Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100 Received for review: 2021-10-07
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2021-11-22
DOI:10.5545/sv-jme.2021.7429 Original Scientific Paper Accepted for publication: 2021-12-16
*Corr. Author’s Address: Shandong University of Technology, School of Transportation and Vehicle Engineering, China, yuyuewei2010@163.com 90
Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System 
and Biomechanical Behaviour Prediction
Zhao, L. – Yu, Y – Cao, J. – Zhou, W.
Leilei Zhao
1 
– Y uewei Y u
1,*
 – Jianhu Cao
1 
– Weiwei Zhou
2
1 
Shandong University of Technology, School of Transportation and Vehicle Engineering, China 
2 
Zibo V ocational Institute, School of Automotive Engineering, China
The biomechanical responses of the driver-truck system in a dynamic environment have become a significant concern in the design and 
control of trucks. When evaluating the riding comfort of the seat-cab system, it is necessary to predict the biomechanical responses of the 
driver’s different parts and directions. However, there is no reliable model and method for effectively predicting the response characteristics 
of the driver-seat-cab system. Aiming at such problem, firstly, based on the 7 DOF (degree-of-freedom) seated human biodynamic model 
established previously, a 10 DOF non-linear dynamic model of the driver-seat-cab system was created, and its vibration differential equations 
were established. Secondly, the vibration signals for simulation and verification were collected through the road test using a truck. Thirdly, 
based on the Newmark-β integration method, the specific solution process of the model was given. The non-linear damping coefficients of 
the front and rear dampers for the cab suspensions were measured with a bench test. Moreover, the simulations were conducted based on 
the measured model parameters, taking the collected frame vibration signals as the inputs. The results show that the simulation results 
agree with the test results, proving that the dynamic model can effectively predict the driver’s biomechanical responses. Finally, some useful 
conclusions were obtained through the simulation analysis. The established model and conclusions can provide technical support for comfort 
evaluation, optimization design, and control of the seat-cab suspension system.
Keywords: driver-seat-cab system, driver’s health, dynamic modelling, biomechanical responses
Highlights
• A 10 DOF non-linear dynamic model of the driver-seat-cab system was created.
• The proposed dynamic model can effectively predict the driver’s biomechanical responses. 
• The driver’s vertical vibration caused by the road excitation mainly concentrates in 0 Hz to 10 Hz.
0  INTRODUCTION
When trucks run on real roads, the vibration caused 
by road roughness directly affects ride comfort 
and human health [1]. When the vibration reaches a 
certain level, it will make drivers feel uncomfortable, 
even cause traffic accidents [2]. Thus, it is imperative 
to study the vibration characteristics of the driver-
vehicle under the dynamic environment. Moreover, 
it has significant application values in truck design, 
dynamic simulation, the auxiliary analysis of the man-
machine interface, and the prediction of the driver’s 
biomechanical responses [3]. 
With the rapid development of the truck industry, 
improving ride comfort has become a significant 
concern of modern truck designers [4] and [5]. The 
seat-cab system is an essential part of modern trucks 
[6] and [7]. Their suspension systems directly affect the 
NVH (noise, vibration, and harshness) performance 
[8] and [9]. The optimal design of the seat-cab system 
through dynamic models is an important and effective 
means to improve ride comfort. It is the premise 
and basis to accurately simulate the biomechanical 
responses of the driver coupled with the seat-cab 
system for the design and optimization of the cab 
and/or seat suspension [10]. If the biomechanical 
responses obtained are inaccurate, it is easy to lead to 
wrong optimization results.
The research methods of the vibration of 
driver-vehicle systems can be roughly divided into 
two categories: the experimental method and the 
theoretical method [11]. In recent years, with the 
development of testing technology, the experimental 
method has been widely applied. However, because of 
the restriction of the high cost of the test conditions, 
the researchers have been seeking to theoretically 
simulate and predict the driver’s biomechanical 
responses [12].
The truck driver’s vibration can be divided 
into local vibration and whole-body vibration. The 
local vibration refers to the vibration of the driver’s 
