*Corr. Author’s Address: Shandong University of Technology, Zibo, China, zhaoleilei611571@163.com 73
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81 Received for review: 2022-09-24
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-12-25
DOI:10.5545/sv-jme.2022.375 Original Scientific Paper Accepted for publication: 2023-01-19
Analytical Formulae and Applications of Vertical Dynamic 
Responses for Railway Vehicles
Yu, Y . – Song, Y . – Zhao, L. – Zhou, C.
Yuewei Yu – Yunpeng Song – Leilei Zhao
*
 – Changcheng Zhou
Shandong University of Technology, School of Transportation and Vehicle Engineering, China
To effectively improve the estimated level of railway vehicles’ vertical dynamic responses and provide a more suitable reference for the 
selection of its secondary suspension damping parameter, this paper has derived the root mean square values analytical formulae of the 
car body vertical acceleration, the secondary suspension vertical stroke, and the axle box vertical action force for railway vehicles under the 
random excitation of the track closer to the actual track characteristics. The correctness of the analytical formulae is verified by testing a 
real vehicle. Then, according to the analytical formulae derived, an analytical design method of the optimal damping ratio for the secondary 
suspension system is constructed based on the multi-objective programming and single-objective interval constraint analysis, which can 
be used to find the best trade-off for conflicting performance indices, such as ride comfort, running smoothness, and running safety, and 
the influences of the system parameters on the optimal damping ratio are analysed. This research can effectively characterize the vertical 
vibration response of railway vehicles and provide an effective reference for the initial design of the railway vehicle secondary suspension 
damping parameter.
Keywords: railway vehicle, vertical dynamic response, model deduction, damping parameter design, optimal compromise
Highlights
• 	 The analytical formulae of the vertical dynamic response for railway vehicles were derived.
• 	 An optimal damping ratio design method for the secondary suspension system was constructed.
• 	 The influences of the system parameters on the optimal damping ratio were analysed.
0  INTRODUCTION
As bogie suspension system parameters are essential 
for the running stability, safety, and comfort of railway 
vehicles, the design of their suspension system 
parameters has become an important part of the bogie 
system design and has become a key concern for 
designers [1].
Generally, a bogie suspension system is composed 
of the elastic element and the damper element; it mainly 
includes two important parameters to be designed, i.e., 
the spring static deflection value and the suspension 
damping parameter value. In order to obtain an 
accurate and reliable design result for the suspension 
damping parameter, one widely used method is to use 
computer simulation technology and vehicle dynamic 
analysis software to optimize and determine the final 
value [2] to [4]. This method can simulate the actual 
operation of railway vehicles well [5] and [6] and can 
achieve a more accurate design result. However, the 
simulation analysis requires much time and cannot 
visually show the one-to-one correspondence between 
parameters and responses, which is not conducive 
to the adjustment and optimization of the structural 
parameters in the initial design stage of suspension 
systems. Meanwhile, the suspension parameter values 
before design are often unknown. The complexity of 
the model makes it complicated to do much work to 
design the suspension damping parameters, which is 
not conducive to the designers making a reasonable 
engineering choice quickly and effectively. In order to 
solve this problem, the most effective method is to use 
the analytical method, i.e., through reasonable model 
simplification, from the theoretical analysis point of 
view, and then obtain the required design value of 
the damping parameters [7] to [9]. For this reason, 
based on the simplified 1/4 vehicle model, using the 
