Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 3, pp. 909-919, Warsaw 2016
DOI: 10.15632/jtam-pl.54.3.909

A SMOOTH MODEL OF THE RESULTANT FRICTION FORCE ON

A PLANE CONTACT AREA

Grzegorz Kudra, Jan Awrejcewicz

Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, Łódź, Poland

e-mail: grzegorz.kudra@p.lodz.pl

A class of smooth approximations of the total friction force occurring on a plane finite con-
tact surface is presented. It is assumed that the classical Coulomb friction law is valid on
any infinitesimal element of the contact region. The models describe the stick phase, the
fully developed sliding and the transition between these two modes. They take into acco-
unt different values of static and dynamic friction coefficients. The models are applied in
simulation of a dynamical system performing translational and rotational stick-slip oscilla-
tions, and then they are verified by comparison with the corresponding results in which an
event-driven discontinuous model of friction is used.

Keywords: friction modelling, Coulomb-Contensoumodel, stick-slip, plane contact

1. Introduction

Contensou (1963) derived the integral model of resultant friction forces between two contacting
bodies on the assumption of validity of the Coulomb friction law on each element of a circular
contact area, Hertzian contact stresses and fully developed sliding. He pointed out that the
normal component of relative angular velocity cannot be neglected in the model of friction
forces for a certain class of mechanical systems.

Howe and Cutkosky (1996), for practical purposes of robotics, proposed an approximation
of this problem in the form of ellipsoid description of the limit surface bounding the zone of
admissible tangential loadingsof the contact.However, it is notadirectdescriptionof the relation
between the components of sliding and friction. Some researchers proposedPadé approximations
as a convenient substitution of the integral model of friction forces (Zhuravlev, 1998; Zhuravlev
and Kireenkov, 2005). Then the Padé approximants (Kireenkov, 2008) and hyperbolic tangent
functions (Kudra and Awrejcewicz, 2011b) were used in the modelling of friction forces in the
case of a circular contact zone and presence of rolling resistance. Kosenko and Aleksandrov
(2009) developed a piece-wise linear approximation of the integral model of friction in the case
of elliptical contact zone and Hertzian contact stresses. Kudra and Awrejcewicz (2012a, 2013)
presented approximations which can be understood as a certain kind of generalizations and
modifications of the Padé approximants, and applied them to the above problems including the
case of the circular contact with uniform contact pressure distribution as well as elliptical one
with the presence of rolling resistance.

The above mentioned models approximate the friction forces during the fully developed
sliding or their limits in the stick mode. The simulation requires also models describing the
transition between these twomodes. The assumption of negligibly short time of duration of this
transition results in a non-smooth dynamical system. They can often bemodelled by the use of
piecewise smooth differential equations (PWS). Then the PWS systems are sometimes divided
in the following way (Leine andNijmeijer, 2004): a) systems with a continuous but non-smooth



910 G. Kudra, J. Awrejcewicz

vector field; b) systemswith a continuous but non-smooth state; c) systemswith a discontinuous
state.

Existence and uniqueness of the solution are guaranteed only for systems belonging to gro-
up (a), known also as Filippov type systems (Filippov, 1964, 1988; Leine and Nijmeijer, 2004).
For example, a mechanical system with dry friction modelled as a discontinuous function of
time belongs to group (b). For instance, the assumption of rigidity of contacting bodies and
the friction force dependent on normal load, which is not known in advance, can lead in some
cases to inconsistency of the solution, known as the Painlevé paradox (Jellet, 1872; Painlevé,
1895; Génot and Brogliato, 1999). Such cases should be interpreted in such a way that the
mathematical model and solution concept do not correspond to the real physical phenomenon.

The techniques of simulation of non-smooth systems can be divided in the following way
(Leine and Nijmeijer, 2004): (i) event-driven integration methods; (ii) time-stepping methods.
Acary and Brogliato (2010) presented a comprehensive review of the event-driven and time-
stepping numerical methods for simulation of non-smooth dynamical systems. Sometimes as
another way of dealingwith the non-smooth dynamical systems are regularization or smoothing
methods.

