2013 multiobjective robust design optimization of rail vehicle moving in short radius curved tracks...

8/13/2019 2013 Multiobjective Robust Design Optimization of Rail Vehicle Moving in Short Radius Curved Tracks Based on th

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Multiobjective robust design optimization of rail vehicle moving

in short radius curved tracks based on the safety and comfort criteria

M. Nejlaoui a,, A. Houidi b, Z. Affi a, L. Romdhane c

a Laboratoire de Gnie Mcanique, Ecole Nationale dIngnieurs de Monastir, Avenue Ibn Eljazzar, 5019 Monastir, Tunisiab Laboratoire de Mcanique de Sousse, Institut Suprieur des Sciences Appliques de Sousse, Sousse, Cit Etaffela, 4003 Sousse Ibn khouldoun, Tunisiac Laboratoire de Mcanique de Sousse, Ecole Nationale dIngnieurs de Sousse, Avenue 18 janvier, Bab jedid, 4000 Sousse, Tunisia

a r t i c l e i n f o

Article history:

Received 29 July 2011

Received in revised form 10 July 2012

Accepted 23 July 2012

Available online 11 October 2012

Keywords:

Robustness

Rail vehicle

Safety

Comfort

a b s t r a c t

This work deals with the multiobjective robust design optimization of rail vehicle systems

moving in short radius curved tracks. Two criteria are considered simultaneously, i.e.,

safety (considered by the derailment risk) and comfort given by noise level. The authors

show that the deterministic optimal solutions, for the nominal design parameters, can

be altered seriously by the design parameters uncertainty. The authors of this paper pro-

pose an original algorithm that combines Genetic Algorithms and Monte Carlo Simulation

in order to be used for the robust multiobjective optimization of the rail vehicle design. The

obtained solutions, presented by design vectors of the rail vehicle, are analyzed in terms of

performances and robustness. The authors show that the robust multiobjective optimiza-

tion can yield solutions less sensitive to the design parameters uncertainties.

2012 Elsevier B.V. All rights reserved.

1. Introduction

It is a common practice in a product design to consider the nominal value only as input variables for design optimization.

The design parameters (DPs) have usually an uncertainty around their nominal values due to the presence of variations in

manufacturing, geometry and material properties. To avoid erroneous results due to these uncertainties, the designer uses

generally an arbitrary safety factor. This practice usually leads to a high product cost. As an alternative, we adopt the robust

design strategy. A robust design is the one that is less sensitive to the DP uncertainties. The pioneer in the robust design do-

main is Taguchi,[1]. The complexity of the Taguchi method is proportional to the number of the DP. Moreover, the robust-

ness is usually evaluated at the end of the deterministic design process.

Several works focused on the deterministic design optimization of mechanical systems. Rajagopal and Ganguli [2] studied

unmanned aerial vehicle (UAV) conceptual design using a multiobjective Genetic Algorithm. The defined objective functions

are the endurance maximization and the wing weight minimization. Based on the dynamics of the railway vehicle duringmotion along a curved track, Zboinski [3]optimized the rail vehicles (RV) vibrations and stability. He and McPhee[4]used

the nominal value of DP to develop a mono-objective optimization design through Genetic Algorithms. The objective func-

tion is a weighted combination of the angle of attack and the ratio of the lateral force to the vertical one applied by each

wheel on the rail. Rejeb et al. [5]optimized the critical speed as a function of the nominal value of design variables of the

RV system in a rectilinear motion using the Genetic Algorithm method. Majka and Hartnett [6]studied the effects of the

nominal value variations of some DP on the dynamic response of the railway bridges.

1569-190X/$ - see front matter 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.simpat.2012.07.012

Corresponding author. Tel.: +216 73 500 511; fax: +216 73 500 514.

E-mail addresses:[email protected](M. Nejlaoui),[email protected](A. Houidi),[email protected](Z. Affi),lotfi.romdha-

[email protected](L. Romdhane).

