· Nonlinear Dyn ORIGINAL PAPER Set-valued anisotropic dry friction laws: formulation, experimental veriﬁcation and

Nonlinear Dynhttps://doi.org/10.1007/s11071-019-04829-6

ORIGINAL PAPER

Set-valued anisotropic dry friction laws: formulation,experimental verification and instability phenomenon

S. V. Walker · R. I. Leine

Received: 24 September 2018 / Accepted: 6 February 2019© Springer Nature B.V. 2019

Abstract Many technical applications, such as brakesand metal forming processes, are affected by aniso-tropic frictional behavior, where the magnitude and thedirection of the friction force are dependent on the slid-ing direction. Existing dry friction laws do not suf-ficiently describe all relevant macroscopic aspects ofanisotropic friction, and the influence on the dynamicsof mechanical systems is largely unknown. Further-more, previous experimental work on anisotropic fric-tion is limited and the fact that the friction force is notalways acting parallel to the sliding direction is oftenneglected. In this paper, an anisotropic dry friction lawwith the capability to describe the nonsmooth behav-ior of stick and slip and allowing for non-convex butstar-shaped sets of admissible friction forces is formu-lated using tools from convex analysis. The formula-tion of the friction law as normal cone inclusion enablesthe direct implementation in numerical time-steppingschemes. The stability of systemswith anisotropic fric-tion is studied and an eigenvalue analysis reveals thatthe anisotropic friction law is in theory capable of caus-ing anisotropic friction-induced instability. In addition,experimental setups for detailed investigations of thefrictional behavior are described. The measurementsreveal complex shaped force reservoirs and confirm the

S. V. Walker (B) · R. I. LeineInstitute for Nonlinear Mechanics, University of Stuttgart,Pfaffenwaldring 9, 70569 Stuttgart, Germanye-mail: [email protected]

R. I. Leinee-mail: [email protected]

validity of the presented friction law. Finally, it is shownthat the presented friction law leads to a more accu-rate prediction of the motion of nonsmooth mechanicalsystems.

Keywords Contact · Friction · Anisotropy · Stability

1 Introduction

In this work, a set-valued anisotropic dry friction lawthat enables the use of non-convex friction force reser-voirs and allows for an accurate representation of thebehavior of stick and slip is formulated in the frame-work of convex analysis. In addition, experiments areperformed to verify the friction law and anisotropicfriction is identified as a possible cause of friction-induced instability.

While friction is indispensable in many everydaysituations and technical applications such as screws,braking systems, clutches and driving wheels, it oftencauses unwanted effects in engineering and a largeamount of effort is put into the control and reductionof friction forces. The most basic laws of friction inuse today are commonly attributed to Amontons andCoulomb (see [1] and [2]) and read as:

– The friction force is directly proportional to the nor-mal load.

– The friction force does not depend on the apparentarea of contact.

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S. V. Walker, R. I. Leine

– The friction force is independent of the magnitudeof the velocity once motion starts.

The work of tribologists has been focused on extend-ing the friction laws to address different aspects of fric-tion especially on the microscale [3]. For many appli-cations, only the macroscopic aspect of friction is ofinterest, and the laws of Amontons and Coulomb ade-quately describe the frictional behavior, which is whythey are frequently implemented in multibody simula-tions. To facilitate a numerical treatment, often regular-ized friction laws are applied [4]. These regularizationsare based on smooth approximations of the discon-tinuous behavior of the friction force at the transitionfrom slip to stick. Commonly, arctangent functions areused to describe the friction force as a smooth single-valued function of the sliding velocity. The system isthen described by an ordinary differential equation andstandard integration techniques can be applied. How-ever, regularized friction laws lead to stiff differentialequations that cause numerical difficulties and lack theability to properly describe stiction since even for smallnonzero external forces motion is initiated. This moti-vates the use of set-valued force laws [5] that allow thefriction force to take a range of values at zero slidingvelocity. The nonsmooth transition from slip to stickinvolves a discontinuity in the time evolution of thefriction force,while the state of the system remains con-tinuous. Such a behavior can be described by extendingthe differential equation to a differential inclusion withset-valued right-hand side [6]. The framework of thisformulation is given by convex and nonsmooth anal-ysis (see [7–10]). Systems with unilateral contact andfrictional impacts can be formulated as measure differ-ential inclusions [11,12]. This formulation gives rise toa numerical discretisation known as the time-steppingmethod [12–14].

Typically, isotropic frictional properties are assu-med, in the sense that the magnitude and direction ofthe friction force are independent of the sliding direc-tion and the location of the contact point on a surface.In reality, the frictional properties ofmany surfaces sig-nificantly vary along different directions of a surface.This anisotropic frictional behavior can be induced bythe crystal structure of a material, occurs on the sur-face of biological, composite or textile materials andmay result from machining or finishing of a surface[15]. In engineering, knowledge of anisotropic frictionis essential since the majority of engineering surfaces

has such properties. Machined surfaces, created, e.g.,by cutting, milling or grinding, have a distinct surfacepattern which determines frictional behavior [16]. Insheet metal forming, the rolling direction of the mate-rial has an influence on the friction forces between theworkpiece and the tool which in turn influences thedeformation of the workpiece and the surface finishof the product [17,18]. Early experiments specificallyperformed to study directional effects of friction onmetallic surfaces have been carried out by Rabinow-icz [19] and Halaunbrenner [20]. Their results indi-cate that the magnitude of the friction force is depen-dent on the direction of sliding and the friction forceis not always acting parallel to the sliding direction.This is confirmed by the experiments in [21] and [22],where anisotropy results fromamacroscopic periodicalwaviness of the surface. Knowledge of both the magni-tude of the friction force and the relationship betweenthe sliding direction and the direction of the frictionforce is crucial for accurate simulations of systemswithanisotropic dry friction.

Different approaches on themodeling of anisotropicdry friction exist. In [23] sliders with wheels or skatesare used to model anisotropic frictional behavior. Incontrast, in [24] and [25], the microstructure of thecontact surface is modeled with unidirectional wedge-shaped asperities, which on the macroscopic scaleleads to friction laws similar to laws in plasticity the-ory. The resulting sets of all friction forces are notnecessarily convex, but for convex shapes a bipo-tential [26] can be applied for finite element com-putations of systems with anisotropic friction [27].Furthermore, tensor formulations of anisotropic fric-tion laws exist (see [28] and [29]) and the conceptof normal cone inclusions has been applied, e.g., forthe simulation of a snake robot [30] and a bobsleigh[31].

Another important aspect of mechanical systemswith friction is the possible occurrence of friction-induced instabilities. The squealing of automotivebraking systems is one of themost prominent examplesdemonstrating the loss of stability of a steady slidingmotion.Many other examples can be found in technicalapplications,where the loss of stability often occurs dueto friction-induced Hopf bifurcations. Despite the factthatmany researchers have extensively studied the fieldof friction-induced vibrations, the underlying mech-anisms are still not fully understood. Especially, theinfluence of anisotropic friction on the stability of slid-

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Set-valued anisotropic dry friction laws

ing motion has not been studied yet. In [32] a historicalreview of the stick-slip phenomenon is given. In addi-tion to stick-slip, in [33] possible mechanisms leadingto chatter are described. The effects of sprag-slip andmode-coupling are described, e.g., in [34] and [35],respectively. The surveys [36–39] review the classicalfrictionmodels used to represent the frictional behaviorin braking systems. Furthermore, in [40], an in-depthstudy of the stability of systems with friction and brakesqueal in particular is given.

From the existing literature in the field of anisotropicdry friction, several unsolved problems can be identi-fied:

– No existing dry friction law sufficiently describesall relevantmacroscopic aspects of anisotropic fric-tional behavior and is formulated such that it allowsfor an implementation in numerical time-steppingschemes.

– The influence of anisotropic friction on the stabilityof equilibria of mechanical systems is unknown.

– Previous experimental work on anisotropic frictionis insufficient.

A fundamental criterion of a meaningful dry fric-tion law is the accurate description of the constitutivebehavior of stick and slip, which requires a set-valuedformulation. Friction laws given in the form of normalcone inclusions are capable of describing this behav-ior. However, they are based on the assumptions thatthe set of all admissible friction forces is convex andthe sliding direction is defined by normality to thatset. Those assumptions do not necessarily hold for realanisotropic frictional behavior. Most of the existing tri-bometers and other experimental setups used for themeasurement of friction forces only record the com-ponent of the friction force directly opposing the slid-ing direction. Anisotropic friction forces, however, aregenerally not collinear to the sliding direction. Infor-mation on the force component acting orthogonal tothe direction of sliding is therefore often lost. To vali-date anisotropic friction laws, accurate measurementresults of the magnitude and direction of the fric-tion as a function of the sliding direction are neces-sary.

In this paper, the properties of systemswith anisotro-pic dry friction are analyzed analytically as well asexperimentally. The paper provides the reader with themathematical background necessary for the formula-tion of anisotropic friction laws in the framework of

convex analysis and for the numerical simulation withtime-stepping methods. Specifically, the aims of thiswork are:

– To formulate a highly general force law that enablesthe description of a large class of anisotropic dryfriction models and that can be implemented inexisting numerical algorithms,

– To classify different isotropic friction-inducedinstability phenomena according to their underly-ing mechanisms and to analyze the possibility ofanisotropic friction being the cause of self-excitedvibrations,

– To experimentally analyze the effects of anisotropicfriction both qualitatively and quantitatively.

Throughout this work, in the sense of Amontons, thefriction force is assumed to be proportional to the nor-mal load and independent of the apparent area of con-tact. The scope of this work is the contact betweenuniform surfaces, i.e., anisotropy due to different fric-tion properties at different locations of a surface is notconsidered. Furthermore, only the macroscopic behav-ior of mechanical systems is of interest. Microscopiceffects and wear at the contact points are neglected.

Section 2 provides the mathematical preliminariesthat are needed in the remainder of this work andthe equation of motion of a system with contacts isintroduced. Anisotropic friction laws are discussed inSect. 3. First, associated friction laws that are basedon normality of the sliding direction to a convex forcereservoir are analyzed. After showing that differentfrictionmodels lead to force laws in which normality tothe force reservoir is no longer given and the force reser-voir is not necessarily convex, an extended normal coneinclusion friction law is formulated. Section 4 is con-cerned with friction-induced instability mechanisms.Two experimental setups are presented in Sect. 5, andexperimental results for different contact partners areshown.

2 Nonsmooth dynamics

This section gives a brief introduction to the dynam-ics of nonsmooth mechanical systems [5,13,41,42].The nonsmooth nature of systems with unilateral con-tact or dry friction can be expressed in the frame-work of convex analysis by means of set-valued forcelaws.

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2.1 Elements from convex analysis

Comprehensive treatises of the topic are given by [7]and [8]. We consider a subset C of R

p and denote itsboundary and interior, bdryC and intC , respectively.

Definition 1 (Convex Set) A set C ⊆ Rp is called

convex if for every pair of points x ∈ C and y ∈ Calso (1 − s)x + s y ∈ C for all s ∈ (0, 1). In addition,it is strictly convex if (1 − s)x + s y ∈ intC for alls ∈ (0, 1).

A special case of possibly non-convex sets are star-shaped sets.

