DESIGN OPTIMIZATION OF VEHICLE SUSPENSIONS WITHA QUARTER-VEHICLE MODEL

Zhongzhe Chi, Yuping He and Greg F. NatererFaculty of Engineering and Applied Science, University of Ontario Institute of Technology

2000 Simcoe Street North, Oshawa, Ontario, L1H 7K4Contact: yuping.he@uoit.ca

Received December 2007, Accepted June 2008No. 07-CSME-68, E.I.C. Accession 3037

ABSTRACTThis paper presents a comparative study of three optimization algorithms, namely Genetic

Algorithms (GAs), Pattern Search Algorithm (PSA) and Sequential Quadratic Program (SQP),for the design optimization of vehicle suspensions based on a quarter-vehicle model. In theoptimization, the three design criteria are vertical vehicle body acceleration, suspension workingspace, and dynamic tire load. To implement the design optimization, five parameters (sprungmass, un-sprung mass, suspension spring stiffness, suspension damping coefficient and tirestiffness) are selected as the design variables. The comparative study shows that the globalsearch algorithm (GA) and the direct search algorithm (PSA) are more reliable than the gradientbased local search algorithm (SOP). The numerical simulation results indicate that the designcriteria are significantly improved through optimizing the selected design variables. The effect ofvehicle speed and road irregularity on design variables for improving vehicle ride quality hasbeen investigated. A potential design optimization approach to the vehicle speed and roadirregularity dependent suspension design problem is recommended.

OPTIMISATION DE DESIGN DES SUSPENSIONS DE VEHICULES AVEC UNMODELE DE VEHICULE D'UN QUART DE TONNE

RESUMEGet article presente une etude comparative de trois algorithmes d'optimisation, soit les

algorithmes genetiques (AG), les algorithmes de recherche de forme (ARF) et les programmessequentiels quadratiques (PSO), pour I'optimisation de design des suspensions de vehiculebase sur un modele de vehicule d'un quart de tonne. Dans I'optimisation, les trois criteres dedesign sont I'acceleration du corps de vehicule vertical, I'espace de travail de la suspension etla charge pneumatique dynamique. Pour appliquer I'optimisation de design, cinq parametres(masse suspendue, masse non suspendue, rigidite des ressorts de suspension, coefficientd'amortissement de suspension et rigidite pneumatique) sont selectionnes comme variables desuspension. L'etude comparative revele que I'algorithme de recherche globale (ARG) etI'algorithme de recherche directe (ARD) sont plus fiables que I'algorithme de recherche local(ARL) base sur Ie gradient. Les resultats de simulation numerique indiquent que les criteres dedesign sont grandement ameliores grace a I'optimisation des variables de design selectionnees.L'effet de la vitesse du vehicule et de I'irregularite de la route sur les variables de design pourameliorer la qualite du voyage en vehicule a fait I'objet d'un examen. Une approched'optimisation de design possible au probleme de design de suspension lie a la vitesse duvehicule et a I'irregularite de la route est recommandee.

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1. INTRODUCTION

Conventionally, the design practice of vehicle suspensions has been based on trial and errorapproaches, where designers iteratively change the values of the design variables and re­analyze the system until acceptable design criteria are achieVed. This is both time-consumingand tedious. Due to advances in computational power and theoretical methods, the focus ofvehicle suspension design has switched from pure numerical analysis to extensive designsynthesis using optimization approaches. A number of optimization algorithms have been usedto determine optimal suspension characteristics. There are numerous methods available andeven the choice of an efficient optimization algorithm is a non-trivial problem [1].

Genetic algorithms (GAs) have been used in various applications such as functionoptimization, system identification and control systems. It is known that GAs offer significantadvantages over traditional methods by using several principles simultaneously and heuristics,whose most important aspects are: a population-wide search, a continuous balance betweenexploitation (convergence) and exploration (maintained diversity), and the principle of building­block combination [2]. GAs are general-purpose stochastic optimization methods for solvingsearch problems to seek a global optimum. However, GAs are characterized by a large numberof function evaluations [3].

The pattern search algorithm (PSA) is typically based on function comparison techniques.Most of these procedures are heuristic in nature and derivative evaluations are not needed.They can be used to solve problems where the objective function is not differentiable andcontinuous [4].