Event drivenmethods (Leine andNijmeijer, 2004) use classical integrationmethods between
two successive events (switches between two different modes) when a system behaves smoothly.
The integration stops if the event is detected. Then the system may undergo a step change
and the next mode is determined. Special rules for the system change can be constructed in
order to avoid inconsistency of the solution mentioned above. Kudra and Awrejcewicz (2012a)
developed an event driven simulation algorithm of a mechanical system with circular frictional
contact with a uniform contact pressure distribution and the Coulomb friction law. The scheme
uses approximations of the integral friction model for both the fully developed sliding and
stick phase (for determination of the end of the stick mode), but different values of static and
(constant) dynamic friction coefficients are used. The combined translational and rotational
stick-slip oscillations are presented.

Time-steppingmethods are special numerical schemes which do not require detection of the
events (Moreau, 1988; Stewart andTrinkle, 1996; Jean, 1999;Awrejcewicz andLamarque, 2003).
Some authors (Leine and Glocker, 2003; Möller et al., 2009) constructed such algorithms with
appropriate approximations of the friction forces for a finite circular contact areawith theHertz
stress distribution and Coulomb friction law on each element of the contact.

The regularization methods are modifications of the mathematical model leading to a lower
degree of “non-smoothness” of the dynamical system. Particularly, they can result in a smooth
dynamical system, allowing for theuse of classical integrationmethods. In the case of dry friction
modelling, a common approach is to replace the set-valued friction force in the stick mode by
the force of action of a stiff spring with some saturation threshold, usually supplemented with
additional damping elements or with additional state variables resulting in better description
of the real phenomenon (Do et al., 2007). In the instance of a one-dimensional dry friction
problem (i.e. either pure translational or rotational relative displacement of the contact), one
can encounter smooth approximations of the friction force,where the signum function is replaced
by such functions like arctangent or hyperbolic tangent. If this function ismultiplied by another
expression describing the changes of the dynamic friction coefficient during motion, then the
phenomenon when the friction coefficient decreases at the beginning of the movement can be
modelled (Awrejcewicz and Olejnik, 2003; Pilipchuk and Tan, 2004; Awrejcewicz et al., 2008).

The drawback of the above mentioned regularizations may be stiffness of the resulting ordi-
nary differential equations. Sometimes theymay change some properties of themodel, especially
may change properties or destroy the equilibrium related to the stickmode. Also, in some regu-
larized models, the loos of stability of the equilibrium and occurrence of artificial oscillations is
possibly. On the other hand, the regularizations in general decrease the risk of the occurrence of



A smooth model of the resultant friction force... 911

problemswith the solution inconsistency. In particular, if the friction force is not a discontinuous
function of sliding velocity (it may be a non-smooth function), the system belongs to the above
mentioned group (a) and the solution uniqueness and existence are guaranteed. There aremany
works in which the influence of the friction model on the properties of the resultingmechanical
system, including stability and bifurcations of the related solutions, is investigated (Bogacz and
Sikora, 1990; Sikora and Bogacz, 1993; Leine et al., 2000).

In the case of combined translational and rotational relative displacement of a finite con-
tact, there are smooth models developed in which singularities of the expressions are avoided
by the addition of a small positive parameter in special places of approximations of the integral
models (Kudra and Awrejcewicz, 2011b, 2012b). However, these constructions do not assume
the possibility for the friction coefficient to decrease in the beginning of slip. Stamm and Fidlin
(2007, 2008) developed a regularized two-dimensional model based on the elastic-plastic the-
ory and being a generalization of similar approaches for one-dimensional contact. It requires,
however, discretization of the contact area (in contrast to the other above-referred models for
two-dimensional contact), leading to much higher computational cost.

In this paper, we present smooth approximations of the friction force and torque appearing
between twobodies contacting on aplane contact area of an arbitrary shape.Theyare applied to
a specialmechanical system inwhich combined translational and rotational stick-slip oscillations
appear and the results are compared to those obtained by the event-driven numerical algorithm
(Kudra and Awrejcewicz, 2012a). A part of the present work was previously presented in a
shortened version (Kudra and Awrejcewicz, 2011a).

2. Integral model of friction

Let us consider the plane contact area F whose dimensionless form is presented in Fig. 1a. The
Cartesian coordinate systemAxyz is introduced,where the axesx and y lie in the contact plane.
Thedimensionless length is related to the real characteristic dimensionof the contact â, therefore
thedimensionless coordinates of thepoint situatedon thecontact planearex= x̂/âandy= ŷ/â,
where x̂ and ŷ are the corresponding real coordinates. The non-dimensional elementary friction
force acting on the element dF of the contact area is defined as dT= dT̂/(µN̂), where dT̂s is its
real counterpart, N̂ is the real normal component of the total force of interaction between two
bodies, while µ is the dry friction coefficient during slip. Themoment of the force dT about the
pointA (center of contact) reads dM=ρ×dT= dM̂/(âµN̂), where dM̂ is the real counterpart
of dM. The dimensionless contact pressure distribution is defined as σ(x,y) = σ̂(x,y)â2/N̂,
where σ̂(x,y) is the real pressure. For the purpose of compatibility with the dimensionlessmodel
of the mechanical system presented in Section 5, we additionally introduce the dimensionless
time t=αt̂, where t̂ denotes its real counterpart.

