Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 289-310, Warsaw 2007

STATIC FRICTION INDETERMINACY PROBLEMS AND

MODELING OF STICK-SLIP PHENOMENON IN DISCRETE

DYNAMIC SYSTEMS

Dariusz Żardecki

Automotive Industry Institute (PIMOT), Warsaw

e-mail: dariuszzardecki@aster.pl; zardecki@pimot.org.pl

The paper presents a new method of modeling of the friction action in
discrete dynamic systems in cases of undetermined distribution of sta-
tic friction forces. This method is based on the Gauss Principle and
the piecewise linear luz(. . .) and tar(. . .) projections with their origi-
nal mathematical apparatus. The derived variable-structure model of a
two-body system with three frictional contacts describes the stick-slip
phenomenon in detail. The model has an analytical form applicable to
standard (without iterations) computational procedures.

Key words: static friction, force indeterminacy, stick-slip, multi-body
systems, mathematical modeling, Gauss principle, piecewise linear
projections

1. Introduction

Modeling and simulation of strongly non-linear dynamic systemswith friction
is an attractive challenge for researchers. We can encounter numerous publi-
cations on sophisticated friction problems in scientific journals dedicated to
theoretical and applied mechanics, physics of continuous and granular media,
tribology, theory ofmechanisms,multi-body andmulti-rigid-body systems, fi-
nite element method, robotics, automatics, biomedicine, non-linear dynamics,
hybrid systems, the numerical methods in simulation, identification and opti-
mization, and even to computer graphics and animation. This interest is com-
prehensible. The friction problems are very important for life and technology,
but a lot of theoretical questions is still without satisfactory solutions. One of
them is a singular problemof the static friction forces indeterminacy in context
ofmodeling of the stick-slip phenomenon.The question is: whether simulation



290 D. Żardecki

of a givenmulti-body system is possible if the spatial distribution of resultant
static friction forces is undetermined? This problem will be discussed in the
paper.

1.1. Bibliographical overview on friction indeterminacy problems

Singular type of problems of friction indeterminacy (nonuniqueness of a
solution) or inconsistency (nonexistence of the solution) are well known as
classic problems of theoretical mechanics. They concern even the simplestmo-
dels which are grounded on the non-smooth but piecewise linear Coulomb (for
kinetic friction) and saturation (for static one) characteristics. Such singula-
rities were firstly noticed by Painlevé in 1895 who analyzed motion of a rigid
rod on a frictional surface. He noticed that for some parameter configuration
with a large friction force coefficient, themotionwasundetermined.Nowadays,
similar frictional problems of indeterminacy or inconsistency being observed
in many planar rigid-body systems are named as the ”Painlevé paradoxes”.
They were studied, for example by Lötsted (1981), Mason and Wang (1988),
Baraff (1991), Genot and Brogliato (1999), Leine et al. (2002).
Uniqueness and existence of solutions appear asmajor problems for univer-

sal computational methods elaborated for simulation and contact analysis of
multi-rigid-bodysystems in the2Dor3Dspacewhen load-dependent force fric-
tions are changing, and object’s topology is varying. Thesemethods and their
algorithms (usually iterative) are intensively used especially for such extreme-
ly difficult tasks as path and grasp steering for arm- or finger-mechanisms of
robots and surgerymanipulators, as physics-based animation for virtual envi-
ronments (including motion of granular materials), and so on. Usually, they
are based on the elementary Coulomb frictionmodel and utilize calculation of
friction forces in every computation step.But, if thenumberof friction forces is
larger than the number of degrees of freedom, some of the forcesmust be unk-
nown! This is especially evident when the stiction states appear. Hence, some
special computational tricks and treats must be applied. General description
of themethod including discussion on the uniqueness and existence is given in
state-of-the-art papers by Armstromg-Helouvry et al. (1994), Joskowicz et al.
(1998), Brogliato et al. (2002), and in many regular papers. Selected articles
are cited below.
Several computational approaches have been perceived: the first one – ba-

sing on the ”penalty method” enables small penetration between contacting
bodies.Because of hard springs added, one has no indeterminacies, but the so-
lutionsmay be numerically instable (stiffness problems). Schwager andPoshel
(2002) described a similar method for granular dynamics studies, basing on
artificially composed infinitely small but linear deformations of contacting par-
ticles. In their opinion, this algorithm is supposedly stable but such statement



