The paper presents a simple friction model containing two parts: dry and viscous friction. Friction model is built inside the model of the guiding device of a mechanical system consisting of a mass, linear spring and damper. System is excited by the movement of the base. Main idea of the presented algorithm is to split up the solution into several parts, which follow one after another in time, and to combine their results subsequently

B. Janecek

J. Buchta

J. Skliba

M. Marek

V. Kracik

Z. Herda

English

Journal of Vibroengineering

  VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
891
708. Simple friction model of the guiding device of a 
mechanical system: mass, spring and damper 
 
B. Janecek1, V. Kracik2, J. Skliba3, Z. Herda4, M. Marek5, J. Buchta6 
Technical University of Liberec, Czech Republic 
E-mail: 1 bedrich.janecek@tul.cz, 2 vladimir.kracik@tul.cz, 3 jan.skliba@tul.cz, 
4 zdenek.herda@gmail.com, 5 martin.marek@email.cz, 6 jaroslav.buchta@tul.cz  
(Received 3 September 2011; accepted 4 December 2011) 
 
Abstract. The paper presents a simple friction model containing two parts: dry and viscous 
friction. Friction model is built inside the model of the guiding device of a mechanical system 
consisting of a mass, linear spring and damper. System is excited by the movement of the base. 
Main idea of the presented algorithm is to split up the solution into several parts, which follow 
one after another in time, and to combine their results subsequently.  
 
Keywords: vibrations, mechanical system, dry friction, software implementation of friction 
model, Matlab. 
 
Introduction 
 
A simple static friction model is considered in this paper. It consists of two components: dry 
and viscose friction. Dry friction is subdivided into static and kinetic one. Static friction is also 
called stiction [1]. The static friction is present in a guiding mechanism that is at rest, while the 
kinetic one - in case of moving mechanism. During movement of a guiding mechanism the 
relative stroke – distance ( ) ( )tztz 12 −  is changed (Fig. 1). 
 
System description 
 
A mechanical system including mass, damper and spring is considered. This system is 
complemented by the guide scissors mechanism (Fig. 1). The system is excited by moving the 
base, whose position is 1z . 
 
Fig. 1. Mass, spring and damper with the scissors guide mechanism 
 
In the system we assume a linear spring and a linear damper. The variables 1z  and 2z  are 
used for the description of the system. Variable 2z  is the deviation from the equilibrium 
position, which is determined by the size of mass m and stiffness of the spring. In initial 
position the variable 1z  will be zero. The initial value of variable 2z  can be, but must not be 
zero. Initial values of the first derivatives of these variables for the described model are assumed 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
892
to be zero. In the following text we will denote the spring force SF , which shall be the deviation 
of spring power from the equilibrium value. The real initial spring power is gm− , g  is the 
gravitational constant. 
As already mentioned, the system in Fig. 1 is excited by moving the base with position 1z . It 
means that we know the time behavior of ( )tz1 , ( )
( )
dt
tdz
tv 11 =  and ( )
( )
2
1
2
1 dt
tzd
ta =  for 0≥t . 
Let us designate ( ) ( ) ( )tztztzR 12 −=  , ( )
( ) ( ) ( )
dt
tdz
dt
tdz
dt
tdz
tv RR
12
−== , where:  
( )tzR  is the relative stroke, 
( )tvR  is the speed of relative stroke, or the relative speed. 
We assume that the friction force in the mechanism is approximately determined by the 
function from Fig. 2. 
 
 
Fig. 2. Friction force as a function of relative velocity 
 
The forces between the mass and the scissors mechanism are drawn in Fig. 3.  
 
 
Fig. 3. Forces acting on the mass m  from Fig. 1 
 
The upward direction of forces and movement are considered as positive.  
( )
( )
2
2
2
dt
tzd
mtFINERTIA −= ,       ( )tFINERTIA  is the inertial force,    (1) 
( ) ( )tzktF RSS −= ,                    ( )tFS         is the force of the spring,   (2) 
( ) ( ) ( )tFtFtF VISDRYFRIC += ,      ( )tFFRIC     is the friction force.    (3) 
In the case of movement of scissors mechanism, ( ) 0≠tvR , 
( ) ( ) KINRDRY FtvsigntF )(−= ,   ( )tFDRY      is the force of dry friction,   (4) 
 
( ) ( )tvktF RVISVIS −= ,                        (5) 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
893
( )tFVIS  is the viscous force of damper and guide mechanism. 
The force ( )tFVIS  consists of two elements: viscous friction of guide mechanism and viscous 
force of  the damper. 
Vector ( )tFINERTIA  in Fig. 3 is drawn for the case 
( ) 02 >
dt
tdv
 , vector ( )tFS  for ( ) 0<tzR  and  
vector ( )tFFRIC  for ( ) 0>tvR . 
( )tFFRIC  is the sum of two forces (see (3)). 
 