It is assumed that the contact F can only operate in the twomodes: the stick and the fully
developed sliding (the transition between these twomodes is infinitely short).During the sliding,
local relativemotion of the contacting bodies can be assumed as planemotion of the rigid body.
Moreover, we assume that the spatial Coulomb friction law is valid on each element dF

dT∈





−σ(x,y)

vP

‖vP‖
dF for vP 6=0

{u∈ R2 : ‖u}¬ ησ(x,y)dF} for vP =0
(2.1)

where η is the ratio of the static friction coefficient to the dynamic one, and vP is velocity of
the element dF (see Fig. 1b).It is a priori assumed that the elementary friction force dT as well
as the relative sliding velocity vP lie in the plane of the contact F, so the problem described by
Eq. (2.1) is in fact the two-dimensional one.



912 G. Kudra, J. Awrejcewicz

Fig. 1. The plane contact area (a) and the magnitude of the elementary friction force as a function of
the magnitude of relative velocity (b)

The system of elementary friction forces can be reduced to the force T = Txex +Tyey =
Tτeτ +Tυeυ acting at the pointA and the correspondingmomentM=Mez, where ex, ey, ez,
eτ and eυ are the unit vectors of the corresponding axes.

During the slip mode, relative motion of the contact area is described by the use of the
following quantities: vs = v̂s/(αâ)= vsxex+vsyey = vseτ – dimensionless linear velocity of the
pole A, ωs = ω̂s/α = ωsez – dimensionless angular velocity of the contact (v̂s and ω̂s denote
the corresponding real counterparts). Then the components of the friction force and moment
can be obtained by the use of the following integrals

Tx =−Tsx(θs,ϕs)=−

∫∫

F

σ(x,y)
cosθscosϕs−y sinθs√

(cosθscosϕs−y sinθs)2+(cosθs sinϕs+xsinθs)2
dxdy

Ty =−Tsy(θs,ϕs)=−

∫∫

F

σ(x,y)
cosθscosϕs+xsinθs√

(cosθscosϕs−ysinθs)2+(cosθs sinϕs+xsinθs)2
dxdy

M =−Ms(θs,ϕs)=−

∫∫

F

σ(x,y)
(x2+y2)sinθs+xcosθs sinϕs−ycosθscosϕs√
(cosθscosϕs−y sinθs)

2+(cosθs sinϕs+xsinθs)
2
dxdy

(2.2)

where the angles θs and ϕs are defined in such a way, that

vs =λscosθs ωs =λs sinθs λs =
√
v2s +ω

2
s

vsx = vscosϕs vsy = vs sinϕs
(2.3)

The signs of the functionsTsx,Tsy andMs are changed in order to simplify the further notation.

Then the full model of the friction forces can be defined as



Tx
Ty
M


∈






−



Tsx(θs,ϕs)
Tsy(θs,ϕs)
Ms(θs,ϕs)


 for λs > 0

{u3 ∈ R3 : ξs′(u)¬ η} for λs =0

(2.4)

where the bottompart of the expression concerns the stickmode. In this case the scalar function
ξs′(u) ­ 0 is defined as a part of the solution (ξs′,θs′,ϕs′) to the following set of algebraic
equations

u=−ξs′




Tsx(θs′,ϕs′)
Tsy(θs′,ϕs′)
Ms(θs′,ϕs′)



 (2.5)