simplified track irregularity power spectral density (of 
the form 1/ω
4
 and 1/ω
2
) as the input excitation of the 
system, the analytical formula for calculating the root 
mean square (RMS) value of the bogie frame vertical 
acceleration for railway vehicles is derived, and an 
analytical design method for the primary vertical 
suspension damping ratio of railway vehicles is given 
in [1] and [10], respectively. In addition, according 
to the simplified 1/4 vehicle model (mainly refers 
to the two-axle bogie railway vehicle), using the 
simplified track irregularity power spectral density 
(of the form 1/ ω
4
 and 1/ ω
2
) as the input excitation of 
the system, the analytical formulae for calculating the 
RMS values of the car body vertical acceleration, the 
secondary suspension vertical stroke, and the axle box 
vertical action force for railway vehicles are derived; 
an analytical design method for the secondary vertical 
suspension damping ratio of railway vehicles is given 
in [11] and [12]. 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
74 Yu, Y. – Song, Y. – Zhao, L. – Zhou, C.
However, compared with the power spectral 
density of the form 1/[(ω
2
+v
2
Ω
c
2
) (ω
2
+v
2
Ω
r
2
)], which 
is widely used in the dynamic simulation of railway 
vehicles, the power spectral density of the form 1/
ω
4
 overestimates and underestimates in the low-
frequency and high-frequency parts of the track 
irregularity, respectively, and the power spectral 
density of the form 1/ω
2
 overestimates in both the 
low-frequency and high-frequency parts of the track 
irregularity.
Thus, in this paper, using the Germany track 
vertical irregularity power spectral density, which is 
closer to the actual track line, as the input excitation 
of the vehicle system, based on the analytical method, 
the RMS values analytical formulae of the vertical 
dynamic response of railway vehicles under the 
excitation of the form of 1/[(ω
2
+v
2
Ω
c
2
)(ω
2
+v
2
Ω
r
2
)] 
are deduced, and a more reasonable analytical design 
method for the secondary vertical suspension damping 
ratio of railway vehicles is given. As an extension and 
supplementing [11] and [12], it will provide effective 
technical support for the engineering selection of the 
bogie vertical suspension system parameters to deeply 
explore the vertical dynamic of railway vehicles on 
actual track lines in the theoretical analysis field.
1  SYSTEM MODEL
1.1  Vertical Vibration Model of Railway Vehicles for 
Analytical Calculation
Because of the weak coupling between the vertical and 
lateral dynamic behaviours of railway vehicles, when 
analysing its dynamic performance, they are often 
modelled separately and analysed separately, which 
can simplify the model establishment and facilitate 
the result analysis [13] on the premise of ensuring 
sufficient analysis accuracy. Numerous studies have 
shown that the commonly used vertical dynamic 
models of railway vehicles mainly include three types 
[14] to [16]: the 1/4 vehicle model, the single vehicle 
model, and the multi-marshalling vehicle model; each 
of the models has its advantages and its application 
occasions. Here, the 1/4 vehicle model is widely used 
in the analytical design of the suspension system 
parameters [10] to [12] and the semi-active and active 
suspension control research [17] and [18], because of 
its advantages of easy qualitative understanding of 
the relationship between the vehicle vertical dynamic 
characteristics and the structural parameters, simple 
solving process and easy engineering application. 
Therefore, in order to construct the analytical 
calculation model of the vertical dynamic response 
of railway vehicles and to provide effective guidance 
for the initial design of the secondary suspension 
damping parameter, this paper takes the 1/4 railway 
vehicle model [11] (which mainly contains two bogies 
per carriage, two wheelsets per bogie) as the model 
reference and carries out various research work, as 
shown in Fig. 1.
Fig. 1.  1/4 vehicle model
In Fig. 1, m
1
 is the half mass of the bogie frame; 
m
2
 is the quarter mass of the car body; K
1
 and K
2
 
