The paper presents the results of simulation research of visco-elastic model of shock-absorber. Tested shock absorber has been in the suspension system. The aim of research was determine values of force of damping for different kinematic input functions. The results are presented in graphical form that allows for their analysis

Ireneusz Celiński

Jakub Młyńczak

Jan Warczek

English

Vibroengineering PROCEDIA

 248 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533  
Simulation studies of a shock absorber model proposed 
under conditions of different kinematic input functions 
Jan Warczek1, Jakub Młyńczak2, Ireneusz Celiński3 
Faculty of Transport, Silesian University of Technology, Krasińskiego Street 8, 40-019, Katowice, Poland 
1Corresponding author 
E-mail: 1jan.warczek@polsl.pl, 2jakub.mlynczak@polsl.pl, 3ireneusz.celinski@polsl.pl 
(Accepted 2 October 2015) 
Abstract. The paper presents the results of simulation research of visco-elastic model of shock-
absorber. Tested shock absorber has been in the suspension system. The aim of research was 
determine values of force of damping for different kinematic input functions. The results are 
presented in graphical form that allows for their analysis. 
Keywords: damping of vibrations, road roughness, model of suspension with visco-elastic 
damper. 
1. Introduction 
Automotive vehicles are driven on roads of diversified pavement condition. A road profile 
may theoretically take different deterministic and stochastic forms, however in practice, only with 
regard to special testing sections of road, one may assume that pavements can be described by 
means of unique mathematical functions [1-5]. Under actual conditions of road pavements on 
which automotive vehicles run, the most convenient method is to characterise the road profile by 
means of power spectral density of irregularities. In such a case, it is envisaged that recording of 
the irregularity profile creates a stationary and ergodic process. Recording a road profile with the 
linear speed of 1 m/s allows for the spectral density of irregularities to be represented in the 
function of e.g. irregularity wavelength ܮ௡ m. A car running on the given road is subject to input 
functions depending on the pavement condition as well as the current driving speed. Power 
spectral densities of road pavement irregularities may be represented as functions of frequency 
having previously taken an additional parameter of driving speed into consideration. In accordance 
with the applicable standards, the road profile is described through a set of points of road 
irregularity height measured along the trace of cooperation between the vehicle’s right wheel and 
the road pavement. Measurements of road pavement irregularity heights are recorded with a 
constant interval of the road longitudinal length equalling 0,1 m. An irregularity profile thus 
recorded provides grounds for calculation of the longitudinal irregularity parameters. Based on 
the irregularity profile, one calculates a set of longitudinal road roughness parameters, the most 
important of which is IRI (International Roughness Index). What one applies in simulation studies 
for modelling of road irregularities is the road irregularity models defined in the ISO 8608 standard, 
where power spectral densities of road irregularities in the form depicted in Fig. 1 are used to 
describe the profile. 
 