Static friction indeterminacy problems... 291

seems not to be convincing enough. The second approach, most intensively
developed – for example by Glocker and Pfeifer (1993), Baraff (1993, 1994),
Stewart and Trinkle (1996), Trinkle et al. (1997), Pang and Trinkle (2000),
Balkom and Trinkle (2002), treats simulation of the friction multi-rigid-body
system as the Linear Complementarity Problem (LCP). The LCP methods
(iterative methods, primary designed for frictionless impact systems) are in-
tensively developed because they use very efficientmatrix subroutines.Apply-
ing the LCP method to friction systems, a special arrangement of constrains
must be done with using a linear approximation. These simplifications and
some ”heuristics” in the numerical code that simplify the problembutmake it
determined, cause that utility of the LCPmethod is practically limited to sys-
temswith small friction coefficients. TheSingularValueDecomposition (SVD)
approach byMirtich (1998) is also based on theLCP-typeprimarymodel. But
for calculation of some of its components, the model is modified. Using the
SVDmethods, one obtains a ”naturally symmetric” force distributionwhich is
admitted as satisfying (heuristics!). Recently, the ”impulse-velocity methods”
worked by Mirtich (1998), Moreau (2003), Kaufman et al. (2005) have been
developed. They also utilize some heuristics to obtain efficient and fast algo-
rithms. For example, only a single pair of contact points is handled at a given
time. Such simplifications allow one to obtain a real-time well-realistic anima-
tion. Concluding this survey, we can confirm the previous note that to resolve
indeterminacy problems in multi-rigid-body simulation, the force-based mo-
dels are conformablymodifiedwith application of heuristic ideas. By the way,
physicists working on ”molecular dynamics” have treated the friction indeter-
minacy as a probability problem – see Unger et al. (2004). By repetition of
simulations with perturbed static friction forces in each step, they obtained
and analyzed some statistics of simulation results.
Letus return toourquestion (now little extended).Whether the simulation

of a multi-body system is possible without some supplementary heuristics, if
accurate calculation of the resultant static friction forces is impossible? In this
paper, we will prove that this is possible for some class of discrete dynamic
systems for which the simulation might be handledwithout calculation of the
friction forces distribution in singular states. In our study, theGauss principle
of least constraint will be used for answering this question.
General presentation of theGaussprinciplebasedapproach for constrained

systems is given in Grzesikiewicz (1990), as well as in the papers by Glocker
(1997, 1999), Redon et al. (2002), and recently by Fan et al. (2005 – a special
paper on the 175 anniversary of Gauss’ work). According to theGauss Princi-
ple (with its extensions), accelerations of a dynamic multi-body system must
fulfill sufficient optimality conditions. The optimization concerns some convex
function which expresses the ”acceleration energy” of the system. Because of
convexity, this problem has a unique solution. So, from the theoretical point



292 D. Żardecki

of view, the accelerations of themodeledmulti-body system should be unique,
even some forces can be undetermined. In Fan et al. (2005), we can find a
special section addressed to the indeterminacy problems inmulti-body system
dynamics. It is shown that calculation of accelerations in such systems can be
supportedby a specialmatrix apparatus (theory of generalized inverses ofma-
trices). However, there are notmany papers describing a concrete application
of the Gauss principle in simulation studies of multi-body systems. Grzesikie-
wicz and Wakulicz (1979) described a numerical iterative matrix method for
simulation of motion of a train modeled as a multi-body series system with
Coulomb dry friction forces in multiple draft gears. This method is very so-
phisticate and seems to explore the theory of generalized inverses of matrices.
Simulation of a braking train seems to be a classical solvable problem with
the indeterminacy of static friction forces (static friction forces distribution is
unknown but the train stops!). Surprisingly, there are no papers found with
explicitly given analytical models for simple dynamic systems with the static
friction force indeterminacy. This absence should be filled.

1.2. Scope of the studies

Strictly analytical models of single or two-body systemswith theCoulomb
friction are well known in scientific literature. Most of them are based on
the Karnopp (1985) concept. The dry friction structures of such systems are
presented below.

Fig. 1. Three simple friction systems having analytical forms ofmathematicalmodels

Mathematical models of systems shown in Fig.1 have variable-structure
forms expressing both kinetic friction (for non-zero velocities) as well as static
friction (for zero velocities) actions, so they are applicable to the stick-slip
models of mechanisms with friction. Their analytical formulas are ready to
use in ODE (Ordinary Differential Equations) computational procedures.



Static friction indeterminacy problems... 293

But the mathematical modeling for the next little more complex struc-
ture of a friction system (presented on Fig.2) is noticed to be absent in the
literature! This two-body object having three frictional contacts is the sim-
plest systemwith static friction indeterminacy (for total zero-velocity stiction
state, the distribution of static forces is unknown). So, our study on friction
indeterminacy problems focuses on such a system.

Fig. 2. A simple friction systemwith static friction indeterminacy

In this study, we use special piecewise linear luz(. . .) and tar(. . .) pro-
jections and their mathematical apparatus. They are very efficient functions
for the modeling of non-smooth mechanical systems. Basing on this appara-
tus, the paper continues an approach presented in many previous authors’
publications. Several papers are cited below. The formalism of luz(. . .) and
tar(. . .) projections was described in details and proofs by Żardecki (2001,
2006a). The method of modeling piecewise-linear dynamical systems having
freeplays (backlashes, clearances) and frictions (kinetic and static) was pre-
sented by Żardecki (2005, 2006b). In the last paper by Żardecki (2006c), all
models relating to systems shown in Fig.1 have been derived with using the
luz(. . .) and tar(. . .)mathematical apparatus and theGauss principle. In this
study, such an approach is continued.