 
Fig. 4. Dry friction force as a function of relative velocity 
 
 
Fig. 5. Viscous friction force as a function of relative velocity 
 
The characteristic of the dry friction is nonlinear. The discussed system in the model 
calculation passes between “basic states” with non–zero time duration. In one of the basic states 
the relative speed is zero, ( ) 0=tvR , system is stuck – locked and in the second state the speed is 
nonzero ( ) 0≠tvR  and ( )( )tvsign R  does not change, system is released. 
Movement of the system in the released state when ( ) 0≠tvR  is described with the equations: 
( ) ( ) ( ) ( ) 0=+++ tFtFtFtF VISDRYSINERTIA ,       (6) 
( )
( ) ( ) ( ) 0)(22
2
=+++ tvkFtvsigntzk
dt
tzd
m RVISKINRRS  .    (7) 
In equation (7), describing the dynamics of the system of Fig. 1, the dynamics of the scissors 
mechanism is neglected. Inertial moments of the arms of the scissors mechanism in the 
mathematical model are considered to be zero. They could be neglected because the loading 
mass  m  is much greater than the mass of the arms of the scissors mechanism (ten times greater 
in the case of the investigated nonlinear system with the air–spring). 
In “basic locked state” when ( ) 0=tvR  movement of the system is also described with (6), 
but:    
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
894
( ) ( )
dt
tdz
dt
tdz 12
=              (8) 
and 
( ) ( )
2
1
2
2
2
2
dt
tzd
dt
tzd
= ,          (9) 
( )tF S  is constant, ( ) constFtF SS = ,    
( ) 0=tFVIS ,      
( ) ( )tFtF DRYFRIC = ,         (10) 
( )
( ) ( )
2
2
2
2
1
2
dt
tzd
m
dt
tzd
mtFINERTIA −=−= . 
We can rewrite (6):  
( ) ( )
( ) ( ) constFtFtF
tFconstFtF
SINERTIADRY
DRYSINERTIA
−−=
=++ 0
 .      (11) 
The development of ( )tz2  is calculated by integration of known 
( )
dt
tdz1
, with initial value, 
which is equal to the final value ( )tz2  from the previous “basic state”. 
First testing of the friction simulating algorithm was performed by the program that used 
harmonic excitation signal ( )tz1  with constant amplitude A   and linearly increasing frequency 
(chirp signal),  ( ) tkt
ω
ω =     for    max,0 Tt∈ , maxmax Tkωω =  . 
( ) ( )21 sin tkAtz ω= ,         (12) 
( ) ( )21 cos2 tktkA
dt
tdz
ωω
= ,        (13) 
( ) ( ) ( )( )2222
1
2
sin2cos2 tktktkkA
dt
tzd
ωωωω
−= .      (14) 
Let us discus the case of chirp excitation signal with zero initial conditions, ( ) 001 =z  and 
( ) 001 =
dt
dz
. The initial acceleration of ( )tz1  is 
( )
ω
kA
dt
zd 20
2
1
2
= . Let the next initial condition of 
system be ( ) 002 =z  and 
( ) 002 =
dt
dz
. The initial value of the spring force is, in this case, 
( ) 00 =SF . Let us consider, that ωk  was chosen in such a way as to have STATFdt
zd
mabs <





2
1
2 )0(
, 
i.e. relative movement of the system at the beginning is zero, ( ) 00 =Rv . In this case, when the 
frequency of the excitation signal increases, system passes through the discussed two “basic 
states” several times, relative movement is several times stuck – locked and then released. 
When the frequency increases the system comes to the states, where signs of ( )tvR  changes and 
the system is still released. System no longer comes to the “locked basic state”. 
Let us describe the system movement with the text structure which will be close to the 
structure of the program. We designate the beginning time 1t . The presented friction model 
starts from initial conditions ( ) 011 =tz , 
( ) 011 =
dt
tdz
 and ( ) 012 =
dt
tdz
. It means that ( ) 01 =tvR . The 
initial condition ( )12 tz  and parameters of the model (i.e.  m  , Sk   , VISk   , STATF   , KINF   , A  , ωk   
and  stop – final time 
max
T ) can be set to arbitrary values in the program. 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
895
The second derivation of 1z  of chirp signal at time 0=t  is non zero, 
( )
ω
kA
dt
zd 202
1
2
= .    
For  0>t    is   ( )
2
1
2
dt
tzd
  smooth – continuous function.  The same property has to valid for any 
other excitation function, which can bee built in the discussed model of the system with friction.  
The friction model which can work with excitation signal with discontinuous ( )
2
1
2
dt
tzd
 will be 
worked out as well. 
For simplicity of explanation, the program behavior is described only with a simpler model 
that is specified as follows. The movement of the system is solved for ( ) 002 =z  and ωk  is 
chosen so that  
STATFdt
zd
mabs <