Fig. 1. Classification of road irregularity profiles in accordance with ISO 8608 
SIMULATION STUDIES OF A SHOCK ABSORBER MODEL PROPOSED UNDER CONDITIONS OF DIFFERENT KINEMATIC INPUT FUNCTIONS.  
JAN WARCZEK, JAKUB MŁYŃCZAK, IRENEUSZ CELIŃSKI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533 249 
A road of the best quality (smallest irregularities) is marked with the letter A. The worse the 
road pavement condition is, the further the designating letter is in the alphabet, with H marking 
the worst road quality. A problem difficult to solve in terms of an automotive vehicle operation is 
to diagnose a shock absorber installed in the suspension system. The main function performed by 
a shock absorber is connected with limiting relative movements of the sprung as well as the 
unsprung mass. The essence of the problem is that, despite the relatively simple design, a car 
shock absorber is a component of complex non-linear dynamic properties. The paper addresses 
results of simulation studies conducted using a dedicated diagnostic shock absorber model 
developed. The shock absorber model in question was introduced into a quarter-vehicle 
suspension system model whose other parameters were selected based on an analysis of the actual 
automotive suspension system. 
2. Simulation studies 
With reference to previous studies undertaken by the authors, a mathematical shock absorber 
model was developed, entailing the apparent shock absorber elasticity and defining damping 
characteristics as a two-argument function. 
The strong dependence of the damping characteristics for the working stroke length determines 
the need to take this phenomenon into account in computational models of automotive shock 
absorbers [6-8]. The relationship describing this phenomenon is generally as follows: 
ܨ஽ = ݂ሺݒ, ݓ௦ሻ, (1)
where: ݒ – current velocity of input, ݓ௦ – total length of the working stroke. 
Characteristics of stiffness of the shock absorber force is a function of frequency and 
displacement. Thus the seeming stiffness of the damper can be described as: 
ܨ௦௦ = ݂ሺ݀, ݂ሻ, (2)
where: ݀ – displacement, ݂ – frequency. 
The mathematical notation of the shock absorber model is as follows. 
The value of damping force ܨ஽ in the conceptual shock absorber model can be determined by 
interpolation: 
ܨ஽ = ݂ሺݒ, ݓ௦ሻ: ሾܽ, ܾ, ܿ, ݀ሿ → ܴ, (3)
where: limits ܽ, ܾ, ܿ and ݀ are determined empirically. 
The approximation of function values ܨ஽ for any arguments: 
ݒ ∈ ሾݒ଴, ݒ௡ሿ ⊂ ሾܽ, ܾሿ, ݓ௦ ∈ ሾݓ଴, ݓ௠ሿ ⊂ ሾܿ, ݀ሿ,
is made for the known values – interpolation nodes which are the results of measurements: 
݂ሺݒ, ݓ௦ሻ = ൦
݂ሺݒ଴, ݓ଴ሻ ݂ሺݒ଴, ݓଵሻ … ݂ሺݒ଴, ݓ௠ሻ
݂ሺݒଵ, ݓ଴ሻ ݂ሺݒଵ, ݓଵሻ … ݂ሺݒଵ, ݓ௠ሻ
⋮ ⋮ ⋱ ⋮
݂ሺݒ௡, ݓ଴ሻ ݂ሺݒ௡, ݓଵሻ … ݂ሺݒ௡, ݓ௠ሻ
൪, (4)
where: ܽ ≤ ݒ଴ ≤ ݒଵ ≤ ݒଶ ≤ ⋯ ≤ ݒ௡ ≤ ܾ  and ܿ ≤ ݓ଴ ≤ ݓଵ ≤ ݓଶ ≤ ⋯ ≤ ݓ௠ ≤ ݀  using a 
spline functions. 
The force of the resulting damper stiffness ܨ௦௦ reads: 
ܨ௦௦ = ݂ሺ݀, ݂ሻ: ሾܽ, ܾ, ܿ, ݀ሿ → ܴ, (5)
SIMULATION STUDIES OF A SHOCK ABSORBER MODEL PROPOSED UNDER CONDITIONS OF DIFFERENT KINEMATIC INPUT FUNCTIONS.  
JAN WARCZEK, JAKUB MŁYŃCZAK, IRENEUSZ CELIŃSKI 
250 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533  
where: limits ܽ, ܾ, ܿ and ݀ are determined by experimental approximation of function ܨ௦௦: 
݀ ∈ ሾ݀଴, ݀௡ሿ ⊂ ሾܽ, ܾሿ,   ݂ ∈ ሾ ଴݂, ௠݂ሿ ⊂ ሾܿ, ݀ሿ,
for interpolation nodes: 
݂ሺ݀, ݂ሻ = ൦
݂ሺ݀଴, ଴݂ሻ ݂ሺ݀଴, ଵ݂ሻ … ݂ሺ݀଴, ௠݂ሻ
݂ሺ݀ଵ, ଴݂ሻ ݂ሺ݀ଵ, ଵ݂ሻ … ݂ሺ݀ଵ, ௠݂ሻ
⋮ ⋮ ⋱ ⋮
݂ሺ݀௡, ଴݂ሻ ݂ሺ݀௡, ଵ݂ሻ … ݂ሺ݀௡, ௠݂ሻ
൪, (6)
where: ܽ ≤ ݀଴ ≤ ݀ଵ ≤ ݀ଶ ≤ ⋯ ≤ ݀௡ ≤ ܾ, ܿ ≤ ଴݂ ≤ ଵ݂ ≤ ଶ݂ ≤ ⋯ ≤ ௠݂ ≤ ݀. 
The simulation studies addressed were conducted using the Matlab/Simulink program. The 
suspension system model developed, with a detailed shock absorber diagnostic model 
implemented, has been illustrated in Figs. 2 and 3. 
 
Fig. 2. Computational model of suspension used in the simulation studies 
 
Fig. 3. The shock absorber model 
z2__
z2
z1__
z1_
z1
z1_
velocity of z1
z2_
velocity of the sprung mass
In1 Out1
unsprung mass
In1 Out1
stiffness of the tire
In1 Out1
sprung mass
In1 Out1
spring
z_r
relative velocity
zr
relative 
displacement
z__r
relative 
accelerations
s
v
Out1
model of shock absorber
zp
input force accelerations
input f orce
input force
z1
displacement of z1
z2
displacement of the sprung mass
In1 Out1
damping of the tire
z1__
accelerations of z1
z2__
acceleration of the sprung mass
1
s
Integrator5
1
s
Integrator4
1
s
Integrator3
1
s
Integrator2
du/dt
Derivative1
du/dt
Derivative
1
scope2
scope1
przem_wy m sr
determination of working stroke
przej_0 f r1
determination of frequency
sr
v r
Ftwy zn
fcn
damping force 
determination module
przem
f r
Fswy zn
fcn
damper stiffness 
determination module 
Input frequency
Hit 
Crossing
1
Gain3
1
Gain2
-K-
Gain1
-K-
Gain
Display
Add
2
v
1
s
 FD [N]skok [mm]
v
 [N]
 Fss [N]
SIMULATION STUDIES OF A SHOCK ABSORBER MODEL PROPOSED UNDER CONDITIONS OF DIFFERENT KINEMATIC INPUT FUNCTIONS.  
JAN WARCZEK, JAKUB MŁYŃCZAK, IRENEUSZ CELIŃSKI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533 251 
The studies included application of different random kinematic input functions generated for 
the given running speed using the spectral description of pavement irregularities provided  
in Fig. 1. 
3. Research results and their analysis  
The aim of the study was to analyze the forces generated in a detailed model of the dynamic 
damper built-in suspension of the vehicle. Examples of kinematic input used in the simulation 
tests is shown in Fig. 4. 
 