Definition 2 (Star-Shaped Set) A setC ⊆ Rp is called

star-shapedwith respect to the origin if for every x ∈ Calso sx ∈ C for all s ∈ [0, 1). In addition, it is calledstrictly star-shaped if sx ∈ intC for all s ∈ [0, 1).Figure 1 shows examples of different sets.

A function f : Rp → R ∪ {∞} is called convex

if its epigraph epi( f (x)) = {(x, z) ∈ Rp × R | z ≥

f (x), x ∈ Rp} is a convex subset of R

p+1. For a con-vex but possibly non-differentiable function having theadditional properties of being lower semi-continuousand proper [7], the subdifferential is defined as

∂ f (x) = {y | f (x∗) ≥ f (x) + yT(x∗ − x), ∀x∗ ∈ R

p}.

(1)

The subdifferential is a set-valued function, ∂ f (x) :R

p ⇒ Rp. An example is given by the set-valued Sign

function being defined by

Sign(x) = ∂|x | =

⎧⎪⎨

⎪⎩

−1 for x < 0,

[−1, 1] for x = 0,

1 for x > 0.

(2)

Definition 3 (Gauge Function) Let C ⊆ Rp be

closed and star-shaped with respect to the origin.The gauge function of C is defined as kC (x) =inf {s > 0 | x ∈ sC }.

It follows that the set C in Definition 3 is given aslevel set of the gauge function, i.e., C = {x ∈ R

p |kC (x) ≤ 1}. Gauge functions are defined for star-shaped sets and are nonnegative and positively homo-geneous functions of degree one [43] as illustrated inFig. 2a.

Definition 4 (Indicator Function) The indicator func-tion of the set C is defined as

ΨC (x) ={0 for x ∈ C ,

+∞ for x /∈ C .(3)

The gauge as well as the indicator function of a set Care convex if and only if the set C is convex.

The convex conjugate f ∗ of a convex function f isdefined as

f ∗(x∗) = supx

{xTx∗ − f (x)

}. (4)

Fig. 1 a Convex set, bstar-shaped and c notstar-shaped set

(a) (b) (c)

Fig. 2 a Gauge functionkC and level set C , bconvex set C with normalcone, proximal point andtangent cone

(a) (b)

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For proper, lower semi-continuous and convex func-tions, it holds that the Fenchel equality

xTx∗ = f (x) + f ∗(x∗) ⇐⇒ x∗ ∈ ∂ f (x)

⇐⇒ x ∈ ∂ f (x∗) (5)

is fulfilled. Of particular importance is the convex con-jugate of the indicator function, which is called supportfunction.

Definition 5 (Support Function) Let C ⊆ Rp be

closed and convex. The support function of C is givenas

Ψ ∗C (x∗) = sup

x

{xTx∗ | x ∈ C

}. (6)

In thiswork, normal cone inclusions are of special inter-est since they allow for the formulation of set-valuedforce laws.

Definition 6 (Normal Cone) Let C ⊆ Rp be closed,

nonempty and convex and let x ∈ C . The set of allvectors y ∈ R

p which do not make an acute angle withany vector x− x∗ for all x∗ ∈ C form the normal coneof C in x,

NC (x) = {y | yT(x∗ − x) ≤ 0, x ∈ C , ∀x∗ ∈ C

}.

(7)

For x /∈ C , it holds that NC (x) = ∅. The subdif-ferential of the indicator function yields exactly thedefinition of the normal cone, i.e.,

∂ΨC (x) = NC (x). (8)

For a nonsingular linear mapping A ∈ Rp×p, the trans-

formation property

y ∈ NC (x) ⇐⇒ AT y ∈ NA−1C (A−1x) (9)

follows from the chain rule of convex analysis [41,44].The transformed set A−1C consists of all the vectorsz with Az ∈ C .

In addition to the normal cone, the tangent cone [45]and proximal point function can be defined as follows.

Definition 7 (Tangent Cone) Let C ⊆ Rp be closed

and let x ∈ C . The tangent cone of C in x is definedas

TC (x)={ y | ∀tk ↓ 0, xk → x with xk ∈ C ,

∃ yk → y with xk+tk yk ∈ C }.(10)

Definition 8 (Proximal Point Function) Let C ⊆ Rp

be closed, nonempty and convex. The proximal pointfunction proxC (z) determines the closest point to z inC ,

proxC (z) = argminx∗∈C

‖z − x∗‖, z ∈ Rp. (11)

The notation proxC (z) has been used here to denote aprojection on a closed convex set, see [12], and has notto be confused with the proximal operator associatedwith a convex function. Normal cone inclusions canbe rewritten as implicit equations using proximal pointfunctions. It holds that

y ∈ NC (x) ⇐⇒ x = proxC (x + r y), r > 0.(12)

In Fig. 2b, normal and tangent cones as well as therelationship between the normal cone and the proximalpoint function are illustrated for a two-dimensional set.

The proximal point function x = proxC (z) yieldsx = z if z ∈ C . For a point z /∈ C , the proximal pointfunction results in a projection of z on the boundary ofthe set C . For convex sets having a smooth boundary,i.e., the gradient of the gauge function kC (x) exists forall x �= 0, it holds that

x=proxC (z) ⇐⇒ x + β∇kC (x)= z, with β > 0.

(13)

In addition, the condition kC (x) = 1 must be fulfilledsince the projected point x has to be on the boundaryof C . This set of equations allows for the calculationof the closest point to z in C .

2.2 Equation of motion

The dynamics of a time-autonomous mechanical sys-tem with n degrees of freedom and no frictional uni-lateral constraints can be described by the equation ofmotion

M(q)q − h(q, q) = 0, (14)

where q = q(t) ∈ Rn denotes the time dependent set of

generalized coordinates, M ∈ Rn×n is the symmetric

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and positive definitemassmatrix and the vector h ∈ Rn

contains all differentiable forces and gyroscopic terms.The restriction to time-autonomous systems is notessential, but is made in this work for simplicity. If, inaddition, m normal and frictional contact forces act onthe system, the equation of motion can be extended as

M(q)u − h(q, u) =m∑

i=1

(wNi (q)λNi + W i (q)λi ) ,

(15)

where the generalized velocities are denoted u, andu = q holds for almost all t . The normal force at contactpoint i is calledλNi ∈ R andλi ∈ R

p is the correspond-ing friction force. The dimension p of the friction forcemay vary depending on the considered system. Forinstance, for planarmechanical systems the sliding fric-tion force is scalar (p = 1). For spatial mechanical sys-tems, the contacting bodies have a contact plane and thesliding friction force is two-dimensional (see Sect. 3.1).Higher dimensions are necessary if combined slidingand drilling or rolling friction is considered [42]. Thegeneralized force directions of the normal and frictionforce are defined by wNi ∈ R

n and W i ∈ Rn×p.

The contact forces and force directions are assem-bled in the vectors and matrices λN , λ,WN ,W and theequation of motion can be rewritten in the form

M(q)u − h(q, u) = WN (q)λN + W(q)λ. (16)

Note that whenever sliding friction in particular is con-sidered, i.e., without additional drilling or rolling fric-tion, the index T is added to the friction force and gen-eralized force direction yielding λT and WT . In thiswork, motion without impacts is considered. There-fore, the generalized coordinates q(t) as well as thevelocities u(t) are assumed to be time-continuous func-tions. However, when a transition from slip to stickoccurs or when the direction of sliding at a contactpoint is reversed, the friction force λ(t) is discontinu-ous.At such time instants, the generalized accelerationsu are not defined. Equation (16) therefore only holdsfor almost all t .

2.3 Time-stepping method

The equation of motion Eq. (16) of a mechanicalsystem with contact points in combination with set-valued force laws, given, e.g., in the form of normal

cone inclusions, yields a differential inclusion. Numer-ical integration of nonsmooth systems is possible withevent-driven or time-stepping methods. In the follow-ing, the time-stepping method developed by [12] isbriefly described.

The equation ofmotion can be replaced by the equal-ity of measures

Mdu − hdt = WNdPN + WdP, (17)

where the dependence of the terms on q and u is omittedfor brevity. This formulation allows for the occurrenceof impacts. In this work, impacts are not consideredand the differential measures are given by du = udt ,dPN = λNdt and dP = λdt . Since the set of timeinstants for which the acceleration u does not existis Lebesgue negligible [42], the time evolution of thevelocity u is given by

u(t) = u(t0) +∫ t

t0du ∀t ≥ t0. (18)

InMoreau’s time-stepping scheme, the equality ofmea-sures is approximated over small time steps Δt . Thesubscripts B, M and E are used to describe values atthe beginning, midpoint and end of a time step, respec-tively. The generalized coordinates and velocities atthe beginning of a time step, qB and uB, are knownand the coordinates at the midpoint are obtained fromqM = qB + 1

2uBΔt . Systemmatrices and vectors eval-uated at the midpoint are denoted with a subscript M.For closed contacts at the midpoint, the inclusion prob-lem

MM(uE − uB) − hMΔt = WNMPN + WMP,

γ NE ∈ NR

−0(−PN ), γ E ∈ NC (PN )(−P)

(19)

needs to be solved for the velocities uE at the end ofthe time step, where the set-valued force laws are for-mulated in terms of the normal and frictional contactefforts, PN and P . Using Eq. (12), normal cone inclu-sions can be rewritten as implicit proximal point func-tions. This allows for an iterative solution of the contactproblem, e.g., with a modified Newton algorithm [46].The scheme

ukE = uB + M−1M

(hMΔt + WNMPk

N + WMPk)

,

(20)

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Pk+1N = − prox

R−0

(−Pk

N + rNγ kNE

), rN > 0,

(21)

Pk+1 = − proxC(Pk+1N

)(−Pk + rγ k

E

), r > 0

(22)

is iterated with an initial guess of the contact efforts,until a stopping criterion is reached. The relative veloc-ities are given as

γ kNE = WT

NMukE and γ kE = WT

MukE + χ . (23)

Finally, the coordinates qE at the end of the time stepcan be calculated by qE = qM + 1

2uEΔt .

3 Anisotropic friction force laws

In the following, λ ∈ Rp describes the friction force

and γ ∈ Rp the relative velocity between two contact-

ing bodies. In the case of combined sliding and drillingor rolling friction, γ contains the tangential slidingvelocity and the spin of the contacting bodies scaled toa velocity. The normal force at the contact is supposedto be known and is calledλN . In the sense ofAmontons,friction forces are assumed to be directly proportionalto the normal force. The dependence of the frictionforce on the normal force is not written explicitly.

Definition 9 (DryFriction Law) LetF : Rp ⇒ R

p bean upper semi-continuous set-valued function and letthe image ofF (γ ) be a compact set for all fixed valuesof γ and a star-shaped set with respect to the origin forγ = 0 with F (0) �= {0}. A dry friction law connectsthe relative sliding velocity to the friction force via

−λ ∈ F (γ ). (24)

A dry friction law is characterized by having a stickphase, i.e., the friction force has for zero relative veloc-ity the character of a constraint force. Hence, a dryfriction law allows for a nonzero friction force at zerorelative velocity (stick).