particular parts (e.g., the head and limbs), including 
the vibration of the driver’s hand or feet. The whole-
body vibration refers to the vibration transmitted to 
the body through the driver’s supporting surface. 
The driver’s vibration mainly belongs to the whole-
body vibration, which is often transmitted to drivers 
through their buttocks and back [13]. Generally, the 
whole-body vibration is the primary vibration form 
that causes bodily harm to drivers. In order to evaluate 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
91 Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction
the effects of the whole-body vibration on people, the 
International Standard Organization (ISO) proposed 
the international standard ISO 2631  [14], which has 
been supplemented and revised many times.
When studying the vibration of the driver-seat-
cab system for trucks, the driver is usually regarded 
as a rigid mass [15]. Marcu [16] studied the cab 
suspension control using a simplified cab model with 
3 DOFs; however, he did not consider the driver-seat-
cab dynamics interaction. Akçay and Türkay [17] 
derived a 3 DOF (degree-of-freedom) cabin model to 
investigate the ride comfort of the cabin system for a 
truck; they regarded the driver-seat-cab as a rigid mass. 
Wang et al. [18] made a fatigue failure assessment 
based on loading reproduction for commercial vehicle 
cab. They also ignored the vibration characteristics of 
the driver-seat system. Liu et al. [19] optimized the 
ride comfort of cab mounts but did not consider the 
drivers’ biomechanical characteristics. Zhao et al. [20] 
simulated the non-linear vibration responses of the cab 
system subject to suspension damper complete failure 
and took the driver-seat-cab as a rigid mass. These 
studies provide useful references for analysing the 
vibration of the driver-seat-cab system. However, few 
studies on integrated modelling and biomechanical 
responses simulation of the driver coupled with a 
seat-cab system for trucks undergoing random roads 
excitation exist.
The present paper’s main objective is to provide a 
reliable model for effectively predicting the responses 
characteristics of the driver-seat-cab system. The 
previous work [21] proposed a 7 DOF seated human 
biodynamic model. The test results show that the 
biodynamic model can effectively reproduce the 
seated human biodynamic responses. In this paper, as 
the extension of the previous work, a 10 DOF non-
linear dynamic model of the driver-seat-cab system 
was proposed to simulate, analyse, and predict 
the driver’s biomechanical responses for trucks 
undergoing random roads. 
In Section 1, combined with the 7 DOF seated 
human biodynamic model, a 10 DOF non-linear 
dynamic model of the driver-seat-cab system was 
created, and its vibration differential equations 
were established. In Section 2, the vibration signals 
for simulation and verification were collected. In 
Section 3, the specific solution process of the model 
was given. In Section 4, the non-linear damping 
coefficients of the front and rear dampers for the 
cab suspensions were measured. Moreover, based 
on the measured model parameters and the collected 
frame vibration signals, the model was validated. In 
Section 5, the biomechanical responses of the driver 
coupled with the seat-cab were simulated. In Section 
6, some useful conclusions were obtained through the 
simulation analysis.
1  INTEGRATED MODELLING OF DRIVER  
COUPLED WITH SEAT-CAB 
1.1  The 10 DOF Driver-seat-cab Model
To simulate, analyse, and predict the driver’s 
biomechanical responses for trucks undergoing 
random roads excitation, a non-linear dynamic model 
of the driver-seat-cab system is proposed, as shown in 
Fig. 1; the driver sub-model is shown in Fig. 2. This 
sub-model is based on the previous work [21]. The 
proposed model has 10 DOFs, with two DOFs for the 
cab (z
c
 and φ), one DOF for the seat (z
s
), and seven 
DOFs for the driver (z
1
, z
2
, z
3
, z
4
, z
5
, x
b
, and θ).
In Fig. 1, variables a and b are the horizontal 
distances from the front and rear suspensions to 
the cab mass centre, respectively; h is the vertical 
distance between the mass centres of the driver and 
the cab; l
x
 is the horizontal distance between the seat 
mounting point to the cab mass centre; K
s
 and C
s
 are 
the vertical stiffness and the damping coefficient of 
the seat suspension, respectively. The cab suspension 
system includes the front spring-damper assembly and 
the rear spring-damper assembly. K
f
 and C
f
 are the 
vertical stiffness and the damping coefficient of the 
front spring-damper assembly, respectively. K
r
 and C
r
 