are the vertical equivalent stiffness of the primary 
suspension and the secondary suspension; C
1
 and C
2
 
are the vertical equivalent damping of the primary 
suspension and the secondary suspension; z
1
 and z
2
 
are the vertical displacements of the bogie frame and 
the car body; z
v
 is the track irregularity.
According to the d'Alembert principle, the 
vibration differential equations of the 1/4 vehicle 
model can be obtained.
 
mz Cz zC zz Kz z
Kz z
mz
11 11 21 21 1
21 2
22
0
  

  
 
() () ()
()
vv
   





Cz zK zz
22 12 21
0 () ()
.

 (1)
By using Fourier transform, the transfer functions 
between  z
2
 and z
v
, f
d
 and z
v
, F
d
 and z
v
 can be solved 
respectively according to Eq. (1), as follows:
H
CC CK CK KK
mm Cm Cm
zz
j
j
v
12 12 21 12
12 12 21

 

 
 


2
43 2
4
~
()
(    

  

Cm KK
CC Km Km Km CK CK
22 12
12 12 21 22 12 21
j
j
)
() ()
,


3
2
 (2)
H
Cm Km
mm Cm Cm Cm KK
fz
j
j
j
dv
12 12
12 12 21 22 12



 

   
~
()
32
43
 
  

() ()
,
CC Km Km Km CK CK
12 12 21 22 12 21
j 
2
(3)
H
CmmC Cm CCmK mm
KC CK
Fz
j
j
dv
1121 21 1221 12
12 1


 
  



~
()
(
54
2 21 21 21 2
12 12 21 22 1
j
j
)( )( )
()
mm KK mm
mm Cm Cm Cm K
 


 


32
43
K K
CC Km Km Km CK CK
2
12 12 21 22 12 21
j



 


() ()
,

2
 (4)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
75 Analytical Formulae and Applications of Vertical Dynamic Responses for Railway Vehicles
where, f
d
 is the secondary suspension vertical stroke 
f
d
 = z
2
 – z
1
; F
d
 is the axle box vertical action force  
FC zz Kz z
dv v
 
11 11
() () . 
1.2  Excitation Model of Track Irregularity
As the main excitation source of the vertical 
vibration of railway vehicles, the random input of 
the track vertical irregularity is basically a stationary 
random process along the track. Usually, according 
to the measured irregularity data of the track, the 
mathematical statistics is made, and the irregularity 
is expressed as the power spectral density form by 
interpolation processing [19]. The research indicates 
that there are mainly three kinds of analytical 
expressions for the track irregularity power spectral 
density commonly used in the research of railway 
vehicles at present, as shown in Eqs. (5), (6), and 
(7). The power spectral densities shown in Eqs. (5) 
and (6) are mainly used in the analytical calculation 
of railway vehicle dynamics and the semi-active 
and active suspension control research. The power 
spectral density shown in Eq. (7) is the closest to the 
actual track line and is widely used in the dynamic 
simulation analysis of railway vehicles [19].
 S
Av
v1
b
()
()
, 



2
3
4
 (5)
where, A
b
 is the track roughness coefficient, 
A
b
=0.928× 10
-10
 m
-1
, v is the vehicle speed.
 S
Av
v2
r
() , 



2
2
 (6)
where, A
r
 is the track roughness coefficient, 
A
r
 = 2.5×10
-7
 m.
 S
Av
vv
v3
vc
2
cr
2
() , 







2
3
22 22 2


 (7)
where, A
v
 is the track roughness coefficient, A
v
= 
4.032×10
-7
 m; Ω
c
 and Ω
r
 are the truncated spatial 
frequencies, Ω
c
=0.824 6 m
-1
, Ω
r
=0.020 6 m
-1
.
According to Eqs. (5) to (7), the power spectral 
density functions of the track irregularity at 300 km/h 
are expressed in the double logarithmic coordinate 
system, as shown in Fig. 2. It can be seen from the 
figure that, the power spectral densities shown in Eqs. 
(5) and (6) are straight lines with slopes of -4:1 and 
-2:1, respectively. The power spectral density shown 
in Eq. (7) can be approximated to a combination of 
three slopes (0:1, -2:1, -4:1). Compared with Eq. 
(7), Eq. (5) overestimates and underestimates in the 
low-frequency and high-frequency parts of the track 
irregularity, respectively; Eq. (6) overestimates in 
both the low-frequency and high-frequency parts 
of the track irregularity, but it is consistent with the 
actual line in the mid-frequency range.
Fig. 2.  Power spectral density curve of track irregularity
2  ANALYTICAL DESCRIPTION OF THE RAILWAY VEHICLE 
VERTICAL DYNAMICS RESPONSE
To quickly and effectively characterize the vertical 
vibration response of railway vehicles in actual 
operation and to enable designers to make reasonable 
judgments and choices on the initial design values 
of the system parameters quickly, using the power 
spectral density of the track irregularity shown in 
Eq. (7) as the input excitation source of the railway 
vehicle, the RMS values analytical formulae of the 
railway vehicle vertical dynamic response will be 
solved in this section. Note that the RMS values 
analytical formulae of the railway vehicle vertical 
dynamic response under the power spectral densities 
shown in Eqs. (5) and (6) can be found in references 
[11] and [12], respectively, which will not be 
introduced here.
2.1 RMS Values of the railway Vehicle Vertical Vibration 
Response
In the study of railway vehicle dynamics, the RMS 
values of the system response are usually used to 
evaluate the vibration characteristics and isolation 
effect of the vehicle system [11], [12] and [14]. 
According to the theory of random vibration, the 
following equation can be obtained.
  
