In this paper different mathematical models of the vehicle load are presented, depending on the one of the moving load types. In general, vehicle load models can be classified into two types: static and dynamic moving load models. Static moving load model is model with nonvarying load value and shape in time. Dynamic moving load model is defined by differential equations of motion where the road roughness function affects the solution. Due to this influence some of the differential equations cannot be solved analytically and the numerical approach has to be performed. Therefore, in this work Matlab Simulink models have been developed to solve this problem. Validation of Simulink models is carried out and presented for all analysed dynamic moving load models

Jočković, Miloš

Petronijević, Mira

Šarkić-Glumac, Anina

English

 GraFar - Repository of the Faculty of Civil Engineering

1 
5th International Congress of Serbian Society of Mechanics 
Arandjelovac, Serbia, June 15-17, 2015    
 
 
 
MODELS RELATED TO THE MOVING LOAD PROBLEMS 
 
Miloš Jočković1, Anina Šarkić1, Mira Petronijević1  
 
1 Faculty of Civil Engineering, 
The University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade, Serbia 
e-mail: mjockovic@grf.bg.ac.rs, anina@grf.bg.ac.rs, pmira@grf.bg.ac.rs 
 
 
 
Abstract 
  
In this paper different mathematical models of the vehicle load are presented, depending on 
the one of the moving load types. In general, vehicle load models can be classified into two types: 
static and dynamic moving load models. Static moving load model is model with nonvarying load 
value and shape in time. Dynamic moving load model is defined by differential equations of 
motion where the road roughness function affects the solution. Due to this influence some of the 
differential equations cannot be solved analytically and the numerical approach has to be 
performed. Therefore, in this work Matlab Simulink models have been developed to solve this 
problem. Validation of Simulink models is carried out and presented for all analysed dynamic 
moving load models.  
 
Key words: moving vehicle load, dynamic, MATLAB Simulink 
 
 
1. Introduction 
 
Moving load problems are very common in engineering. Such problems are related to the 
soils subjected to load which moves in space and excite the soils into vibration. These vibrations 
affect the surrounding buildings which might have negative effects not only to buildings, but also 
to humans or sensitive equipment. In order to obtain numerical model for traffic induced vibration 
prediction, the appropriate vehicle load mathematical models are required. In addition, these 
models can be used in vehicle design, since the ride quality depends on vehicle vertical 
displacement, velocity and acceleration. Therefore, different types of vehicle load mathematical 
models for static, as well as for dynamic moving load are developed and presented in this paper.  
 
 
2. Different moving load model 
 
A mathematical models used to describe the behavior of the moving load, in our case vehicle 
load, can be classified into two groups, static and dynamic. Which mathematical model will be 
applied depends on the type of the requested output data. In the following, different static as well 
as the dynamic moving load models will be presented. 
 
 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
2 
2.1 Static moving load model 
 
First mathematical model, called the static mathematical model is analysed. The main 
assumption in this model is that only static component with constant load shape and constant 
velocity exists. Load shape in plane can be described as function [1]: 
2
0
2sin
2
x
LS p x
L


 
 
  
 
 
 (1) 
where p0 is amplitude of vehicle wheel force and L is the length of wheel load transfer to road. In 
this model the dynamic and inertial part of loading are neglected.   
Spatial representation of static load can be presented in the similar way: 
2
0
2 2sin sin
2 2
x y
x y
x y
L L
S p x y
L L
 
 
  
 
  
 
 (2) 
where p0 also presents an amplitude of vehicle wheel force, Lx and Ly are the length of wheel load 
transfer to road respectively toward and perpendicular to the vehicle motion. 
 
2.2 Dynamic moving load model 
 
Due to the road roughness the dynamic and inertial parts of loading occur which have to be 
taken into account. Therefore as a second type, dynamic mathematical model which incorporates 
the dynamic behavior of the vehicle is considered. In Figure 1 several well known dynamic 
models of vehicle load are presented:  1-DOF model, the Quarter car model (QCM) with 2-DOF, 
as well as the Half car model with 2-DOF (HCM 2DOF) and 4-DOF (HCM 4DOF). 
 