2. Theoretical background for piecewise-linear approach with

luz(. . .) and tar(. . .) projections

Definition

For x,a∈R, a­ 0

luz(x,a) =x+
|x−a|− |x+a|

2
tar(x,a) =x+asgh(x)

where

sgh(x)=





−1 if x< 0
s∗ ∈ [−1,1] if x=0
1 if x> 0



294 D. Żardecki

Fig. 3. Geometric interpretations of luz(. . .) and tar(. . .) projections

Theseprojections have simplemathematical apparatus containing algebra-
like formulas, formulas for some compositions and transformations, theorems
on disentanglement of some algebraic equations as well as theorems for diffe-
rential inclusions and equation transformations – Żardecki (2001, 2006a). We
will explore only peculiar formulas and theorems. They will be recalled when
necessary.
Below we present some formulas and statements useful for minimization

problemswith constraints (and for theGauss principle application in the stick-
slip modeling in Sec. 3 and 4). The following ”saturation function” (Fig.4) is
used in our studies

x=






−x0 for x<−x0

x for |x| ¬x0

x0 for x>x0

Fig. 4. Piecewise linear saturation characteristics

Remark: In the next points, a simple notation of saturation is applied for
variables, e.g.

vi means vi = vi(vi,v0i)

Corollary 1

x=x− luz(x,x0)



Static friction indeterminacy problems... 295

Lemma 1

Let x,x0 ∈R, x0 ­ 0, f(x) – a convex function.
If x̃ solves the minimization problem without constraints x̃ : minxf(x),

then the minimization problem with constraints x̂ : minxf(x) ∧ |x| ¬ x0,
has the solution x̂= x̃= x̃− luz(x̃,x0).

Proof

Fig. 5. Minimization of a convex function y= f(x) with limitation |x| ¬x0. The
pictures are representative: for |x̃| ¬x0 (a) and for x̃ > x0 (b)

– If x̃ <−x0, then x̂= x̃− (x̃+x0) =−x0. Because of the convexity of
f(x), for any δ > 0 f(−x0+ δ)>f(−x0), so x̂=−x0.

– If |x̃| ¬x0, then x̂= x̃− luz(x̃,x0)= x̃, so x̂= x̃.

– If x̃ > x0, then x̂= x̃−(x̃−x0)=x0. Because of the convexity of f(x)
for any δ > 0 f(x0− δ)>f(x0), so x̂=x0.

Hence x̂= x̃= x̃− luz(x̃,x0).

Corollary 2

Let x,g1,g2,x0,k1,k2,p∈R, k1+k2 > 0, x0 ­ 0

f(x)= k1(g1−x)
2+k2(g2−x)

2+p

Because f(x) is convex andhasminimum in the point x̃=(k1g1+k2g2)/(k1+
+k2), so theminimization problemwith constraints x̂ : minxf(x) ∧ |x| ¬x0
has the solution

x̂= x̃=
k1g1+k2g2
k1+k2

− luz
(k1g1+k2g2
k1+k2

,x0
)



296 D. Żardecki

Lemma 2

Let x1,x2,g1,g2,x01,x02,k1,k2 ∈R, k1,k2 > 0, x01,x02 ­ 0

f(x1,x2)= k1(g1−x1)
2+k2(g2−x2)

2

Theminimizationproblem x̂1, x̂2 : minx1,x2 f(x1,x2)∧ |x1| ¬x01, |x2| ¬x02,
has the solution

x̂i = x̃i = gi− luz(gi,x0i) i=1,2

Proof
First, we resolve the problemwithout constraints

∂f(x1,x2)
∂xi

=−2ki(gi−xi)= 0
∂f2(x1,x2)
∂x2i

=2ki > 0 i=1,2

∂f2(x1,x2)
∂x1∂x2

=
∂f2(x1,x2)
∂x2∂x1

=0

Because ki > 0 so f(x1,x2) is convex for all xi,gi (i = 1,2) and has the
minimum: x̃i = gi (i=1,2). Thus

x̃i = x̃i− luz(x̃i,x0i)= gi− luz(gi,x0i) i=1,2

Now we check whether x̂i = x̃i, for |xi| ¬x0i (i=1,2). We know that

x̃i =





−x0i if gi <−x0i
gi if −x0i ¬ gi = x̃i ¬x0i
x0i if g1 >x01

i=1,2

We solve 6 new simpler optimization tasks for functions with a single varia-
ble. Note that for a function of a single variable we can use Lemma 1 and
Corollary 2.