Fig. 4. The example of input force generated by road roughness 
 
a) The damping force 
 
b) The spring force 
 
c) The total force of shock-absorber d) Total force of shock-absorber for the velocity 
Fig. 5. The results of simulation studies for good road  
0 5 10 15 20 25 30 35 40-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
time [s]
inp
ut 
dis
pla
ce
me
nt 
[m
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-30
-20
-10
0
10
20
30
40
50
60
70
time[s]
FD
 [N
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.5
0
0.5
1
1.5
2
time[s]
FS
S 
[N
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-30
-20
-10
0
10
20
30
40
50
60
70
time[s]
Fs
um
 [N
]
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30
-20
-10
0
10
20
30
40
50
60
70
v [m/s]
Fs
um
 [N
]
SIMULATION STUDIES OF A SHOCK ABSORBER MODEL PROPOSED UNDER CONDITIONS OF DIFFERENT KINEMATIC INPUT FUNCTIONS.  
JAN WARCZEK, JAKUB MŁYŃCZAK, IRENEUSZ CELIŃSKI 
252 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533  
Input of road roughness can be interpreted as a kinematic input to the computations dynamic 
model. The work of damper exhausted in turn derived from to force dynamic unsprung mass. This 
is because the damping force affects the movement of the wheels and thus occurs between shock 
and unsprung mass interaction. Relative movement of the sprung and unsprung masses are the 
primary cause of the forces in the damper. The scope of the forces generated by the shock absorber 
is wide. Examples of results of research work suspension damper model are presented in Figs. 5-6. 
 
a) The damping force 
 
b) The spring force 
 
c) The total force of shock-absorber d) Total force of shock-absorber for the velocity 
Fig. 6. The results of simulation studies for bad road 
The larger the road surface roughness including greater operating range automotive shock 
absorber [9, 10]. Work damper improves the comfort and safety. The model allows us to analyze 
complex phenomena associated with dampened suspensions vehicles. 
4. Conclusions 
The study of mechanical systems with nonlinear damping are the subject of numerous 
scientific papers. The diagnostic tests simulation assumed that the mathematical model should be 
compatible with the real object. In the cited study we used the results of the developed model of 
automobile shock absorber. Analysis of the images work damper allows for the development of 
specific measures of a technical condition. Based on the results achieved can, going one step 
further, to develop a method of diagnosing shock absorbers under their normal operation. That 
task will be the subject of further work of the authors. 
References 
[1] Dixon C. J. The Shock Absorber Handbook. John Wiley and Sons, Ltd., 2007. 
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50
0
50
100
150
200
time[s]
FD
 [N
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
-1
0
1
2
3
4
5
time[s]
FS
S 
[N
]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50
0
50
100
150
200
time[s]
Fs
um
 [N
]
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-50
0
50
100
150
200
v [m/s]
Fs
um
 [N
]
SIMULATION STUDIES OF A SHOCK ABSORBER MODEL PROPOSED UNDER CONDITIONS OF DIFFERENT KINEMATIC INPUT FUNCTIONS.  
JAN WARCZEK, JAKUB MŁYŃCZAK, IRENEUSZ CELIŃSKI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2015, VOLUME 6. ISSN 2345-0533 253 
[2] Warczek J. Application of time shift for nonlinear damping characteristic identification. Transport 
Problems, Vol. 4, Issue 3, 2009. 
[3] Warczek J., Burdzik R., Peruń G. The method for identification of damping coefficient of the trucks 
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[4] Burdzik R. Identification of Sources, Propagation and Structure of Vibrations Affecting Humans in 
Means of Transport Based on the Example of Automotive Vehicles. JVE Book Series on 
Vibroengineering, Vol. 1, JVE International Ltd., Kaunas, Lithuania, 2014. 
[5] Burdzik R., Konieczny Ł. Vibration issues in passenger car. Transport Problems, Vol. 9, Issue 3, 
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[6] Warczek J., Burdzik R., Konieczny Ł. The concept of multi-parametric diagnostic model of car 
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[7] Warczek J., Burdzik R. Visco-elastic model of dynamic of hydraulic damper as a basis for 
determining the measurement condition. Scientific Papers of Transport. Silesian University of 
Technology, Issue 66, 2010. 
[8] Warczek J., Młyńczak J., Burdzik R., Konieczny Ł. Simulation of a visco-elastic damper based on 
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Simulation studies of a shock absorber model proposed under conditions of different kinematic input functions

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Simulation studies of a shock absorber model proposed under conditions of different kinematic input functions

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