xx z
HS
2
2




() () .
~
jd
v
v
 (8)
Here, x represents the response of the system,   
H
xz
()
~
j
v
ω is the transfer function, S
v
(ω) is the power 
spectral density of the system input.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
76 Yu, Y. – Song, Y. – Zhao, L. – Zhou, C.
For a linear system, the following relationship 
exists between its amplitude-frequency characteristics:
 HH H
xz xz xz
() () () .
~~ ~
jj j
vv v
 
2
 (9)
Therefore, according to Eq. (9), Eqs. (2) to (4) 
can be expressed as follows
 H
NN
DD
xz
()
() ()
() ()
,
~
j
jj
jj
v



2



 (10)
where, N(jω) and D(jω) are the numerator and the 
denominator of the Eq. (2), Eq. (3) and Eq. (4), 
respectively.
In addition, the power spectral density function 
shown in Eq. (7) can be rewritten as follows:
S
Av
vvvv
v3
vc
2
ccrr
jjjj
() , 



       
2
3


 (1 1)
Thus, according to Eqs. (10) and (11), the RMS 
values of the car body vertical vibration acceleration, 
the secondary suspension vertical stroke, and the 
axle box vertical action force can be obtained by 
using the method of integral solution of the complex 
variable function [20], respectively.


 z
Avaa aaaa aa aa ba aa aa
2
22 3
63
2
4352 5
2
1650 05
2
13

   2
vc
 () (
6 61 45 11
2
60 35 1252
0
2
5
3
03
2
45
  




aaab aa aaaa aa b
aa aaaa a
)( )
0 03
3
60 235
2
013560 145
2
1
3
6
2
1
2
25
32
2
aa aaaa aaaaaa aaa
aa aaaa
 


6 61
2
3461
2
4
2
51 2
2
5
2
123
2
61 2345
  

aaaa aaa aaaa aaaa aaaa
, (12)


f
Avaa aaaa aa da aa aaaa a
d
2
vc

  
22 3
05
2
1451 36 00 1
2
60 35 12
 () ( a ad
aa aaaa aaaa aaaa aaaa
51
0
2
5
3
03
2
45 03
3
60 235
2
01356
32
) 



   a aaaa
aa aaaa aaaa aaa aaa
0145
2
1
3
6
2
1
2
2561
2
3461
2
4
2
51 2
2
5
2
2

   a aaaa aaaaa
123
2
61 2345


, (13)


F
Avaaaa aa aaaa aaaa aa
d
2
vc

  
22 3
1346 1
2
6
2
1256 14
2
52
2
5
2
2
2  ( a aa aaaa aaaa aaae
aa aa aa aa
3
2
62 3450 3560 45
2
0
03 45 25
2
3
2
 

 
)
(
6 61 56 10 1450 5
2
1362 01 25 1
2
60
   aaae aa aa aa aaae aa aa aa aa )( )(
3 35 3
00
2
5
3
03
2
45 03
3
60 235
2
01356
3
ae
aa aa aaaa aa aaaa aaaaa
) 

   

  
2
2
0145
2
1
3
6
2
1
2
2561
2
3461
2
4
2
51 2
2
5
aaaa
aa aaaa aaaa aaa aaa
2 2
123
2
61 2345


aaaa aaaaa
, (14)
where,
b
0
 = C
1
2
 C
2
2
,
b
1
 = (C
1
K
2
 + C
2
K
1
)
2
 – 2C
1
C
2
K
1
K
2
,
b
2
 = K
1
2
K
2
2
;
d
0
 = C
1
2
m
2
2
,
d
1
 = K
1
2
m
2
2
;
e
0
 = C
1
2
m
1
2
m
2
2
,
e
1
 = [C
1
C
2
(m
1
 + m
2
)+K
1
m
1
m
2
]
2
  