In the following, friction laws are considered whereno dependence on the magnitude of the sliding velocityis assumed to exist, as stated by Coulomb. Coulomb-type friction laws are defined based on Definition 9 ofdry friction laws.

Definition 10 (Coulomb-Type Friction Law) A dryfriction law is called Coulomb-type friction law if itholds that

F (γ ) = F (aγ ) ∀a ∈ R+. (25)

Hence, the magnitude of the sliding velocity has noinfluence on the flow rule.

The set of all admissible friction forces C ⊂ Rp

is called force reservoir. It holds that F (γ ) ⊆ C and−λ ∈ C . During stick, the force reservoir C and theimage of F (0) coincide, i.e., C = F (0). Since theimage of F (γ ) is compact for all γ and star-shapedfor γ = 0, the force reservoir has the same properties.A bounded force reservoir excludes friction forces withan infinite magnitude which would completely preventsliding in the corresponding direction, e.g., a one-wayclutch. Force reservoirs that are not star-shaped are notsuitable to describe the stick phase. Consider a bodyunder an external load being in static equilibrium suchthat the negative friction force equals the external load.If the magnitude of the external load is reduced, thebody remains in the stick phase and the magnitudeof the friction force reduces correspondingly. This isimpossible for a force reservoir that is not star-shapedwith respect to the origin as seen in Fig. 3a since frictionforces with a smaller magnitude are not always part ofthe force reservoir.

During slip, the friction force is defined by the pos-sibly set-valued flow ruleF (γ ) with γ �= 0. A funda-mental requirement for a physicallymeaningful frictionforce law is that the friction force does not create energyin the system. Let the rate of dissipation D(−λ, γ ) begiven by

D(−λ, γ ) = −λTγ with − λ ∈ F (γ ). (26)

During stick, D(−λ, 0) = 0. IfF (γ ) is single-valuedfor γ �= 0, then the rate of dissipation is solely afunction of the sliding velocity and we write D(γ ).The same holds if the function is multi-valued if for afixed γ the condition −λTγ = const. is fulfilled for all−λ ∈ F (γ ). This case is illustrated in Fig. 3b.

Definition 11 (Dissipativity of Dry Friction Law) Adry friction law is called

(i) dissipative if

D(−λ, γ ) ≥ 0 ∀γ ∀ − λ ∈ F (γ ),

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Fig. 3 a Not star-shapedand therefore nonadmissiblefriction force reservoir. bConstant rate of dissipationfor different friction forcesλ. c Range of admissiblesliding directions for agiven λ

(a) (b) (c)

(ii) strictly dissipative if

D(−λ, γ ) > 0 ∀γ �= 0 ∀ − λ ∈ F (γ ).

Figure 3c shows a graphical representation of theadmissible sliding directions for a specific friction forcefor dissipative friction laws. In the following, restric-tions on the relationship between the sliding velocityand the friction force and on the force reservoir aregiven.

Definition 12 (Associated Coulomb Friction Law) Letthe set C ⊂ R

p be nonempty, compact and convex.A Coulomb-type friction law is called associated ifF (γ ) = ∂Ψ ∗

C (γ ), i.e.,

−λ ∈ ∂Ψ ∗C (γ ) ⇐⇒ γ ∈ NC (−λ). (27)

The associated friction law is thus defined by normalityof the sliding velocity γ to the boundary of the forcereservoir during slip. For all friction forces in the inte-rior of the force reservoir, the contacting bodies stickand the relative sliding velocity is zero. This behavioris described by the normal cone inclusion where thesliding velocity is in the normal cone NC (γ ) to theforce reservoir C .

The inverse relation of the normal cone inclusion isthe friction force being in the subdifferential of the sup-port function of the force reservoirΨ ∗

C (γ ). It is derivedusing the Fenchel equality Eq. (5) and the fact that thesubdifferential of the indicator function of a convex setis the normal cone to the set.

Proposition 1 (Dissipativity of Associated CoulombFriction Law) An associated Coulomb friction law isdissipative. It is even strictly dissipative if 0 ∈ intCfor all γ .

Proof From the definition of the rate of dissipationEq. (26) and the Fenchel equality Eq. (5), it follows that

D(−λ, γ ) = −λTγ = ΨC (−λ) + Ψ ∗C (γ ). (28)

The indicator functionΨC (−λ) vanishes since the fric-tion force is always in the force reservoir. It follows thatthe rate of dissipation is only a function of the slidingvelocity. Because the force reservoir must contain theorigin, the support function is nonnegative [42] whichgives

D(γ ) = Ψ ∗C (γ ) ≥ 0. (29)

In addition, if 0 ∈ intC , the support function is strictlypositive, Ψ ∗

C (γ ) > 0 for all γ �= 0. ��Next, we consider the principle of maximum dissipa-tion introduced in [47].

Proposition 2 (Principle of Maximum Dissipation)For a given compact and convex force reservoir C ,the flow rule which maximizes the rate of dissipation isthe associated flow rule.

Proof Let the friction force during slip (γ �= 0) begiven by the flow rule −λ ∈ F (γ ) such that the rateof dissipation yields

D(−λ, γ ) = −λTγ ≥ −λ∗Tγ ∀γ ∀ − λ ∈ F (γ )

(30)

for all other flow rules −λ∗ ∈ F ∗(γ ) with −λ∗ ∈ C .Rewriting Eq. (30) results in the definition of a normalcone (see Definition 6),

−(λ − λ∗)Tγ ≥ 0 ∀ − λ∗ ∈ C �⇒ γ ∈ NC (−λ).

(31)

Therefore, if the principle of maximum dissipationis assumed to hold, then the associated flow rule isobtained. ��Definition 13 (Maximal Monotone Coulomb-TypeFriction Law) A Coulomb-type friction law is calledmaximal monotone ifF (γ ) is maximal monotone, i.e.,

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(i) ∀(γ ,−λ)∈Graph(F ), ∀(γ ∗,−λ∗)∈Graph(F ),

�⇒ −(λ − λ∗)T(γ − γ ∗) ≥ 0,(ii) Graph(F ) ⊆ Graph(F ) �⇒ Graph(F ) =

Graph(F ) for all functions F fulfilling Condition(i).

Definition 13(i) ensures the monotonicity ofF , whileCondition(ii) (see [48]) adds thatF is called maximalmonotone if there exists no other monotone set-valuedfunction whose graph strictly contains the graph ofF .In other words, no enlargement of the graph is possiblewithout destroying the monotonicity of F [8].

Proposition 3 Every maximal monotone Coulomb-type friction law is an associated Coulomb friction law.

Proof Substitution ofγ ∗ = 0 in the definition ofmono-tonicity (Definition 13(i)) leads to the definition of anormal cone, i.e.,

−(λ − λ∗)Tγ ≥ 0 ∀ − λ∗ ∈ C �⇒ γ ∈ NC (−λ).

��Therefore, for a given force reservoir, the only exist-ing maximal monotone Coulomb-type friction law isthe associated Coulomb friction law. Vice versa, thefollowing proposition holds.

Proposition 4 Every associated Coulomb friction lawis maximal monotone.

Proof For associated Coulomb friction laws it holdsthatF (γ ) = ∂Ψ ∗

C (γ ). The support function Ψ ∗C (γ ) is

a lower semi-continuous convex function. The subdif-ferential of such a function is maximal monotone [7].��Hence,maximalmonotoneCoulomb-type friction lawsare exactly associated Coulomb friction laws.

Let a system be described by a differential inclusionconsisting of the equation of motion together with aset-valued associated Coulomb friction law. From themaximal monotonicity ofF (γ ), existence and unique-ness of solutions follows [42].

3.1 Planar friction

The definitions and propositions formulated above arevalid in R

p. If tangential friction forces in a contactplane are considered, then p is equal to two. Higherdimensions are necessary if in addition to sliding fric-tion forces a friction torque due to combined slidingand drilling or rollingmotion occurs [42,49]. In the fol-lowing sections, sliding in the tangential contact planespanned by eT1 and eT2 is considered (see Fig. 4a). Thetangential friction force and the relative sliding velocityare defined by

λT =[λT1λT2

]and γ T =

[γT1γT2

]. (32)

The normal force at the contact is called λN . A spe-cific force reservoir C ⊂ R

2 in combination with aflow rule defines the friction law at the contact. Theterm planar friction is used to emphasize the existenceof two-dimensional friction forces in the contact plane.Other authors sometimes refer to such behavior as spa-tial friction (see, e.g., [5]).

3.2 Associated Coulomb friction laws

Associated Coulomb friction laws according to Defi-nition 12 are valid for all compact and convex force

Fig. 4 a Contact plane andcontact forces on body 1. bGraphical representation ofisotropic friction law withstick and slip

(a) (b)

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Fig. 5 a Body sliding onanisotropic surface. bAssociated Coulombfriction law

(a) (b)

reservoirs C . The shape of the force reservoir has agreat influence on the frictional behavior at a contactpoint. In this section, different shapes of the force reser-voir are discussed.

Isotropic Friction

The constitutive behavior of stick and slip for isotropicfriction in a plane is given by the following definition.

Definition 14 (Isotropic Coulomb Friction Law) Letμ > 0 be the isotropic friction coefficient. An isotropicCoulomb friction law fulfills

(i) γ T = 0 , ‖λT ‖ ≤ μλN ,(ii) γ T �= 0 , −λT = μλN

γ T‖γ T ‖ ,

where the fraction represents the normalized slidingvelocity, i.e., the unit vector in the plane pointing in thedirection of γ T .

This frictional behavior can be expressed with theassociated Coulomb friction law (Definition 12) anda circular force reservoir C called Coulomb-Moreau’sdisk [41]:

γ T ∈ NC (−λT ) (33)

with C = {−λT ∈ R2 | ‖λT ‖ ≤ μλN

}, see Fig. 4b.

For friction forces inside the force reservoir, the slid-ing velocity is zero. For friction forces on the boundaryof the set, the vector of the sliding velocity γ T is nor-mal to the set and therefore points in the direction of−λT . The magnitude of the sliding friction force isindependent of the sliding direction.

Anisotropic Friction with Various Force Reservoirs

The associated Coulomb friction law is not only use-ful for the description of isotropic friction but can alsobe applied in the anisotropic case if the principle ofmaximum dissipation (Proposition 2) is assumed tohold. This is an assumption that is frequently made

but lacks an experimental verification. In the case ofanisotropic friction, the force reservoir is changed froma circle to any convex and compact shape. For symmet-ric force reservoirs, orthotropic force laws are obtained.For orthotropic friction, often an ellipsoidal force reser-voir of the form

C ={

−λT ∈ R2∣∣∣∣

( −λT1

μ1λN

)2

+( −λT2

μ2λN

)2

≤ 1

}

(34)

is assumed. For a non-circular force reservoir, −λT

and γ T are generally not collinear. The force compo-nent acting orthogonal to the sliding direction causes abody sliding on a horizontal plane without the presenceof external forces to deflect (see Fig. 5a, b). Only if thesliding velocity is parallel to the principal axes of theellipsoidal force reservoir, the friction force is directlyopposing it.