are the vertical stiffness and the damping coefficient of 
the rear spring-damper assembly, respectively. C
f
 and 
C
r
 nonlinearly depend on the relative motion velocities 
of the front damper and the rear damper, respectively. 
q
f
 and q
r
 are the frame displacements at the mounting 
positions of the front spring-damper assembly and the 
rear spring-damper assembly, respectively.
 
Fig. 1.  The 10 DOF nonlinear dynamic model of the driver-seat-cab system 
 
K
s
 
C
s
 
x
c
 
 
z
c
 
 
l
x
 
q
f
 
 
q
r
 
 
h 
K
f
 
C
f
 
C
r
 
K
r
 
a 
b 
φ 
 
O
c 
 
O
b 
 Driver 
Seat 
Cab 
Fig. 1.  The 10 DOF non-linear dynamic model  
of the driver-seat-cab system

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
92 Zhao, L. – Yu, Y – Cao, J. – Zhou, W.
 
 
x
b
 
Seat m
s 
z
5
 
z
4
 
z
3
 
z
2
 
z
1
 
Spine 
K
3 
C
3 
 
K
1 
 
C
1 
 
K
5 
 
K
41 
 
C
41 
 
z
s
 
C
5 
 
k
t
, c
t 
 
K
4 
 
C
4 
 
K
2 
 
C
2 
 
Lower torso m
1 
Upper torso m
4 
Arms- 
m
3 
Viscera m
2 
Head and neck m
5 
K
bx 
 
C
b 
 
θ 
x
cs
 
Fig. 2.  The 7 DOF driver biodynamic model
In Fig. 2, the driver model is composed of two 
sub-models: the vertical model and the fore-and-
aft model [20]. The vertical model consists of five 
segments: head and neck, upper torso, arms, viscera, 
and lower torso. The spring (K
41
) and damper (C
41
) 
connecting the upper and lower torsos represent 
the body spine. In the fore-and-aft model, the main 
body mass (m
1
+ m
2
+ m
3
+ m
4
) is treated as a single 
combined mass. The head and neck and the main 
body are connected by a rotational degree. These rigid 
masses are coupled by linear elastic and damping 
elements.
1.2  Non-linear Motion Equations
1.2.1  Differential Equations
The differential equation of the cab vertical vibration 
can be expressed as
 
mz Kz aqCz aq
Kz bqCz b
cc fc ff cf
rc rr c
   

  
  
() ()
() (

   


 
q
Kz zC zz
r
sc ss sc ss
)
[( )( )], (1)
where, C
f
 and C
r
 can be further expressed as:
 C
Cz aq
Czaq
f
f+ cf
f- cf






  
  


0
0
, (2)
 C
Cz bq
Czbq
r
r+ cr
r- cr






  
  


0
0
. (3)
The differential equation of the cab pitch 
vibration can be expressed as
 
IK zaqC zaqa
Kz bqCz
   

 

  
  
[( )( )]
[( )(
fc ff cf
rc rr c
b bq b
Kz zC zz l
Kx xC x
x
 



 
 
r
sc ss sc ss
bc sb bc
)]
[( )( )]
[( )(
s sb
tt

 



xh
kc
)]
() () ,   (4)
where, I is the cab moment of inertia.
The differential equation of the seat vertical 
vibration can be expressed as
 
mz Kz zC zz
Kz zC zz
ss ss cs ss cs
s1 1s 1
 

  
 
() ()
() () .
1
 (5)
The differential equation of the lower torso 
vertical vibration can be expressed as
 
mz Kz zC zz Kz z
Cz zK z
11 1s 1s 22
22
 

   
 
() () ()
() (
11 1
14 11
   zC zz
44
)( ).
41 1
 (6)
The differential equation of the viscera vertical 
vibration can be expressed as
 
mz Kz zC zz
Kz zC zz
22 22 22
44
 

  
 
() ()
() () .
11
42 42
 (7)
The differential equation of the arms vertical 
vibration can be expressed as
 mz Kz zC zz
33 33 3
 
34 4
   () () . (8)
The differential equation of the upper torso 
vertical vibration can be expressed as
 
mz Kz zC zzKz z
Cz zK z
43 33 34 2
42 41
 

44 44
44
   
 
() () ()
() (   
 
zC zz
Kz zC zz
14 11
55 55
)( )
() () .


4
44
 (9)
The differential equation of the head and neck 
vertical vibration can be expressed as
 mz Kz zC zz
55 55 5
 
54 4
   () () . (10)
The differential equation of the whole-body 
longitudinal vibration can be expressed as:
 
mx ml ml
Kx xC xx
nn bb
bb cs bb cs
coss in 
 




 
55
2
 
() () , (11)
where, m
b
=m
1
+m
2
+m
3
+m
4
.
The differential equation of the head and neck 
pitch vibration can be expressed as
 
ml mlxm lg k
c
nn n 5
2
55




 



bt
t
coss in ()
() . (12)

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
93 Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction
In addition, the cab vertical displacement z
cs
 at 
the seat installation position can be expressed as:
 zz l
x cs c
 . (13)
The longitudinal (x-direction) displacement x
cs
 of 
the seat backrest can be expressed as:
 
xh
cs
.
 (14)
1.2.2  Matrix Equation
The displacement, velocity, and acceleration response 
vectors of the proposed model can be expressed as
 u[,,,,,,,,, ], zz zzzzzx
cs 12345b
T
 (15)
     

u[,,,,,,,,, ], zz zzzzzx
cs 12345b
T
 (16)
          

u[,,,,,,,,, ]. zz zzzzzx
cs 12345b
T
 (17)
The displacement and velocity excitations of the 
proposed model can be expressed as
 q =[,], qq
fr
T
 (18)
   q =[,]. qq
fr
T
 (19)
The matrix equation of the proposed model can 
be expressed as
 Mu CuKu Cq Kq    ++
uu qq
, (20)
where, M is the mass matrix, K
u
 is the system 
stiffness matrix, C
u
 is the system damping matrix, K
q
 
is the excitation stiffness matrix, C
q
 is the excitation 
damping matrix.
The matrix C
u
 can be expressed as 
C
u
ccc
cccc c
cccc