– 2C
1
m
1
m
2
(K
1
C
2
+K
2
C
1
) (m
1
 + m
2
),
e
2
 = [(K
1
C
2
 + K
2
C
1
)
2
 – 2K
1
K
2
C
1
C
2
](m
1
 + m
2
)
2
 
– 2K
1
2
K
2
m
1
m
2
(m
1
 + m
2
),
e
3
 = K
1
2
K
2
2
(m
1
 + m
2
)
2
;
a
0
 = m
1
m
2
,
a
1
 = C
1
m
2
 + C
2
m
1
 + C
2
m
2
 + m
1
m
2
v(Ω
c
 + Ω
r
),
a
2
 = C
1
C
2
 + v(Ω
c
+Ω
r
)(C
1
m
2
 + C
2
m
1
 + C
2
m
2
)
 + K
1
m
2
 + K
2
m
1
 + K
2
m
2
 + m
1
m
2
Ω
c
Ω
r
v
2
,
a
3
 = C
1
K
2
 + C
2
K
1
 + v(Ω
c
+Ω
r
)
 ·(C
1
C
2
 + K
1
m
2
 + K
2
m
1
 + K
2
m
2
)
 + (C
1
m
2
 + C
2
m
1
 + C
2
m
2
)Ω
c
Ω
r
v
2
,
a
4
 = K
1
K
2
 + v(C
1
K
2
 + C
2
K
1
)(Ω
c
 + Ω
r
)
 +(C
1
C
2
 + K
1
m
2
 + K
2
m
1
 + K
2
m
2
)Ω
c
Ω
r
v
2
,
a
5
 = K
1
K
2
(Ω
c
 + Ω
r
)(C
1
K
2
 + C
2
K
1
)Ω
c
Ω
r
v
3
,
a
6
 = K
1
K
2
Ω
c
Ω
r
v
2
.
2.2  Test Verification of the Analytical Formulae
In order to verify the correctness of the RMS values 
analytical formulae for the vertical dynamic response 
of railway vehicles, a CRH2 EMU, which is widely 
used in China, is taken as an example to test its 
vibration, and the result of the vibration test and the 
analytical calculation is analysed. The vehicle running 
speed is 200 km/h, the sampling frequency is 500 Hz, 
and the sampling time length is 120 s. The vehicle 
parameter values (1/4 vehicle equivalent parameters) 
of the CRH2 EMU are shown in Table 1.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
77 Analytical Formulae and Applications of Vertical Dynamic Responses for Railway Vehicles
Table 1.  Equivalent parameters of 1/4 CRH2 EMU
Parameters Unit Values
m
1
kg 1,300
m
2
kg 8,150
K
1
N/m 2,352,000
K
2
N/m 189,140
C
1
N·s/m 39,200
C
2
N·s/m 20,000
Fig. 3 shows a picture of the vehicle vibration 
test. Table 2 gives the comparisons results of the RMS 
values of the car body vertical acceleration and the 
secondary suspension vertical stroke obtained from 
the test and the analytical calculation.
a)
b) c)
Fig. 3.  Vehicle vibration test; a) test vehicle,  
b) acceleration sensor installed on the upper end of the secondary 
suspension, and c) acceleration sensor installed on the lower end 
of the secondary suspension
Table 2.  Comparisons of the test and calculation results
RMS values Calculation results Test results Absolute deviation
σ
 z
2
 [m/s
2
]
0.440 0.487 -0.047
σ
f
d
 [m]
0.013 0.012 0.001
As can be seen from Table 2, the analytical results 
are in good agreement with the actual vehicle test 
results, and the relative deviations of the RMS values 
of the car body vertical acceleration and the secondary 
suspension vertical stroke between the analytical 
results and the vehicle test results are only 9.65 % 
and 8.33 %, respectively. This shows that the RMS 
values analytical formulae for the vertical dynamic 
response of railway vehicles established are correct 
and reliable.
3  INFLUENCE OF SECONDARY VERTICAL SUSPENSION ON 
RAILWAY VEHICLE VERTICAL VIBRATION RESPONSE
In the parameter design of railway vehicles suspension 
system, in order to make the design result be of more 
practical value in engineering, the stiffness parameter 
is usually converted to frequency, and the damping 
coefficient is usually converted to the damping ratio. 
Therefore, according to the definition of the frequency 
and the damping ratio, it is known that the frequency 
and the damping ratio of the secondary suspension 
system can be written as [1]:
 f
K
m
C
Km
2
2
2
2
2
22
1
2 2