Fig. 1. Dynamic mathematical models of vehicle load: a-1DOF, b-2DOF (1/4 car model, QCM), c-2DOF 
(HCM 2DOF), d-4DOF (HCM 4DOF) 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
3 
A mathematical model of a vehicle dynamic system is defined as a set of equations that 
represents the dynamics of the system. The equations of motion are the system of n coupled 
second-order ordinary differential equations, where n denotes number of degrees of freedom 
(DOF) of the system. In the following, the differential equations of motion of above mentioned 
dynamic moving load models are presented. 
 
2.2.1 Dynamic 1DOF moving load model 
 
The simplest dynamic moving load model represents 1DOF model (Figure 1a) [2,9], where 
vertical position of the vehicle body (ms) is defined with one parameter, xs. The differential 
equation of motion is given as: 
s s s s s s s sm x c x k x c y k y       (3) 
where cs is vehicle damping, ks vehicle stiffness and y road roughness. 
 
2.2.2 Dynamic 2DOF moving load model (Quarter Car Model (QCM)) 
 
Quarter Car Model (QCM) [3,7,9] analyses the behavior of the single wheel (mu) and vehicle 
body (ms) connected with main suspension (Figure 1b). Main suspension consists of vehicle 
damping (cs) and stiffness (ks). Single wheel is in turn connected to the ground via tyre damping 
(cu) and stiffness (ku). The differential equations of motion that describe the behavior of this 
model are given as: 
   
0s s s s s u s s s u
u u s s s u u s u s u u u u
m x c x c x k x k x
m x c x c c x k x k k x k y c y
    
       
  
     (4) 
which can be presented in the following matrix form: 
0 0
0
s s s s s s s s
u u s s u u s s u u u u
m x c c x k k x
m x c c c x k k k x k y c y
              
                               
 
    (5) 
 
2.2.3 Dynamic 2DOF moving load model (Half car model with 2-DOF (HCM 2DOF)) 
 
Half car model with 2-DOF [4], vertical displacement xs and angular movement θ of the 
vehicle body at the center of gravity, is presented in Figure 1c. The model is defined by vehicle 
body mass (ms) and inertia moment (Jo), front and rear vehicle damping, cf and cr, and front and 
rear vehicle stiffness kf and kr. The differential equations of motion of this model are given as: 
       
       
1 2 1 2
2 2 2 2
1 2 1 2 1 2 1 2
1 2 1 2
s s f r s f r f r s f r
f f r r f f r r
f r s f r f r s f r
f f r r f f r r
m x c c x c l c l k k x k l k l
c y c y k y k y
J c l c l x c l c l k l k l x k l k l
c y l c y l k y l k y l
 
  
        
   
        
   
 
 
 
 
 (6) 
 
 
 
 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
4 
The matrix form of the Equation (6) is: 
1 2 1 2
2 2 2 2
1 2 1 2 1 2 1 2
1 2 1 2
0
0
f r f r f r f rs s s s
f r f r f r f r
f f r r f f r r
f f r r f f r r
c c c l c l k k k l k lm x x x
c l c l c l c l k l k l k l k lJ
c y c y k y k y
c y l c y l k y l k y l
  
             
                         
    
      
 
 
 
 
 
(7) 
 
 
 
 
where l1 and l2  are the lengths between the front and rear axles and the vehicle center of gravity, 
respectively. 
 
2.5 Dynamic 4DOF model (Half car model with 4-DOF (HCM 4DOF)) 
 
Half car model with 4-DOF [5, 6], which is also called pitch-bounce model, typically consists 
of either the left or right half of the vehicle. Also, either the front or rear half of the vehicle can be 
taken into account. In this paper the left (right) half of the vehicle, presented in Figure 1d, is 
considered. This vehicle model is composed of three rigid bodies: vehicle body, which is defined 
by vehicle body mass (ms) and inertia moment (Jo), and two wheels, which masses are m1 and m2. 
This model has four degrees of freedom: the heave and pitch motion of vehicle body, xs and θ, 
and heave motion of wheels, x1 and x2. The differential equations of motion are: 
   
   
     