On the other hand, traditional methods, such as sequential quadratic programming (Sap),are well known to exploit all local information in an efficient way, provided that certain conditionsare met and the function to be minimized is 'well-conditioned' in the neighborhood of a uniqueoptimum. These methods require adequate local information to be known (such as the gradientand Hessian matrix) [5]. If the basic requirements are not satisfied, the reliability of the sapmethod is greatly jeopardized [6].

The design optimization of vehicle suspensions requires the best trade-off solutions, involvingthe vertical vehicle body acceleration, suspension working space, and dynamic tire load, byfinding optimal design variables. These design variables may involve suspension springstiffness, damping coefficient, geometry, and inertial parameters [7]. By means of the ride qualityanalysis in the frequency domain, the vertical vehicle body acceleration, suspension workingspace and dynamic tire load can be obtained [8]. In this design optimization, the main objectiveis to minimize the vertical vehicle body acceleration. In the meantime, the suspension workingspace and dynamic tire load are constrained. If the suspension working space is too small, thesprung mass will strike the un-sprung mass and this may lead to damage of the vehicle. If thedynamic tire load is greater than the static tire load, the vehicle's tires will bounce off the road [1]and this will result in unstable modes of vehicle motion. Therefore, it is necessary to optimizethe suspension working space and dynamic tire load as well.

In order to fully investigate the potential design criterion improvement in the vehiclesuspension design, in the current research, the sprung mass, un-sprung mass, suspensionspring stiffness, suspension damping coefficient and tire stiffness are selected as the designvariables. To find effective optimization algorithms for the design synthesis of vehiclesuspensions, three optimization algorithms, Le., GAs, PSA, and SOP, will be compared andevaluated through the vehicle suspension optimization using a quarter-vehicle model.

2. VEHICLE SYSTEM MODELING

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2.1 Mathematic Model

Figure 1 shows a simplified 2 degrees of freedom (OOF) quarter-vehicle model [9]. It consistsof a sprung mass (m2) supported by a primary suspension, which in tum is connected to the un­sprung mass (m1). The tire is represented as a simple spring, although a damper is oftenincluded to represent the small amount of damping inherent to the visco-elastic nature of the tire[8]. The road irregularity is represented by q, while m1, m2, Kt. K and C are the un-sprung mass,sprung mass, suspension stiffness, suspension damping coefficient and tire stiffness,respectively. The nominal vehicle parameters of this 2 OOF model are provided as follows:m1=104 kg, m2= 637 kg, kt=6.993x 105 (N/m), K=1.006x 105 (N/m) and C=3,200 (N/m/s) [1]. Thegoverning equations of motion of the 2 OOF quarter-vehicle model are

{m2l 2+C(i2 -i l )+K(Z2 -ZI)=O

mIll +C(iI-i2)+K(ZI- ZJ+K,(zl-q)=0(1 )

K

Kt

_tc

__t

_t q

Figure 1. 2 OOF quarter-vehicle model

Performing a Fourier transform of equation (1) yields:

{Z2(-eo 2m2+ jeoC +K) = z\ (jeoC +K)

z\(-eo2ml + jeoC + K + K,) = Z2(jeoC +K)+qK,(2)

Equation (2) is a complex frequency expression which consists of real and imaginarycomponents, denoted by the 'j' operator. The amplitude ratio between the un-sprung massdisplacement, Z1, and the road excitation, q, is given as follows:

\

~ = r[ (1- -")'6.+ 4';'-'']' (3)

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where,

The stiffness ratio, mass ratio, sprung mass angular frequency, and damping ratio are defined

as r =K( / K, J.1 =m2 / m1 , W o=~K / m2 ' and q= ./f;;;K' respectively. The amplitude ratio2 m2K

between the sprung mass displacement, Z2, and the road excitation, q, is

Therefore, the amplitude ratio between the sprung mass acceleration, Z2' and the roadexcitation, q, can be expressed as:

The suspension working space is the allowable maximum suspension displacement, (d. Thesuspension working space in response to the road displacement input is:

(7)

The dynamic tire load is defined as Fd = K( (Zl - q), and the static tire load is

G=(~ +m2)g=~(J.1+1)g,where g is the acceleration of gravity. Thus, the amplitude ratio

between the relative dynamic tire load, I~ I, and the road input, q, becomes:

1

(2)2 "2

IF

dI= yaJ ~-l +4eIf

Gq g /),.