11 12 13
21 22 23 29
31 32 33 34
0000000
00000
000000
0
t
0 00 000
0000 000
000000 00
000
43 44 45 47
54 55 57
66 67
74 75
cccc
cc c
cc
ccc
7 76 77 78
87 88
92 99
00
000000 00
00 000000
00 000000
cc
cc
cc
cc 






tt
 

























,(21)
where, c
11
 = C
f
 + C
r
 + C
s
, c
44
 = C
1
 + C
2
 + C
41
, 
c
66
 = C
3
, c
22
 = a
2
C
f
 + b
2
C
r
 + l
x
2
C
s
 + h
2
C
b
 + c
t
, 
c
33
 = C
s
 + C
1
, c
55
 = C
4
 + C
2
, c
77
 = C
3
 + C
4
 + C
41
 + C
5
, 
c
88
 = C
5
, c
99
 = C
b
, c
21
 = c
12
 = aC
f
 + bC
r
 + l
x
C
s
,  
c
13
 = c
31
 = –C
s
, c
23
 = c
32
 = l
x
C
s
, c
34
 = c
43
 = –C
1
, 
c
45
 = c
54
 = –C
2
, c
47
 = c
74
 = –C
41
, c
57
 = c
75
 = –C
4
, 
c
67
 = c
76
 = –C
3
, c
78
 = c
87
 = –C
5
, c
29
 = c
92
 = hC
b
.
The matrix C
q
 can be expressed as
   C
q
Ca C
Cb C








ff
rr
T
00000000
00000000
. (22)
From Eqs. (21) and (22), it can be seen that both 
C
q
 and C
u
 contain C
f
 and C
r
. From Eqs. (2) and (3), it 
can be seen that C
f
 and C
r 
are non-linear. Thus, there 
are non-linear terms in C
u
 and C
q
.
Therefore, matrix Eq. (20) is a set of non-linear 
dynamic equations.
2  VIBRATION SIGNALS ACQUISITION
To obtain the vibration signals for simulation and 
validation, according to the national standard GB/T 
4970-2009 (China) [22], the road test was carried out, 
and the vibration signals were collected. The test was 
conducted on an automobile comprehensive test field. 
An asphalt road and a gravel road were selected as 
the test roads. For each road, the pavement is straight, 
and the longitudinal slope is not more than 1 %. A 
medium-sized truck was selected as the test vehicle. 
The driver is a male, and his weight is 72.0 kg. 
The test equipment is LMS Test.Lab test 
system, as shown in Fig. 3. Before the test, one 
accelerometer was installed on the seat surface and 
one accelerometer was installed on the seat backrest. 
In addition, two accelerometers were installed on 
the frame at the mounting positions of the front 
spring-damper assembly and the rear spring-damper 
assembly. The four accelerometers are PCB 356A26 
accelerometers. During the test, the driver fastened 
his seat belt and put his hands on the steering wheel 
and remained still. Moreover, the running speed was 
60 km/h. The sample frequency was set as 512 Hz. 
The total duration of the collected signal is 120 s for 
each measurement. The measured vibration signals on 
the asphalt road are shown in Fig. 4. The measured 
vibration signals on the gravel road are similar to 
those on the asphalt road.
Fig. 3.  The LMS Test.Lab system for the test  
of the vibration signals acquisition

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
94 Zhao, L. – Yu, Y – Cao, J. – Zhou, W.
a) 
0 20 40 60 80 100 120
-3
-2
-1
0
1
2
3
Time  [s]
Acceleration [m/s  
2
]
b) 
0 20 40 60 80 100 120
-1.0
-0.5
0.0
0.5
1.0
Time  [s]
Acceleration  [m/s 
2
]
c) 
0 20 40 60 80 100 120
-12
-8
-4
0
4
8
12
Time [s]
Acceleration  [m/s 
2
]
d) 
0 20 40 60 80 100 120
-12
-8
-4
0
4
8
12
Time  [s]
Acceleration  [m/s 
2
]
Fig. 4.  The measured acceleration signals on the asphalt road: 
a) the seat surface vertical acceleration, b) the seat backrest 
longitudinal acceleration, c) the vertical frame acceleration at the 
mounting position of the front spring-damper assembly, and d) the 
vertical frame acceleration at the mounting position of the rear 
spring-damper assembly
3  NUMERICAL SOLUTION METHOD  
OF THE NON-LINEAR EQUATION SET
The solution of Eq. (20) is the solving problem of the 
non-linear differential equation set. The Newmark-β 
method has good stability and high accuracy [23]. It 
can easily fulfil the needs of solving complex non-
linear dynamic differential equation set in a large 
number of projects. Therefore, this method is used to 
solve the numerical solution of Eq. (20). The iterative 
equations used in the Newmark-β method are as 
follows:
    uu uu
tt tt tt
t