 ,. (15)
According to Eq. (15), substitute it into Eqs. (12) 
to (14), then the influence of the secondary vertical 
suspension on the vertical vibration response of the 
train can be obtained, as shown in Figs. 4 to 6. Here, 
the vehicle parameter values are shown in Table 1.
Fig. 4.  Influence of secondary vertical suspension  
on the car body vertical vibration acceleration  
and the secondary suspension vertical stroke
As can be seen from Figs. 4 to 6, under a certain 
natural frequency f
2
, if the damping ratio ξ
2
 is too 
small, the secondary suspension vertical stroke will 
be too large, which is not conducive to the train 
operation. With the increase of the damping ratio ξ
2
, 
the secondary suspension vertical stroke decreases 
gradually, while the car body vertical vibration 
acceleration and the axle box vertical action force first 
decrease and then increase, that is, there are minimum 
extreme points for both. It can be seen that, when 
the actual frequency value of the high-speed train 
is adopted, i.e., f
2 
= 0.7 Hz to 1.2 Hz [4], selecting 
an appropriate damping ratio ξ
2
 can make the car 
body vertical vibration acceleration, the secondary 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
78 Yu, Y. – Song, Y. – Zhao, L. – Zhou, C.
suspension vertical stroke, and the axle box vertical 
action force reach a low compromise effect at the 
same time.
Fig. 5.  Influence of secondary vertical suspension on the car body 
vertical vibration acceleration and the axle box vertical action force
Fig. 6.  Influence of secondary vertical suspension  
on the secondary suspension vertical stroke  
and the axle box vertical action force
4  ANALYTICAL DESIGN OF THE SECONDARY SUSPENSION 
DAMPING PARAMETER
It can be seen from Section 3 that the secondary 
vertical suspension damping parameter has a 
highly significant influence on the car body vertical 
acceleration, the secondary suspension vertical stroke, 
and the axle box vertical action force for railway 
vehicles. Therefore, in order to make the train have 
good running quality, when designing the secondary 
suspension damping parameter, the influence of these 
three indexes should be considered comprehensively. 
Based on this, a design method of the secondary 
suspension damping parameter for railway vehicles 
will be studied in this section by using the established 
RMS values analytical formulae.
4.1 Damping Ratio Design of the Secondary Suspension 
System
According to Eq. (15), substituting CK m
22 22
2   
into Eq. (12), solving the partial derivative of the 
car body vertical acceleration RMS value σ
 z
2
 with 
respect to the secondary suspension damping ratio 
ξ
2
, and let dd 
 z
2
2
0
   / , then, the analytical 
design equation of the optimal damping ratio for the 
secondary suspension system based on the minimum 
RMS of the car body vertical acceleration can be 
established, that is
 


ccccc
cccc
02
8
12
7
22
6
32
5
42
4
52
3
62
2
72 8



  0, (16)
where, Λ
c0 
to
 
Λ
c8
 are the coefficients of the 
analytical design equation expressed by vehicle 
parameters and vehicle speed, respectively.
Similarly, if let the partial derivative of the axle 
box vertical action force RMS value σ
f
d
, Eq. (14), 
with respect to the secondary suspension damping 
ratio ξ
2
 equal to zero, the analytical design equation of 
the optimal damping ratio for the secondary 
suspension system based on the minimum RMS of the 
axle box vertical action force can be established.
 


