Spectral Optimization of the Suspension System of High-speed Trains

D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
ISSN: 0975-3060 (Print), 0975-3540 (Online)
doi: 10.4273/ijvss.1.4.07
© 2009. MechAero Foundation for Technical Research & Education Excellence
98
International Journal of
Vehicle Structures & Systems
Available online at www.ijvss.maftree.org
Spectral Optimization of the Suspension System of High-speed Trains
Davood Younesian a , Amir Nankali
School of Railway Engineering,
Iran University of Science and Technology, Tehran, 16846-13114, Iran
a Corresponding Author, Email: younesian@iust.ac.ir
ABSTRACT:
Based on the spectral analysis in frequency domain, an optimal suspension system for high-speed passenger trains is
proposed in this paper. In optimization procedure, two main objective functions of the ride comfort and the fatigue life
of the suspension system are simultaneously taken into consideration. Spectral densities for the shear stress and also
vertical acceleration are obtained using the spectral approach. A multi-variable optimization is carried out using the
Genetic Algorithm. Four design parameters including the damping properties of the secondary and primary suspension
as well as the wire diameter of their coil springs are obtained. A comprehensive parametric study is carried out and
effects of different parameters consist of the travelling speed, level of irregularity and eccentricity of the wagon body on
the performance of the optimized system are then evaluated. Furthermore, influences of the positive and negative
deviation with respect to the optimal design parameters on the dynamic responses are studied.
KEYWORDS:
Spectral density, suspension system, high-speed train, optimization, Genetic Algorithm, Monte-Carlo
CITATION:
D. Younesian and A. Nankali. 2009. Spectral Optimization of the Suspension System of High-speed Trains, Int. J.
Vehicle Structures & Systems, 1(4), 98-103.
1. Introduction
Railway operators always seek methods to offer their
passengers with shorter journey time and better ride
comfort to make the railways more competitive with the
other means of transportation. Increasing the speed of a
railway vehicle is the direct way to shorten the journey
time but the ride quality would be deteriorated as the
high speed of the train will cause significant car body
vibrations. Therefore, an effective suspension system is
necessary to maintain or even improve the passenger’s
ride comfort while increasing the running speed of the
railway vehicle. For this goal, numerous methods have
been proposed in recent years.
Passive methods are generally grouped under the
two classes of a) Vibration Isolation and b) Vibration
Absorption. For several years, optimization of passive
suspension systems has been one of the most challenging
objectives in this area. As the train travelling speeds are
increasing, this engineering problem becomes more
complicated to solve. In addition to the ride quality,
shear failure of the suspension springs is nowadays a
challenging problem. There is always an alternation
between better ride quality and longer fatigue life. In
other words, higher stiffness with larger strength of the
coil spring always leads to poorer ride quality.
Consequently there exists an optimal value for the
properties of the suspension system.
A literature survey shows that many efforts have
been done so far in this area. Kim and Young-Guk tried
to optimize the suspension system of high speed trains
using Neural Network and Design of Experiment (DOE)
to build a meta-model for the system with 29 design
variables and 46 responses. The results show that the
proposed methodology yields to an improved design in
the ride comfort, unloading ratio and stability index [1].
Xu, Zhixiang and his colleagues studied the optimization
of suspension system of maglev vehicles. They
simplified the repulsive magnetic levitation vehicles with
secondary suspension into a two-degree-of-freedom
vibration model, and then analyzed the vibration
properties of the system [2].
Burchak and Savos'kin presented some methods for
choosing optimal suspension parameters of transport
facilities, and for predicting some aspects of its
reliability. In that paper, optimization of the suspension
parameters of transport facilities with the stochastic
properties of the system and disturbances were taken into
account [3]. Gao used a semi-active control of highspeed
train suspension system with magneto rheological
(MR) damper and showed that application of the semiactive
control system can effectively suppress the
vibration of suspension system [4]. Shieh and Niahn-
Chung proposed a systematic and effective optimization
process for the design of vertical passive suspension of
light rail vehicles (LRVs) using a new constrained multiobjective
evolution algorithm. They tried to attain the
best compromise between ride quality and suspension
deflections due to irregular gradient tracks [5].
