shape optimization of clutch disc using differential evolution method
DESCRIPTION
An interesting article about shape optimization. (N.Kaya, S. Kartal, T. Cakmak,F.Karpat, A.Karaduman)TRANSCRIPT
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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
S. KARTAL Valeo A..
Bursa, Turkey [email protected]
T. AKMAK Valeo A..
Bursa, Turkey [email protected]
F. KARPAT Department of Mechanical Engineering
Uludag University, Bursa, Turkey
A.KARADUMAN Valeo A..
Bursa, Turkey [email protected]
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
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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
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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.
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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.
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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
. . .
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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
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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
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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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