sssss
ssss
02
8
12
7
22
6
32
5
42
4
52
3
62
2
72 8



   0, (17)
where, Λ
s0 
to
 
Λ
s8
 are the coefficients of the analytical 
design equation expressed by vehicle parameters and 
vehicle speed, respectively.
Thus, solving Eqs. (16) and (17) with respect 
to the positive real root of ξ
2
, the optimal damping 
ratios of the secondary suspension system based on 
the minimum RMS value of the car body vertical 
acceleration and the axle box vertical action force can 
be obtained, respectively, i.e., the design values of ξ
c
, 
and ξ
s
.
Furthermore, in order to effectively avoid the 
probability of the suspension hitting the elastic stop 
block, according to the relationship between the 
probability distribution and the standard deviation, 
the relationship between the RMS value of the vertical 
stroke and the limit stroke [f
d
] of the secondary 
suspension system can be obtained, that is:
 3
f
f
d
d
[] . (18)
Therefore, according to Eq. (13), solving Eq. 
(18) with respect to the positive real root of ξ
2
, the 
minimum damping ratio of the secondary suspension 
system, i.e., ξ
b
 can be obtained based on the maximum 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
79 Analytical Formulae and Applications of Vertical Dynamic Responses for Railway Vehicles
a)          b) 
c)             d) 
e)            f) 
Fig. 7.  Influence of system parameters on the damping ratio of the secondary suspension;  
a) bogie mass m
1
, b) car body mass m
2
, c) vertical stiffness of the primary suspension K
1
, d) vertical stiffness  
of the secondary suspension K
2
, e) vertical damping of the primary suspension C
1
, and f) vehicle running speed v
RMS value of the secondary suspension vertical 
stroke.
It can be seen that the design of the secondary 
suspension damping parameter is a multi-objective 
optimization problem. In order to improve a railway 
vehicle’s comprehensive performance and to simplify 
the design process, this paper transforms the multi-
objective optimization problem into a single-
objective interval constraint problem by using the 
linear weighting method. Based on this, the optimal 
compromise between the minimum RMS value of 
the car body vertical acceleration and the minimum 
RMS value of the axle box vertical action force can 
be obtained. Also, the optimal damping ratio can 
effectively avoid the suspension impact limit stroke:
 
  
 
2
1

 