– If g1 < −x01, h1(x2) = f(−x01,x2) = k1(g1 +x01)2 +k2(g2 −x2)2, so
minimization of h1(x2) gives x̂2 = x̃2− luz(x̃2,x02)= g2− luz(g2,x02)

– If |g1|<x01, h2(x2)= f(g1,x2)= k1(g1−x2)2 – results in the same

– If g1 >x10, h3(x2)= f(x01,x2)= k1(g1+x01)2+k2(g2−x2)2 – results
in the same

– If g2 < −x02, h4(x1) = f(x1,−x02) = k1(g1 −x1)2 +k2(g2 +x02)2, so
minimization of h4(x1) gives x̂1 = x̃1− luz(x̃1,x01)= g1− luz(g1,x01)

– If |g2|<x02, h5(x1)= f(x1,g2)= k1(g1−x1)2 – results in the same



Static friction indeterminacy problems... 297

– If g2 >x02, h6(x1)= f(x1,x02)= k1(g1−x1)2+k2(g2−x02)2 – results
in the same

So x̂i = x̃i = gi− luz(gi,x0i) (i=1,2), indeed.

Lemma 3

Let x1,x2,g1,g2,w,x01,x02,w0,k1,k2 ∈R, k1,k2 > 0, x01,x02,w0 ­ 0

f(x1,x2,w) = k1(g1− (x1+w))
2+k2(g2− (x2−w))

2

The solutions x̂1, x̂2, ŵ to the minimization problem

x̂1, x̂2, ŵ : min
x1,x2,w1

f(x1,x2,w) ∧ |x1| ¬x01, |x2| ¬x02, |w| ¬w0 fulfill

x̂1+ ŵ= g1− luz(g1,x01+w0) x̂2− ŵ= g2− luz(g2,x02+w0)

Proof
Note that direct resolution of the task without limitation is impossible

∂f(. . .)
∂x1

=−2k1(g1−x1−w)= 0
∂f(. . .)
∂x2

=−2k2(g2−x2+w)= 0

∂f(. . .)
∂w

=−2k1(g1−x1−w)+2k2(g2−x2+w)= 0

x1,x2,w are linearly dependent (indeterminacy problem!).
Settingnewvariables v1 =x1+w,v2 =x2−w,wecan redefinetheproblem.

Now f(v1,v2)= k1(g1−v1)2+k2(g2−v2)2. The constraints fulfill the relations
|v1|= |x1+w| ¬ |xi|+ |w| ¬x01+w0, |v2|= |x2−w| ¬ |x2|+ |w| ¬x02+w0.
We resolve the new problemwith constraints applying Lemma 2

v̂1, v̂2 : min
v1,v2
f(v1,v2) ∧ |v̂1| ¬x01+w0, |v̂2| ¬x02+w0

Because the solution to the problem without constraints is ṽi = gi (i=1,2).
So v̂i = ṽi = gi− luz(gi,x0i+w0) and finally x̂1+ŵ= g1− luz(g1,x01+w0),
x̂2− ŵ= g2− luz(g2,x02+w0) �
Note that this lemma does not give solutions (they are indeterminate) but

some relations between them.

3. A method of modeling of friction forces and stick-slip

phenomena

The luz(. . .) and tar(. . .) projections and theirmathematical apparatus sim-
plify a synthesis and analysis of stick-slip phenomena in multi-body systems



298 D. Żardecki

with friction(s) expressedby piecewise linear characteristics. Itmeans that the
range ofmethodusability is limited to objectswhichhave constant friction for-
ce topology and friction forces not load-dependent. Themethod is commented
below for the simplest one-mass systemwith friction.

Fig. 6. One-mass systemwith friction; M –mass, F – external force, which
expresses conjunctions with other elements of the multi-body system

The synthesis of the model is done in several steps.
➢ Firstly, friction force characteristics are assumed.Typical friction force cha-
racteristics FT(V ) (Fig.7) are presented in an extended form (with ”hidden”
but limited static friction force for V =0). Such characteristics can be descri-
bed directly or piecewise-linear approximated by the luz(. . .) and tar(. . .)
projections.

Fig. 7. Typical friction force characteristics: (a) exactly Coulomb’s, (b) Coulomb’s +
static friction augmented, (c) Coulomb’s + static friction augmented + Stribeck’s
effect; area V =0 for static friction action denoted by double line; FT – friction
force, V – relative velocity of elements, FT0K – kinetic dry friction force,
FT0S – maximum static friction force, FT0 – maximum dry friction force (for
Coulomb’s characteristics FT0 =FT0K =FT0S), C – damping factor

In our studies, we use the Coulomb extended characteristic which is usu-
ally treated as basic for friction problems. Such a characteristic is written
directly as FT(V ) = C tar(V,FT0/C) (Żardecki. 2006b,c). For V 6= 0, they
express the kinetic friction force FTK. For V = 0, FT(0) = FTS = FT0s∗

(s∗ ∈ [−1,1]), so the static friction force FTS should be additionally deter-
mined by the resultant force FW (in one-mass system FW = F). Generally,
FTS(FW) are like saturation characteristics, but the forces FW mayhave com-
plex forms or be even undeterminable.



