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DESIGN OPTIMIZATION OF VEHICLE SUSPENSIONS WITH
A QUARTER-VEHICLE MODEL
Zhongzhe Chi, Yuping He and Greg F. Naterer
Faculty of Engineering and Applied Science, University of Ontario Institute of Technology
2000 Simcoe Street North, Oshawa, Ontario, L1H 7K4
Contact: yuping.he@uoit.ca
Received December 2007, Accepted June 2008
No. 07-CSME-68, E.I.C. Accession 3037
ABSTRACT
This 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 the
optimization, the three design criteria are vertical vehicle body acceleration, suspension working
space, and dynamic tire load. To implement the design optimization, five parameters (sprung
mass, un-sprung mass, suspension spring stiffness, suspension damping coefficient and tire
stiffness) are selected as the design variables. The comparative study shows that the global
search algorithm (GA) and the direct search algorithm (PSA) are more reliable than the gradient
based local search algorithm (SOP). The numerical simulation results indicate that the design
criteria are significantly improved through optimizing the selected design variables. The effect of
vehicle speed and road irregularity on design variables for improving vehicle ride quality has
been investigated. A potential design optimization approach to the vehicle speed and road
irregularity dependent suspension design problem is recommended.
OPTIMISATION DE DESIGN DES SUSPENSIONS DE VEHICULES AVEC UN
MODELE DE VEHICULE D'UN QUART DE TONNE
RESUME
Get article presente une etude comparative de trois algorithmes d'optimisation, soit les
algorithmes genetiques (AG), les algorithmes de recherche de forme (ARF) et les programmes
sequentiels quadratiques (PSO), pour I'optimisation de design des suspensions de vehicule
base sur un modele de vehicule d'un quart de tonne. Dans I'optimisation, les trois criteres de
design sont I'acceleration du corps de vehicule vertical, I'espace de travail de la suspension et
la charge pneumatique dynamique. Pour appliquer I'optimisation de design, cinq parametres
(masse suspendue, masse non suspendue, rigidite des ressorts de suspension, coefficient
d'amortissement de suspension et rigidite pneumatique) sont selectionnes comme variables de
suspension. L'etude comparative revele que I'algorithme de recherche globale (ARG) et
I'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 de
design 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 pour
ameliorer la qualite du voyage en vehicule a fait I'objet d'un examen. Une approche
d'optimisation de design possible au probleme de design de suspension lie a la vitesse du
vehicule 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 error
approaches, where designers iteratively change the values of the design variables and reanalyze the system until acceptable design criteria are achieVed. This is both time-consuming
and tedious. Due to advances in computational power and theoretical methods, the focus of
vehicle suspension design has switched from pure numerical analysis to extensive design
synthesis using optimization approaches. A number of optimization algorithms have been used
to determine optimal suspension characteristics. There are numerous methods available and
even the choice of an efficient optimization algorithm is a non-trivial problem [1].
Genetic algorithms (GAs) have been used in various applications such as function
optimization, system identification and control systems. It is known that GAs offer significant
advantages over traditional methods by using several principles simultaneously and heuristics,
whose most important aspects are: a population-wide search, a continuous balance between
exploitation (convergence) and exploration (maintained diversity), and the principle of buildingblock combination [2]. GAs are general-purpose stochastic optimization methods for solving
search problems to seek a global optimum. However, GAs are characterized by a large number
of 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 and
continuous [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 conditions
are met and the function to be minimized is 'well-conditioned' in the neighborhood of a unique
optimum. These methods require adequate local information to be known (such as the gradient
and Hessian matrix) [5]. If the basic requirements are not satisfied, the reliability of the sap
method is greatly jeopardized [6].
The design optimization of vehicle suspensions requires the best trade-off solutions, involving
the vertical vehicle body acceleration, suspension working space, and dynamic tire load, by
finding optimal design variables. These design variables may involve suspension spring
stiffness, damping coefficient, geometry, and inertial parameters [7]. By means of the ride quality
analysis in the frequency domain, the vertical vehicle body acceleration, suspension working
space and dynamic tire load can be obtained [8]. In this design optimization, the main objective
is to minimize the vertical vehicle body acceleration. In the meantime, the suspension working
space and dynamic tire load are constrained. If the suspension working space is too small, the
sprung mass will strike the un-sprung mass and this may lead to damage of the vehicle. If the
dynamic 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 optimize
the suspension working space and dynamic tire load as well.