2.2 Stochastic Road Modeling and PSD Response

(8)

Road irregularity or unevenness represents the main disturbing source for either the rider or

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vehicle structure itself. The road profile elevation is usually expressed in terms of the powerspectral density (PSD). The PSD of the road profile elevation is expressed as

(9)

where n is the spatial frequency and n = 2;r / A. (radls) , A is the wavelength (m), no is thereference spatial frequency, generally no=1.0 (rad/m). When the vehicle is traveling at a speedof V (mls) on the road, the spatial frequency of the road excitation w is

OJ = Vn (10)

In the temporal frequency domain, the power spectrum density of the road excitation isexpressed as

(11)

If w= 2, then R=no2Gq(no)(n/nofw and substituting equations (9) and (10) into (11) yields

(12)

Based on the theory of stochastic vibration [10], the PSD of response Z with respect to q is

Therefore the mean square of the response of z is

'"O"z2 = JGzq(OJ)dOJ

a

By means of numerical integration, o"z2

can be calculated and equation (14) becomes:

(J"Z2 = tl z(nL\0J)1

2

Gq(nL\OJ)L\OJ11=1 q(nL\OJ)

(13)

(14)

(15)

where, n=1, 2, 3... N. For the purposes of design optimization, according to James' principle,

the root mean square (RMS) of the sprung mass acceleration Z2 can be expressed as

(16)

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The RMS of the suspension working space fd is

The RMS of the relative dynamic tire load can be calculated as

(17)

KtKm1 + CK ]}IIZCmZ(m1 +mz) 2m1mZ

(18)

3. DESIGN OPTIMIZATION IMPLEMENTATION

In this section, the sprung mass vertical acceleration is minimized, while the designconstraints on the suspension working space and dynamic tire load should be satisfied. Toimplement the design optimization, the three optimization algorithms, Le., GAs, PSA, and SOP,will be applied, respectively.

3.1 Optimization Implementation Using SQP

The SOP algorithm is a non-linear programming technique that is used for the purpose ofminimizing a smooth non-linear function subjected to a set of constraints with upper and lowerbounds. The objective function and the constraint functions are assumed to be at least twice­continuously differentiable. This algorithm is a gradient-based search method [11, 12]. Thisalgorithm is well-suited for constrained design optimizations. Vehicle suspension designsynthesis is a typical multi-criteria optimization problem with a variety of constraints. Toinvestigate the effectiveness of this algorithm for coordinating the trade-off relationships amongthe design criteria and various constraints of the vehicle suspension design problem, thefollowing optimization is formulated and implemented.

The RMS of the acceleration of a sprung mass O"z is frequently used to evaluate the riding2

quality of a vehicle. A rider's comfort improves as the acceleration decreases. Ride comfort ischosen to be the design criterion. The suspension working space 0"fd and dynamic tire load

0"FdlG are selected as the design constraints. The design variables are m1, m2, Kt, K and C,

respectively. Thus, the design optimization problem can be described as:

Minimize:

Subject to

CfFdlG(m l ,mz,K, ,K,C):::; a

Cfjd(ml,mZ,Kt,K,C) :::;b

83.2:::; ml :::; 124.8

509.6:::; mz :::; 764.4

559440 :::;K, :::;839170

80480 :::; K :::; 120720

2560 :::; C :::; 3840

(19)

(20)

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In equation (20), the value of a is selected as 0.4472, so the possibility for the tire to bounceout of the road is 2.51%. If b=O.3333hd , where hd is the maximum suspension dynamicdeflection, the possibility for the un-sprung mass to strike the sprung mass is less than 0.3%. Inthe current study, b is chosen as 0.05 (m).

To investigate the effects of vehicle speed on the optimal design variables, in the case study,three speeds and corresponding road irregularity coefficients are selected. These vehiclespeeds and the corresponding coefficients of the power spectrum of road irregularity Gqo arelisted in table 1. The sap algorithm from the MATLAB toolbox is used to implement this designoptimization.

Table 1. Vehicle speeds and coefficients of the power spectrum of roads

Vehicle Speed (m/s) G"o(no) (m3)

Case 1 40.0 6.5x10-oCase 2 30.0 1.2x10-5

Case 3 21.0 2.0x10-:>

3.2 Optimization Implementation Using GA

GAs are global search methods that are based on the Darwin's principle of natural selectionand genetic modification. The GA operates with a population of possible solutions (individuals)of the optimization problem. These solutions are evaluated with respect to their degree of fitnessthat indicates how well the individual will fit the optimization problem. The selection of theappropriate candidates (designs) will be related to a fitness based on the objective functionformulated by the designer. The GA works using three operators: selection, crossover andmutation [2, 4].