Static friction indeterminacy problems... 299

Having assumed a type of friction force extended characteristics, their pa-
rameters must be given. Sometimes (for example when contact surfaces have
heterogeneous properties), calculation of friction force parameters can require
some additional assumptions (even heuristics!). In our studies,we assume that
the friction force parameters are known.

➢ In the second step, the primary inclusionmodel is written. In our case, this
is

Mz̈(t)∈F(t)−C tar
(
ż(t),
FT0
C

)

The inclusion model must be translated to the ODE form. The problem con-
cerns only the state ż(t) = 0, because for ż(t) 6= 0 the tar(. . .) describes
friction characteristics one to one. So:
— if ż(t) 6=0

Mz̈(t)=F(t)−C tar
(
ż(t),
FT0
C

)

— if ż(t)= 0
Mz̈(t)∈F(t)−s∗FT0 s

∗ ∈ [−1,1]

➢ The inclusion model is analyzed for the state ż(t) = 0. The static fric-
tion force FTS = s∗FT0 is unknown but limited (FTS ∈ [−FT0,FT0]). For
application of the Gauss principle, the acceleration energy Q is defined. Here

Q=Mz̈2 =
(F −FTS)2

M

According to the Gauss principle, the function Q(. . .) (here Q(FTS)) is mini-
mized. For the one-mass system, the optimization tasks has a form

FTS : min
FTS

(F −FTS)2

M
∧ |FTS| ¬FT0

According to Corollary 2, the optimal solution is

FTS = s
∗FT0 =F − luz(F,FT0)

➢ Finally, the inclusionmodel is translated to theODEform.Hereoneobtains

Mz̈(t)=




F(t)−C tar

(
ż(t),
FT0
C

)
if ż(t) 6=0

luz(F(t),FT0) if ż(t)= 0

This formula strictly corresponds to the one-mass Karnoppmodel (Karnopp,
1985) and clearly describes the stick-slip phenomenon. Note, when ż(t) = 0



300 D. Żardecki

and |F(t)| ¬ FT0, then luz(F(t),FT0) = 0, then z̈(t) = 0. This means
stiction. When |F(t)|>FT0, we have luz(F(t),FT0) 6=0 and z̈(t) 6=0 – the
state ż(t)= 0 is temporary.

Advantages of using the luz(. . .) and tar(. . .) projections concern not
only brief analytic forms of the friction characteristics and clear the stick-
slip description. Using their mathematical apparatus, we can transform the
stick-slipmodels by parametric operations, and this seems to be an important
benefit too (more details in the paper by Żardecki (2006c)).

4. A model of the two-mass system with three frictional contacts

– the simplest indeterminacy problem for static friction forces

The two-mass system with three friction sources, which is shown in Fig.2, is
representative for several physical object configurations. In such a case, the
mass blocks rub with each other as well as with amotionless base surface (or
casing).

Fig. 8. Exemplary physical configurations of the two-mass systemwith three
frictional contacts

➢ One assumes that all kinetic friction forces fulfill the Coulomb characteri-
stics. The following denote: M1, M2 – masses of bocks, FT012, C12 – coeffi-
cients of the Coulomb characteristics for friction existing between the blocks,
FT010, C10 – coefficients for friction between the top block and base surfa-
ce, FT020, C20 – coefficients for friction between the bottom block and base
surface, F1, F2 – external forces.



Static friction indeterminacy problems... 301

➢ The primary inclusion model is

M1z̈1 ∈F1−C12 tar
(
ż1− ż2,

FT012
C12

)
−C10 tar

(
ż1,
FT010
C10

)

M2z̈2 ∈F2+C12 tar
(
ż1− ż2,

FT012
C12

)
−C20 tar

(
ż2,
FT020
C20

)

where s∗12,s
∗

10,s
∗

20 ∈ [−1,1] (see definition of the tar(. . .)).