 
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
1 2 1 2 1 2
1 2
1 1
0
f f s f f f s f
r r s r r r s r
s s f r f r s f r f r f r s
f r
o f
m x c c x c x l c k k x k x l k c y k y
m x c c x c x l c k k x k x l k c y k y
m x c x c x c c x l c l c k x k x k k x
l k l k
J l c x
 
 



         
         
          
  

   
   
   
       
 
2 2
2 2 1 2 1 2 1 1 2 2 1 2
2 2
1 2 0
r f r s f r f r f r s
f r
l c x l c l c x l c l c l k x l k x l k l k x
l k l k


         
  
 
 (8) 
The matrix presentation of Equation (8) can be written in the following form: 
1 11 1 1
2 22 2 2
1 23
2 2
1 2 1 2 1 2
1 1
2
00 0 0
00 0 0
0 0 0
0 0 0
0
0
f f f
r r r
f r f r f rs s
f r f r f ro
f f f
r
c c c l cm x x
c c c l cm x x
c c c c l c l cm x x
l c l c l c l c l c l cJ
k k k l k
k k
 
        
                                         
  
 

 
 
 
 
1 1 1 1 1
2 2 2 2 2 2
1 2
2 2
1 2 1 2 1 2
0
0
r r
f r f r f r s
f r f r f r
x c y k y
k l k x c y k y
k k k k l k l k x
l k l k l k l k l k l k 
     
                                  


 (9) 
Where cf and cr are front and rear vehicle damping, kf and kr front and rear vehicle stiffness, c1 
and c2 front and rear tyre damping, k1 and k2 front and rear tyre stiffness. Lengths between the 
front and rear axles and the vehicle center of gravity are given with l1 and l2, respectively. 
 
3. Simulink dynamic moving load models 
 
Since the equations of motion, with arbitrary road roughness function (y), cannot be solved 
analytically the Matlab Simulink model is developed to solve this problem. The Simulink [9] is a 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
5 
software package that enables user to model, simulate, and analyse systems whose outputs change 
over time, i.e. the dynamic systems. User creates a block diagram, using the Simulink model 
editor that graphically presents time-dependent mathematical relationships among the system’s 
inputs and outputs. The block diagram consists of the basic building blocks from the Simulink 
library. In the following, Simulink models for previously specified dynamic moving load models 
are presented. 
 
Fig. 1. Matlab Simulink model for 1DOF dynamic system (Equation (3)) 
 
 
Fig. 2. Matlab Simulink model for QCM (Equation (4)) 
 
 
Fig. 3. Matlab Simulink model for HCM 2DOF (Equation (6)) 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
6 
 
Fig. 4. Matlab Simulink model for HCM 2DOF (Equation (8)) 
 
4. Validation of Simulink models 
 
In order to validate Simulink models the solutions of differential equations of motion, for all 
cases presented in Figure 1, obtained by Simulink models are compared with analytical solutions. 
The dynamic input is a steady state harmonic excitation 3500.002 i ty e  , presented in Figure 5. 
The vehicle properties used for simulations are shown in Table 1. 
 
 1DOF [2] 2DOF (QCM) [3] 
HCM 
2DOF [5] 
HCM 
4DOF [5] 
ms [kg] 236.124 241.5 1200 1200 
cs [Ns/m] 1385.4 300 - - 
ks [N/m] 12394 6000 - - 
Jo [kgm2] - - 2100 2100 
l1 [m] - - 0.847 0.847 
l2 [m] - - 1.513 1.513 
cf [Ns/m] - - 2500 2500 
kf [N/m] - - 28000 28000 
cr [Ns/m] - - 2000 2000 
kr [N/m] - - 21000 21000 
mu [kg] - 41.5 - - 
cu [Ns/m] - 1500 - - 
ku [N/m] - 140000 - - 
m1 [kg] - - - 62.2 
c1 [Ns/m] - - - 700 
k1 [N/m] - - - 134000 
m2 [kg] - - - 60 
c1 [Ns/m] - - - 700 
k1 [N/s] - - - 134000 
 
Table 1. Dynamic moving load model properties  
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
7 
 
Fig. 5. Road roughness function 3500.002 i ty e   
In Figure 6 the vehicle body displacement obtained for 1DOF dynamic model using the 
Simulink as well as the analytical solution are presented. 
 