Simulation Modelling Practice and Theory 30 (2013) 2134

Contents lists available at SciVerse ScienceDirect

Simulation Modelling Practice and Theory

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e/ s i m p a t
http://dx.doi.org/10.1016/j.simpat.2012.07.012mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.simpat.2012.07.012http://www.sciencedirect.com/science/journal/1569190Xhttp://www.elsevier.com/locate/simpathttp://www.elsevier.com/locate/simpathttp://www.sciencedirect.com/science/journal/1569190Xhttp://dx.doi.org/10.1016/j.simpat.2012.07.012mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.simpat.2012.07.012


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To estimate the effect of DP uncertainty on the performance of a product, several methods have been described in the

literature. In particular, the Monte Carlo simulation (SMC) is a popular tool because of its relative precision and simplicity

[7]. Chandrashekhar and Ganguli [8] developed a method based on fuzzy logic for damage detection in the steel beam having

material uncertainty. To evaluate statistical properties of the frequency variations of a beam, they performed a probabilistic

analysis using the SMC. In order to optimize the crash box and the dumper beam of passenger cars, Hilmann et al. [9]used

the SMC by varying all the DP under the assumption of uniform statistical distributions. Murugan et al.[10]illustrated the

need to consider uncertainty of composite material properties in the helicopter aeroelastic analysis by using a random elastic

modulus and Poissons ratio.

Other works have studied the robust product design where uncertainties of the DP are considered. Xiaoping et al. [11]

developed a strategy for the robustness assessment and robust mechanism synthesis when random and interval variables

are involved. They used the SMC to quantify the variability of performances. Ouisse and Cogan [12]presented a robustness

methodology of spot weld resistance, based on energetic consideration, in the vehicle design process. Doltsinis and Kang [13]

developed a methodology for robust structures design. They used a gradient based optimization algorithm to minimize both

the mean value and the standard deviation of a given objective function. Guedri et al. [14] proposed a method that quantifies

the effects of uncertainties of design parameters on the variability of dynamic behavior of mechanical structures. The goal is

to find optimal and robust solutions resulting from numerical simulations based on GA multiobjective optimization. Al-Ao-

mar [15] developed robust design approach by incorporating the Taguchi method and the GA. The robustness is quantified by

the signal-to-noise ratio. Bouazizi et al. [16] studied the robust optimization of a vibration absorber using the GA. The robust-

ness, defined by the ratio of the mean value to the standard deviation, is treated as an objective function. Ghanmi et al. [17]

Nomenclature

i index of wheelsetj index of wheel

k index of bogieg the gravity constantGki wheelset center of mass

Gk bogie center of massG car body center of massm half wheelset mass

m axle box body massM bogie mass

M car body mass

N a vertical contact forceH vertical distance between the primary and the secondary suspension

h0 vertical distance between the primary suspension and the bogie center of massd the rail inclination

cnc lateral acceleration of the rail vehiclec0 inclination of the tangent plan of contact wheelrail with the horizontale0 half of the track gauge

ce

equivalent conicityRc radius of curveyki transversal displacement of the wheelset i of bogiek

yk transversal displacement of the bogieky transversal displacement of the car bodyaki yaw angle of the wheelset i of the bogie kak yaw angle of the bogieka yaw angle of the car body

hk roll angle of the bogiekh roll angle of the car body

hki roll angle of the wheelsetski

V the speed of the vehicleKu spring stiffness of the primary suspension in the directionu (u = x,y)

Ku spring stiffness of the secondary suspension in the direction u

d transversal distance between the primary suspension and the bogie center of massd transversal distance between the secondary suspension and the car body center of massr0 radius of the wheelset circle of rolling in centered position

R curvature radius of the wheel profile

R0 curvature radius of the Rail profile

22 M. Nejlaoui et al./ Simulation Modelling Practice and Theory 30 (2013) 2134


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presented an original contribution to multiobjective optimization. The proposed method takes into account the uncertainties

in the design parameters. Taking these uncertainties into account allowed the authors to obtain robust optimal solutions.

The goal of this work is the Multiobjective robust design optimization of a rail vehicle moving in short radius curved

tracks based on the safety and comfort criteria. In Section 2, the dynamic model of the RV is reviewed and the safety and

comfort criteria are defined. In Section 3, the deterministic multiobjective design optimization is presented for simulta-

neously maximizing the safety and the comfort. Then, the authors show that the deterministic optimal solutions can be seri-

ously altered by the DP uncertainties. The performance variability is quantified using the SMC. In Section 4, an original

algorithm is developed and used in the multiobjective robust design optimization of a RV. Then, the obtained solutions

are discussed. In Section5, some concluding remarks are presented.