Orthotropic Coulomb friction during slip can also bemodeled byusing two independent sign functions alongthe principal directions of the orthotropic surface [50].To account for the set-valued nature of Coulomb fric-tion, the force law can be rewritten using set-valuedSign functions:

−λT1 ∈ −μ1λNSign(γT1),

−λT2 ∈ −μ2λNSign(γT2).(35)

Here, stick and slip along the two principal direc-tions are independent of each other. A physical modelexhibiting such a behavior is the X–Y table shown inFig. 6a. Each of the two slide bearings, with frictioncoefficient μ1 and μ2, respectively, can move indepen-dently of the other. The inclusions (35) are equivalentto the associated Coulomb friction law with the rect-angular force reservoir

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Fig. 6 a Sketch of a X–Ytable. b Graphicalrepresentation of associatedCoulomb friction law withrectangular force reservoir

(a) (b)

Fig. 7 a Superellipsoidalfriction force reservoirs withdifferent roundness factors.b Friction force reservoircomposed of a rectangle andtwo half-ellipses

(a) (b)

C = {−λT ∈ R2 | ∣∣−λT1

∣∣ ≤ μ1λN ,∣∣−λT2

∣∣ ≤ μ2λN},

(36)

see Fig. 6b. The force reservoir is convex, but notstrictly convex. If sliding in one of the principal direc-tions is considered, then the slide bearing orthogonalto that direction is in the stick phase. Therefore, for agiven sliding direction, a set of possible friction forcesexists, namely all friction forces on the correspondingstraight segment of the boundary of the force reservoir.The rate of dissipation in the case of such a convex butnot strictly convex force reservoir is still unique sincethe component of the friction force orthogonal to thesliding velocity does not contribute to the dissipation.A unique friction force is found if the dynamics of thesystem is considered.

For friction forces in the corner of the rectangularforce reservoir, the normal cone returns the set of allsliding directions in the corresponding quadrant. Thismeans that for a range of sliding directions, the mag-nitude and direction of the friction force is the same.

Besides the elliptical and rectangular force reser-voir, other convex force reservoirs for orthotropic fric-tion are possible. A useful mathematical description ofsets with two semi-axes are superellipses. The generalexpression for a superellipsoidal set with roundnessfactor s is given as

C ={−λT ∈ R

2∣∣∣∣

∣∣∣∣λT1

μ1λN

∣∣∣∣

s

+∣∣∣∣

λT2

μ2λN

∣∣∣∣

s

≤ 1

}.

(37)

The roundness factor determines the shape of thesuperellipse as depicted in Fig. 7a. For s ≥ 1 the set isconvex. A rhombus is obtained for s = 1. The standardellipse has a roundness factor of s = 2. If, in addition,μ1 = μ2 holds, then the force reservoir is circular andthe force law reduces to the form of isotropic Coulombfriction.

The associated Coulomb friction law can be furtheradjusted to specific anisotropic frictional behavior byimplementing suitable force reservoirs. An example ofanisotropic frictional behavior is the contact of a slideron ice, as plays an important role in the numerical simu-lation of a bobsleigh on the surface of an ice track [51].Without anisotropic friction, the bobsleigh would beunable to steer. In [31], it is assumed that the forcereservoir consists of two half-ellipses connected to arectangle, see Fig. 7b.

Capability and Limitations of the Friction Law

The formulation of the friction force law as normalcone on the friction force reservoir C is an effectivemethod to describe the set-valued nature of stick andslip. By adjusting the shape of the force reservoir, the

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magnitude of the friction force for different slidingdirections is specified and the relationship between thesliding direction and the direction of the friction forcefollows from normality to the set. In order to guaran-tee dissipativity of the friction law, the force reservoirnecessarily has to contain the origin. The formulationis capable of describing many aspects of anisotropicfrictional behavior. However, all chosen friction forcereservoirs have to fulfill the condition of being convex.Non-convex sets are not allowed, since the definition ofnormal cone inclusions (see Definition 6) is only validfor convex sets. This limitation makes it impossible toincorporate more complex non-convex force reservoirsthat are discussed later in this work.

3.3 Non-associated anisotropic friction laws

The assumption of maximum dissipation which holdsfor the associated Coulomb friction law is not neces-sarily valid for real anisotropic frictional behavior. Inthis section, other possible anisotropic friction laws arebriefly discussed. Since the sliding direction in thosecases is not directly related to the shape of the forcereservoir, the force laws are called non-associated forcelaws.

Collinear Friction Law

The associated Coulomb friction law described abovehas for non-circular force reservoirs, a limited numberof sliding directions where the friction force directlyopposes the direction of sliding. A specific anisotropicfriction law which is often encountered in the litera-ture (e.g., [52]) is based on friction forces that alwaysdirectly oppose the sliding direction. In this work, sucha friction law is called collinear friction law.

Definition 15 (Collinear Friction Law) A collinearfriction law is characterized by a friction force dur-ing slip that is always parallel to the sliding directionand opposes the relative sliding velocity.

The friction coefficient and with that, the magnitudeof the friction force is dependent on the sliding direc-tion. During slip, the force law can be expressed in theform

γ T �= 0 ; −λT = μ(γ T )λNγ T

‖γ T ‖ , (38)

where the scalar magnitude of the friction force is mul-tiplied by the normalized vector of the relative sliding

velocity. As before, the shape of the force reservoir hasto be defined. In the following, an ellipsoidal shape asgiven in Eq. (34) is assumed. The friction coefficientsalong the major axes are μ1 and μ2. With the diagonalmatrix

T =[

1μ1λN

0

0 1μ2λN

]

(39)

the force reservoir C can be expressed as

C ={−λT ∈ R

2∣∣∣∣

√λTT T

2λT ≤ 1

}. (40)

The sliding friction force is then given by

−λT = 1√

γ TT T

2γ T

γ T , (41)

i.e., the friction coefficient as a function of the slidingdirection is

μ(γ T ) = 1√

γ TT T

2γ T

‖γ T ‖λN

=√

γ 2T1

+ γ 2T2

√(γT1μ1

)2 +(

γT2μ2

)2.

(42)

The force law can be transformed into a set-valued nor-mal cone inclusion force lawbymakinguse of the trans-formation property Eq. (9). Considering the fact thatunder the linear mapping T the collinear vectors −λT

and γ T remain collinear, the force law can be illus-trated by the graph shown in Fig. 8a. The transforma-tion matrix T is chosen such that the elliptical set C istransformed to the circular set TC with radius one andthe force law is therefore represented by the inclusion

Tγ T ∈ NTC (−TλT ). (43)

This inclusion can be transformed to a normal coneinclusion on the ellipsoidal force reservoir C usingEq. (9),

T2γ T ∈ NC (−λT ). (44)

Figure 8b shows a graphical representation of thiscollinear force law with ellipsoidal force reservoir.Since the friction force is directly opposing the slidingvelocity, a sliding body experiencing such a force law

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Fig. 8 a Collinear frictionlaw transformed to circle. bCollinear friction law withelliptical force reservoir

(a) (b)

Fig. 9 a Relative slidingvelocity of a spring incontact with an inclinedplane. b Contact forcesacting on a spring and theinclined plane depicted inlocal and global referenceframe. Adapted from [25]

(a) (b)

is not deflected. Writing the force law as normal coneinclusion allows for an efficient numerical treatment ofthe set-valued nature of Coulomb friction.

Asperity Model

Microscopic surface asperities play an important rolein the frictional behavior of contacting bodies. Ran-domly distributed asperities result in isotropic friction.Anisotropic friction can be modeled by the superposi-tion of a random distribution and a systematical orderof asperities. In [24], such a model is introduced todescribe anisotropic friction on machined metal sur-faces. The macroscopic frictional behavior is deter-mined by investigating the behavior of a mass slidingon an isotropic inclined plane. Sliding along the planeon a fixed level curve leads to a macroscopic frictionforce equal to the force induced by the isotropic frictionof themass on the plane. However, pushing themass upthe inclined plane requires a higher force which leadsto anisotropic macroscopic behavior. The combinationof inclined planes to parallel wedge-shaped asperitiesresults in a model capable of specifying the frictionforce for all sliding directions. In general, the modelresults in non-convex force reservoirs. The relationshipbetween the friction force and the sliding direction isnot defined by normality of the sliding direction to theforce reservoir C , but by normality to a different con-vex direction set D . The model therefore leads to anon-associated friction law.

In a more advanced model [25], one of the con-tacting surfaces is still modeled with parallel wedge-shaped asperities. The second surface, however, nowconsists of uniformly distributed springs with uniformstiffness attached to a moving plane. Contact compli-ance in normal and tangential direction is neglected.The springs are used to model the friction force whenthe two surfaces move parallel to each other with aconstant distance. Figure 10a shows an example of afixed wedge-shaped asperity with the same inclinationangle β on both sides in contact with a moving platewith spring asperities. No separation of the springs andthe wedge asperities is assumed and the springs arealways compressed. Following the calculations of [25],the set of admissible macroscopic friction forces act-ing in the contact plane can be obtained. The macro-scopic relative sliding velocity is γ T . We denote therelative sliding velocity of a single spring in contactwith an inclined plane by γ T as shown in Fig. 9a. Theprojection of γ T on the macroscopic contact plane isγ T , being under the angle ϕ to the eIT1 -axis. The con-tact forces acting on a single spring and an inclinedplane are depicted in Fig. 9b. Along the inclined eKT1 -eKT2 -plane, isotropic friction with friction coefficient μ

is present. The magnitude λT of the friction force act-ing on the spring is therefore given by λT = μλN ,where λN represents the normal force, orthogonal tothe inclined plane. The contact forces expressed in theK -frame acting on the spring are in equilibrium with

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Fig. 10 a Asperity modelwith wedge-shapedasperities in contact withsprings. b Resultingnon-convex force reservoir

(a) (b)

the contact forces acting on the inclined plane λT1 , λT2and λN expressed in the I -frame of the macroscopiccontact plane. It holds that

λN = λN cosβ − λT1 sin β, (45)

λT1 = λN sin β + λT1 cosβ, (46)

λT2 = λT2 . (47)

The maximum friction force in eKT1 -direction is λT1 =μλN . Substitution of the friction force inEq. (45) yields

λN = λN (cosβ − μ sin β) . (48)

With the conditions λN > 0 and λN > 0, the admis-sible range of the friction coefficient is obtained asμ < cot(β). From Eqs. (45)–(47), functions describ-ing the macroscopic friction forces of a single springcan be calculated in the form λT1 = λN HT1(μ, β, ϕ)

and λT2 = λN HT2(μ, β, ϕ), where HT1 and HT2 arefunctions of μ, β and ϕ, see [25]. The vertical forceλN depends on the displacement of the correspondingspring. If a set of springs in contact with both sides ofparallel wedge-shaped asperities with the same incli-nation angle β on both sides is considered, then themacroscopic friction forces are found as

λT1 = λN

2

(HT1(μ, β, ϕ) + HT1(μ,−β, ϕ)

),

λT2 = λN

2

(HT2(μ, β, ϕ) + HT2(μ,−β, ϕ)

).