This model can be rewritten as:
— if ż1 6=0, ż2 6=0, ż1 6= ż2

M1z̈1 =F1−C12 tar
(
ż1− ż2,

FT012
C12

)
−C10 tar

(
ż1,
FT010
C10

)

M2z̈2 =F2+C12 tar
(
ż1− ż2,

FT012
C12

)
−C20 tar

(
ż2,
FT020
C20

)

— if ż1 = ż2 6=0

M1z̈1 ∈F1−FT012s
∗

12−C10 tar
(
ż1,
FT010
C10

)

M2z̈2 ∈F2+FT012s
∗

12−C20 tar
(
ż2,
FT020
C20

)

— if ż1 =0, ż2 6=0

M1z̈1 ∈F1+C12 tar
(
ż2,
FT012
C12

)
−FT010s

∗

10

M2z̈2 =F2− (C12+C20)tar
(
ż2,
FT012+FT020
C12+C20

)

— if ż1 6=0, ż2 =0

M1z̈1 =F1− (C12+C10)tar
(
ż1,
FT012+FT010
C12+C10

)

M2z̈2 ∈F2+C12 tar
(
ż1,
FT012
C12

)
−FT020s

∗

12

— if ż1 =0, ż2 =0

M1z̈1 ∈F1−FT012s
∗

12−FT010s
∗

10

M2z̈2 ∈F2+FT012s
∗

12−FT020s
∗

20

For the state ż1 = 0, ż2 6= 0 as well as for ż1 6= 0, ż2 = 0 the equations
and inclusions have been little compressed. The formulas

tar(−x,a) =−tar(x,a)

k1 tar(x,a1)+k2 tar(x,a2)= (k1+k2)tar
(
x,
k1a1+k2a2
k1+k2

)

for k1,k2 ­ 0 were used.



302 D. Żardecki

➢ Nowwe analyze the inclusion forms. They concern four velocity conditions:

1) When ż1 = ż2 6=0 (then ż1− ż2 =0)
– problem of FTS12 (FTS12 =FT012s∗12)

2) When ż1 =0, ż2 6=0 (then ż1− ż2 6=0)
– problem of FTS10 (FTS10 =FT010s∗10)

3) When ż1 6=0, ż2 =0 (then ż1− ż2 6=0)
– problem of FFS20 (FFS20 =FT020s∗20)

4) When ż1 = ż2 =0 (then ż1− ż2 =0)
– problem of FTS12, FTS10, FTS20

Note, there is no problem of double singularities, for example a pair of
FTS12, FTS10. The problem concerns of the FTS12, FTS10, FTS20 triplet at
once.
In each case, the acceleration energy Q=M1z̈21 +M2z̈

2
2 is defined and an

appropriateminimization task is resolved. Calculations of every static friction
force (cases 1, 2, 3) can be realised in a standard way. Analysis of the triplet
FTS12, FTS10, FTS20 will be a task with indeterminacy!

Case 1 (ż1 = ż2 6=0)

M1z̈1 ∈FW1−FTS12 where FW1 =F1−C10 tar
(
ż1,
FT010
C10

)

M2z̈2 ∈FW2+FTS12 where FW2 =F2−C20 tar
(
ż2,
FT020
C20

)

The acceleration energy Q as function of FTS12 is

Q(FTS12)=
(FW1−FTS12)2

M1
+
(FW2+FTS12)2

M2
=

=
(FW1−FTS12)2

M1
+
(−FW2−FTS12)2

M2

So the optimization problem FTS12 : minFTS12Q(FTS12) ∧ |FTS12| ¬FT012 is
compatible to the task inCorollary 2.Note, in our case k1 =1/M1, g1 =FW1,
k2 =1/M2, g2 =−FW2, p=0, and

k1g1+k2g2
k1+k2

=
FW1
M1
− FW2
M2

1
M1
+ 1
M2

=
M2FW1−M1FW2
M1+M2

= F̃TS12

By application of Corollary 2, one finally obtains

FTS12 =
M2FW1−M1FW2
M1+M2

− luz
(M2FW1−M1FW2

M1+M2
,FT012

)



Static friction indeterminacy problems... 303

Note that

FW1−FTS12 =
M1(FW1+FW2)
M1+M2

+ luz
(M2FW1−M1FW2

M1+M2
,FT012

)

FW2+FTS12 =
M2(FW1+FW2)
M1+M2

− luz
(M2FW1−M1FW2

M1+M2
,FT012

)

Case 2 (ż1 =0, ż2 6=0)

M1z̈1 ∈FW1−FTS10 where FW1 =F1+C12 tar
(
ż2,
FT012
C12

)

M2z̈2 =FW2 where FW2 =F2− (C12+C20)tar
(
ż2,
FT012+FT020
C12+C20

)

The acceleration energy Q as function of FTS10 is

Q(FTS10)=
(FW1−FTS10)2

M1
+
F2W2
M2

So the optimization problem FTS10 : minFTS10Q(FTS10) ∧ |FTS10| ¬ FT010
is compatible to the task in Corollary 2. In this case k1 = 1/M1, g1 = FW1,
k2 =0, g2 =0, p=F2W2/M2, so

k1g1+k2g2
k1+k2

=FW1 = F̃TS10

According to Corollary 2, we have FTS10 =FW1− luz(FW1,FT010).
Note that FW1−FTS10 = luz(FW1,FT010).