2. The rail vehicle model

The RV system is made of a car body, two bogies and four wheelsets. The car body C is connected to the bogies Ck by 4

secondary suspensions elements. Each bogie is connected to 2 wheelsetsSki using 4 primary suspensions elements. In the

real design, each such element is formed by a vertical spring and a damper in parallel (Fig. 1). Each one works in the three

directions with different stiffness and damping coefficients. Generally, each suspension is modeled by three systems, formed

by springs and dampers in parallel, acting in the three directions [1821].

In lateral dynamics, generally, the RV system has 21 degrees of freedom, which are assumed independent and they are

summarized inTable 1[22].

Based on the RV symmetry and due to the fact that we focus on the transversal dynamics, it is proved that the study of the

dynamic model of the system can be reduced to modeling of its quarter as presented by ( Fig. 1)[22]. Hence, the RV systemhas only 8 degrees of freedom.

For the dynamic model of the RV system, the following reference frames are considered:

Rg=(Og,Xg,Yg,Zg) is attached to the center of curve.

R i=(Gi,Xi,Yi,Zi) is a frame attached to the body Si of the RV system.

R0= (O0, X0,Y0,Z0) is an inertial reference frame in motion with the vehicle.

Using Lagrange formalism, with respect toR0, yield:

d

dt

@T

@_qi

@T

@qiQi

X4j1

@

@_qif#Sj gR0 fTSj gR0=Rg 1

0Z

0X

A

0Y

G

0h

1h 1h

The quarter

of the RV

Primary suspension

Secondary suspension2a

Bogie2

0O

Rail

Wheelset

0Z

0O

Bogie1

Carbody

Fig. 1. The rail vehicle CAD model (the RV quarter is in the dashed cub).

M. Nejlaoui et al./ Simulation Modelling Practice and Theory 30 (2013) 2134 23
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where Tis the total kinetic energy expressed in R0; Sjthe components of the RV system; qithe generalized coordinates; Qithe

generalized forces;fTSjgR0=Rgthe dynamic screw (the screw of inertia forces);f#SjgR0 the twist at Gj, the gravity center ofSj,

expressed inR0 (formed by the angular and the linear velocities at Gj e Sj).

The frame R0is obtained from Rgby two rotations dandr, aroundX0andZ0=Zg, respectively, and they are assumed to beconstant angles.d is the rail elevation andr is an angular parameter, which defines the inclination ofR0toRgin the curvedtrack. Thus, the rotation rate of the frame R0 with respect to Rgis zero,~xR0=Rg

~0. Consequently, the screw of Coriolis forces is

equal to zero and the inertia forces are reduced to only centrifugal forces of theSjcomponentsmsjV2

Rc. Since the RV has a con-

stant and a low traveling speed in curves, we assume that the roll motion and the damping forces are not significant com-

pared to the elastic ones. Therefore, the dissipative energy, which depends on the velocities, is neglected. Therefore, in the

final model, only the generalized coordinates remain as variables. The generalized coordinates vector q is given by:

q y;a;y1;a1;y

11;a11;y

12;a12

TR0

2

Vectorq contains the parameters of the different displacements representing the degrees of freedom of the quarter RV sys-tem at the equilibrium. These displacements do not include the constant lateral displacement produced in curved tracks (for

more detail see[18,22]).

The analytical model of the RV system is obtained based on the dynamic model derived with use of the Lagrange formal-

ism. Then it is reduced making use of the reduction of variables and can be expressed as follows:

Aqq b 3

For more details see[22,18]. The matrixA and the vector b are given inAppendix A.

2.1. Safety and comfort

In curved tracks with a low radius (Rc < 500 m), there is usually a flange-rail contact. This contact leads to a lateral force F1i(Fig. 2) that if it exceeds a certain limit, it can cause the derailment of the RV system. Moreover, this kind of motion is usually

accompanied by the relative slip between the wheel and the rail. This slip generates the wear of the rail as well as the wheel,and it is accompanied by a high noise level. This is considered as a general comfort criterion.

In what follows, the authors will give, in a closed form, the forces responsible for the derailment and the creep.