Feasibility for improving the ride quality of railway
vehicles with semi-active secondary suspension systems
using magneto rheological (MR) dampers was studied by
Liao and Wang in [6]. A nine degree-of-freedom railway
vehicle model, which includes a car body, two trucks

D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
four wheel sets, was proposed to cope with vertical,
pitch and roll motions of the car body and trucks. The
LQG control law using the acceleration feedback was
adopted as the system controller, in which the state
variables were estimated from the measurable
accelerations with the Kalman estimator. Inter-vehicle
active suspension for railway vehicles was proposed by
Mei et al [7]. They developed a new optimization
process for the design of vertical active suspension
controllers using multi-objective genetic algorithm.
Performance of neural network for the identification
and optimal control of active pneumatic suspensions of
high-speed railway vehicles was studied by Nagai et al.
in [8]. It was shown that neural networks can be
efficiently trained to identify the dynamics of nonlinear
pneumatic suspensions, as well as being trained to work
as optimal nonlinear controllers. Zolotas et al. [9]
presented a work on a set of novel strategies for
achieving local tilt control, i.e. applied independently for
each vehicle rather than the whole train precedence
approach that is being commonly used. A linearized
dynamic model was developed for a modern tilting
railway vehicle with a tilt mechanism (tilting bolster)
providing tilt below the secondary suspension.
Surveying the literature shows that in most of the
published works just one of the objective functions of
ride quality or static strength of the suspension system
have been solely taken into account because of the
simplification. The present paper is aimed to propose a
technique to optimize the system simultaneously
obtaining the desired ride quality and static strength.
Furthermore, it should be noted that most of the works
already done in this area are mainly dealing with the
optimization in the time domain. Consequently, in such
optimization procedures, the obtained results severely
become dependent on the nature of irregularly input
data. In the present research, the corrugation profile is
simulated with its spectral density which intrinsically
contains several sample data. In other words, the selected
approach makes the procedure more general and reliable
for variety of the irregularity inputs.
A multi-variable optimization is carried out using
the Genetic Algorithm. Four design parameters of
damping coefficient of the secondary and primary
suspension as well as the wire diameter of their coil
springs are obtained. Contribution of the paper is mainly
directed to the new concepts of a) Optimization in
random frequency domain which is more realistic, b)
Combinational objective function simultaneously dealing
with minimum body acceleration and maximum fatigue
life. A parametric study is carried out and performance
of the proposed optimal suspension system is evaluated
for different values of the operational speed, corrugation
level and also eccentricity of the wagon body. Effects of
positive and negative deviation with respect to the
optimal design parameters on the dynamic responses are
then studied.
2. Mathematical Modelling
The wagon model is schematically illustrated in Fig. 1,
in which the carriage body is connected with two bogies,
via secondary suspension system indicated by Kss and
99
Css. The coach body has vertical and pitch motions in
the vertical plane. Wheel-sets are connected to bogies
via primary suspension system denoted by Kps and Cps.
Contact between wheel and rail is modelled by linear
Hertzian contact spring and is shown by KH.