2
1
2 )0(
 . The system at the beginning time 01 =t  is stuck – 
locked. 
1) We set initial values ( ) 011 =tz , ( ) 011 =dt
tdz
 , ( ) 012 =tz , 
( ) 012 =
dt
tdz
. The variables ( )
2
11
2
dt
tzd
 and 
( )
2
12
2
dt
tzd
 must be known for model computation. Their values at time 1t  are  
( ) ( )
ω
kA
dt
tzd
dt
tzd 22
12
2
2
11
2
== , see (9) and (14), 
2) if ( ) ( ) STATRS Ftzkdt
tzd
mabs <





+ 12
11
2
      (15) 
Then starts the solution of movement of the system, which is in “locked basic state” i.e. 
( ) 0=tvR , ( )tzR  is constant. The solution of the locked system will be performed until time 2t , 
when the condition   
( ) ( ) STATRS Ftzkdt
tzd
mabs =





+ 22
21
2
,       (16) 
is fulfilled. 
After finishing the solution of   “locked basic state” we will compute the variable 
( )
( ) KINRSKINLAST Ftzkdt
tzd
msignF 





+= 22
22
2
,      (17) 
which will be used in part 3) of the program, we set 21 tt = , otherwise the system does not come 
in “locked basic state”. 
For the use in 3) we will compute  
( )
( ) KINRSKINLAST Ftzkdt
tzd
msignF 





+−= 12
12
2
,      (18) 
and 
3) the system comes in “basic released state” ( ) 0≠tvR  and  during this  “basic state” the sign of 
( )tvR  does not change, ( )tFDRY  is constant,      
( ) KINLASTDRY FtF = .           (19) 
Eq. (4) for computing of ( )tFDRY  cannot be used. Therefore ( ) 01 =tvR  at the beginning of 
“released state”. Explanation of (17) and (18) will be made in the following text. Movement of 
the system is described with the equation   
( )
( ) ( ) 022
2
=+++ tvkFtzk
dt
tzd
m RVISKINLASTRS .      (20) 
Basic state ( ) 0≠tvR  will be solved until time 3t , when speed comes to zero, 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
896
( ) 03 =tvR ,          (21)  
put 31 tt =  and go to 2). 
The algorithm will terminate at the specified final time 
max
T . 
Conditions (16) and (21) are tested in Simulink procedures with the use of “Hit Crossing” 
blocks. If they are satisfied, the simulation is stopped using “Stop Simulation” blocks. For 
proper function of these procedures we can not use “Derivative” blocks. “Hit Crossing” block 
uses iterative procedure to find an exact instant of time when the condition is satisfied. If we 
would like to excite the system with external data, the time interpolation in external data must 
be made in Matlab after finishing the Simulink procedure. 
In the previous description of the program, the use of condition (15) in case of ( ) 01 =tvR  is 
physically obvious. Depending on whether this condition is fulfilled or not, the system comes to 
“locked state” or “released state”. 
The computation of “released basic state” ( ) 0≠tvR  of the system begins at time 1t  , see part 
3) of program description. Let us consider that we use eq. (4) for computation of ( )1tFDRY . At 
time 1t , ( ) 01 =tvR , ( ) 00 =sign . This results in ( ) 01 =tFDRY , which is wrong. ( )1tFDRY  must be 
computed from the end of the previous “basic state”. Previous “basic state” could be locked, or 
released. 
a)  At first let us discuss, that previous “basic state” was locked. In the case of “locked basic 
state” ( )tFDRY  changes in interval  ( )STATSTAT FF ,− . In Fig. 4, ( )tFDRY  moves only on vertical axis, 
eq. (11) holds. The time, when ( )tFDRY  comes on borders of the mentioned interval was 
designated 2t  in part 2) of program description. At time t2 either ( ) STATDRY FtF −=2  or  
( ) STATDRY FtF =2 . In next time, 2tt >  the system comes to “released basic state” and ( )tFDRY   starts 
to be either ( ) KINDRY FtF −=  or  ( ) KINDRY FtF = . In the discussed time development, in time 2tt ≥ , 
the sign of ( )tFDRY  does not change. For this reason ( )( )tFsign DRY  in the “released basic state” is 
the same as ( )( )2tFsign DRY  at the end of the previous “locked basic state”. This fact is expressed 
in eq. (17) and (19).  
b)  Let us discuss the second variant, when the system comes to the “released basic state” from 
the previous “released basic state”. If ( )tvR  was positive in the previous “released basic state”, it 
becomes negative in the next “released basic state” and vice versa. The change of signs of 
( )tFDRY  is expressed in eq. (18) and (19). 
 