 [( )] 1 , (23)
     uu uu u
tt tt tt t
tt

  

    [( )] ()
1
2
2
 . (24)
where, u
t
 is the displacement vector at time t,  u
t
 and 
 u
t
 are the velocity vector and the acceleration vector 
at time t, respectively; u
t+Δt
,  u
tt 
, and  u
tt 
 are the 
vectors of the displacement, the velocity, and the 
acceleration at time t+Δt, respectively. α and β are 
constant.
Based on the Newmark-β method, combined 
with the Eq. (20) and its non-linear characteristics, 
the specific solution process was given to solve Eq. 
(20) for the time length [0, T]. T is divided into N parts 
(nΔt = T), meaning the integral step is Δt. In this study, 
△t was set as the reciprocal of the sampling frequency 
512 Hz. The solution process based on Newmark-β 
method is shown in Fig. 5.
 
Fig. 5. The specific solution process for the equation set of the proposed model. 
 
Load the excitation signals (T=n △ t) 
Based on Eqs. (23) and (24), solve u
i 
and 
i
u 
i=n? 
End 
Yes 
Begin 
 
No 
Initialization setting of the Eq. (20) 
 
C
u
 
 
0
u 
 
0
u 
 
0
u 
Update C
u
 and C
q
 
Let i=1, start up the Newmark-β algorithm 
i=i+1 
 
C
q
 
Fig. 5.  The specific solution process for the equation set of the 
proposed model

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
95 Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction
4  MODEL VALIDATION
In order to verify the effectiveness of the proposed 
model and equations for the driver-seat cab system, the 
numerical simulations and comparison of the driver’s 
vibration responses were carried out according to the 
specific solution process.
4.1.  Model Parameters
To obtain the damping coefficients of the dampers for 
the front suspension and the rear suspension of the 
cab, using the ZJS-30 hydraulic test bench produced 
by Changchun Test Machine Research Institute, the 
damping characteristic tests of the dampers were 
carried out by bench test, as shown in Fig. 6a. The 
master computer is shown in Fig. 6b.
a)    b) 
Fig. 6.  Damper test; a) the test bench, and b) the master computer
Using the bench test, a sine wave excitation with 
an amplitude of 30 mm and a frequency of 1.0 Hz was 
exerted on each damper. At the same time, the damper 
damping force was measured by an NS-WL1 tension-
compression sensor. The damper displacement was 
measured with a SDVG20 displacement sensor. 
Based on the measurement data, the master computer 
automatically calculated the damping coefficients 
of the tested dampers, as shown in Table 1. Other 
parameters of the seat-cab system provided by the 
manufacturer, as shown in Table 2. 
Table 1.  The values of the damping coefficients
Moving stroke
Front 
damper
Rear 
damper
Compression damping coefficient [Ns/m] 2317 1574
Rebound damping coefficient [Ns/m] 6256 4013
Table 2.  The parameters values of the seat-cab system
Parameter Value Parameter Value
a [m] 0.87 I [kg·m
2
] 751
b [m] 1.00 C
s
 [Ns/m] 592
h [m] 0.25 K
s
 [N/m] 15000
l
x
 [m] 0.03 K
f
 [N/m] 52000
m
c
 [kg] 810 K
r
 [N/m] 32000
According to the curve fitting method in reference 
[21], by minimizing the errors of the seat-to-head 
transmissibility function between the computed values 
using the model and the tested values, the parameters 
of the human biomechanics model were re-estimated. 
The previous model parameter values in reference 
[21] were selected as the initial values in this research. 
When the root mean square errors converged to 0.01, 
the parameter values were determined, as shown in 
Table 3. 
Table 3.  The parameters values of the driver model
Parameter Value Parameter Value
C
1
 [Ns/m] 2375.4 K
1
 [N/m] 120103.1
C
2
 [Ns/m] 676.8 K
2
 [N/m] 5301.9
C
3
 [Ns/m] 144.8 K
3
 [N/m] 13172.5
C
4
 [Ns/m] 1791.7 K
4
 [N/m] 9150.8
C
41
 [Ns/m] 4020.2 K
41
 [N/m] 128193.0
C
5
 [Ns/m] 979.4 K
5
 [N/m] 292014.6
C
b
 [Ns/m] 621.2 K
b
 [N/m] 9925.3
c
t
 [Ns/m] 18.3 k
t
 [N/m] 779.0
m
1
 [kg] 10.2 m
4
 [kg] 20.2
m
2
 [kg] 9.0 m
5
 [kg] 5.8
m
3
 [kg] 8.1 l
n
 [m] 0.19
4.2  The Effectiveness Validation of the Proposed Model
In this section, the simulations were conducted and 
the simulation results were verified with the tested 
results. For the simulations, the measured frame 
vertical acceleration signals at the mounting positions 
of the front spring-damper assembly and the rear 
spring-damper assembly were taken as the inputs. The 
simulation time T was set as 120 s. The lower torso 
vertical frequency-weighted RMS (root mean square) 
acceleration values were calculated from the simulated 
 z
1
 and the measured  z
1
, respectively. The driver body 
longitudinal frequency-weighted RMS acceleration 
values were calculated from the simulated 
b
x  and the 
measured 
b
x  , respectively. A comparison of the 
frequency-weighted RMS acceleration values 
calculated from the measured data and from the 
simulated data is shown in Table 4.
From Table 4, it can be seen that the relative 
errors of the driver body longitudinal frequency-
weighted RMS acceleration values calculated from 
the simulated 
b
x  and the measured 
b
x  are less than 20 
%; the relative errors of the lower torso vertical 
frequency-weighted RMS acceleration values 
calculated from the simulated  z
1
 and the measured  z
1
 