Case 3 (ż1 6=0, ż2 =0)

M1z̈1 =FW1 where FW1 =F1− (C12+C10)tar
(
ż1,
FT012+FT010
C12+C10

)

M2z̈2 ∈FW2−FTS20 where FW2 =F2+C12 tar
(
ż1,
FT012
C12

)

The acceleration energy Q as function of FTS20 is

Q(FTS20)=
F2W1
M1
+
(FW2−FTS20)2

M2

So the optimization problem FTS20 : minFTS20Q(FTS20) ∧ |FTS20| ¬ FT020
is compatible to the task in Corollary 2. In this case k1 = 1/M2, g1 = FW2,
k2 =0, g2 =0, p=F2W1/M1, so

k1g1+k2g2
k1+k2

=FW2 = F̃TS20



304 D. Żardecki

According to Corollary 2, one finally obtains the static friction force
FTS20 =FW2− luz(FW2,FT020).
Note that FW2−FTS20 = luz(FW2,FT020).

Case 4 (ż1 = ż2 =0)

M1z̈1 ∈F1− (FTS10+FTS12)

M2z̈2 ∈F2− (FTS20−FTS12)

The acceleration energy Q as function of FTS10, FTS20, FTS12 is

Q(FTS10,FTS20,FTS12)=
[F1− (FTS10+FTS12)]2

M1
+
[F2− (FTS20−FTS12)]2

M2

The optimization problem

FTS10,FTS20,FTS12 :

min
FTS10,FTS20,FTS12

Q(FTS10,FTS20,FTS12) ∧ |FTS10| ¬FT010,

|FTS20| ¬FT020, |FTS12| ¬FT012

is compatible to the task in Lemma 3 (appropriate for the indeterminacy
problem).
Here k1 =1/M1, g1 =F1, k2 =1/M2, g2 =F2.
By application of Lemma 3, we know that the solutions fulfill

FTS10+FTS12 =F1− luz(F1,FT010+FT012)

FTS20−FTS12 =F2− luz(F2,FT020+FT012)

We have not calculated the static friction forces (they are undetermined), but
their necessary combinations have been found. Note that

F1− (FTS10+FTS12)= luz(F1,FT010+FT012)

F2− (FTS20−FTS12)= luz(F2,FT020+FT012)

➢ Finally, the inclusion model is translated to the variable structure ODE
form. Such amodel is convenient for analysis of the stick-slip phenomena. For
the two-mass systemwith three frictional contacts, we obtain:

—When ż1 6=0, ż2 6=0, ż1 6= ż2

M1z̈1 =F1−C12 tar
(
(ż1− ż2),

FT012
C12

)
−C10 tar

(
ż1,
FT010
C10

)
(4.1)

M2z̈2 =F2+C12 tar
(
ż1− ż2,

FT012
C12

)
−C20 tar

(
ż2,
FT020
C20

)
(4.2)



Static friction indeterminacy problems... 305

No stiction states, only slipping
—when ż1 = ż2 6=0

M1z̈1 =
M1

M1+M2

[
F1−C10 tar

(
ż1,
FT10
C10

)
+F2−C20 tar

(
ż2,
FT20
C20

)]
+
(4.3)

+luz

(
M2
[
F1−C10 tar

(
ż1,
FT10
C10

)]
−M1

[
F1−C20 tar

(
ż2,
FT20
C20

)]

M1+M2
,FT012

)

M2z̈2 =
M2

M1+M2

[
F1−C10 tar

(
ż1,
FT10
C10

)
+F2−C20 tar

(
ż2,
FT20
C20

)]
+
(4.4)

− luz

(
M2
[
F1−C10 tar

(
ż1,
FT10
C10

)]
−M1

[
F1−C20 tar

(
ż2,
FT20
C20

)]

M1+M2
,FT012

)

If

∣∣∣∣∣
M2
[
F1−C10 tar

(
ż1,
FT10
C10

)]
−M1

[
F1−C20 tar

(
ż2,
FT20
C20

)]

M1+M2

∣∣∣∣∣ ¬FT012

then luz(. . .) = 0, and the equations for z̈1, z̈2 have identical forms. Since
z̈1− z̈2 =0 and ż1− ż2 =0, itmeans that the blocks are stuck. In other cases,
the state ż1− ż2 =0 is temporary (without stiction).

—When ż1 =0, ż2 6=0

M1z̈1 = luz
(
F1+C12 tar

(
ż2,
FT012
C12

)
,FT010

)
(4.5)

M2z̈2 =F2− (C12+C20)tar
(
ż2,
FT012+FT020
C12+C20

)
(4.6)

The stiction state between the mass M1 and the base surface appears when

∣∣∣F1+C12 tar
(
ż2,
FT012
C12

)∣∣∣ ¬FT010

In other cases, the state ż1 =0 is temporary.