The sap is very suitable for the general constrained minimization problem described in thestandard form of equations (19) and (20). However, if the GA is used for the constrainedminimization problem, the objective function and constraints expressed in equations (19) and(20) cannot be used directly since the GA is suited for unconstrained optimization problems. Butone can use the penalty methods, which degrade the fitness ranking in relation to the degree ofconstraint violation. With these methods, a constrained problem in optimization is transformedinto an unconstrained optimization problem by associating a cost or penalty with all constraintviolations. Therefore, the solution to the problem is to find an appropriate fitness function to beminimized, which depends on the objective function and constraints.

Based on equations (19) and (20), the fitness function is defined as

Fitness = CT" +amax[O,(CTF• 1G -a)]+ ,Bmax[O,(CTfd -b)] (21 )

where the first term on the right side corresponds the objective function expressed in equation(19), which should be minimized. The last two terms on the right side correspond to the first twoconstraints from equation (20). These two constraints are introduced into the fitness function as

penalty terms. When the values of <TFdlG and <Tfd are less than a and b, respectively, these

terms will be zero, so the design constraints are satisfied. If the constraints are violated, thesetwo terms will take positive values and they are "penalized". To let the penalty terms be sensitive

to the change of <TF• 1G and <Tfd , the penalty multipliers a and 13 should be assigned large

values. In the current research, both a and 13 take the values of 400.0.For complex constrained optimization problems, such as the design synthesis of vehicle

suspensions described in equations (19) and (20), if the GA is applied, it will result in a lengthy

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fitness function with a number of penalty terms. The more complicated the fitness function, themore difficult the GA can manipulate and coordinate the relationships among the objectivefunction term and various penalty terms representing the constraints. To avoid this dilemma andeliminate the design variable bond constraints expressed in equation (20), the functiony=tanh(x) shown in figure 2 is introduced. With this function, when the independent variable xvaries between _00 to 00, the dependent variable y is constrained within the range of -1 to 1.Thus, the original design variables "X are transformed to a set of new design variables "y' andthe corresponding constraints are eliminated. In this case, the resulting design variables are

y(l) = 20.8 x tanh(m,) +104

y(2) = 127.4x tanh(m2) + 637

y(3) = 139860 x tanh(K1) + 699300

y(4) = 20120 x tanh(K) +100600

y(5) =640 x tanh(C) + 3200

(22)

The GA from MATLAB is used to implement the design optimization problem expressed inequations (21) and (22).

y=tanh(x)

y

Figure 2. Function y=tanh(x)

3.3 Optimization Implementation Using PSA

Direct search algorithms are typically based on function comparison techniques. Most suchprocedures are heuristic in nature and derivative evaluations are not needed. Direction searchalgorithms can be used to solve problems when the objective function is not differentiable andcontinuous. The pattern search algorithm (PSA) provided by MATLAB [4] is an example of directsearch algorithms. The PSA searches a set of points, called a mesh, around the current pointcomputed at the previous step of the algorithm. The mesh is formed by adding the current pointto a scalar multiple of a set of vectors called a pattern. If the PSA finds a point in the mesh thatimproves the objective function at the current point, the new point becomes the current point atthe next step of the algorithm. The PSA can be used to solve general problems described inequations (19) and (20).

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With the application of the PSA to the design synthesis of the vehicle suspension" theoptimization problem can be stated as:

Minimize:

Subject to

Bonds

{

CYFdIG(mpm2,K"K,C)::; a

cy/Ampm2,K"K,C)::; b

83.2::; m1 ::; 124.8

509.6::; m 2 ::; 764.4

559440 ::; K, ::; 839170

80480 ::; K ::; 120720

2560 ::; C ::; 3840

(23)

(24)

(25)

4. OPTIMIZATION RESULTS AND DISCUSSION

In this section, to examine the performance of the algorithms for optimizing road vehiclesuspensions, the optimization results based on the sap, GA, and PSA will be compared andanalyzed.