2.1.1. The derailment angle

When we have a flange-rail contact the lateral displacement y1i becomes a known constant and the corresponding dis-

placement appears as a reaction to force F1i. These unknown forces become additional variables, instead of the corresponding

displacement y1i, which can be identified through the algebraic equations of the analytical model (Eq. (3))[18].

F1iXi2i1

Kyy1y

11 1

iaa1y

12y0 1

iKx

d2

2aa

y1A

y11y

12

2a a1

A

Rc

!Xi2i1

1i

Kxd

2

2a

a1 a1iy11y

12

2a

1i a

Rc 2vC22a1iX

i

m mcncWfy0y1i 4

Table 1

Degrees of freedom of the RV system.

Lateral displacement Roll Yaw

Car bodyC y h a

BogiesCk (k= 1, 2) yk hk akWheelsetsSki (k,i = 1, 2) yki hki aki

F1i T1ij

S

N

X1ij

M1ij

(a) Initial flange-rail contact

F1i

T1ij S

N

X1ij

M1ij

(b) Flange-rail contact

F1i

N

Fig. 2. Direction of contact forces during flange-rail.

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whereC22 is the Kalkers coefficient[23]andP

i, v,W, f andcncare defined inAppendix A.In this case, the rail exerts two reaction forces on the wheel; the force Snormal to the contact surface and the creep force

T1ij tangent to the contact surface (Fig 2).

The analysis of the equilibrium forces in the vertical and lateral directions yields:

F1iN

tan h

T1ijS

1 T1ijtan h

S

5

The left-hand side term in Eq. (5) can be considered as a criterion of safety [24]. The maximum of safety is obtained when thisterm is minimal. According to the Coulomb law, the maximum of (T1ij/S) corresponds to the friction coefficientl. Therefore,at a given maximum contact angle, to avoid the derailment, the Nadals criterion [24]has to be verified:

F

N>>>>>>>>>>:

12

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whereb is the derailment angle and R is the creep force. For hmax= 75[24]andl= 0.31[18], the derailment angleb shouldbe less than bmax= 58(Eq.(7)) in order to avoid the derailment risk. The GA tool is used to resolve this problem.

3.1. The optimization method

Genetic Algorithms have been shown to solve linear and non-linear problems by exploring all regions of the state space

and exploiting promising areas through selection, crossover and mutation applied to individuals in a population[26]. More-

over, Genetic Algorithms have the advantage to be used in multiobjective optimization [27].

In the case of a multiobjective optimization, the flowchart of the used GA is illustrated in Fig. 4.X0 is the initial design

vector (population) to be evaluated[28].Xpar, andXi are, respectively, the Paretian (non-dominated) population and the dy-

namic population for the ith iteration. For each evolutionary period, the Paretian population individuals are selected from the

q= q+ q

Computation of andR

Yes

Design parameters

Calculation of C ij

Initialization (q = 0)

Res = b - A q

q = A-1

Res

0,005q q

q q

T

Tp

=

q

No

.

Fig. 3. The solving algorithm.

Table 2

Design variables and their search domains.

DP D(DP) DP D(DP)

Kx (N/m) [105, 108] ce [0.02,0.25]

Ky (N/m) [105, 108] a (m) [1,2]

KxN=m [103, 107] Mkg [35,000,46,000]

KyN=m [103, 107] M(kg) [2500, 4000]

Table 3

The DP of the RV system[4,5].

The suspension parameters The geometric parameters

u=x u=y

e0(m) 0.75 dm 0.58

d 0.06 Am 8.23

Ku (N/m) 3.15 107 3.96 106 c0 0.025 R(m) 0.52

Ku (N/m) 6.87 105 1.97 105 H(m) 0.46 R0 (m) 0.3

The masses (kg) ce 0.1 h0 0.88

m m M M a (m) 1.04 h1 0.21

1190 32,820 3072 r0(m) 0.356 h0 0

d (m) 0.813 h1(m) 0.12



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current dynamic ones. This dynamic population is generated by the random selection from the initial population, in order to

form a new Paretian one. The evolutionary process is stopped after a given maximum number of generations.

The parameters of the optimization algorithm are the crossover probability PC = 0.9, the mutation probability PM = 0.1

and the maximum number of generations Gmax= 600.