A smooth model of the resultant friction force... 913

InEq. (2.5), the angles θs′ andϕs′ can be interpreted in such away thatλs′ cosθs′ cosϕs′ = vs′x,
λs′ cosθs′ sinϕs′ = vs′y and λs′ sinθs′ = ωs′ (λs′ > 0) are the components of virtual (possible
sliding) in the event that the friction coefficient is sufficiently small. The above construction of
the part of the model concerning the stick phase is motivated by the fact that during the stick
mode the components of the friction force andmoment (Tx,Ty,M) lie inside the zone bounded
by a surface parametrically described by the functions Tx =−ηTsx(θs,ϕs), Ty =−ηTsy(θs,ϕs)
andM =−ηMs(θs,ϕs). Since this zone is convex, there always exists one solution to equations
(2.4) for ξs′ ­ 0. The solution to Eq. (2.5) can be obtained numerically by the use of Newtons’
method using a proper starting point. Then the first component of the solution (ξs′,θs′,ϕs′)
is used for detection of the possible end of the stick mode for ξs′(u) = η. The two remaining
components (θs′,ϕs′) define the direction of slip in the beginning of the possible sliding. The
criterion ξs′(u)= η can also be interpreted as the yield surface.

3. Approximations of the integral model of sliding friction

An exact integral model of friction forces (2.2) is computationally expensive and inconvenient
for use in numerical simulations. Kudra and Awrejcewicz (2013) proposed different families of
its approximations. Some of them can be presented in the following form

f(In)(θs,ϕs)=

n∑
i=0
af,icos

n−iθs sin
iθs

(
|cosθs|mn+ bm|sinθs|mn

)m−1 (3.1)

where f = Tsx, Tsy, Ms, and n is the order of the approximation. For the assumed model of
the contact pressure distribution σ(x,y), the functions af,i = af,i(ϕs, sgn(cosθs), sgn(sinθs))
(it occurs that for an even numbern they do not depend on signs of cosθs and sinθs) are found
in such a way, that the following conditions are fulfilled

∂if(In)

∂ cosiθs
=
∂if

∂ cosiθs
for cosθs =0 and i=0,1, . . . ,n1

∂if(In)

∂ siniθs
=
∂if

∂ siniθs
for sinθs =0 and i=0,1, . . . ,n2

(3.2)

where n1+n2 =n−1. The other constants m­ 0 and b­ 0 do not influence conditions (3.2)
and they can be found in the process of optimization of fitting the approximate model to the
integral components, or identified experimentally.
Taking into account relations (2.3), approximations (3.1) take the form as follows

f(In)(vs,ωs,ϕs)=

n∑
i=0
af,iv

n−i
s ω

i
s

(
|vs|mn+ bm|ωs|mn

)m−1 (3.3)

For n1 =0, n2 =0 (n=1), one gets the following form of the approximation

T
(I0,0)
sx =

vsx− bc
(x,y)
0,1,1ωs

(
|vs|m+ bm|ωs|m

)m−1 T
(I0,0)
sy =

vsy + bc
(x,y)
1,0,1ωs

(
|vs|m+ bm|ωs|m

)m−1

M
(I0,0)
s =

bc
(x,y)
0,0,−1ωs− c

(x,y)
0,1,0vsx+ c

(x,y)
1,0,0vsy

(
|vs|
m+ bm|ωs|

m
)m−1

(3.4)



914 G. Kudra, J. Awrejcewicz

where

c
(x,y)
i,j,k
=

∫∫

F

xiyj(x2+y2)−
k
2σ(x,y) dxdy

and the alternative notation f(In1,n2) have been introduced.
For a circularly symmetric contact pressure distribution Tυ = 0, we use notation

Ts = Tsτ = −Tτ. In this case, we can also write that Tsx = Tscosϕs and Tsy = Ts sinϕs.

Now c
(x,y)
0,1,1 = c

(x,y)
1,0,1 = c

(x,y)
0,1,0 = c

(x,y)
1,0,0 =0, and approximations (3.4) take the following form

T
(I0,0)
s =

vs
(
|vs|m+bm|ωs|m

)m−1 M
(I0,0)
s =

bc
(x,y)
0,0,−1ωs

(
|vs|m+ bm|ωs|m

)m−1 (3.5)

where T
(I0,0)
s is the approximation of Ts (T

(I0,0)
sx =T

(I0,0)
s cosϕs and T

(I0,0)
sy =T

(I0,0)
s sinϕs).