4.1 Optimization Results Based on sap. GA and PSA

In this sub-section, the optimization results are derived for a vehicle traveling at the speed of40 m/s on the road with an irregularity coefficient of power spectrum taking the value of 6.5x10-6m3

, as shown in table 1 for case 1. This high speed and road condition are intended to representa typical scenario for passenger vehicles running on highways. Thus, it is expected that highspeed vehicle ride quality can be evaluated.

Table 2 shows that with the design variables (m1, m2, kt, K and C) taking five sets of initialvalues, different local optimum points are identified by the sap. A close observation disclosesthat most of these local optimum points are located on the bonds. This algorithm frequently getstrapped at the local optimum points. Figures 3, 4, 5 and 6 show the relationships of the sprungmass vertical acceleration with the five design variables and the results indicate that there isonly one global optimum point.

Table 3 shows numerical results based on five runs of the GA. The results indicate that theGA is robust for finding the global optimum point since all of the five runs reach the samesolutions.

Table 4 shows the search results by the PSA with five different initial points. The numericalresults demonstrate that the PSA is not sensitive to the initial points selected and this algorithmis reliable in finding the global optimum point.

It is reported that the sap was applied to the design optimization of a vehicle suspension,based on the same model, where the number of design variables was three. The sapsuccessfully identified the optimum point [1]. In the current research, the number of designvariables is five instead of three, but the reliability for finding the optimum decreases with theincrease of the number of design variables. This occurs because sap largely depends ongradient information to find the optimum points. In contrast, whether the number of designvariables is three or five, the GA can still reliably find the optimum. This can be explained by thefact that the GA works on a population of design variables in parallel, not on a unique point.Thus, the GA has higher reliability to find the global optima. However, the GA achieves this high

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reliability at the expense of large fitness function evaluations. Compared with the GA, the PSA isalso featured with a high reliability for identifying the optimum, but with higher computationalefficiency.

Table 2. Optimal design variables based on the SQP for minimizing the sprung massvertical acceleration with the vehicle speed of 40 m/s

1st 2nd 3rd 4th 5thInitialpointsm1(kg) 10 80 70 80 100m2(kg) 10 800 900 100 2000Kt(N/m) 10 500000 400000 900000 1300000K(N/m) 10 700000 60000 70000 90000C(N/m/s) 10 4000 4000 1000 5000optimalpointm1(kg) 124.75 83.2 83.2 124.8 124.8m2(kg) 764.4 764.4 764.4 764.4 764.28Kt(N/m) 559440 559440 559440 722900 701800K(N/m) 80480 120720 80480 80480 8525.5C(N/m/s) 2564.3 3840 3840 2560 2560

CTi , (m/s2) 1.2913 1.5388 1.0703 1.3033 1.3725

(Jfd(m) 0.038119 0.033658 0.030414 0.038152 0.038104

(JF/G 0.36679 0.42545 0.42545 0.4472 0.43675

~, ;

,s,p,run~:ma.s~t~g2

Figure 3. CTi , vs. m1 and m2 with Kt, K and C fixed at optimal values

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suspensiond.~inpe.r ..coeffiCient ~(NImlS). : .

< 0.8

5:x·.10

s.tispens!oi'LStiJfne,S~XWrp)

Figure 4. eJ'i2

vs. C and K with m1, m2 and Kt fixed at optimal values

Figure 5. eJ'i vs. C and Kt with m1, m2 and K fixed at optimal values2

With the optimal design variables shown in tables 3 and 4, the corresponding naturalfrequency of both of the sprung and un-sprung masses can be calculated. The sprung massheave natural frequency is reduced to 1.633 Hz from the baseline value of 2.0 Hz, while the un­sprung mass heavy natural frequency maintains its baseline value of 13.958 Hz. The optimal

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suspension stiffness is reduced to 8.048x104 NIm from the baseline value of 1.006x105• These

optimal results can explain the improvement of the ride quality: The softer suspension of theoptimal design transmits less high-frequency force from the tires to the vehicle body than thebaseline suspension; The larger the distance between the vehicle body natural frequency andthat of the tire, the better the softer suspension filters the high-frequency force from the tires tothe vehicle body.