Fig. 1(a): SKS300 High-speed train
Fig. 1(b): Schematic model of the passenger train wagon
According to the model, the equations of motion of
the carriage components are derived as follow:
[ m ] { & x&
} + [ C]
{ x&
} + [ K]
{ x}
= { f }
In which:
(1)
{f} = External force vector
[m] = Mass matrix
[c] = Damping matrix
[k] = Stiffness matrix
{x} = Displacement vector
H ω can be then obtained by
Harmonic transfer matrix ( )
[ ] [ ] [ ] 1
2
H( −
) = [ −ω
M + i C ω + K ]
ω
Consequently:
(2)
⎡ 2
2 2 2 2
−Ibdω
+ iω
( Css
1( + e)
+ Css(
1−e)
) + 1( −e)
Kss
+ ( 1+
e)
Kss
2iωeC
ss+
2eK
ss −iω(
Css(
1+
e))
−Kss
( 1+
e)
⎢
2
⎢
iω
( 2eC
ss)
+ 2eK
ss
−Mbdω
+ 2iωC
ss+
2Kss
−iωC
ss−Kss
⎢
−iω
( Css(
1+
e))
−Kss(
1+
e)
−iωC
ss−Kss
−Mbg
+ iω(
Cps+
Css)
+ Kps
+ Kss
H(
ω)
= ⎢
⎢
iω
Css(
1−e)
+ Kss(
1−e)
−iωC
ss−Kss
0
⎢
0
0
−iωC
_
⎢
ps Kps
⎢⎣
0
0
0
−1
iωC
ss ( 1 − e)
+ K ss ( 1 − e)
0
0
⎤
− iωC
0
0
⎥
ss − K ss
⎥
0
− iωC
_
0
⎥
ps K ps
⎥
− M bg _ iω(
Css
+ C ps ) + K ps + K ss
0
− iωC
ps _ K ps ⎥
0
− M + 2 + +
0
⎥
w KH K ps iωC
ps
⎥
− iωC
ps − K ps
0
− M w _ iωC
ps + 2KH
+ K ps ⎥⎦
(3)
In which, MBD and MBG are respectively the body and
bogie mass and IBD represents the body mass inertia.
Subscripts ss and ps denote secondary and primary
suspension systems. The main excitation of such a
dynamical system arises from the rail corrugation. Rail
corrugation consists of undesirable fluctuations in wear
on railway track and costs the railway industry
substantially for its removal by regrinding.

D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
Using above equation spectral density matrix of
displacement can be calculated as:
T*
S ( ω ) = H ( ω)
S ( ω)
H ( w)
(4)
xx
ff
In which S ff is density matrix of forces:
S
y
ff
⎡0
⎢
⎢
0
⎢0
= ⎢
⎢0
⎢0
⎢
⎢⎣
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
KH
KH
0
0
0
0
2
2
S
S
y
y
0 ⎤
0
⎥
⎥
0 ⎥
⎥
0 ⎥
2
KH S ⎥
y
⎥ 2
KH S y ⎥⎦
S is spectral density of the base displacement:
S(
Ω)
S y ( ω ) =
(6)
v
In which, ν is the train speed.
Track irregularity is assumed to be a random
function characterized by the following power spectral
density (PSD) function S(Ω) [7-8]:
(5)
AΩ2
S ( Ω)
=
(7)
2 2 2 2
( Ω + Ω )( Ω + Ω )
1
2
In which ⎛ 2π × ω ⎞
Ω = ⎜ ⎟ denotes the spatial frequency
⎝ v ⎠
(rad/m), and A(m), Ω1 (rad/m) and Ω2 (rad/m) are
relevant parameters indicated in Table 1. This spectrum
is based on the measurements made on the railways of
the USA [12]. ‘Class 6’ and ‘Class 1’ correspond to the
best and the worst quality respectively.
