Spectral Optimization of the Suspension System of High-speed Trains

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.