(49)

Here, the macroscopic normal force λN is a function ofthe spring stiffness and the constant mean spring dis-placement. Equation (49) describes the friction forcesfor all possible sliding directions and allow for the cal-

culation of the set of admissible macroscopic frictionforces acting in the contact plane. The force reservoircan be expressed in polar coordinates by

rC (θ) = μλN

cosβ

√cos2 β cos2 θ + sin2 θ

sin2 θ+[1−(

1 + μ2)sin2 β

]cos2 θ

,

(50)

where θ describes the angle between the eIT1 -axis andthe direction of the friction force (see [25]). The inclina-tion angle lies in the range 0◦ ≤ β < 90◦. For β = 0◦,the isotropic friction lawwith circular force reservoir isrecovered. In Fig. 10b, the force reservoir C is shownfor the parameters μ = 1.2 and β = 36◦. For the givenparameters, the set is clearly non-convex. However, thesetC is always star-shaped since rC (θ) > 0 for all θ , ifadmissible values ofμ andβ are considered in Eq. (50).

The sliding direction is not associated with the fric-tion force reservoir. Instead, it is defined by normalityto an ellipsoidal direction set D at the intersection ofthe boundary of the set and the direction of the negativefriction force −λT . The ellipsoidal set is given by

D ={

−λT ∈ R2∣∣∣∣

(−λT1

p1

)2

+(−λT2

p2

)2

≤ 1

}

,

(51)

with the semi-axes ratio

p2p1

= cosβ. (52)

For the parameters given above, the set is also shown inFig. 10b. The scaling of the direction setD is irrelevantand can be chosen arbitrarily.

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Fig. 11 Graphicalrepresentation ofnon-associated friction lawa with ellipsoidal sets bwithnon-convex force reservoir

(a) (b)

In the literature, the non-convexity of the force reser-voir is sometimes neglected and the two sets C and Dare assumed to both be of ellipsoidal or superellipsoidalshape with the same roundness factor [25,53,54]. Agraphical representation of the friction law for ellip-soidal sets is shown in Fig. 11a. Similar to Sect. 3.3,the force law in that case can be formulated as normalcone inclusion. Here, the transformation matrix T isgiven by

T =[

p1μ1λN

00 p2

μ2λN

]

(53)

and the force law is written as

T sγ T ∈ NC (−λT ), (54)

where s represents the roundness factor of the superel-lipsoidal sets. The collinear friction lawwith ellipsoidalforce reservoir (s = 2) is contained in that formulationand is obtained for a circular set D for which Eq. (53)and (54) equal Eqs. (39) and (44), respectively. If the setD is taken equal to the force reservoirC , then the asso-ciated Coulomb friction law (Definition 12) is recov-ered.

To analyze the existence of a convex pseudo-potential, the force law has to be checked for cyclicmaximal monotonicity [8]. From Proposition 3 it fol-lows that the force law is not maximal monotone andconsequently not cyclically maximal monotone if Cand D differ in shape. Thus, in general, no convexpseudo-potential exists. Instead, following [26,54,55],a bipotential can be found. A force law capable of han-dling non-convex force reservoirs is derived in the fol-lowing section.

3.4 Extended normal cone inclusion friction law

The asperity model motivated by [25] and described inSect. 3.3 aswell as the tensor formulation of anisotropicfriction by [56] both result in non-associated frictionforce laws with possibly non-convex force reservoirs.The force law Eq. (54), however, is only valid for con-vex force reservoirs. In this section a set-valued forcelaw allowing for different shapes of the force reservoirC and the direction set D as well as for non-convexforce reservoirs is developed in the framework of con-vex analysis.

Definition 16 (ExtendedNormal Cone Inclusion Fric-tion Law) Let C ⊂ R

2 be a compact and star-shapedset and let D ⊂ R

2 be a compact and convex set withD ⊆ C and 0 ∈ intD . The gauge functions of the setsare given by kC and kD . ACoulomb-type friction law iscalled an extended normal cone inclusion friction lawif

γ T ∈ ND (−αλT ) (55)

with

α = 1

kD (−λT ) − kC (−λT ) + 1. (56)

A graphical representation of such a friction law isshown in Fig. 11b. The nonempty sets C and D canbe represented as the level sets of their gauge functionskC and kD by

C = {−λT | kC (−λT ) ≤ 1} , (57)

D = {−λT | kD (−λT ) ≤ 1} . (58)

The origin is required to be contained in the interior ofD to guarantee that kD (−λT ) is defined for all frictionforces. The sliding direction is given by normality to

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the direction set D . Instead of formulating a force lawas normal cone inclusion on the force reservoir as inEq. (54), a normal cone inclusion on set D is defined.Here, the normality condition is directly fulfilled andnon-convex force reservoirs are no longer excluded.The challenge is the accurate description of the stickphase since the force reservoir is independent ofD andthe condition γ T = 0 for friction forces inside the forcereservoir is not automatically fulfilled.a) −λT ∈ intC : For all friction forces in the interiorof the force reservoir, the sliding velocity must be zero.Since gauge functions return values less than one forpoints in the interior of the corresponding set, it musthold that γ T = 0 for kC (−λT ) < 1.b) −λT ∈ bdryC : For friction forces at the boundaryof the force reservoir, the sliding velocity is defined bythe normal cone to the setD at the point of the frictionforce scaled to the boundary of D by the parameter α,i.e.,

γ T ∈ ND (−αλT ) with − αλT ∈ bdryD . (59)

With the gauge function being one for points at theboundary of the set, Eq. (59) can be rewritten as:

γ T ∈ ND (−αλT ) for kC (−λT ) = 1 (60)

with kD (−αλT ) = 1. The scaling function α =f (−λT ) thereby scales the friction force to the inte-rior of D if the friction force lies in the interior of theforce reservoir and to the boundary of D if the force isat the boundary of C . It is therefore designed as

α :{

< 1kD (−λT )

for kC (−λT ) < 1

= 1kD (−λT )

for kC (−λT ) = 1.(61)

These conditions are combined in the function

α = 1

kD (−λT ) − kC (−λT ) + 1, (62)

where the argument of the scaling function α(−λT ) isomitted for brevity. To ensure that the scaling functionα is always positive, the condition

kD (−λT ) ≥ kC (−λT ) (63)

is imposed. Therefore,D ⊆ C must apply. This can bereached for all star-shaped C with 0 ∈ intC since it

is always possible to scale the compact set D withoutloss of generality.

One of the advantages of the described force law asnormal cone inclusion on the set D is that the forcereservoir is no longer restricted to convex shapes. Thedefinitions of gauge functions and corresponding levelsets Eq. (58) are also valid for sets that are star-shapedwith respect to the origin [43]. Star-shaped sets arepossibly non-convex. Limiting the force reservoir Cto star-shaped and the set D to convex shapes ensuresthat for a given γ T and known dynamics of a system,a unique friction force λT is assigned. For star-shapedforce reservoirs, the magnitude of a friction force act-ing in a certain direction during slip is well defined.During stick the friction force in that direction can takeany value between the value of the sliding friction forceand zero.

If the direction set D is even strictly convex, thena given nonzero relative sliding velocity γ T is in thenormal cone of the compact set D at a unique bound-ary point z = −α(−λT )λT (see Fig. 11). From thescaled friction force z, the friction force −λT can becalculated by expressing the scaling parameter α as afunction of z, i.e.,

−λT = 1

αz, (64)

= (kD (−λT ) − kC (−λT ) + 1)z. (65)

= 1

α(kD (z) − kC (z) + α)z. (66)

Herein, α from Eq. (62) is substituted, then the argu-ments of the gauge function are replaced with Eq. (64)and the fact that the gauge functions are positivelyhomogeneous of degree one is used. If again Eq. (64)is substituted in Eq. (66), the expression

α(z) = kC (z) − kD (z) + 1 (67)

is obtained, which together with Eq. (64) allows for thecalculation of the friction force for known values of z.For z on the boundary of D , the gauge function kD (z)is equal to one.

In analogy to Proposition 1, in which the dissipativ-ity of the associated friction law is proven, we discussthe dissipativity of the extended normal cone inclusionfriction law.

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Proposition 5 (Dissipativity of Extended NormalCone Inclusion Friction Law) The extended normalcone inclusion friction law is strictly dissipative.

Proof Rewriting the normal cone inclusion with thesubdifferential of the indicator function and again usingthe Fenchel equality results in

γ T ∈ ∂ΨD (−αλT ) ⇐⇒ −αλT ∈ ∂Ψ ∗D (γ T )

⇐⇒ −αλTT γ T = ΨD (−αλT ) + Ψ ∗

D (γ T ).(68)

The indicator function ΨD (−αλT ) is zero because thescaled friction force is always in the setD . It holds thatΨ ∗D (γ T ) > 0 for all γ T �= 0 since 0 ∈ intD . There-

fore, using α > 0, the dissipation rate for the extendedfriction force law is given as

D(−λT , γ T ) = −λTT γ T = 1

αΨ ∗D (γ T ) > 0. ��

Implementation in a Time-Stepping Method

As described in Sect. 2.3, set-valued force laws given inthe form of normal cone inclusions can be used for thenumerical simulation of nonsmooth systems by rewrit-ing them as implicit proximal point functions. In orderto implement the extended normal cone inclusion fric-tion law in the iteration scheme, Eq. (22) has to bereplaced by

Pk+1T = − 1

αkproxD (−αk Pk

T + rγ kTE), r > 0,

(69)

where the scaling parameter is determined by

αk = 1

kD (−PkT ) − kC (−Pk

T ) + 1(70)

in each iteration [57].

Capability and Limitations of the Friction Law

The associated force law, based on normality of thesliding direction to the friction force reservoir C , islimited to convex shapes of the force reservoir. Theextended normal cone inclusion force lawdoes not havethis limitation. It allows for the implementation of non-convex but star-shaped force reservoirs, which is moti-vated, e.g., by the tensor formulation described by [28]and the asperity model (see Sect. 3.3). Non-convex and

not star-shaped force reservoirs are not permitted sincegauge functions are not defined for such sets. However,force reservoirs that are not star-shaped are not allowedfor dry friction laws since for such sets a decrease inthe static friction force may lead to a stick-slip tran-sition. While the force reservoir may be non-convex,the direction set D in this work is limited to convexsets.

The formulation of the force law as normal coneinclusion makes it possible to represent the set-valuednature of friction with the transition of stick and slip.Thereby, the advantage of the associated Coulomb fric-tion law is retained. The extended friction law reducesto the associated Coulomb friction law if both sets areconvex and C = aD for some a ∈ R

+ which can betaken equal to one, i.e., C = D . Collinear force lawsare given for a circular shape of the direction set D .Unlike in Sect. 3.3, where only ellipsoidal force reser-voirs are considered for the collinear force law, here,the shape of the force reservoir is arbitrary as long as itis star-shaped. If both sets have a circular shape, thenthe isotropic Coulomb friction force law is recovered.

In the following, the force laws given in tensorformulation by [28] are considered as examples. Fig-ure 12 shows the graphical representation of a numberof non-associated force laws. In this work, the direc-tion sets D are added. The sets are determined numer-ically under the condition of normality of the slidingdirection to the boundary of the set D . The methodis similar to the processing of the experimental datawhich is described in more detail in Sect. 5.2. Theboundary of D is described in polar coordinates by

rD (θ) = rD (0)e∫ θ0 tan(θ−ϕ)dθ , (71)

where θ is the angle between the friction force and afixed axis and ϕ is the angle between the sliding direc-tion and the same axis (see Fig. 12c). The initial valuerD (0) is chosen such that D ⊆ C .