are less than 10 %. In the longitudinal direction 
(x-direction), the relative errors are larger than those 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
96 Zhao, L. – Yu, Y – Cao, J. – Zhou, W.
in the vertical direction. Moreover, the driver body 
longitudinal frequency-weighted RMS acceleration 
values calculated from the simulated 
b
x  are less than 
those calculated from the measured 
b
x  . The main 
reasons are as follows: firstly, in the modelling 
process, the cab longitudinal acceleration  x
c
 was not 
considered. According to Eq. (14), the longitudinal 
acceleration  x
cs
 of the seat backrest can be expressed 
as   xh
cs
. In fact,  x
cs
 should be written as 
   xh x
cs c
   . Secondly, there are some errors in the 7 
DOF driver biodynamic model. From the previous 
work [21], the model shows a higher average 
goodness-of-fit in the vertical direction than in the 
fore-and-aft direction. Moreover, in the fore-and-aft 
direction, the average goodness-of-fit is slightly larger 
than 80 %. It is noted that, although the driver model 
accuracy is not ideal, the fitting results show an 
improved fit compared with other existing human 
models.
Table 4.  A comparison of the frequency-weighted RMS acceleration 
values
Road Response
Measured
[m/s
2
]
Simulated
[m/s
2
]
Absolute 
error [m/s
2
]
Relative 
error [%]
Asphalt 
road
 z
1
0.34 0.37 0.03 8.8
b
x 
0.16 0.13 0.03 18.7
Gravel 
road
 z
1
0.56 0.51 0.05 8.9
b
x 
0.26 0.21 0.05 19.2
By comparison, the results show that the 
simulation values agree with the test values. Although 
the proposed model should be further improved, 
the model is workable, and the model accuracy is 
acceptable. The proposed model can effectively 
predict the driver’s biomechanical responses and basic 
dynamic characteristics for trucks running on random 
roads.
5  BIOMECHANICAL SIMULATION  
OF DRIVER COUPLED WITH SEAT-CAB
5.1  Simulation Analysis of the Time Domain Responses
In the prediction and analysis of ride comfort, it 
is helpful to understand the changes of the human 
vibration in time series. The time-domain analysis 
of the driver’s biomechanical responses is conducive 
to the design and the control of vehicle suspension 
systems. Moreover, it can help designers to determine 
the problems in design and to make corresponding 
improvements.
The simulations were conducted using 
the proposed model to investigate the driver’s 
biomechanical responses for trucks running on random 
roads, taking the collected frame vibration signals 
as the inputs. The simulated force and acceleration 
responses for trucks running on the asphalt road are 
shown in Fig. 7. The simulated driver’s biomechanical 
responses for trucks running on the gravel road 
are similar to those on the asphalt road. The RMS 
force values and the frequency-weighted RMS 
acceleration values of the driver’s different parts and 
different directions are shown in Table 5 and Table 6, 
respectively.
Table 5.  The RMS force values
Position
Asphalt 
road
Gravel 
road
The head for the vertical motion [N] 3.79 4.55
The upper torso for the vertical motion [N] 13.71 17.82
The arms for the vertical motion [N] 5.87 7.04
The viscera for vertical motion [N] 6.25 7.50
The lower torso for the vertical motion [N] 6.56 8.20
The whole-body for longitudinal motion [N] 6.15 7.38
The head for the pitch motion [Nm] 0.10 0.13
Table 6.  The frequency-weighted RMS acceleration values
Position
Asphalt 
Road
Gravel 
road
The head for the vertical motion [m/s
2
] 0.37 0.53
The upper torso for the vertical motion [m/s
2
] 0.37 0.52
The arms for the vertical motion [m/s
2
] 0.43 0.57
The viscera for vertical motion [m/s
2
] 0.37 0.53
The lower torso for the vertical motion [m/s
2
] 0.36 0.51
The whole-body for longitudinal motion[m/s
2
] 0.13 0.24
The head for the pitch motion [rad/s
2
] 0.24 0.32
From Fig. 7, it can be seen that the time histories 
of the forces of the lower torso, the viscera, the arms, 
the upper torso, and the head are very similar. Table 
5 shows that the RMS force values of the upper 
torso are larger than those of other organs. For trucks 
running on the asphalt road, the RMS force value of 
the upper torso is nearly twice as large as that of the 
lower torso. The RMS force values illustrate a similar 
rule for trucks running on the gravel road. Table 6 
shows that the frequency-weighted RMS acceleration 
values of the viscera, the arms, the upper torso, and 
the head are all larger than that of the lower torso. 
For trucks running on the asphalt road, the frequency-
weighted RMS acceleration value of the arms is 
19.4 % larger than that of the lower torso. For trucks 
running on the gravel road, the frequency-weighted 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
97 Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction
a) 
0 20 40 60 80 100 120
-30
-20
-10
0
10
20
30
Time  [s]
Force  [ N]
b) 
0 20 40 60 80 100 120
-30
-20
-10
0
10
20
30
Time  [s]
Force  [ N]
c) 
0 20 40 60 80 100 120
-30
-20
-10
0
10
20
30
40
Time  [s]
Force [ N]
d) 
0 20 40 60 80 100 120
-60
-40
-20
0
20
40
60
Time  [s]
Force  [ N]
e) 
0 20 40 60 80 100 120
-20
-10
0
10
20
30
Time  [s]
Force [ N]
f)
0 20 40 60 80 100 120
-40
-20
0
20
40
Time  [s]
Force  [ N]
g) 
0 50 100 150
-0.5
0.0
0.5
Time  [s]
Torque  [ Nm ]
h) 
0 20 40 60 80 100 120
-40
-20
0
20
40
Time  [s]
Force  [ N]
i) 
0 20 40 60 80 100 120
-3
-2
-1
0
1
2
3
Time  [s]
Acceleration  [m/s  
2
]
 