In order to fully investigate the potential design criterion improvement in the vehicle
suspension design, in the current research, the sprung mass, un-sprung mass, suspension
spring stiffness, suspension damping coefficient and tire stiffness are selected as the design
variables. To find effective optimization algorithms for the design synthesis of vehicle
suspensions, three optimization algorithms, Le., GAs, PSA, and SOP, will be compared and
evaluated 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 consists
of 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 often
included 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:
5
m1=104 kg, m2= 637 kg, kt=6.993x 10 (N/m), K=1.006x 105 (N/m) and C=3,200 (N/m/s) [1]. The
governing equations of motion of the 2 OOF quarter-vehicle model are
m2l 2+C(i2 -i l )+K(Z2 -ZI)=O
{ mIll +C(iI-i )+K(ZI- ZJ+K,(zl-q)=0
2
(1 )
_t
c
K
_ _t
Kt
_t
q
Figure 1. 2 OOF quarter-vehicle model
Performing a Fourier transform of equation (1) yields:
Z2(-eo 2 m2 + jeoC + K) = z\ (jeoC + K)
{ z\(-eo 2m + jeoC + K + K,) = Z2(jeoC +K)+qK,
l
(2)
Equation (2) is a complex frequency expression which consists of real and imaginary
components, denoted by the 'j' operator. The amplitude ratio between the un-sprung mass
displacement, Z1, and the road excitation, q, is given as follows:
\
~ = r[ (1- -")'6.+ 4';'-'']'
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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 ratio
2 m2 K
between the sprung mass displacement, Z2, and the road excitation, q, is
Therefore, the amplitude ratio between the sprung mass acceleration, Z2'
excitation, q, can be expressed as:
and the road
The suspension working space is the allowable maximum suspension displacement, (d. The
suspension working space in response to the road displacement input is:
(7)
The dynamic tire load is defined as
G=(~ +m2)g=~(J.1+1)g,where
and the static tire load is
F d = K( (Zl - q),
g is the acceleration of gravity. Thus, the amplitude ratio
between the relative dynamic tire load,
I~
I,
and the road input, q, becomes:
2)2
~-l +4e If
Fd I= yaJ (
I
Gq
g
1
"2
(8)
/),.
2.2 Stochastic Road Modeling and PSD Response
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 power
spectral 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 the
reference spatial frequency, generally no=1.0 (rad/m). When the vehicle is traveling at a speed
of 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 is
expressed 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
(13)
Therefore the mean square of the response of z is
'"
O"z2 = JGzq(OJ)dOJ
(14)
a
2
By means of numerical integration, o"z can be calculated and equation (14) becomes:
(J"Z2
=
tl
2
z
(nL\0J)1 Gq(nL\OJ)L\OJ
11=1 q(nL\OJ)
(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
(17)
The RMS of the relative dynamic tire load can be calculated as
KtKm 1
+ CK ]}IIZ
CmZ(m 1 +m z) 2m1mZ
(18)
3. DESIGN OPTIMIZATION IMPLEMENTATION
In this section, the sprung mass vertical acceleration is minimized, while the design
constraints on the suspension working space and dynamic tire load should be satisfied. To
implement 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 of
minimizing a smooth non-linear function subjected to a set of constraints with upper and lower
bounds. 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]. This
algorithm is well-suited for constrained design optimizations. Vehicle suspension design
synthesis is a typical multi-criteria optimization problem with a variety of constraints. To
investigate the effectiveness of this algorithm for coordinating the trade-off relationships among
the design criteria and various constraints of the vehicle suspension design problem, the
following optimization is formulated and implemented.
The RMS of the acceleration of a sprung mass O"z 2 is frequently used to evaluate the riding
quality of a vehicle. A rider's comfort improves as the acceleration decreases. Ride comfort is
chosen 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:
(19)
Minimize:
CfFdlG(m l ,mz,K, ,K,C):::; a
Cfjd(ml,mZ,Kt,K,C) :::;b
Subject to
83.2:::; m l :::; 124.8
509.6:::; m z :::; 764.4
(20)
559440 :::;K, :::;839170
80480 :::; K :::; 120720
2560 :::; C :::; 3840
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In equation (20), the value of a is selected as 0.4472, so the possibility for the tire to bounce
out of the road is 2.51%. If b=O.3333hd , where hd is the maximum suspension dynamic
deflection, the possibility for the un-sprung mass to strike the sprung mass is less than 0.3%. In
the 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 vehicle
speeds and the corresponding coefficients of the power spectrum of road irregularity Gqo are
listed in table 1. The sap algorithm from the MATLAB toolbox is used to implement this design
optimization.