Structure of Matlab – Simulink program 
 
It may seem from the previous text that the discussed algorithm can be implemented in 
Simulink as consisting of two subroutines. First subroutine for solving “basic locked state”  
when ( ) 0=tvR  and the second subroutine for solving “basic released state” when  ( ) 0≠tvR . But 
the problem is, that second subroutine ought to start and finish, see (21), with zero relative 
movement. For this reason it was necessary to divide this subroutine into two subroutines (but 
with very similar structure). For the solution of the discussed algorithm we worked out three 
subroutines in Simulink.   
 
Parameters for test of the system with chirp excitation signal 
 
The system with friction was tested by a program in MATLAB with parameters: 
kgm 100= ,  
m
NskVIS 200= . 
Spring constant Sk  was chosen so that the natural frequency of the undamped system is: 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
897
Hzf
n
5.1= , 
nn
fpiω 2= ,   
m
N
mk
nS 8883
2
≈= ω  , NFSTAT 26= ,    NFKIN 18= . 
The calculation was finished at time sT 9.5
max
= . 
For excitation the amplitude of “chirp signal” was mA 05.0=  and maximal frequency 
Hzf 0.825
max
= .   
Coefficient 
ω
k  ought to be computed from equation: 
max
max
max
max
2 T
f
T
k piω
ω
==   .        (22) 
For our test: 
0.2728
2
max
max
max
max
≈==
T
f
T
k piω
ω
.    
 
Simulation results 
 
 Simulation results are presented in Fig. 6-9. 
 
0 1 2 3 4 5 6 7 8 9
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
 t [s]
 
z 
[m
]
 excitation Chirp signal
 
 
 z1
 z2
 zr
 starts, stops or cont.of rel.mov.
 
Fig. 6. System excited with „chirp“ signal, 1z  is dashed black, 2z  is full black, rz  is grey, circles are time 
borders between the basic states 
 
Conclusions 
 
The friction model proposed in this paper is a static model with stiction, dry and viscous 
frictions. This model was used as a part of simulation of a system that was otherwise linear. The 
excitation was executed with the smooth time function generated in Simulink. In the future we 
would like to use this friction model as part of a more complicated nonlinear model including 
an air–spring. This model will be excited with the measured data for the purpose of 
identification. 
We offer the software (in Matlab 2007) that was shortly described in this paper as a 
freeware. It can be sent to interested persons upon e-mail request. 
 
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
898
0 1 2 3 4 5 6 7 8 9
-150
-100
-50
0
50
100
150
 t [s]
 
F 
[N
]
 excitation Chirp signal
 
 
 -m d2z1/dt2
  FINERTIA
  FFRIC
  FDRY
  FS
starts, stops or cont.of rel.mov.
 
Fig. 7. System excited with the „chirp“ signal, INERTIAF  is thick, FRICF  is thin, DRYF  is doted, SF  is dashed, 
force  
2
1
2
dt
zd
m−   is grey, circles are time borders between the basic states 
4 5 6 7 8 9
-150
-100
-50
0
50
100
150
 t [s]
 
F 
[N
]
 excitation Chirp signal
 
 
 -m d2z1/dt2
  FINERTIA
  FFRIC
  FDRY
  FS
starts, stops or cont.of rel.mov.
 
Fig. 8. Time zoom of Fig. 7  
 
Acknowledgment 
 
This research was supported by the Ministry of Education of the Czech Republic under 
contract No. MSM 4674788501 “Optimization of Machine Properties in Interaction with 
Working Processes and Human Aspects”.  
708. SIMPLE FRICTION MODEL OF THE GUIDING DEVICE OF A MECHANICAL SYSTEM: MASS, SPRING AND DAMPER. 
B. JANECEK V. KRACIK, J.  SKLIBA, Z. HERDA, M. MAREK, J. BUCHTA 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2011. VOLUME 13, ISSUE 4. ISSN 1392-8716 
899
0 1 2 3 4 5 6 7 8 9
-25
-20
-15
-10
-5
0
5
10
15
20
25
 t [s]
 
F D
RY
 
[N
],  
 
v
R[c
m
/s
]
 excitation Chirp signal
 
 
  FDRY
  vR
 starts, stops or cont.of rel.mov.
 
Fig. 9. System excited with „chirp“ signal, DRYF  is doted, Rv  is grey, circles are time borders between the 
basic states 
 
References 
 
[1] H. Olsson, K. J. Aström, C. Canudas de Wit, M. Gäfvert, P. Lischinsky Friction Models and 
Friction Compensation. European Journal of Control, Dec. 1998, No. 4, p. 176-195. 
[2] C. Canudas de Wit, H. Olsson, K. J. Aström, P. Lischinsky A New Model for Control of Systems 
with Friction. IEEE Transactions on Automatic Control, March 1995, Vol. 40, No. 3, p. 419-425. 


Simple friction model of the guiding device of a mechanical system: mass, spring and damper

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Simple friction model of the guiding device of a mechanical system: mass, spring and damper

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