Fig. 6. Vehicle body displacement xs, 1DOF model 
 
The results obtained using both methods are the same, which means that the Simulink model 
for 1DOF system is well-set and can be used for any type of road roughness functions.  
The same procedure is repeated for QCM, using the same road roughness function presented 
in Figure 5. Analytical solution as well as numerical solution obtained using the Simulink model 
for QCM is presented in Figures 7 and 8 for vehicle body displacement and wheel displacement, 
respectively. As shown in figures, displacements obtained with both methods are identical. 
 
Fig. 7. Vehicle body displacement xs, QCM 
 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
8 
 
Fig. 8. Wheel displacement xu, QCM 
 
In order to validate HCM 2DOF, analytical and numerical solutions using the Simulink are 
compared. The model is excited by the same road roughness function with a time delay, which 
induces the diferences between front and rear suspension movements, y1 and y2. Vehicle body 
displacements xs and vehicle body rotation   of this model is presented in Figure 9 and Figure 
10, respectively. Obtained results show that developed Simulink model is adequate. 
 
Fig. 9. Vehicle body displacement xs, HCM 2DOF 
 
 
Fig. 10. Vehicle body rotation , HCM 2DOF 
 
At the end, HCM 4DOF is analysed. Analytical and numerical solutions obtained using 
Simulink for all four displacements of half car model subjected to harmonic excitation are 
presented in Figures 11 to 14. The same solutions indicate that the Simulink model can be used to 
solve differential equations for arbitrarily road roughness function. 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
9 
 
Fig. 11. Front wheel displacement x1, HCM 4DOF 
 
Fig. 12. Rear wheel displacement x2, HCM 4DOF 
 
Fig. 13. Vehicle body displacement xs, HCM 4DOF 
 
Fig. 14. Vehicle body rotation , HCM 4DOF 
M. Jočković, A. Šarkić, M. Petronijević, Models related to the moving load problems 
10 
5. Conclusion 
 
The main objective of this paper is to find the solution for different moving load models, 
which can be used to describe behavior of vehicle moving on the road with arbitrary roughness. 
Static moving load model is based on assumption that dynamic and inertial part of loading can be 
neglected, while these components of loading in dynamic moving load models have to be taken 
into account due to the effects of road roughness. Four dynamic vehicle models are analysed in 
this paper (1DOF, QCM, HCM 2DOF, HCM 4DOF). Dynamic moving load models cannot be 
solved analytically for all cases of road roughness functions. Due to this, the Simulink models are 
developed in order to define the behavior of vehicle subjected to arbitrarily road roughness 
function. Validation of Simulink models via harmonic excitation follows presented dynamic 
vehicle models. 
 
 
Acknowledgement 
 
The authors are grateful to the Ministry of Science and Technology of the Republic of Serbia 
for the financial support (Project TR36046). 
 
 
References  
 
[1] Petronijevič M., Brčić S., Nefovska-Danilović M., Train-induced vibrations: Moving load 
modeling, MASE 2009, 678-683, 2009. 
[2] Sawant S.,H., Belwalkar M., V., Kamble M., A., Khot P.,B., Patil D., D., Vibration analysis 
of quarter car vehicle dynamic system subjected to harmonic excitation by road surface, 
International Journal of Instrumentation, Control and Automation, Vol. 1, 14-16, 2012. 
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suspension for passenger cars using Matlab, Journal of mechanical and civil engineering, 6 – 
14, 2012 
[4] Rehnberg A., Vehicle dynamic analysis of wheel loaders with suspension axles, Royal 
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[5] Patel C., B., Gohil P., P., Borhade B., Modelling and vibration analysis of road profile 
measuring system, International Journal automotive and Mechanical Engineering, Vol. 1, 13-
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[6] Barbosa R., S., Vehicle vibration response subjected to longwave measured pavement 
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[8] Jazar, R., N., Vehicle Dynamics: Theory and Application, Springer, New York, 2009. 
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