Figure 12a shows an orthotropic force lawwith ellip-soidal force reservoir. The setD is also an ellipse withthe same orientation of the major axes but with a differ-ent semi-axes ratio which causes the non-associativityof the force law. The force laws shown in Fig. 12b, calso allow for a direct calculation of a closed convexset D . In the case of the force law with nonconstantfriction tensor shown in Fig. 12b, the set D is a circlewhich corresponds to a collinear force law.

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(a) (b) (c)

Fig. 12 Graphical representation of non-associated anisotropic friction force laws with parameters from [56] and calculated directionsets D . a Orthotropic force law. b Force law with nonconstant friction tensor. c Force law with fourth order tensor

4 Friction-induced instability

This section is concerned with stability properties ofmechanical systems with friction. Despite the fact thatmany researchers have extensively studied friction-induced vibrations, the underlying mechanisms arestill not fully understood. Especially, the influenceof anisotropic friction on the stability of a slidingmotion has not been studied yet. In this work, sys-tems with a single frictional contact point are consid-ered. First, different isotropic friction-induced instabil-ity phenomena are classified according to their under-lying mechanisms. In Sect. 4.2, anisotropic friction isshown to be a possible cause of self-excited vibra-tions that previously has been ignored by the researchcommunity.

4.1 Classification of isotropic friction-inducedinstability phenomena

A n-DOF mechanical system with generalized posi-tions q ∈ R

n and velocities u = q ∈ Rn and a single

frictional contact point is described by the equation ofmotion

Mu − h(q, u) = wNλN + WTλT , (72)

a friction law −λT ∈ F (γ T , λN ) and a unilateralconstraint gN (q) ≥ 0. The generalized force direc-tions in normal and tangential direction are denoted bywN ∈ R

n and WT ∈ Rn×p. For the relative sliding

velocity γ T ∈ Rp it holds that

γ T = WTT u − χ , (73)

where χ ∈ Rp represents a tangential velocity at the

contact point, e.g., originating from the movement ofa belt. We assume that there is no explicit time depen-dence of the normal contact distance and the relativesliding velocity (χ = const.).

In Fig. 13 isotropic friction-induced instability phe-nomena are summarized. The three underlying mecha-nisms of instability are (a) a nonconstant friction coeffi-cient causing stick-slip vibrations, (b+c) a nonconstantnormal force leading to geometrically induced insta-bility and (d) a follower force behavior of the frictionforce. In all cases, the friction force during forwardslip is given by an isotropic friction law. Two of theparameters: friction coefficient, normal force and gen-eralized force direction are held constant in each case,while the third is given as a function of the generalizedcoordinates or velocities.

The archetype model of friction-induced vibrationsis the one degree of freedom oscillator in contact with amoving surface often referred to as block-on-beltmodel(see Fig. 13a). The stick-slip behavior of the modelis extensively studied in the literature, e.g., by [59–61]. The key aspect is that the friction coefficient isassumed to be a function of the magnitude of the slid-ing velocity. A decreasing friction coefficient for anincreasing sliding velocity, known as Stribeck effect,causes a negative damping coefficient in the linearizedequation of motion, which leads to Hopf-type instabil-ity of the equilibrium of the system.

For Coulomb friction, the magnitude of the frictionforce is not only dependent on the friction coefficient

123


Fig. 13 Isotropicfriction-induced instabilityphenomena. a Stick-sliposcillator and Stribeckfriction characteristic. bMechanical model toillustrate mode-coupling. cFrictional impact oscillatorand simplified model(adapted from [58]). dMechanical model of asystem with frictionalfollower force

(a)

(b)

(c)

(d)

μ, but also on the normal force λN . Depending on thegeometry of a mechanical system, the normal forcemay be a function of the generalized coordinates orvelocities of the system. Such a dependence of the nor-mal force is responsible for two instability phenomenathat occur for isotropic friction without Stribeck effect.Mode-coupling instability is characterized by the con-vergence of oscillation frequencies of structural modesunder the influence of a parameter. When they merge,a pair of an unstable and stable mode is created [35].

In Fig. 13b a planar three degree of freedom modelis proposed. It consists of a mass being constrainedby springs that are always horizontal, vertical or diag-onal, respectively, and a belt attached to the groundvia vertical springs. The belt is constrained to moveonly in the vertical direction. Small vibrations are con-sidered, i.e., geometric nonlinearity due to the springsis neglected. The model differs from the models pre-sented by [62] and [63] in that the mass is in directcontact with the belt, instead of the contact between amassless slider and the belt. If the system with closed

123


contact is considered, in addition to the symmetricstructural coupling terms, displacement dependent fric-tional coupling terms occur in the equation of motionthat can cause instability of the equilibrium. Besidesmode-coupling, sprag-slip is a friction-induced insta-bility phenomenon occurring due to a nonconstant nor-mal force.This effect hasfirst beendescribed in the con-text of brake squeal in [34] using a system of inclinedrigid rods pressed on a moving surface. In [64], anelastic beam in contact with a belt is used to modelsprag-slip, while [58] made use of the multibody sys-tem shown in Fig. 13c to model a similar effect, whichis closely related to the Painlevé paradox [65]. The fric-tional impact oscillator, consisting of a rigid rod, twopointmasses and linear as well as rotational springs anddampers, can be approximated for small angles withlinear springs and dampers as shown in the same Fig.[58]. Unlike in the example of mode-coupling, wherethe dependence of the normal force on the displacementis responsible for the instability of an equilibrium, inthis case the feedback of the velocity dependent partof the normal force causes the equilibrium to becomeunstable.

If the friction coefficient as well as the normal forceare assumed to be constant, the follower force char-acteristic of a friction force can be a third cause offriction-induced instability. Such a force is acting on abody and changing the direction according to the dis-placement of the body. Frictional follower forces occur,e.g., between a mass and a disk due to the deforma-tion of the disk [66]. In [67], experimental evidenceof instability caused by a frictional follower force isprovided. The follower force characteristic of the fric-tion force causes the generalized force direction to be afunction of the generalized coordinates q. A frictionalfollower force can be realized as shown in Fig. 13d.The model consists of a double pendulum with mass-less rods connected via rotational springs. At the tip ofthe pendulum, a wheel with negligiblemass is mountedto the pendulum such that it can spin freely around itsaxis. The wheel is in contact with a belt moving withconstant velocity. The isotropic Coulomb friction forceλr acting at the contact point of the wheel and the beltis assumed to be transmitted to the pendulum only inaxial direction of the wheel, which causes the general-ized force direction wr to be nonconstant. In the fol-lowing, systems without the Stribeck effect and with aconstant normal force as well as a constant generalizedforce direction are considered.

4.2 Stability of systems with anisotropic friction

Consider the autonomous differential inclusion

x(t) ∈ F (x(t)) , (74)

with the state vector x(t) ∈ Rr . The set of admissible

states is calledA . A solution x(t) = ϕ(t, t0, x0) of thedifferential inclusion with initial condition x0 ∈ A isan absolutely continuous function x : R → R

r whichfulfills Eq. (74) for almost all t ≥ 0. Statements onthe stability of all solutions can be made. If all solutioncurves in forward time remain close to their neighbor-ing solutions, it is referred to as incremental stability(see [42]).

Definition 17 (Incremental Stability of a DifferentialInclusion) The differential inclusion Eq. (74) is calledincrementally stable if for all t0 ∈ R, arbitrary admis-sible initial conditions x1(t0), x2(t0) ∈ A and for allcorresponding solution curves x1(t) = ϕ(t, t0, x1(t0))as well as x2(t) = ϕ(t, t0, x2(t0)), it holds that foreach ε > 0 there exists a δ = δ(ε) such that‖x1(t0) − x2(t0)‖ < δ implies ‖x1(t) − x2(t)‖ < ε

for almost all t ≥ 0.

In this section, the effect of anisotropic friction lawson the stability of sliding motion is analyzed. To date,no literature exists that specifically studies the stabil-ity properties of systems with anisotropic friction. Toeliminate all factors being capable of causing friction-induced instability for isotropic friction, a system hav-ing one frictional contact with

– Constant normal force,– Constant generalized force direction,– No Stribeck effect,

is considered, which excludes all mechanisms ofself-excitation mentioned before. In the following, aminimal mechanical model to analyze the effect ofanisotropic friction is presented.

Figure 14a shows a mass constrained by two linearsprings, sliding on a horizontal belt having anisotropicfriction properties. The top view of the model isdepicted in Fig. 14b. The sliding body is modeled as apoint mass on a surface moving constantly with veloc-ity v. The generalized coordinates q = [x y]T are ori-ented parallel to the axes eT1 and eT2 . The equation ofmotion of the system is given as

Mq + Kq = WTλT , (75)

123


Fig. 14 a Mass on belt withanisotropic frictionproperties. b Two degree offreedom model of the masson belt system

(a) (b)

with the diagonal system matrices and belt velocityvector χ

M =[m 00 m

], K =

[k 00 k

], WT =

[1 00 1

],

χ =[v cosϕ

v sin ϕ

]. (76)

The relative sliding velocity γ T is given by the motionof the mass and the motion of the belt as γ T =WT

T q − χ . The matrix of the generalized force direc-tion WT , defined by WT = (∂γ T /∂ q)T, in this case issimply the identity matrix. In the following, differentanisotropic friction laws are implemented for the fric-tion force λT . In all cases, the normal force resultingfrom gravity is constant, λN = mg. For vanishing gen-eralized velocities and accelerations, i.e., q = q = 0,the system is at the equilibrium qeq = K−1WTλT .Since the belt is moving continuously (χ �= 0), the rel-ative sliding velocity γ T is nonzero for q = 0 so theequilibrium is not in the stick phase. We will now con-sider different types of anisotropic friction and discussthe stability properties.

First, the associated Coulomb friction law specifiedin Definition 12, γ T ∈ NC (−λT ), is considered. Here,no additional restrictions on the shape of the force reser-voir C other than convexity are made.

Theorem 1 (Stability for the associated Coulomb fric-tion law) The differential inclusion given by themechanical system Eq. (75) in combination with theassociated Coulomb friction law is incrementally sta-ble. Consequently, equilibria of the system are stable.

Proof From Proposition 4, it is known that the asso-ciated Coulomb friction law is maximal monotone.