3.2. Results and discussions

The application of the optimization program gives the results presented in the Pareto front (Fig. 5.).

One can note (from Fig. 5) that when the creep force Rincreases, the derailment angleb decreases. In fact, the increase of

the creep forces generates a larger slipping area in the elliptical contact zone. Consequently, the ratio of the tangential to the

Start

Create X0

Selection

Crossover

Mutation

yes

No Ending conditions

End

For i=1 to Gmax

Evaluate X0

Filter Xpar

Evaluate Xi

Fig. 4. The genetic algorithm for a multiobjective optimization[29].

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

x 104

0

10

20

30

40

50

60

R (N)

(degree)

S

Fig. 5. The deterministic optimal solutions.



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normal force (T1ij

S ) increases [30,32]. On the other hand, we remark (from Eq. (5)) that at a given contact angle, high value of

T1ijS

term corresponds to low value of the derailment angle b tan FN

.

All the solutions presented in the Pareto front correspond to safe design vectors of the RV system (b< bmax). However, for

a more realistic analysis, the DP can present fluctuations around their nominal values. These uncertainties can generate

fluctuations of the RV performance evaluated by band R. Therefore, the performances represented through the optimal solu-

tions, specifically the safety, are often altered by the DP uncertainties.

3.3. Performance variability as a function of the design parameters uncertainties

In what follows, the variability of the optimized RV system performance generated by the DP uncertainties will be

estimated. Thus, the case of optimal solution S, given in Table 4, will be studied.

Table 4

Design vector corresponding to solution S.

DP Kx (N/m) Ky (N/m) Kx (N/m) Ky (N/m) ce a(m) M(kg) M(kg)

Nominal value 8.61 107 7.23 107 7.96 105 6.67 105 0.054 1.92 38,625 3015

n


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Each uncertain design parameter is represented by a statistical distribution. For a normal distribution this requires a

mean value and a standard deviation. An SMC is performed for every DP where each evaluation consists of a specified num-

ber of runs (Fig. 6). For each SMC, all the deterministic design variables are fixed at their nominal values and the uncertain

design variables are selected randomly from their statistical distributions supposed to vary within 10% of the specified

nominal values[4].

With the SMC, we perform n= 104 simulations to determine the mean values bandRand their respective standard devi-

ationsrb andrR. The evolution of the derailment angleb is presented inFig. 7. One can note that the derailment angle canreach up to 59.5and the imposed constraint b< b

maxis not satisfied. Consequently, by considering the DP uncertainties the

safety given by the deterministic optimal solution S is no longer guaranteed. Therefore, the optimal solutions, obtained by

the multiobjective optimization presented previously, are not robust to the DP uncertainties.

In order to have a robust and optimal design of the RV system, an original algorithm that combines the GA and the SMC

(GASMC) will be developed.

4. Robust optimum design of the rail vehicle

The robust optimal design of the RV system should have the maximum of safety and comfort levels (defined by the min-

imum of b and R) and also these two criteria should have the minimum variability generated by the DP uncertainty. It is

worth mentioning that all the solutions should satisfy all the physical constraints imposed by the designer. Consequently

the robust optimal design strategy can be expressed as follows:

minimize bDP

minimize rbDP

minimize RDP

minimize rRDP

Subject to :b 3rb 6 bmaxDP2 DDP

8>>>>>>>>>>>>>>>>>>>>>:

13

The constraint b 3rb 6 bmaximposes that the solutions are considered only if we have 99% confidence that the derailmentangle does not exceed the maximum allowed derailment angle.

Start

Create X0

Selection

Crossover

Mutation

Yes

No

G


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4.1. The GASMC Algorithm

The GASMC Algorithm (Fig. 8) combines the GA and the SMC methods. For each research iteration of non-dominated

solutions, during the evaluation stage, the GA sends a generation of solutions to the SMC. The SMC performs n simulations

for every solution and sends back the mean values, band R, and the standard deviations,rbandrR(see Fig. 8). Then the non-dominated solutions undergo the selection, crossover, mutation and reinsertion operations. The execution stops when the

maximum number of generations is reached. The parameters of GA are the crossover probability, PC = 0.9, the mutation

probability, PM = 0.1, the maximal number of generations,Gmax

= 1200, and the SMC number of simulationsn = 104.