uc sb u
bb u
() ,
,
. (19)
Here, α is a weighting factor, and its value can be 
determined according to the importance of each sub-
target, α ∈ [0, 1]. Note that, in order to meet the needs 
of the engineering and improve the design efficiency, 
the golden section method [21] can be used to let 
α = 0.618.
4.2  Influence of System Parameters on the Optimal Ration 
of the Secondary Suspension
To determine the influence of each parameter on the 
optimal damping ratio of the secondary suspension 
system, the optimal damping ratios of the secondary 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
80 Yu, Y. – Song, Y. – Zhao, L. – Zhou, C.
suspension system under each parameter are analysed 
with the example of the railway vehicle shown in 
Table 1. The curves of the secondary suspension 
damping ratio with the variation of the system 
parameters are obtained, as shown in Fig. 7. Here, the 
weighting factor α = 0.618, the vehicle running speed 
v = 300 km/h, and the limit stroke [f
d
] = 60 mm.
As can be seen from Fig. 7, the optimal damping 
ratio ξ
c
: almost unchanged with the increase of m
1
; 
decreases with the increase of m
2
; decreases with 
the increase of K
1
; increases with the increase of K
2
; 
increases first and then decreases with the increase of 
C
1
, but the overall change is not obvious; decreases 
with the increase of v. The optimal damping ratio ξ
s
: 
increases with the increase of m
1
; decreases with the 
increase of m
2
; decreases first and then increases with 
the increase of K
1
; decreases first and then almost 
unchanged with the increase of K
2
; decreases first 
and then increases with the increase of C
1
; increases 
gradually with the increase of v, but the overall 
change is not obvious. The minimum damping ratio 
ξ
b
: increases with the increase of m
1
; increases with 
the increase of m
2
; first increases with the increase of 
K
1
 and then remains unchanged; decreases with the 
increase of K
2
; almost unchanged with the increase of 
C
1
; increases with the increase of v. The compromised 
damping ratio ξ
u
: increases with the increase of m
1
; 
decreases with the increase of m
2
; decreases first and 
then increases with the increase of K
1
; decreases first 
and then increases with the increase of K
2
; almost 
remains unchanged with the increase of C
1
; decreases 
with the increase of v, but the overall change is small. 
It can be seen that, among the system parameters, 
the car body mass m
2
 and bogie frame mass m
1
 have 
the greatest influence on the optimal damping ratio 
of the secondary suspension system, followed by 
the secondary suspension vertical stiffness K
2
, the 
primary suspension vertical stiffness K
1
, the vehicle 
running speed v, and the primary suspension vertical 
damping C
1
. Therefore, when choosing the damping 
parameter of the secondary suspension system for 
railway vehicles, the influence of vehicle running 
speed should be taken into account in addition to 
the parameters of the vehicle itself, in which this 
phenomenon has not been found in previous studies.
4.3  Engineering Design Example
Taking the train shown in Table 1 as an example, the 
damping ratio of its secondary vertical suspension is 
designed by using the established analytical design 
method. The design results of the damping ratio at 
different vehicle running speeds as shown in Table 3. 
Here, the secondary suspension vertical limit stroke 
[f
d
] = 60 mm.
Table 3.  Design results of the damping ratio
Damping 
ratio
Design value
v = 200 km/h v = 250 km/h v = 300 km/h v = 350 km/h
ξ
c
0.206 0.195 0.187 0.180
ξ
s
0.307 0.321 0.334 0.345
ξ
b
0.107 0.131 0.154 0.175
ξ
u
0.244 0.243 0.243 0.243
ξ
2
0.244 0.243 0.243 0.243
As can be seen from Table 3, the design 
values of the secondary suspension damping ratio 
corresponding to the CRH2’s operation speed range 
(200 km/h to 350 km/h) are all around 0.24, which 
is basically the same. Therefore, in order to take into 
account the running quality of the vehicle at different 
running speeds, the damping ratio of the secondary 
suspension system can be chosen as ξ
2 
= 0.24. It can 
be seen that the design result is close to the original 
vehicle design value 0.25; moreover, it is within the 
feasible design range (0.2 to 0.4) of the damping ratio 
of the secondary vertical suspension system given in 
literature [1], which indicates that the design value of 
the damping ratio obtained by this method is reliable.
5  CONCLUSIONS
1. According to the 1/4 railway vehicle model, 
using the Germany track vertical irregularity 
power spectral density as the input excitation of 
the vehicle system, the RMS values analytical 
formulae of the car body vertical acceleration, 
the secondary suspension vertical stroke, and the 
axle box vertical action force are derived, and the 
correctness of the analytical formulae is verified 
by the real vehicle test. The analytical formulae 
can more reasonably and accurately estimate 
the dynamic characteristics of the actual vehicle 
running on the track. 
2. According to the analytical calculation formulae 
derived, an analytical design method of the 
optimal damping ratio for the secondary 
suspension system is proposed based on the multi-
objective programming and single-objective 
interval constraint analysis, which can be used to 
find the best trade-off for conflicting performance 
indices such as ride comfort, running smoothness 
and running safety.
3. The influences of the system parameters on the 
optimal damping ratio are analysed. It can be seen 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)1-2, 73-81
81 Analytical Formulae and Applications of Vertical Dynamic Responses for Railway Vehicles
that, when choosing the secondary suspension 
damping parameter, the influence of the vehicle 
running speed should be taken into account in 
addition to the parameters of the vehicle itself. 
This study can provide an effective theoretical 
reference for the selection of the initial design 
value of the secondary suspension damping 
parameter for railway vehicles.
Note that this paper has deduced the analytical 
formulae of the vertical dynamic responses for 
railway vehicles and proposed an analytical design 
method of the damping parameter for its secondary 
suspension system. Although the research is based on 
the simplified model of the railway vehicle, it can help 
to have a qualitative understanding of the phenomena 
referring to suspension dynamics and enable designers 
to make reasonable engineering choices quickly 
and effectively. Moreover, it can greatly simplify 
the inconvenience of analysis and solution caused 
by many unknown parameters in the early stage of 
design.
6  ACKNOWLEDGEMENTS
This work is supported by the Natural Science 
Foundation of Shandong Province under Grant 
ZR2021QE082 and ZR2020ME127.
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