Table 1: Parameters of different tracks [11-12]
Track Class 4 5 6
A (m) 2.39X10 -5 9.35X10 -6 1.5X10 -6
Ω1 (rad/m) 2.06X10 -2 2.06X10 -2 2.06X10 -2
Ω2 (rad/m) 0.825 0.825 0.825
2
Acceleration mean square then can be derived by:
[ ] = ∫ ( )
+∞
ω ω
ii
2 4
x&
i Sxx
E & (8)
−∞
The other objective of the optimization procedure is
shear stress in coil springs with spectral density of:
2
⎛ 8D
⎞
τ a ( ω)
= ⎜ K B ⎟ 3
⎝ πd
⎠
Ssf
( ω)
(9)
4C
+ 2
K B =
4C
− 3
(10)
In which D and d are respectively spring and coil
diameter and C = d
S ω is the
D / is Spring Index. ( )
force spectral density and can be calculated for primary
suspension springs as:
2 ii ij ji jj
S ω = K ( S ω − S ω − S ω + S ω (11)
sf
( ) ( ) ( ) ( ) ( ))
spring
xx
xx
and for secondary suspension springs of the front bogie
as:
S
sf
( l +
2 ii ij ji jj
( ω)
= K spring [ Sxx
( ω)
− S xx ( ω)
− S xx ( ω)
+ S xx ( ω)
+
ik ki jk kj
2 kk
e)(
S ( ω)
+ S ( ω)
− S ( ω)
− S ( ω)
) + ( l + e)
S ( ω)]
xx
xx
xx
xx
xx
sf
xx
xx
(12)
100
and for secondary suspension springs of the rear bogie
as:
S
sf
( l −
2 ii ij ji jj
( ω)
= Kspring[
Sxx
( ω)
− Sxx(
ω)
− Sxx
( ω)
+ Sxx(
ω)
−
ik ki jk kj
2 kk
e)(
S ( ω)
+ S ( ω)
− S ( ω)
− S ( ω ) + ( l − e)
S ( ω)]
In which
xx
xx
xx
xx
xx
(13)
ij
S xx is i th row and j th column of S xx matrix. i
denote vertical motion of body, j denotes vertical motion
of rear and front bogies and k denotes degree of freedom
associated with the pitch motion of body.
Using Monte Carlo simulation algorithm [13], the
applied random surface can be generated. According to
this simulation algorithm, the random profile of the rail
surface can be generated by:
N
f ( x)
∑ 2 PSD ( ω ) ∆ω
cos( ω x + ε )
=
j=
1
f
j
j
j
(14)
In which εj is a random number between 0 to 2π and
with normal probability density function and
1
ω j = ω1
+ ( j − ) ∆ω
; j = 1,
2,...
N
(15)
2
( ωN
− ω1)
∆ ω =
(16)
N
and N is sufficiently large integer (1000 in this paper).
Using Monte-Carlo simulation, the irregularity profile
has been generated for three types of the rail classes and
illustrated in Fig. 2.
Y (m)
0.25
0.2
0.15
0.1
0.05
0
0 200 400 600 800 1000 1200
-0.05
-0.1
-0.15
-0.2
X(m)
Fig. 2: Track irregularities generated for three types of tracks
track 1
track 2
track 3
Genetic algorithm is used as the optimization
technique and a multi-variable optimization is carried
out. The algorithm is directed to simultaneously optimize
both objective functions i.e. RMS of the body
acceleration as well as the shear stress. Four design
parameters of damping coefficients of the secondary and
primary suspension systems as well as the wire diameter
of their coil springs are obtained. Flow-chart of the
implemented genetic algorithm is illustrated in Fig. 3.
3. Numerical Results
For a real high-speed wagon having mechanical
properties listed in Table 2, numerical optimization is
carried out and optimal values of the design parameters
are obtained. Variation of the non dimension
acceleration and shear stress are plotted with respect to
values of secondary suspension damping and its coil
spring diameter in Fig. 4. For a given primary
suspension system, minimum value of the curve obtained

D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
by intersection of the surfaces is taken as the optimal
values. The Genetic Algorithm is seeking for four
optimal parameters of the primary and secondary
suspension so that the level of acceleration becomes
minimum and simultaneously the level of shear stress in
coil spring remains less than a specific value (500Mpa).
Fig. 3: Flow chart of the implemented Genetic Algorithm
Table 2: Mechanical Properties of the wagon body
Parameter Symbol Value
body length
body mass
body mass inertia
L BD
20 m
M 40 ton
BD
I 1.3.3X10
BD
6 kgm 2
Bogie mass
M BG
1200 kg
Wheel set mass M w
1180 kg
Number of spring rings N 10
Springs shear module G 80 GPa
Spring diameter D 0.2 m
Non-dimension accel-
-eration and shear stress
Primary suspension
spring diameter (m)
Secondary suspension
damping (NS/m)
Optimal point
Fig. 4: Geometrical depiction of the optimization procedure
In order to validate the optimization procedure,
effects of positive and negative deviations with respect
to the optimal values of the properties of the suspension
systems obtained for v=20 m/s and e=2 m are illustrated
in Fig. 5 and 6. These Figures are numerically verifying
the optimization procedure. In all the Figures deviation σ
denotes actual value divided by the optimal value.