Therefore, for all pairs (γ T , λT ) and (γ ∗T , λ∗

T ), themonotonicity condition

−(λT − λ∗T )T(γ T − γ ∗

T ) ≥ 0 (77)

from Definition 13(i) holds. We choose two arbitrarysolutions of the differential inclusion, qI(t) and qII(t).Incremental stability (see Definition 17) can be provenby analyzing the distance between the two solutions.Introducing the incremental Lyapunov candidate func-tion with terms similar to the kinetic and potentialenergy of the system as a function of the position andvelocity error between the two solutions gives

V = 1

2

(qI − qII

)T M(qI − qII

)

+1

2

(qI − qII

)T K(qI − qII

). (78)

Only if the two solutions are identical, it holds that V =0. Since M and K are positive definite, the Lyapunovcandidate function V is positive definite. Taking thetime-derivative and substitution of Eq. (75) leads to

V = (qI − qII

)T (M

(qI − qII

) + K(qI − qII

) )

= (qI − qII

)T (WTλTI − WTλTII

)

= (WT

T

(qI − qII

))T (λTI − λTII

)

= (γ TI − γ TII

)T (λTI − λTII

),

(79)

where in the last step by subtracting the two solutionsthe velocity of the belt χ cancels out. From the mono-tonicity condition of the force law Eq. (77) it followsthat V ≤ 0. Therefore, V is a Lyapunov function whichcannot increase over time. This means that the distance

123


between two solutions is never increasing, i.e., the sys-tem is incrementally stable. Since for one of the twosolutions the equilibrium qeq can be taken, stability ofthe equilibrium is proven. Note that attractivity of theequilibrium does not directly follow and depends onadditional damping in the system. ��

Anisotropic friction modeled with the associatedCoulomb friction law is therefore never responsiblefor friction-induced instability of mechanical systemsgiven in the form of Eq. (75). Of course, instability canstill arise if in addition one of the effects described inSect. 4.1 is taken into account. The incremental stabil-ity result in the associated case directly follows fromthe maximal monotonicity property and an equivalentconclusion can also be found in [68] inwhichLur’e sys-tems with a maximal monotone operator in the feed-back loop are considered. In the following, we willconsider friction laws which do not enjoy the maximalmonotonicity property and show that this can lead toinstability.

Weanalyze the stability of the equilibriumof the sys-tem given in Eq. (75) in combination with the extendednormal cone inclusion friction law. Anisotropic fric-tion is shown to be a possible cause of friction-inducedinstability. In the following discussion, the occurrenceof instability is demonstrated for sets having a smoothboundary. In addition, a condition of the relationshipbetween the sliding direction and the shape of the forcereservoir is given that is responsible for anisotropicfriction-induced instability.

FromDefinition 16, the extended normal cone inclu-sion friction law is known as

γ T ∈ ND (−αλT ) (80)

with

α = 1

kD (−λT ) − kC (−λT ) + 1. (81)

The formulation is equivalent to −αλT ∈ ∂Ψ ∗D (γ T )

(see Eq. (68)). We assume a set D being strictly con-vex and having a smooth boundary, i.e., the set has novertices. The force reservoirC is assumed to be strictlystar-shaped with a smooth boundary. The subdifferen-tial of the support function Ψ ∗

D (γ T ) for γ T �= 0 canthen be replaced by the gradient with

∇Ψ ∗D (γ T ) =

(∂Ψ ∗

D (γ T )

∂γ T

)T

, (82)

and the friction law is single-valued in the sliding state.For a constant velocity χ at the contact, an equilibriummust occur during slip. Therefore, the friction forcetakes a value at the boundary of the force reservoir C ,and for the gauge function it holds that kC (−λT ) = 1.Substitution of the friction force

−λT = 1

α∇Ψ ∗

D (γ T ) (83)

in the positively homogeneous gauge function yieldsthe scaling parameter α during slip as a function of thesliding velocity as α = kC (∇Ψ ∗

D (γ T )). Consequently,during slip, the friction force is described by the explicitfunction

−λT = 1

kC (∇Ψ ∗D (γ T ))

∇Ψ ∗D (γ T ). (84)

For an explicit function of the friction force, the equa-tion of motion of the system given by Eq. (75) with aconstant matrix of the generalized force direction WT

can be linearized around the equilibrium qeq giving

Mq + Bq + K(q − qeq

) = 0, (85)

where

B = −WT∂λT

∂γ T

∣∣∣∣q=0

WTT . (86)

For the given example of a pointmass on a belt, thematrix WT is the identity matrix. The matrix B is thusdefined by the Jacobian of the friction force,

− ∂λT

∂γ T= 1

kC (∇Ψ ∗D )

[I − L

]H, (87)

where the argument of the support function is sup-pressed for brevity. Herein, the matrix L and the Hes-sian matrix H are given as

L = 1

kC (∇Ψ ∗D )

∇Ψ ∗D

∂kC (x)

∂x

∣∣∣∣∇Ψ ∗D

, H = ∂2Ψ ∗D

∂γ 2T

.

(88)

123


To determine the stability properties of the equilibrium,we analyze the eigenvalues of the matrix B. Each partof the matrix B is considered separately.

First, the eigenvalues of the matrix L ∈ R2×2 are

determined. The matrix consists of an outer productof nonzero vectors multiplied by a scalar factor beinggreater than zero. Consequently, the matrix is of rankone. One eigenvalue is zero, l L1 = 0 and if L isdiagonalizable, then the second eigenvalue is givenby the trace of the matrix. In general, it holds thattr(x yT

) = xT y, and since the gauge function is posi-tively homogeneous of degree one, it holds that

∂kC (x)

∂x

∣∣∣∣yy = kC ( y). (89)

It follows that l L2 = kC (∇Ψ ∗D )−1kC (∇Ψ ∗

D ) = 1.

A corresponding eigenvector is given by v L2 = λT

which is verified using Eqs. (84), (88) and (89) giv-ing LλT = λT . An eigenvector to the zero eigenvalueis called v L

1 = w. It holds that

Lw = 0 (90)

�⇒ ∂kC (x)

∂x

∣∣∣∣∇Ψ ∗D

w = 0. (91)

Since the gradient of the gauge function ofC is orthog-onal to the boundary of the set, the vector w must betangent to the boundary.

Next, the matrix L = I − L is considered.

Proposition 6 Let v be an eigenvector of the squarematrix L with the corresponding eigenvalue l. It holdsthat v is also an eigenvector of the matrix L = I − Lwith the corresponding eigenvalue (1 − l).

Proof From Lv = lv it follows that

Lv = (I − L)v = v − lv = (1 − l)v. (92)

��With Proposition 6, the eigenvalues and eigenvectors ofL are found to be lL1 = 1, lL2 = 0, vL

1 = w, vL2 = λT .

The support function of the convex setD is convex,and the Hessian matrix of a convex function is positivesemidefinite. Furthermore, the support function Ψ ∗

D ispositively homogeneous of degree one, i.e., it holds that

Ψ ∗D (aγ T ) = aΨ ∗

D (γ T ) for all a > 0, and differentia-tion yields

∇Ψ ∗D (aγ T ) = ∇Ψ ∗

D (γ T ) ∀a > 0. (93)

A Taylor series expansion of the gradient of the supportfunction is given by

∇Ψ ∗D ((1+ε)γ T ) = ∇Ψ ∗

D (γ T )+∂2Ψ ∗D

∂γ 2T

∣∣∣∣γ T

εγ T+O(ε2).

(94)

In addition, from Eq. (93) it follows that

∇Ψ ∗D ((1 + ε)γ T ) = ∇Ψ ∗

D (γ T ) ∀ε > 0. (95)

From Eqs. (94) and (95) we can conclude that

∂2Ψ ∗D

∂γ 2T

∣∣∣∣γ T

εγ T + O(ε2) = 0 ∀ε > 0, (96)

which implies

∂2Ψ ∗D

∂γ 2T

γ T = 0. (97)

Hence, γ T is an eigenvector to the zero eigenvalue ofthe Hessian matrix. Since the Hessian matrix is sym-metric, all eigenvalues are real and the eigenvectors areorthogonal. Therefore, a second eigenvalue lH2 > 0 andan eigenvector γ ⊥

T orthogonal to γ T must exist.The scalar factor kC (∇Ψ ∗

D ) > 0 in Eq. (87) hasno influence on the sign of the eigenvalues and can beneglected in the stability analysis. In summary, so far,we have

LλT = 0, Lw = w, Hγ T = 0, Hγ ⊥T = lH2 γ ⊥

T

(98)

with lH2 > 0. Finally, the eigenvalues of thematrix LHare studied. From Hγ T = 0, it directly follows thatLHγ T = 0. Thus, the first eigenvalue of the matrixproduct is zero and the corresponding eigenvector isγ T . Since the extended normal cone inclusion frictionlaw is strictly dissipative, i.e., −λT

T γ T > 0 holds forall γ T �= 0, the condition

123


�k ∈ R : γ ⊥T = kλT (99)

is fulfilled. In addition, from the first two equations ofEq. (98), we obtain that

�k ∈ R : w = kλT , (100)

which describes that the friction force λT can not becollinear to the vector w which is nonzero and tangentto the force reservoir. This is related to the assumptionthat the force reservoir is strictly star-shaped.

Proposition 7 Let L ∈ R2×2 and H ∈ R

2×2 be squarematrices and let Eqs. (98)–(100) be fulfilled. It holdsthat the nullspaceof thematrix product LH is of dimen-sion one.

Proof The vector γ T is in the nullspace of LH . Con-sider another vector v = a1γ T + a2γ ⊥

T with a2 �= 0.With Eq. (98) it follows that

LHv = LH(a1γ T + a2γ⊥T )

= a2lH2 Lγ ⊥

T

= a2lH2 L(b1λT + b2w)

= a2b2lH2 w �= 0,

(101)

where the vector γ ⊥T can be expressed by the com-

ponents in λT and w direction due to Eq. (100), andb2 �= 0 because of Eq. (99). ��In the case of γ T and w being parallel, i.e., the relativesliding velocity γ T is tangent to the boundary of theforce reservoir, the following statement on the algebraicmultiplicity of the zero eigenvalue can be made.


2×2 be squarematrices and let Eqs. (98)–(100) be fulfilled. If ∃c �=0 : γ T = cw, then the zero eigenvalue of the matrixLH has the algebraic multiplicity two.

Proof From Proposition 7, we know that the geometricmultiplicity of the zero eigenvalue must be one. Thevector γ T is an eigenvector of LH with eigenvaluezero. From Eq. (101) and γ T = cw, with c �= 0, itfollows that for every vector v ∈ R

2

LHv = βw = β

cγ T (102)

and, hence, (LH)2 v = 0. If γ T = cw, every vector isa generalized eigenvector. ��

In the following, the tangent coneTC of the force reser-voir C , which is given by Definition 7, is used.


2×2 be squarematrices and let Eqs. (98)–(100) be fulfilled. The vectorw is an eigenvector of the matrix LH . For the corre-sponding eigenvalue it holds that if γ T ∈ TC (−λT ),then the eigenvalue lLH

2 is non-positive. The eigenvalueis negative if

γ T ∈ intTC (−λT ) (103)

and strictly positive if γ T /∈ TC (−λT ).

Proof We first consider the case ∃c �= 0 : γ T =c (−λT ), i.e., the relative sliding velocity is collinearto the friction force. Therefore, it holds thatγ T /∈ TC (−λT ). From the dissipativity of the frictionlaw it is known that c > 0. Let

w = a1γ T + a2γ⊥T = −a1cλT + a2γ

⊥T (104)

�⇒ γ ⊥T = 1

a2w + a1

a2cλT , (105)

where a2 �= 0 holds due to Eq. (100). Similar reasoningas shown in Eq. (101) results in LHw = lH2 w. Thus, inthe case of collinear sliding velocity and friction force,w is an eigenvector with strictly positive eigenvalue.