4.2. Results and discussion

Using the GASMC algorithm, we obtain the robust optimal solutions presented in the Pareto front (Fig. 9). Each point

represents a non-dominated RV system design vector with the corresponding values of the four objective functions. All

the solutions are non-dominated ones.

The first remark is that band Rtend to vary in the opposite direction ofrbandrR, respectively. This means that reducingthe average of an objective function leads inevitably to increasing its variability [13,31].

A Pareto optimal set, for different configurations ofrb, with the deterministic optimization is plotted in Fig. 10. Threeareas can be identified; the first one is characterized by a high level of safety where the different fronts are close to each

other, the second area is identified by lower creep forces and the distance between the fronts vary in the opposite direction

of b, and the third one is represented by high creep forces where the fronts seem to be parallel to each other.

From Fig. 10, one can note that the designer should choose one of the solutions presented in area 1. In fact, these solutions

are characterized by a high safety level even at relatively high creep force level where the fluctuation is less than 8%. More-

over, in this area the creep force is the most robust to the DP uncertainties. Even if this area presents the maximum of the

derailment angle vulnerability; the evolution of the three solutions (S5, S6 and S7chosen from area 1 in Fig. 10) confirms that

the safety level is guaranteed (Fig. 11). This can be explained by the fact that the standard deviation rb is considered as aseparate objective function that has been minimized.

To better understand the particularity of the solutions presented by area 1, we need to compare them to the rest of the

solutions presented by the global Pareto front (Fig. 9). To achieve this goal, four solutions are selected, two of them are from

the area 1 (the encircled part of the global Pareto front in Fig. 9). The DP of these four solutions are shown inTable 5.

An interesting result is that the high level of security is guaranteed by a more flexible structure (low spring stiffnesses)

(Table 5). In fact, if the structure is more flexible, a great part of the energy generated by the force applied by the car body,

the bogies and the wheelset to the rail is converted to a potential energy which is proportional to the square of the allowable

spring deformation.

( )NR

( )R N

( )degree

Fig. 9. The robust optimal solutions.



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Fig. 10. Some configurations of robust optimal solutions.

6 6.5 7 7.5 8 8.5 9 9.5 100

500

1000

1500

2000

2500

3000

(degree)

Numb

erofsequences

S5

S6

S7

Fig. 11. The distribution of the chosen solutions.

Table 5

The design parameters corresponding to the chosen solutions.

DP # 1 # 2 # 3 # 4

Kx (N/m) 8.31 107 5.05 107 3.68 106 1.62 106

Ky (N/m) 5.38 107 2.93 107 6.71 106 3.54 106

KxN=m 7.88 105 4.72 105 1.08 105 7.57 104

KyN=m 4.79 105 3.66 105 2.57 104 2.04 104

ce 0.058 0.099 0.11 0.17a (m) 1.83 1.5 1.48 1.26

M(kg) 40,180 42,500 43,220 44,855

M(kg) 3064 3181 3250 3466b () 49.44 49.52 2.93 3.04

rb () 0.41 0.37 0.59 0.45

R (N) 11,300 11,600 12,400 13,400

rR (N) 316 314 211 207



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Table 6

The design parameters corresponding to SD and SR.

DP SD SR

Kx (N/m) 125 107 1.33 107

Ky (N/m) 8.49 106 8.68 106

Kx (N/m) 2.18 105 2.9105

Ky (N/m) 5.22 104 5.34 104

ce 0.1 0.098

a (m) 1.48 1.48

M(kg) 42,869 42,937

M(kg) 3189 3214b () 15.54 15.54

rb () 0.6 0.47

R (N) 12,030 12,030

rR (N) 400 280

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

x 104

0

500

1000

1500

2000

2500

3000

R (N)

Numberofsequences

SR

SD

(a) The distribution of the creep forces

13 14 15 16 17 18 190

500

1000

1500

2000

2500

3000

(degree)

Numberofseque

nces

SD

SR

(b) The distribution of the derailment angle

Fig. 12. Comparison between the performance vulnerability of deterministic and robust solutions.