101
Non dimension acceleration and
shear stress
8
7
6
5
4
3
2
1
acceleration
stress
0
0.5 0.7 0.9 1.1 1.3 1.5
Fig. 5: Effects of deviations with respect to the optimal coil spring
diameter of the secondary suspension system
Acceleration (m/s 2 )
0.0366
0.0364
0.0362
0.036
0.0358
0.0356
0.0354
0.0352
Primary Suspension
Secondary Suspension
0.035
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
Fig. 6: Effects of deviations with respect to the optimal damping of
both suspension systems
In Figs. 7 and 8 the effects of deviation with respect
to optimal spring diameter of secondary suspension
system on acceleration and stress in time domain are
shown. As it is seen, the acceleration values obtained for
σ = 0.
5 (half of the optimal diameter) are less than
values of optimal system ( σ = 1 ) however the spring
shear stress is remarkably larger than values of optimal
system and vice versa for σ = 1.
5 . So it can be concluded
that the optimized system leads to minimum acceleration
with a shear stress less than the allowable stress (500
MPa).
Acceleration (m/s 2 )
1.
0.
0
0 0. 1 1. 2 2.
-0.5
-
2
1
-1.5
Time(s)
Fig. 7: Effect of deviations with respect to the optimal coil spring
diameter of the secondary suspension (acceleration)
Tables 3 and 4 are demonstrating how much error
may happen in case of any positive or negative deviation
with respect to the optimal values. It is seen that RMS of
the shear stress increases with decreasing of the spring
diameter up to 500 MPa and contrarily, RMS value of
the acceleration decreases.
σ
σ
1
σ=0.5
σ=1.0
σ=1.5

Shear stress (MPa)
395
375
355
335
315
295
275
D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
× . 13
× 3
σ=0.5 0.5
σ=1.0 1
σ=1.5 1.5
255
0 0.5 1
Time (s)
1.5 2 2.5
Fig. 8: Effects of deviations with respect to the optimal coil spring
diameter of the secondary suspension system (shear stress)
Table 3: Effects of deviations with respect to the optimal values of
the spring coil diameter of the primary suspension system
Coefficient Acceleration
(σ ) RMS(m/s 2 Stress RMS
) (MPa)
0.5 0.03 2605.5
0.6 0.0304 1362.0
0.7 0.0310 1215.3
0.8 0.0321 844.2671
0.9 0.0335 615.1957
1.0 0.0351 465.5598
1.1 0.0369 363.3230
1.2 0.0387 290.8722
1.3 0.0401 237.9541
1.4 0.0413 198.3112
1.5 0.0421 167.9688
Table 4: Effects of deviations with respect to the optimal values of
the damping of the primary suspension system
Coefficient Acceleration
(σ ) RMS(m/s 2 Stress RMS
) (MPa)
0.5 0.0365 499.43
0.6 0.0359 499.404
0.7 0.0355 499.3790
0.8 0.0353 499.3790
0.9 0.0352 499.3401
1.0 0.0351 499.3250
1.1 0.0352 499.3119
1.2 0.0352 499.3006
1.3 0.0353 499.2907
1.4 0.0354 499.2820
1.5 0.0355 499.2743
Optimum values for the damping of primary and
secondary suspensions are illustrated respectively in
Figs. 9 and 10 for different eccentricity values. As it is
seen, optimal damping values for the primary and
secondary suspensions are both decreasing functions of
the operational speeds. It is also seen that for large
values of eccentricity (e> 4 m) trend of variations of the
optimal values becomes significantly different. In other
words, for eccentricity values lower than 4 meters one
can assume that eccentricity has no significant effect on
the optimal values of primary suspension damping.