Table 1. Vehicle speeds and coefficients of the power spectrum of roads
Vehicle Speed (m/s)
40.0
30.0
21.0
Case 1
Case 2
Case 3
3
G"o(no) (m )
6.5x10-o
1.2x10-5
2.0x10-:>
3.2 Optimization Implementation Using GA
GAs are global search methods that are based on the Darwin's principle of natural selection
and 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 fitness
that indicates how well the individual will fit the optimization problem. The selection of the
appropriate candidates (designs) will be related to a fitness based on the objective function
formulated by the designer. The GA works using three operators: selection, crossover and
mutation [2, 4].
The sap is very suitable for the general constrained minimization problem described in the
standard form of equations (19) and (20). However, if the GA is used for the constrained
minimization 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. But
one can use the penalty methods, which degrade the fitness ranking in relation to the degree of
constraint violation. With these methods, a constrained problem in optimization is transformed
into an unconstrained optimization problem by associating a cost or penalty with all constraint
violations. Therefore, the solution to the problem is to find an appropriate fitness function to be
minimized, 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 two
constraints from equation (20). These two constraints are introduced into the fitness function as
penalty terms. When the values of <T FdlG and <T fd are less than a and b, respectively, these
terms will be zero, so the design constraints are satisfied. If the constraints are violated, these
two terms will take positive values and they are "penalized". To let the penalty terms be sensitive
to the change of <T F• 1G and <T fd , 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, the
more difficult the GA can manipulate and coordinate the relationships among the objective
function term and various penalty terms representing the constraints. To avoid this dilemma and
eliminate the design variable bond constraints expressed in equation (20), the function
y=tanh(x) shown in figure 2 is introduced. With this function, when the independent variable x
varies 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' and
the 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
(22)
y(4) = 20120 x tanh(K) + 100600
y(5) = 640 x tanh(C) + 3200
The GA from MATLAB is used to implement the design optimization problem expressed in
equations (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 such
procedures are heuristic in nature and derivative evaluations are not needed. Direction search
algorithms can be used to solve problems when the objective function is not differentiable and
continuous. The pattern search algorithm (PSA) provided by MATLAB [4] is an example of direct
search algorithms. The PSA searches a set of points, called a mesh, around the current point
computed at the previous step of the algorithm. The mesh is formed by adding the current point
to a scalar multiple of a set of vectors called a pattern. If the PSA finds a point in the mesh that
improves the objective function at the current point, the new point becomes the current point at
the next step of the algorithm. The PSA can be used to solve general problems described in
equations (19) and (20).
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With the application of the PSA to the design synthesis of the vehicle suspension" the
optimization problem can be stated as:
(23)
Minimize:
Subject to
CYFdIG(mpm2,K"K,C)::; a
{ cy/Ampm2,K"K,C)::; b
(24)
83.2::; m 1 ::; 124.8
509.6::; m 2 ::; 764.4
Bonds
559440 ::; K, ::; 839170
(25)
80480 ::; K ::; 120720
2560 ::; C ::; 3840
4. OPTIMIZATION RESULTS AND DISCUSSION
In this section, to examine the performance of the algorithms for optimizing road vehicle
suspensions, the optimization results based on the sap, GA, and PSA will be compared and
analyzed.
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 of
40 m/s on the road with an irregularity coefficient of power spectrum taking the value of 6.5x10-6
3
m , as shown in table 1 for case 1. This high speed and road condition are intended to represent
a typical scenario for passenger vehicles running on highways. Thus, it is expected that high
speed vehicle ride quality can be evaluated.
Table 2 shows that with the design variables (m1, m2, kt, K and C) taking five sets of initial
values, different local optimum points are identified by the sap. A close observation discloses
that most of these local optimum points are located on the bonds. This algorithm frequently gets
trapped at the local optimum points. Figures 3, 4, 5 and 6 show the relationships of the sprung
mass vertical acceleration with the five design variables and the results indicate that there is
only one global optimum point.