j) 
0 20 40 60 80 100 120
-1.0
-0.5
0.0
0.5
1.0
Time  [s]
Acceleration  [m/s  
2
]
Fig. 7.  The simulated vibration responses for trucks running on the asphalt road: a) the lower torso vertical force, b) the viscera vertical 
force, c) the arms vertical force, d) the upper torso vertical force, e) the head vertical force, f) the whole-body longitudinal force, g) the head 
pitch torque, h) the seat vertical force, i) the upper torso vertical acceleration, and j) the whole-body longitudinal acceleration

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
98 Zhao, L. – Yu, Y – Cao, J. – Zhou, W.
RMS acceleration values illustrate the similar rule. 
The results show that the driver body can amplify the 
vertical vibration of the lower torso for trucks running 
on the asphalt road and the gravel road.
5.2  Simulation Analysis of the Frequency Domain 
Responses
To analyse the relationship between the vibration 
energy and the vibration frequency of each part of the 
driver body, based on the simulated force responses 
in the time domain, the PSD (power spectral density) 
curves were plotted, as shown in Fig. 8.
From Fig. 8, it can be seen that when the truck 
drives on the asphalt road, the driver’s body vertical 
vibration energy caused by the road excitation mainly 
concentrates in the low-frequency section (0.5 Hz 
to 10.0 Hz). The driver’s body vibration energy in 
the high-frequency segment (10.0 Hz to 100.0 Hz) 
is almost zero. The frequency section of the most 
extreme concentration for the vibration energy is about 
from 1.0 Hz to 5.0 Hz. The largest resonance peak 
appears at about 1.5 Hz. This is related to the natural 
frequencies of the cab system. From the vibration in 
Eq. (20), it can be obtained that the vertical vibration 
natural frequency is 1.35 Hz and the pitch vibration 
natural frequency is 1.70 Hz for the cab system. Figs. 
8a to e show that the PSD curves of the vertical forces 
of the lower torso, the viscera, the arms, the upper 
torso, and the head are very similar. Fig. 8f illustrates 
that the longitudinal vibration energy of the driver’s 
body concentrates in the low-frequency section 
from 1.0 Hz to 8.0 Hz. These simulation results are 
consistent with the results of the human vibration 
characteristics test [24] and [25].
a) 
10
-1
10
0
10
1
10
2
50
100
150
200
Frequency  [Hz]
PSD  [N
2
/Hz]
b) 
10
-1
10
0
10
1
10
2
50
100
150
200
Frequency  [Hz]
PSD  [N
2
/Hz]
c) 
10
-1
10
0
10
1
10
2
20
40
60
80
100
120
140
Frequency  [Hz]
PSD  [N
2
/Hz]
d) 
10
-1
10
0
10
1
10
2
100
200
300
400
500
600
700
800
Frequency  [Hz]
PSD  [N
2
/Hz]
e) 
10
-1
10
0
10
1
10
2
0
20
40
60
80
Frequency  [Hz]
PSD  [N
2
/Hz]
f) 
10
-1
10
0
10
1
10
2
0
20
40
60
80
100
Frequency  [Hz]
PSD  [N
2
/Hz]
Fig. 8.  The PSD curves for trucks running on the asphalt road: a) lower torso vertical force, b) viscera vertical force,  
c) arms vertical force, d) upper torso vertical force, e) head vertical force; f) whole-body longitudinal force

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)2, 90-100
99 Nonlinear Coupled Dynamic Modelling of Driver-seat-cab System and Biomechanical Behaviour Prediction
6  CONCLUSIONs
This paper aims to provide a reliable model and 
method for effectively predicting the responses 