13/14

FromFig. 10, one can note also the coincidence between two optimal solutions; the robust and the deterministic one

(SR + SD). The first remark is that the robust solution (SR) is characterized by a relatively high nominal value of the design

variables (Table 6). To compare the performance vulnerability of the two solutions, the corresponding DP are introduced un-

der the SMC and the obtained results are given in Fig. 12.

It is clear fromFig. 12 that the deterministic solution (SD) presents higher variability of the derailment angle and the

creep force than the robust solution (SR), which validates our developed GASMC algorithm.

Table 6shows the obtainedRand bfor the two coinciding solutions. One can notice that for the same values for R and b,

the presented algorithm succeeded in reducing the variability of the output for a given variation of the DP. Therefore, one can

conclude that finding a deterministic optimal solution is not sufficient. Additionally, one has to be sure that the obtained

solution is robust enough to the uncertainty of the DP. When he knows it is true, he can think of his design as of much better

than those unchecked for the robustness to the uncertainty of their DP.

Figs. 9 and 10 can be a valuable tool for the designer since they can be used to choose the level of confidence of the chosen

solution. If the designer has low confidence in his DP, he can favor the robustness of the solution by choosing low values for

the standard deviation.

5. Conclusions

This paper deals with multiobjective robust design optimization of rail vehicle based on safety and comfort criteria. The

dynamic model of the rail vehicle moving in short radius curved tracks is reviewed. A multiobjective optimization design

that considers simultaneously the safety and the comfort is conducted. The authors showed that the deterministic optimal

solutions can be altered seriously by the design parameters uncertainties. An original algorithm, that combines the GeneticAlgorithm and the Simulation Monte Carlo method, is developed and used for the robust multiobjective optimization of the

rail vehicle design. By minimizing the mean values and the standard deviation of the objective functions, the authors added

to the solution the valuable information about its level of confidence.

Indeed, the presented algorithm turns out to be a valuable tool for the designer since he can have an idea on the level of

confidence of the chosen solution.

Appendix A

Ky KyA Ky 0 0 0 0 0

KyA Kx d2 KyA

2 Kxd2

A KyA Kxd

2 Kxd2

2ab 0 Kx

d2

2a 0

Ky Kx

d2

A KyA

Kx

d2

A2 Ky 2Ky

Kx

d2

A Kx

d2

2aA 2Ky

0 Kx

d2

2aA 2Ky

0

0 Kxd2 Kx

d2

A

Kxd2 2Kxd

2

2Ky a2

! Kx

d2

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14/14

References

[1] K. Tsui, An overview of Taguchi method and newly developed methods for robust design, IIE Transactions 24 (1992) 4457.

[2] S. Rajagopal, R. Ganguli, Conceptual design of UAV using Kriging based multi-objective genetic algorithm, Aeronautical Journal 112 (1137) (2008) 653

662.

[3] K. Zboinski, Dynamical investigation of railway vehicles on a curved track, European Journal of Mechanics A: Solids 17 (1998) 10011020.

[4] Y. He, J. McPhee, Optimization of curving performance of rail vehicles, Vehicle System Dynamics 43 (2005) 895923.

[5] H. Rejeb, Z. Affi, H. Bettaieb, Optimization of the design variables of rail vehicles system in rectilinear motion, in: The Second International Congress

Design and Modeling of Mechanical Systems, CMSM2007, 2007.

[6] M. Majka, M. Hartnett, Effects of speed, load and damping on the dynamic response of railway bridges and vehicles, Computers and Structures 86(2008) 556572.

[7] M. Schinozuka, Monte Carlo solution of structural dynamics, Computers and Structures 2 (1972) 855874.

[8] M. Chandrashekhar, R. Ganguli, Uncertainty handling in structural damage detection using fuzzy Logic and probabilistic simulation, Mechanical

Systems and Signal Processing 23 (2009) 384404.

[9] J. Hilmann, M. Paas, A. Haenschke, T. Vietor, Automatic concept model generation for optimization and robust design of passenger cars, Advances in

Engineering Software 38 (2007) 795801.

[10] S. Murugan, R. Ganguli, D. Harursampath, Aeroelastic response of composite helicopter rotor with random material properties, Journal of Aircraft 45

(1) (2008) 306322.