Figure 6. (JZ2 vs. K and Kt with m1, mz and C fixed at optimal values

Table 3. Optimal design variables based on the GA for minimizing the sprung mass verticalacceleration with the vehicle speed of 40 m/s

1st 2nd 3rd 4th 5thoptimalpointm1(kg) 83.2 83.2 83.2 83.2 83.2mz(kg) 764.4 764.4 764.4 764.4 764.4Kt(N/m) 559440 559440 559440 559440 559440K(N/m) 80480 80480 80480 80480 80480C(N/m/s) 3840 3840 3840 3840 3840(JZ2 (m/sz) 1.0703 1.0703 1.0703 1.0703 1.0703

ofd(m) 0.030414 0.030414 0.030414 0.030414 0.030414

OF/G 0.42545 0.42545 0.42545 0.42545 0.42545

As shown in figure 7, decreasing the un-sprung mass m1 from 104 to 83.2 kg only leads to areduction of the sprung mass acceleration by 1.07%, while optimizing the tire stiffness Kt andsuspension damping coefficient C to reduce the sprung mass acceleration by 1.59% and 7.39%,respectively. The most effective ways are to increase the sprung mass mz and to decrease thesuspension spring stiffness K, which lead to a decrease of the acceleration by 11.15% and21.55%, respectively.

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Table 4. Optimal design variables based on the PSA for minimizing the sprung mass verticalacceleration with the vehicle speed of 40 m/s

I 1st I 2110 I 3m I 41n I 51n

initialpointsm1(kg) 1 104 200 100 1000m2(kg) 1 637 800 500 5000Kt(N/m) 1 699300 800000 600000 1000000K(N/m) 1 100600 200000 200000 20000C(N/m/s) 1 3200 4000 2000 10000optimalpointm1(kg) 83.2 83.2 83.2 83.2 83.2m2(kg) 764.4 764.4 764.4 764.4 764.4K1(N/m) 559440 559440 559440 559440 559440K(N/m) 80480 80480 80480 80480 80480C(N/m/s) 3840 3840 3840 3840 3840

25.00%

20.00%

15.00%

10.00%

5.00%

0.00%m1 m2 Kt K c

Figure 7. Effect of design variables on reducing the sprung mass vertical acceleration

Figure 8 illustrates the ratio of the sprung mass acceleration (J". to road input q against2

frequency. It can be seen that the optimized suspension system significantly decreases thepeaks of the original system.

4.2 Effects of Vehicle Speed on Optimized Design Variables

To investigate the effects of vehicle speeds on design variables for improving vehicle ridequality, and satisfying constraints on suspension working space and relative dynamic tire load,the optimization is also implemented for the vehicle traveling at the speed of 30 m/s on a roadwith an irregularity coefficient of power spectrum taking the value of 1.2x10-5 m3

, as shown intable 2 for case 2. Table 5 shows the optimization results based on five runs of the GA.

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l20o;1-----r.~,------",:T""x------,....---~------.

100Q',

,:~Oft '' ........,~

"~'600~ ,cr '.....N.

i1b ;;;lr: ..!3Q .AiJ:~ :&1fre,q~ency; ff;li}.

Figure 8. Amplitude ratio between the sprung mass vertical acceleration (7'2 and

excitation input q vs. road excitation frequency

Table 5. Optimal design variables based on the GA for minimizing the sprung mass verticalacceleration with the vehicle speed of 30 m/s

1st 2nd 3rd 4th 5thoptimalpointM1(kg) 121.95 121.95 121.95 121.95 121.95M2(kg) 764.4 764.4 764.4 764.4 764.4KtCN/m) 559440 559440 559440 559440 559440K(N/m) 80480 80480 80480 80480 80480C(N/m/s) 3840 3840 3840 3840 3840

(7'2 (m/s2) 1.2843 1.2843 1.2843 1.2843 1.2843

O'fim) 0.036597 0.036597 0.036597 0.036597 0.036597

O'F/G 0.4472 0.4472 0.4472 0.4472 0.4472

Comparing the results shown in table 5 (case 2) to those in table 4 (case 1) illustrates that inboth cases the design variables m2, kt, K and C take the same optimized values. However, incase 1 and 2, the design variable m1 takes the optimized value of 83.2 kg and 121.95 kg,respectively. Compared with case 1, in case 2 the ride quality degrades due to the increasedRMS value of the sprung mass vertical acceleration.