4. Regularized model

In the special case of ωs = 0 or â = 0, models(2.2) and (3.3)-(3.5) are Ts = T
(In)
s = sgnvs

(Ts = −Tτ, Tυ = 0, Ms = M
(In)
s = 0). For the purpose of numerical simulations, often the

regularization of the sign function is performed by the use of arctan or tanh functions (Awrej-
cewicz and Lamarque, 2003; Awrejcewicz and Olejnik, 2003; Kudra and Awrejcewicz, 2011b).
We propose another way of regularization of the sign function

Tsε =T
(In)
sε =

vs√
v2s +ε

2
(4.1)

where ε is a small numerical parameter.
In order tomodel the stick-slip oscillations, when the static friction coefficient is higher than

the corresponding coefficient during the slip, one can develop model (4.1) in the following way

Tsε2 =T
(In)
sε2
= vs

(
1

√
v2s +ε

2
+η′

ε3

(v2s +ε
2)2

)
(4.2)

where the parameter η′ is a certain function of the coefficient η=max |Tsε2|. The function η
′ =

η′(η) doesnotdependonεandcanbeapproximatedasη′ ≈−13.607+30.893η−22.01η2+5.878η3

for η∈ [1,1.3] and η′ ≈−2.41+3.985η−0.3581η2 +0.0493η3 for η∈ [1.3,2.7], where the error
is |∆η|< 0.001. Figure 2 exhibits the exemplary plots of model (4.2).
Trying to generalize the above results and applying them tomodel (3.3), we assume in what

follows

f(In)ε2 (vs,ωs,ϕs)=
n∑

i=0

af,iv
n−i
s ω

i
s

(
1

√
λ2
sb
+ε2
+η′

ε3

(λ2
sb
+ε2)2

)
(4.3)

where

λsb =
(
|vs|
mn+ bm|ωs|

mn
)m−1

In the special case of n=1 (n1 =0, n2 =0, see Eq. (3.4)), one gets

T
(I0,0)
sxε2 =(vsx− b,c

(x,y)
0,1,1ωs)λ

(1)
sbε2

T
(I0,0)
syε2 =(vsy + bc

(x,y)
1,0,1ωs)λ

(1)
sbε2

M
(I0,0)
sε2 =(bc

(x,y)
0,0,−1ωs− c

(x,y)
0,1,0vsx+ c

(x,y)
1,0,0vsy)λ

(1)
sbε2

(4.4)



A smooth model of the resultant friction force... 915

Fig. 2. The regularizedmodel of friction (4.2) for η′ =0, η′ =1, η′ =2, η′ =4 and η′ =8

where

λ
(1)
sbε2
=

1
√
(
|vs|
m+ bm|ωs|

m
) 2
m
+ε2

+η′
ε3

((
|vs|m+ bm|ωs|m

) 2
m
+ε2

)2

For a circularly symmetric contact pressure distribution (see Eq. (3.5)) and m = 2, relations
(4.4) take the following form

T
(I0,0)
sε2 = vs

(
1

√
v2s +b

2ω2s +ε
2
+η′

ε3

(v2s + b
2ω2s +ε

2)2

)

M
(I0,0)
sε2 = bc

(x,y)
0,0,−1ωs

(
1

√
v2s + b

2ω2s +ε
2
+η′

ε3

(v2s +b
2ω2s +ε

2)2

) (4.5)

5. Example of application

Here we recall the model of a mechanical system described and analysed in the work by Kudra
and Awrejcewicz (2012a). The system, shown in Fig. 3, consists of a disk of radius r̂ = â,
mass m̂ and moment of inertia B̂. The disk is situated on a moving belt of velocity v̂b. The
position of the disk is defined by two co-ordinates: x̂ and ϕ describing the linear and angular
positions, respectively. The disk is joined with the support by the use of four elasto-damping
cordswinding the disk as shown inFig. 3, where k̂1/2, k̂2/2, ĉ1/2 and ĉ2/2 are the corresponding
coefficients of stiffness and damping.

It is assumed that the contact pressure distribution is circularly symmetric and T̂=µN̂Teτ
(T =Tτ =Tx, Tυ =Ty =0) as well as M̂=µN̂r̂Mez. Dynamics of the system is then governed
by the following non-dimensional equations

[
1 0
0 m

]{
ẍ
ϕ̈

}
+

[
c c12
c12 c

]{
ẋ
ϕ̇

}
+

[
1 k12
k12 1

]{
x
ϕ

}
=

{
Γ
Ω

}
(5.1)

where the symbol (•̇) stands for derivative with respect to the non-dimensional time t.