characteristics of the driver-seat-cab system. Based 
on the 7 DOF seated human biodynamic model 
established previously, a 10 DOF non-linear dynamic 
model of the driver-seat-cab system was created, and 
its vibration differential equations were established. 
The vibration signals for simulation and verification 
were collected through the road test. Based on the 
Newmark-β integration method, the specific solution 
process of the model was given. The non-linear 
damping coefficients of the front and rear dampers 
for the cab suspensions were measured with a bench 
test. The proposed model was validated, and the 
simulations were numerically conducted. Some useful 
conclusions were drawn: 
(1)  By comparison, the results show that the 
simulation values agree with the test values. The 
model is workable, and the model accuracy is 
acceptable. The proposed model can effectively 
predict the driver’s biomechanical responses 
and basic dynamic characteristics for trucks 
undergoing random roads excitation.
(2)  The time histories of the vertical forces and 
accelerations of the lower torso, the viscera, the 
arms, the upper torso, and the head are similar for 
trucks undergoing random roads excitation. 
(3)  When the truck runs on the asphalt road, the 
driver’s body vertical vibration energy caused 
by the road excitation mainly concentrates in 
the low-frequency section (0 Hz to 10.0 Hz). 
The driver’s body vibration energy in the high-
frequency segment (10.0 Hz to 100.0 Hz) is 
almost zero.
To reasonably evaluate the design effect of the 
cab suspension system, it is necessary to assess the 
cab comfort objectively. According to the ISO 2631-1 
standard [14], the cab comfort is usually evaluated on 
the basis of the vibration accelerations at the driver’s 
backrest and hip. However, the existing cab model can 
only simulate the vibration of the cab floor. Therefore, 
based on the existing cab model, the cab comfort 
cannot be accurately and comprehensively simulated. 
The proposed model in this paper can overcome the 
shortcomings of the existing model, and can more 
effectively simulate the vibration accelerations at 
the driver’s backrest and hip. It can be seen that the 
proposed model is more helpful to guide the matching 
and designing of the cab suspension system than 
the existing model. In addition, the proposed model 
can also be used to analyse the driver’s whole-
body vibration when the truck is impacted, which 
can provide a theoretical basis for taking effective 
measures to avoid the injury of the driver’s viscera. 
It should be noted that the model is not suitable for 
the vibration fatigue analysis of the human body. 
The shortcoming of the proposed model is that it 
can not accurately reproduce the real longitudinal 
acceleration. Thus, to overcome the shortcoming, the 
longitudinal DOF of the cab should be considered, 
and the guide mechanism of the cab suspension 
system should be built into the dynamic model in the 
following study.
7  ACKNOWLEDGEMENTS
This work is supported by Nature Science Foundation 
of Shandong (ZR2020ME127) and the National 
Natural Science Foundation of China (11802338).
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