Table 3 shows numerical results based on five runs of the GA. The results indicate that the
GA is robust for finding the global optimum point since all of the five runs reach the same
solutions.
Table 4 shows the search results by the PSA with five different initial points. The numerical
results demonstrate that the PSA is not sensitive to the initial points selected and this algorithm
is 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 sap
successfully identified the optimum point [1]. In the current research, the number of design
variables is five instead of three, but the reliability for finding the optimum decreases with the
increase of the number of design variables. This occurs because sap largely depends on
gradient information to find the optimum points. In contrast, whether the number of design
variables is three or five, the GA can still reliably find the optimum. This can be explained by the
fact 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 is
also featured with a high reliability for identifying the optimum, but with higher computational
efficiency.
Table 2. Optimal design variables based on the SQP for minimizing the sprung mass
vertical acceleration with the vehicle speed of 40 m/s
1st
Initial
points
m1(kg)
m2(kg)
Kt(N/m)
K(N/m)
C(N/m/s)
optimal
point
m1(kg)
m2(kg)
Kt(N/m)
K(N/m)
C(N/m/s)
2
CT i , (m/s )
(Jfd(m)
(JF/G
2nd
3rd
4th
5th
10
10
10
10
10
80
800
500000
700000
4000
70
900
400000
60000
4000
80
100
900000
70000
1000
100
2000
1300000
90000
5000
124.75
764.4
559440
80480
2564.3
83.2
764.4
559440
120720
3840
83.2
764.4
559440
80480
3840
124.8
764.4
722900
80480
2560
124.8
764.28
701800
8525.5
2560
1.0703
1.2913
1.5388
0.030414
0.038119 0.033658
0.36679
0.42545
0.42545
1.3033
0.038152
0.4472
1.3725
0.038104
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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5
:x·.10
suspensiond.~inpe.r
..coeffiCient ~(NImlS)
.
:
.
<
0.8
s.tispens!oi'LStiJfne,S~XWrp)
Figure 4. eJ'i 2 vs. C and K with m1, m2 and Kt fixed at optimal values
Figure 5. eJ'i 2 vs. C and Kt with m1, m2 and K fixed at optimal values
With the optimal design variables shown in tables 3 and 4, the corresponding natural
frequency of both of the sprung and un-sprung masses can be calculated. The sprung mass
heave 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 the
optimal design transmits less high-frequency force from the tires to the vehicle body than the
baseline suspension; The larger the distance between the vehicle body natural frequency and
that of the tire, the better the softer suspension filters the high-frequency force from the tires to
the 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 vertical
acceleration with the vehicle speed of 40 m/s
1st
2nd
3rd
4th
5th
83.2
764.4
559440
80480
3840
83.2
764.4
559440
80480
3840
83.2
764.4
559440
80480
3840
83.2
764.4
559440
80480
3840
83.2
764.4
559440
80480
3840
ofd(m)
1.0703
0.030414
1.0703
0.030414
1.0703
0.030414
OF/G
0.42545
0.42545
0.42545
optimal
point
m1(kg)
mz(kg)
Kt(N/m)
K(N/m)
C(N/m/s)
(JZ2
(m/s z )
1.0703
1.0703
0.030414
0.42545
0.030414
0.42545
As shown in figure 7, decreasing the un-sprung mass m1 from 104 to 83.2 kg only leads to a
reduction of the sprung mass acceleration by 1.07%, while optimizing the tire stiffness K t and
suspension 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 the
suspension spring stiffness K, which lead to a decrease of the acceleration by 11.15% and
21.55%, respectively.
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Table 4. Optimal design variables based on the PSA for minimizing the sprung mass vertical
acceleration with the vehicle speed of 40 m/s
I
1st
I
2110
I
3m
I
41n
initial
points
104
200
100
m1(kg)
1
800
500
m2(kg)
1
637
Kt(N/m)
1 699300 800000 600000
1 100600 200000 200000
K(N/m)
C(N/m/s)
4000
2000
1
3200
optimal
point
83.2
83.2
83.2
83.2
m1(kg)
764.4
764.4
764.4
764.4
m2(kg)
559440 559440 559440 559440
K1(N/m)
80480
80480
80480
80480
K(N/m)
C(N/m/s)
3840
3840
3840
3840
51n
I
1000
5000
1000000
20000
10000
83.2
764.4
559440
80480
3840
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
m1
m2
Kt
c
K
Figure 7. Effect of design variables on reducing the sprung mass vertical acceleration
Figure 8 illustrates the ratio of the sprung mass acceleration
(J".