916 G. Kudra, J. Awrejcewicz

Fig. 3. The investigatedmechanical system

The dimensionless variables used in Eq. (5.1) are defined in the following way

x=
x̂

r̂
m=

B̂

m̂r̂2
k12 =

k̂2− k̂1

k̂1+ k̂2
Γ =µT Ω=µM

c=
ĉ1+ ĉ2√
m̂(k̂1+ k̂2)

c12 =
ĉ2− ĉ1√
m̂(k̂1+ k̂2)

vb =
v̂b
αr̂

t=αt̂
(5.2)

where

α=

√
k̂1+ k̂2
m̂

µ=
µm̂g

r̂(k̂1+ k̂2)

and where Γ and Ω are the non-dimensional friction force and moment, respectively. Since
vs = α

−1r̂−1v̂s and ωs = α
−1ω̂s, we get vs = ẋ− vb and ωs = ϕ̇, where ẋ and ϕ̇ are the

dimensionless velocities of the disk.

In thework byKudra andAwrejcewicz (2012a), a special event-driven scheme for numerical
solution of equations (5.1) with the set-valued elements Γ and Ω was developed. The solution
is composed of segments obtained by the use of classical integrationmethods for stick or sliding
modes. Each segment starts and ends with the events, i.e. switches between the two successive
modeswhich are detected during simulation.The integralmodel of frictionused for simulation of
the slidingmode aswell as detection of the end of the stickmode (see Section 2) is approximated

by theuse ofEq. (3.5) for c
(x,y)
0,0,−1 =2/3 (uniformcontact pressuredistribution),m=2and b=1.

In the present work, we test simulation of the above presented dynamical system using
the regularized model of friction (4.5). Figure 4 exhibits two examples of periodic stick-slip
attractors obtained for the following parameters of the system: m= 90, k12 = 0.85, c = 10

−4,
c12 = 0, vb = 0.15, µ = 5. The friction coefficient during the stick mode is defined by the
use of η = 4.98 (η′ = 13.7627) for the first attractor (a)-(b) and η = 2.7 (η′ = 6.7627) for
the second orbit (c)-(d). Each attractor is obtained two times through the event-driven scheme
(Kudra and Awrejcewicz, 2012a) and presented in the current work as a regularized model,
and then plotted two times. Since there is no visible difference between them (the plots cover
each other), one can conclude that the proposed smooth model of friction works correctly.
In the event-driven numerical algorithm, the threshold of detection of the singularity λs = 0
during the sliding mode equals 10−7. For integration of the differential equations between the
two successive events, we use the explicit Runge-Kutta (4,5) formula (Dormand-Prince) with
relative and absolute tolerances equal to 10−10. In the case of integration of stiff differential
equationswith the regularized frictionmodel, inwhich it is assumed ε=10−5,weuse amultistep
implicit scheme based on numerical differential formulas with relative and absolute tolerances
set to 10−10. The value of the parameter ε should be chosen based on numerical experiments.



A smooth model of the resultant friction force... 917

Fig. 4. Two examples of periodic attractors for η=4.98 (a)-(b) and η=2.7 (c)-(d) obtained from two
different methods: event-driven scheme (Kudra and Awrejcewicz, 2012a) and regularization of the

frictionmodel

6. Concluding remarks

In the work, a new kind of regularized approximate model of the coupled friction force and
torque for the general plane contact has been presented and tested. The developed here model
approximates the “exact” integral model in which the classical Coulomb friction law on each
element of the contact area and sudden switches between the stickmode and the fully developed
sliding are assumed.

The main properties of the new model are: i) avoidance of the problem of time consuming
integration over the contact area at each time step of integration of the differential equations,
ii) avoidance of singularity problems or set-valued friction forces in the originalmodel of friction
and possibility of the use of classical algorithms for integration of the differential equations,
iii) possibility to model static friction higher than the sliding one and stick-slip oscillations,
iv) parameters of the approximate model can be found based on optimization of the fitting to
the integral model or can be identified experimentally (independently of the integral friction
model). The above pointed out properties can be understood as advantages. The new model
possesses also a drawback, i.e. itmakes the resulting differential equations stiff and thus requires
special numerical methods.

The proposedmodel of friction has been applied to themodelling and simulation of a special
mechanical system in which two-dimensional (linear and rotational) stick slip oscillation occur.
The comparison of the results with those obtained by the event-driven method leads to the
conclusion that the newmodel operates correctly.



918 G. Kudra, J. Awrejcewicz

Acknowledgments

This work has been supported by the Polish National Science Centre, MAESTRO 2, No.

2012/04/A/ST8/00738.A part of this workwas presented at the 11thConference onDynamical System-

sTheory and Applications, December 5-8, 2011, Łódź, Poland.

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Manuscript received January 29, 2015; accepted for print December 18, 2015