Variation of the optimal values of the coil spring
diameter of the primary and secondary suspensions is
illustrated in Fig. 11 for different operational speeds. As
it is seen dependency of the optimal diameters on the
operational speed is much lower than optimal damping
102
values. It is also seen that the optimal value of the
primary suspension coil spring diameter is generally
larger than secondary suspension in the whole speed
range. It is also seen that dependency of the secondary
suspension optimal stiffness on the operational speed is
lower than that of primary suspension system.
Optimal Cps (Ns/m)
210000
200000
190000
180000
170000
160000
150000
140000
130000
15 20 25 30 35 40 45 50 55
Fig. 9: Optimal value of the primary suspension system damping
Optimal Css (Ns/m)
190000
180000
170000
160000
150000
140000
130000
120000
15 20 25 30 35 40 45 50 55
Fig. 10: Optimal value of the secondary suspension system
damping
dss and dps (m)
0.055
0.0545
0.054
0.0535
0.053
0.0525
0.052
Speed (m/s)
Speed (m/s)
dps
dss
0.0515
18 28 38 48 58 68 78 88
Speed (m/s)
Fig. 11: Optimal value of the coil spring diameter for both
suspension systems
Influences of the track quality on the optimal values
of damping are illustrated in Fig. 12.”Track 1” denotes
class 4 and “track2” denotes class 5 in Table 1. As it is
seen, optimal value of the primary suspension damping
decreases when the track quality becomes worse and in
contrary, optimal values of the secondary suspension
damping increases. This conclusion is very well adopted
with reality because locking effect may happen for large
values of the primary suspension damping because it is
directly connected to the rail irregularities. Locking
phenomena (Large damping forces at high frequency) is
e=2
e=4
e=6
e=2
e=4
e=6

D. Younesian et al. 2009. Int. J. Vehicle Structures & Systems, 1(4), 98-103
one of disadvantages of viscose dampers which can
dramatically increase the transmitted accelerations.
Css and Cps (Ns/m)
220000
210000
200000
190000
180000
170000
160000
150000
140000
130000
track1_cps
track2_cps
track1_css
track2_css
120000
15 20 25 30 35 40 45 50 55
Speed (m/s)
Fig. 12: Effect of the track quality on optimal values of damping
4. Conclusions
An optimal suspension system for high-speed passenger
trains was proposed in this paper. Two main objective
functions of the ride comfort and the fatigue life of the
suspension system were simultaneously taken into
account. A multi-variable optimization was performed
using the Genetic Algorithm. Four design parameters
including the damping properties of the secondary and
primary suspension as well as the wire diameter of their
coil springs were obtained for a real high-speed wagon.
It was found that optimal damping values for the primary
and secondary suspensions are both decreasing functions
of the operational speeds. For eccentricity values lower
than 4 meters one can assume that eccentricity has no
significant effect on the optimal values of damping. It
was also found that optimal value of the primary
suspension damping decreases when the track quality
becomes worse and in contrary, optimal values of the
secondary suspension damping increases.
Dependency of the optimal diameters on the
operational speed is much lower than optimal damping
values. It was also observed that the optimal value of the
primary suspension coil spring diameter is generally
larger than secondary suspension in the whole speed
range. Dependency of the secondary suspension optimal
stiffness on the operational speed is lower than that of
primary suspension system. Validity of the optimization
was verified using numerical simulations both in
frequency and time domains. It was proved that any
deviations with respect to optimal value lead to
significant deviations with respect to the optimal
situation.
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EDITORIAL NOTES:
Edited paper from 2 nd Int. Conf. on Recent Advances in
Railway Engineering, 27-28 September 2009, Tehran, Iran.
GUEST EDITOR: Prof. Javad M. Sadeghi, School of Railway
Engineering, Iran University of Science and Technology,
Farjam St, Tehran 16846-13114, Iran.