2
to road input q against
frequency. It can be seen that the optimized suspension system significantly decreases the
peaks 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 ride
quality, 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 road
with an irregularity coefficient of power spectrum taking the value of 1.2x 10-5 m 3 , as shown in
table 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
fre,q~ency; ff;li}.
.AiJ:~
:&1
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 vertical
acceleration with the vehicle speed of 30 m/s
1st
2nd
3rd
4th
5th
121.95
764.4
559440
80480
3840
121.95
764.4
559440
80480
3840
121.95
764.4
559440
80480
3840
121.95
764.4
559440
80480
3840
121.95
764.4
559440
80480
3840
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
optimal
point
M1(kg)
M 2(kg)
KtCN/m)
K(N/m)
C(N/m/s)
2
(7'2
(m/s )
Comparing the results shown in table 5 (case 2) to those in table 4 (case 1) illustrates that in
both cases the design variables m2, kt, K and C take the same optimized values. However, in
case 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 increased
RMS value of the sprung mass vertical acceleration.
These phenomena can be explained by the following observations. Firstly, in case 1, the
natural 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 and
that of the tire is smaller. Thus, as stated above, the ride quality in case 2 is worse than that in
case 1. Secondly, in case 2, the road irregularity coefficient is larger than that in case 1. To
satisfy the design constraints on the suspension working space and relative dynamic tire load in
particular, 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 power
spectrum take the values of 21m/s and 2.0x10·5 m 3 , 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 vertical
acceleration with the vehicle speed of 21 mls
1st
2nd
3rd
4th
5th
C(N/m!s)
124.8
764.4
559440
80480
2699.1
124.8
764.4
559440
80480
2699.3
124.8
764.4
559440
80480
2698.9
124.8
764.4
559440
80480
2699.3
124.8
764.4
559440
80480
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
optimal
point
ml(kg)
m2(kg)
KtCN/m)
K(N/m)
For all of the three cases, results shown in tables 4, 5 and 6 show that the design variables
m2, kl , and K take the same optimized values. Among the three cases, in case 3, the RMS value
of the sprung mass vertical acceleration takes the largest value. Compared to case 2, in case 3
the 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 design
variable 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 specified
suspension working space and relative dynamic tire load, different vehicle speed and road
irregularity have different requirements on the design variables, in particular, the un-sprung
mass m1.
5. CONCLUSION
A comparative study of three optimization algorithms (genetic algorithms, GAs, pattern search
algorithm, PSA, and sequential quadratic programming, Sap), has been conducted through
minimizing the vertical sprung mass acceleration subjected to a suspension working space and
dynamic tire load. A typical quarter-vehicle model was used to implement the design
optimization of the vehicle suspension systems. In the design optimization using the GA, to
improve the performance of the algorithm, the function y=tanh(x) was introduced for eliminating
the penalty terms in the fitness function resulting from the constrained bonds on design
variables. Among the three optimization algorithms, the sap has very strong theoretical and
local convergence properties. The numerical results demonstrate these features of the
algorithm, since the sap is trapped at local optimal points. The GA and PSA are more powerful
to find global optimal points, without restrictive requirements on the gradient and Hessian
matrix.
By optimizing the sprung mass, un-sprung mass, tire stiffness, suspension stiffness and
suspension damping coefficient, compared with the original design, in case 1 the sprung mass
acceleration decreases by 32.8%. The suspension working space and the dynamic tire load
satisfy 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 tire
load, different vehicle speed and road irregularity have different requirements on the design
variables, in particular, the un-sprung mass. It is recommended that to solve this vehicle speed
and road irregularity related vehicle suspension design optimization problem, a multi-level
optimization approach be applied. At a lower level, the vehicle system is optimized with vehicle
speed 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 constraints
are manipulated and coordinated by a multi-criteria optimization method. It is expected that by
means of this multi-level optimization approach, the resulting solutions will compromise the
conflicting requirements on design variables for vehicles traveling at different speed and on
roads with different irregularity.
ACKNOWLEDEGMENTS
Financial support of this research from the Natural Sciences and Engineering Research
Council of Canada is gratefully acknowledged.
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