of calculated by finite

element analysis.

In this study, minimization of chi-square was selected as

objective function.

Fig. 6: Chi-square calculation points

Generally, the RSM consists of three steps. First, a series of

experiments, i.e., designs of experiments (DOE), which will

yield adequate and reliable measurements of the response of

interest, are obtained. Then a mathematical model that best fits

the data collected from the execution of the experimental design

is determined. Using a sufficient number of values, which

depends on the number of design variables and the type of

function used in curve fitting, the RSM defines a surface that

approximates the behaviour of the objective function inside a

certain design space. Finally, the optimum setting of the

experimental factors that produces the maximum (or minimum)

value of the response is found. In this work, third order

polynomials are used as fitting curves.

DIFFERENTIAL EVOLUTION BASED SHAPE OPTIMIZATION

DOE studies are defined as a series of tests in which input

variables of a process or a system are intentionally changed so

that the causes for changes in the output response can be

identified and observed. In a CAE model, the factors such as

thickness, shape design variables and material properties can be

changed to study the output responses of the model

The full factorial DOE method is applied in this study. This

method investigates all possible combinations of the factor levels

(L) and consequently enables the study of all possible

interactions between factors (N). A full factorial study requires

LN runs. Such a design is beneficial for calculating all main and

interaction effects. The use of a full factorial design is only

practical when the number of factors and the number of factor

levels are small. In this study, three shape parameters (factors)

were selected as shown in Figure 7. These are Bx and By (wide

of the folds) and A is the height of the profile.

0

1000

2000

3000

4000

5000

6000

7000

0 0.2 0.4 0.6 0.8 1

Load

[N

]

Deflection [mm]

ucalc_2

ucalc_4

ucalc_n

. . .

ucalc_1

ucalc_3

utarg_4

utarg_1

utarg_2

utarg_3

utarg_n

. . .


Fig. 7: Design parameters (height of the profile A is not

shown here)

Two case studies were given below for shape optimization. Chi-

square value was calculated using actual and target stiffness

curves for objective function. The shape optimization problem is

defined as:

objective: min chi-square

subject to

1.0 1.2, 8.3 10.3, 22.2 24.2

Case Study 1: As a first case study, a target stiffness curve is

given between maximum and minimum stiffness curves in

Figure 8.

Fig. 8: Desired curve for case study 1

According to full factorial design with three levels and three

parameters, 33 = 27 finite element runs were executed, and the

chi-square are obtained as given in Table 1.

Based on the DOE results in Table 1, the response surface model

for chi-square was constructed using a third degree polynomial

as follows (x: A, y: Bx, z:By):

= 5.684 3.339 4.2642 + 0.4823 + 0.186 0.730 + 0.6902 + 0.002042

0.02212 + 0.001073 0.00416+ 0.01086 + 0.19412 0.00276 0.0002272 0.002682 0.006792

+ 0.000182 + 0.0001253

How well the estimated response function fits the design of

experiments is determined by the coefficient of determination,

r2, calculated as 0.99. This optimization problem was solved with

developed software based on differential evolution algorithm.

User interface, input parameters and results are given in Figure

9.

Run # A (mm)

Bx

(mm) By

(mm) Chi-square

1 1 8.3 22.2 0.04293

2 1 9.3 22.2 0.02156

3 1 10.3 22.2 0.01068

4 1 8.3 23.2 0.03439

5 1 9.3 23.2 0.01709

6 1 10.3 23.2 0.00848

7 1 8.3 24.2 0.02853

8 1 9.3 24.2 0.01399

9 1 10.3 24.2 0.00724

10 1.1 8.3 22.2 0.01080

11 1.1 9.3 22.2 0.02367

12 1.1 10.3 22.2 0.04200

13 1.1 8.3 23.2 0.01454

14 1.1 9.3 23.2 0.02956

15 1.1 10.3 23.2 0.04946

16 1.1 8.3 24.2 0.01901

17 1.1 9.3 24.2 0.03610

18 1.1 10.3 24.2 0.05793

19 1.2 8.3 22.2 0.12386

20 1.2 9.3 22.2 0.18700

21 1.2 10.3 22.2 0.24964

22 1.2 8.3 23.2 0.14632

23 1.2 9.3 23.2 0.20743

24 1.2 10.3 23.2 0.26924

25 1.2 8.3 24.2 0.16565

26 1.2 9.3 24.2 0.22738

27 1.2 10.3 24.2 0.28931

Tab. 1: Full factorial design table for case study 1.

By Bx


Fig. 9: Differential evolution parameters and optimum

results for case study 1

After the solution phase completed, finite element model was

solved and stiffness curve was obtained according to optimum

parameters. As seen in Figure 10, optimum curve is very close to

target curve. Thus, optimization methodology was successfully

applied to shape optimization problem in this case study.

Fig. 10: Optimization result for target curve 1

Case study 2: In the second case study, a different characteristic

curve is expected. Target stiffness curve is given in Figure 11.

Fig. 11: Desired curve for case study 2

As in case study 1, a new DOE table is constructed according to

new target curve. Again, the response surface model for chi-

square was constructed using a third degree polynomial function.

DE parameters were selected same as for case study 1. After

solving the optimization problem with DE algorithm, optimum

shape parameters were found as in the Figure 12.

Fig. 12: Differential evolution parameters and optimum

results for case study 2

As seen in Figure 13, optimum curve is very close to target curve.

In this case study, optimum curve is the best one that matches the

target curve2.

Fig. 13: Optimization result for target curve 2

Proposed methodology was successfully applied to shape

optimization problem of cushion disc. It may be considered that

this gives a systematic guidance to the designer of spring

elements. By a similar method, this approach can be used in the

design of other types of spring products in the automotive

industry.

CONCLUSION

Today, the use of optimization methods in the automotive

industry is not yet fully integrated into the design process. CAE

analysis and optimization tools save development time and

reduce costs in the conceptual design phase for new and failed

parts. Therefore, robust and innovative design proposals must be

developed early. The product development process becomes

faster and more efficient by using optimization methods. In this

study, a differential evolution algorithm based shape

optimization is presented. A Pascal code based on DE algorithm

was developed to solve shape optimization problems. DE


algorithm was successfully applied to shape optimization of

cushion disc to obtain target stiffness curves. Ansys software

was used for the FE calculation of objective function. The

proposed method can shorten the cushion disc design cycle and

decrease the trial-and-error efforts.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of Turkish

Technology and Science Minister under grant San-Tez project

0634.STZ.2014 ongoing with collaboration between Uludag

University and Valeo Company in Turkey.

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Shape Optimization of Clutch Disc Using Differential Evolution Method

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