—When ż1 6=0, ż2 =0

M1z̈1 =F1− (C12+C10)tar
(
ż1,
FT012+FT010
C12+C10

)
(4.7)

M2z̈2 = luz
(
F2+C12 tar

(
ż1,
FT012
C12

)
,FT020

)
(4.8)



306 D. Żardecki

The stiction state between the mass M2 and the base surface appears when
∣∣∣F2+C12 tar

(
ż1,
FT012
C12

)∣∣∣ ¬FT020

In other cases, the state ż2 =0 is temporary.

—When ż1 =0, ż2 =0

M1z̈1 = luz(F1,FT012+FT010) (4.9)

M2z̈2 = luz(F2,FT012+FT020) (4.10)

The total stiction state appears when |F1| ¬ FT012 + FT010 and
|F2| ¬ FT012 +FT020. In this case z̈1 = z̈2 = 0 and ż1 = ż2 = 0. When
|F1| > FT012 +FT010 but |F2| ¬ FT012 +FT020 we have z̈1 6= z̈2 = 0, but
ż1 = ż2 =0, so theblock M2 is stuckwith thebase surfacebut the state ż1 =0
of the block M1 and the state ż1−ż2 =0of the blocks M1,M2 are temporary.
Ananalogical situation iswhen |F1| ¬FT012+FT010 but |F2|>FT012+FT020.
When both |F1| > FT012 + FT010 and |F2| > FT012 + FT020, the state
ż1 = ż2 =0 is without any stiction.
The final variable structuremodel of the two-mass systemwith three fric-

tion sources in the form ready to use with standardODE (without iterations)
procedures is presented below

M1z̈1 =






Eq. (4.1) for ż1 6=0, ż2 6=0, ż1 6= ż2
Eq. (4.3) for ż1 = ż2 6=0
Eq. (4.5) for ż1 =0, ż2 6=0
Eq. (4.7) for ż1 6=0, ż2 =0
Eq. (4.9) for ż1 =0, ż2 =0

M2z̈2 =






Eq. (4.2) for ż1 6=0, ż2 6=0, ż1 6= ż2
Eq. (4.4) for ż1 = ż2 6=0
Eq. (4.6) for ż1 =0, ż2 6=0
Eq. (4.8) for ż1 6=0, ż2 =0
Eq. (4.10) for ż1 =0, ż2 =0

5. Final remarks

Anewmethod formodeling of friction actions and stick-slip phenomena in di-
screte dynamic systems including the static frictiondistribution indeterminacy
has been presented in the paper. Themethod is based on theGauss least con-
straints principle and the piecewise linear luz(. . .) and tar(. . .) projections



Static friction indeterminacy problems... 307

with their original mathematical apparatus. Details ofmodel derivations have
been shown on an example of a two-mass system with three friction sources
(mathematical model of such a systemhas not been noticed in others publica-
tions). Thanks to the luz(. . .) and tar(. . .) projections, the model has been
given a clear analytical form ready to use with standard ordinary differential
equations procedures (without iteration). It can be useful even for real time
processing (steering). The idea presented here on the exemplary system can
be applied formore complex systems having a bigger number of friction forces
than the number of degrees of freedom.
The presented method has been discussed in the case of the Coulomb

friction. But the piecewise linear approximation basing on the luz(. . .) and
tar(. . .) projections is applicable also to more sophisticated friction characte-
ristics (expressing the Stribeck effect, non-symmetry and so on). Even though
the stick-slip models have been derived here for simple Coulomb’s friction
characteristics, their final forms can be easily adapted to other, more complex
ones. For example, when themagnitudes of kinetic and static dry friction for-
ces are not identical, two different parameters FT0K and FT0S can be applied
in the variable-structure model. The essence of the model is not changing.

Acknowledgments

The work has been sponsored by the Ministry of Science and Higher Educa-
tion within the grants 9T12C07108, 9T12C05819 and 4T07B05928 realised during
2005-2007.

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Problemy nieokreśloności tarcia statycznego i modelowanie zjawiska

stick-slip w układach dyskretnych

Streszczenie

W artykule przedstawia się nowąmetodęmodelowania procesów stick-slipwdys-
kretnych układach dynamicznych z tarciami dopuszczającą nieokreśloność rozkładu
sił tarcia statycznego.Metoda opiera się na zasadzieGaussa orazwykorzystaniu spe-
cjalnych przedziałami liniowych odwzorowań luz(. . .) i tar(. . .) z ich oryginalnym



310 D. Żardecki

aparatem matematycznym. W pracy prezentowane jest szczegółowe wyprowadzenie
modelu opisującego stick-slip w układzie 2 masowym z 3 miejscami tarcia. Dzięki
zastosowaniu odwzorowań luz(. . .) i tar(. . .) modele układów z tarciemmają anali-
tyczne formy przystosowane do standardowych procedur symulacyjnych.

Manuscript received August 2, 2006; accepted for print February 22, 2007