These phenomena can be explained by the following observations. Firstly, in case 1, thenatural frequencies of both the sprung and un-sprung masses are 1.633 Hz and 13.958 Hz,respectively, while in case 2 the corresponding frequencies are 1.633 Hz and 11.529 Hz.Compared with case 1, in case 2 the distance between the vehicle body natural frequency andthat of the tire is smaller. Thus, as stated above, the ride quality in case 2 is worse than that incase 1. Secondly, in case 2, the road irregularity coefficient is larger than that in case 1. Tosatisfy the design constraints on the suspension working space and relative dynamic tire load inparticular, the un-sprung mass has to be increased.

To further investigate the effects of vehicle speeds on design variables, the optimization is

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implemented for case 3, where the vehicle speed and the irregularity coefficient of the powerspectrum take the values of 21m/s and 2.0x10·5 m3

, respectively, as shown in table 1 for case 3.Table 6 shows the resulting optimization results based on five runs of the GA.

Table 6. Optimal design variables based on the GA for minimizing the sprung mass verticalacceleration with the vehicle speed of 21 mls

1st 2nd 3rd 4th 5thoptimalpointml(kg) 124.8 124.8 124.8 124.8 124.8m2(kg) 764.4 764.4 764.4 764.4 764.4KtCN/m) 559440 559440 559440 559440 559440K(N/m) 80480 80480 80480 80480 80480C(N/m!s) 2699.1 2699.3 2698.9 2699.3 2699.7

O"Z2 (m!s~ 1.6048 1.6048 1.6049 1.6048 1.6046

Grim) 0.047225 0.047223 0.047227 0.047222 0.047219

GF/G 0.46585 0.46585 0.46585 0.46585 0.46585

For all of the three cases, results shown in tables 4, 5 and 6 show that the design variablesm2, kl , and K take the same optimized values. Among the three cases, in case 3, the RMS valueof the sprung mass vertical acceleration takes the largest value. Compared to case 2, in case 3the vehicle ride quality further degrades with respect to case 1. Accordingly, for case 3, the un­sprung mass m1 is further increased with respect to case 1. Moreover, in case 3, the designvariable C takes an optimized value that is different from the value in cases 1 and 2.

Therefore, it can be concluded that to improve vehicle ride quality and satisfy the specifiedsuspension working space and relative dynamic tire load, different vehicle speed and roadirregularity have different requirements on the design variables, in particular, the un-sprungmass m1.

5. CONCLUSION

A comparative study of three optimization algorithms (genetic algorithms, GAs, pattern searchalgorithm, PSA, and sequential quadratic programming, Sap), has been conducted throughminimizing the vertical sprung mass acceleration subjected to a suspension working space anddynamic tire load. A typical quarter-vehicle model was used to implement the designoptimization of the vehicle suspension systems. In the design optimization using the GA, toimprove the performance of the algorithm, the function y=tanh(x) was introduced for eliminatingthe penalty terms in the fitness function resulting from the constrained bonds on designvariables. Among the three optimization algorithms, the sap has very strong theoretical andlocal convergence properties. The numerical results demonstrate these features of thealgorithm, since the sap is trapped at local optimal points. The GA and PSA are more powerfulto find global optimal points, without restrictive requirements on the gradient and Hessianmatrix.

By optimizing the sprung mass, un-sprung mass, tire stiffness, suspension stiffness andsuspension damping coefficient, compared with the original design, in case 1 the sprung massacceleration decreases by 32.8%. The suspension working space and the dynamic tire loadsatisfy the specified design constraints. Numerical experiments reveal the fact that to improve

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vehicle ride quality and satisfy the specified suspension working space and relative dynamic tireload, different vehicle speed and road irregularity have different requirements on the designvariables, in particular, the un-sprung mass. It is recommended that to solve this vehicle speedand road irregularity related vehicle suspension design optimization problem, a multi-leveloptimization approach be applied. At a lower level, the vehicle system is optimized with vehiclespeed and road irregularity coefficient taking typical values, as the case presented in this paper.At a higher level, the vehicle speed and road irregularity related design criteria and constraintsare manipulated and coordinated by a multi-criteria optimization method. It is expected that bymeans of this multi-level optimization approach, the resulting solutions will compromise theconflicting requirements on design variables for vehicles traveling at different speed and onroads with different irregularity.

ACKNOWLEDEGMENTS

Financial support of this research from the Natural Sciences and Engineering ResearchCouncil of Canada is gratefully acknowledged.

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