[11] Du. Xiaoping, Pavan Kumar Venigella, Deshun Liu, Robust mechanism synthesis with random and interval variables, Mechanism and Machine Theory

44 (2009) 13211337.

[12] M. Ouisse, S. Cogan, Robust design of spot welds in automotive structures: a decision-making methodology, Mechanical Systems andSignal Processing

24 (2010) 11721190.

[13] I. Doltsinis, Z. Kang, Robust design of structures using optimization methods, Computer Methods in Applied Mechanics and Engineering 193 (2004)

22212237.

[14] M. Guedri, S. Ghanmi, R. Majed, N. Bouhaddi, Robust tools for prediction of variability and optimization in structural dynamics, Mechanical Systems

and Signal Processing 23 (2009) 11231133.

[15] R. Al-Aomar, Incorporating robustness into genetic algorithm search of stochastic simulation outputs, Simulation Modelling Practice and Theory 14(2006) 201223.

[16] M.L. Bouazizi, S. Ghanmi, R. Nasri, N. Bouhaddi, Robust optimization of the non-linear behavior of a vibrating system, European Journal of Mechanics A:

Solids 28 (2009) 141154.

[17] S. Ghanmi, M. Guedri, M.-L. Bouazizi, N. Bouhaddi, Robust multi-objective and multi-level optimization of complex mechanical structures, Mechanical

Systems and Signal Processing 25 (7) (2011) 24442461.

[18] R. Joly, Circulation dun vhicule ferroviaire en courbe de faible rayon. Inscription des essieux. Efforts dinteraction, Journal de mcanique thorique et

appliqu 4 (1985) 813832.

[19] S. Sezer, A.E. Atalay, Dynamic modeling and fuzzy logic control of vibration of a railway vehicle for different track irregularities, Simulation Modelling

Practice and Theory 19 (2011) 201223.

[20] Y. Li, Sh. Qiang, H. Liaoa, Y.L. Xu, Dynamics of windrail vehiclebridge systems, Journal of Wind Engineering and Industrial Aerodynamics 93 (2005)

483507.

[21] Y.-C. Cheng, S.Y. Lee, H.H. Chen, Modeling and nonlinear hunting stability analysis of high-speed railway vehicle moving on curved tracks, Journal of

Sound and Vibration 324 (2009) 139160.

[22] M. Nejlaoui, Z. Affi, A. Houidi, L. Romdhane, Analytical modeling of rail vehicle safety and comfort in short radius curved tracks, Compte Rendu

Mecanique 337 (2009) 303311.

[23] J.J. Kalker, The Tangential force transmitted by two elastic bodies rolling over each other with pure creepage, Laboratorium voor Technische

Mechanica, Technische Hogeschool Delft, The Netherlands, January 31, 1968.

[24] J. Elkins, H. Wu, Angle of attack and distance based criteria for flange climb derailment, Vehicle System Dynamics Supplement 33 (1999) 293305.

[25] J. Nocedal, S.J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999.

[26] J. Horn, A Niched Pareto genetic algorithm for multiobjective optimization, IEEE Service Center 1 (1994) 8287.

[27] P.G. Busacca, M. Marseguerra, E. Zio, Multiobjective optimization by genetic algorithms: application to safety systems, Journal of Reliability

Engineering and System Safety 72 (2001) 5974.

[28] K. Deb, A. Pratap, S. Agarwal, T. Meyrivan, A fast andelitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation

6 (2002) 182197.

[29] Z. Affi, B. EL-Kribi, L. Romdhane, Advanced mechatronic design using a multi-objective genetic algorithm optimization of a motor-driven four-bar

system, Mechatronics 17 (2007) 489500.

[30] S. Datoussaid, Optimisation du comportement dynamique et cinmatiques des systmes multicorps structure cinmatique complexe, Ph.D these,

Service de mcanique rationnelle, dynamique et vibration, Facult polytechnique de Mons, 1998.

[31] G.C. Marano, S. Sgobba, R. Greco, M. Mezzina, Robust optimum design of tuned mass dampers devices in random vibrations mitigation, Journal of

Sound and Vibration 313 (2008) 472492.

[32] Jean-Bernard Ayasse, Hugues Chollet, Handbook of Railway Vehicle Dynamics, Taylor & Francis Group, 2006.


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