Next, we consider the case c2 �= 0 and

γ T = c1(−λT ) + c2w (106)

�⇒ w = 1

c2γ T + c1

c2λT . (107)

The friction force can be written as:

− λT = d1γ T + d2γ⊥T (108)

�⇒ γ ⊥T = − 1

d2λT − d1

d2γ T , (109)

where d2 �= 0 follows from Eq. (106) with c2 �= 0.The signs of c2 and d2 depend on the arbitrarily chosenpositive direction of the vectorsw and γ ⊥

T (see Fig. 15a)and have no influence on the following consideration.From the dissipation rate−λT

T γ T = d1‖γ T ‖2 > 0, weobtain d1 > 0. Consecutive substitution of Eqs. (106)–(109) together with Eq. (98) yields

123


Fig. 15 a Relationshipbetween the direction of thefriction force λT , slidingvelocity γ T and eigenvectorw of the damping matrix. bEllipsoidal sets with rotatedforce reservoir C

(a) (b)

LHw = LH(1

c2γ T + c1

c2λT

)

= LH(1

c2γ T − c1

c2

(d1γ T + d2γ

⊥T

))

= − d2c1c2lH2 Lγ ⊥

T

= − d2c1c2lH2 L

(− 1

d2λT − d1

d2γ T

)

= − c1c2lH2 L (−λT − d1 (−c1λT + c2w))

= c1d1lH2 w. (110)

It follows thatw is an eigenvector of LH . Since d1 > 0and lH2 > 0, the sign of the corresponding eigenvalueonly depends on the parameter c1. The eigenvector w

is tangent to the boundary of the force reservoir C (seeFig. 15a). Hence, for c1 < 0 in Eq. (106), the slidingdirection is in the interior of the tangent cone of theforce reservoir at −λT and the eigenvalue being theproduct c1d1lH2 is negative. In contrast, if c1 is strictlypositive, then the sliding direction is not in the tangentcone and the eigenvalue is strictly positive. If the slidingdirection is parallel to the boundary of the force reser-voir (c1 = 0), then the zero eigenvalue has algebraicmultiplicity two (see Proposition 8). ��The statement on the sign of the nonzero eigenvaluegiven in Proposition 9 also holds true for the sign ofthe nonzero eigenvalue of the Jacobian matrix givenin Eq. (87). In conclusion, we have that the matrix Bdefined by Eq. (86), which is in general nonsymmetric,is always singular, even if WT has full rank. This isthe case because, due to the assumption of Coulombfriction without Stribeck effect, the friction force onlydepends on the direction of sliding and not on its mag-nitude. The zero eigenvalue always corresponds to theeigenvector γ T . Furthermore, the nonzero eigenvalue

of B is shown to be negative for sliding directions in theinterior of the tangent cone of the force reservoir. Thisis impossible in the case of an associated or collinearfriction law. However, with the extended normal coneinclusion friction law, it is possible, even for convexsets C and D as shown later in this work.

We can separate the generally nonsymmetric matrixB into the symmetric damping matrix D and skew-symmetric gyroscopic matrix G by

D = 1

2

(B + BT

), G = 1

2

(B − BT

). (111)

In the case of B having a nonzero eigenvalue, i.e., γ Tis not tangent to the boundary of the force reservoir,the matrix B is diagonalizable and the trace of B isgiven as the sum of the eigenvalues of B. For B havinga negative eigenvalue, we have tr B = tr D < 0. Ingeneral, no statement on the stability can be made fromthe trace of the damping matrix. However, it is possibleto formulate the following sufficient condition for theinstability of an equilibrium (see [69]).

Proposition 10 (Sufficient Condition for Instability)Let the equation of motion of a mechanical system begiven by Eq. (85). The equilibrium of the system isunstable if the condition

tr(M−1B

)= tr

(M−1D

)< 0 (112)

is fulfilled.

Proof The system can be rewritten in first-order formwith the state vector x = [

qT qT]T and the system

matrix

A =[

0 I−M−1K −M−1B

]. (113)

123


as x = Ax. The equilibrium is unstable if the realpart of at least one eigenvalue of A is strictly positive[70]. If Eq. (112) holds, then trA = −tr

(M−1B

) =−tr

(M−1D

)> 0 . With the trace being the sum of all

eigenvalues, there must be at least one eigenvalue withstrictly positive real part. ��Wehave that the symmetricmassmatrix is positive def-inite and the eigenvalues of B are zero and a negativescalar. But since B is in general nonsymmetric and itssymmetric part D can have positive and negative eigen-values, it is problematic to exploit trace inequalities orthe similarity-invariance of the trace for proving gen-eral statements on the stability. However, for the gen-eral class of systems having a mass matrix in the formof a scalar multiple of the identity matrix, instabilityof the equilibrium can be proven if B has a negativeeigenvalue. This is the case in the mass on belt systemgiven in Eq. (75).

Proposition 11 Let a linearized mechanical system bedescribed by Eq. (85) and let M = mI with m > 0. Ifthe only nonzero eigenvalue of B is negative, then theequilibrium qeq is unstable.

Proof The negative eigenvalue of the matrix B causesthe trace of B to be negative.

For the admissible form of the mass matrix, we canconclude that the instability condition Eq. (112) is ful-filled. ��Finally, a theorem stating the instability of an equilib-rium for a system with anisotropic frictional behaviorcan be given if the extended normal cone inclusion fric-tion law is considered.

Theorem 2 (Anisotropic Friction-Induced Instability)Let a mechanical system with a single contact pointwith anisotropic Coulomb friction be described byEq. (75) with M = mI and m > 0. Let the frictionforce be defined by the extended normal cone inclusionfriction law (see Definition 16). If for the relative slid-ing velocity γ T �= 0 at an equilibrium of the system thefriction law yields

γ T ∈ intTC (−λT ), (114)

then the equilibrium is unstable.

Proof The theorem follows fromPropositions 9 and 11and the above discussion. ��

Theorem 2 proves that even for a constant normalforce, a constant matrix of generalized force directionand a constant force reservoir, an anisotropic frictionlaw can cause friction-induced instability.

Numerical Example

To illustrate the occurrence of anisotropic friction-induced instability with the extended normal coneinclusion friction law, the friction law with ellipsoidalsets C and D is considered as an example [71]. Thedynamics is given by themass on belt system describedby Eqs. (75) and (76). For ellipsoidal sets centeredaround the origin and having parallel principal axes,the eigenvalues of the matrix B, defined by Eq. (86),never have a negative real part. Instead, we consideran ellipsoidal force reservoir rotated around the originwith angle ζ as shown in Fig. 15b. The gauge and sup-port functions of the sets are given with their positiveaxis intercepts n1, n2, p1, p2 and rotation parameter ν

as

kC (−λT ) =√(

λT1

n1

)2

+(

λT2

n2

)2

+ νλT1λT2 , (115)

kD (−λT ) =√(

λT1

p1

)2

+(

λT2

p2

)2

, (116)

Ψ ∗D (γ T ) =

√(p1γT1

)2 + (p2γT2

)2. (117)

For an ellipsoidal force reservoir, the condition |ν| <

2/(n1n2) has to be fulfilled. An eigenvalue analysisof the resulting singular damping matrix B of thelinearized system yields that the nonzero eigenvaluebecomes negative for certain relative sliding directionsif

|ν| >p1 p2

p21 + p22

4

n1n2. (118)

For p1 = p2, the two conditions for ν coincide. Thus,for the collinear friction law for which the setD is cir-cular, anisotropic friction-induced instability is impos-sible. Whenever p1 �= p2, the conditions can be ful-filled. Let the semi-axes ratios of both ellipsoidal setsbe 3/1. The critical range of the rotation angle ζ of theforce reservoir, allowing for a negative eigenvalue isthen found to be approximately 17◦ < ζ < 73◦. In thefollowing, ζ = 30◦ is considered. The friction law doesnot guarantee instability for all sliding directions. Sta-

123


Fig. 16 Stability analysisfor a rotation angle of theforce reservoir ζ = 30◦. aBifurcation diagram withthe angle of the beltdirection as parameter. bGraphical representation ofthe friction law

10

20

30

0° 40° 80° 120° 160°

(a) (b)

Fig. 17 a Limit cycle of thedisplacement of the mass onthe moving belt forϕ = 50◦. b Friction forceand relative sliding velocityof the mass during one cycleand friction forcereservoir C

(b)(a)

bility of the equilibrium depends on the direction of thebelt movement defined by the angle ϕ (see Fig. 14b). Abifurcation diagram showing the stability of the equi-librium as a function of the bifurcation parameter ϕ

is plotted in Fig. 16a. Instability occurs in the criticalrange of approximately 44◦ < ϕ < 76◦. Figure 16bdemonstrates that in the critical range of ϕ, the rela-tive sliding velocity at the equilibrium γ T = χ is inthe tangent cone of the force reservoir, which agrees toTheorem 2. For the limit values of the critical range ofϕ, the relative sliding direction is tangent to the bound-ary of the force reservoir at the corresponding value ofthe friction force.

We consider the mass on belt system with unit massand stiffness matrix and an ellipsoidal force reservoirwith semi-axes 30 N and 10 N rotated around the ori-gin with ζ = 30◦. For angles in the critical range, theequilibrium is unstable. When a small perturbation isapplied, themass on the belt spirals away from the equi-librium position and starts to oscillate. For the exampleof ϕ = 50◦, a numerical simulation using Moreau’stime-stepping scheme as described in Sect. 2.3, withthe extension given in Sect. 3.4, shows that after some

time, a limit cycle is reached. The limit cycle in the x-ydisplacement plane is plotted in Fig. 17a. Since the beltvelocity is chosen low (v = 1m/s), the mass sticks tothe belt during each cycle, which results in the straightsegment of the limit cycle. For higher belt velocities,limit cycles without stiction are observed. In the plot,the position for every eighth time step is marked with across. In Fig 17b, the corresponding friction forces andrelative sliding velocities are plotted together with theforce reservoir C . During the stick phase, the frictionforce increases and linearly approaches the boundaryof the force reservoir, while the relative sliding velocityis zero. When the friction force reaches the boundary,the mass starts to slide and the friction force movesalong the boundary of C . At the end of the slip phase,the friction force jumps to a value in the interior of theforce reservoir, and the cycle repeats.

5 Experimental analysis of anisotropic friction

Accurate modeling of anisotropic friction relies ondetailed experimental data of the frictional behavior

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Fig. 18 Tracking of thesliding path of a disk on ahorizontal anisotropicsurface. a Schematic viewof the experiment. b Topview of the sliding path of apolymer disk on surfaceground steel

(a) (b)

of materials. This section starts with a discussion ofthe surface conditions of the surface ground steel spec-imens considered in the experiments. In Sect. 5.1, anexperiment used to study the qualitative behavior of adisk sliding on an anisotropic surface is presented. Sec-tion 5.2 deals with the measurement of friction forceswith a novel tribometer. Finally, the experimentallyobtained data is used to compare different friction lawsin Sect. 5.3.

Due to the broad field of application of machinedmetals in engineering, the experimental work in thispaper is focused on the frictional behavior of sur-face ground steel. Case hardening steel with the mate-rial number 1.7139 and the chemical composition16MnCrS5 is utilized. Since this tool steel has a highwear resistance, it is often used for camshafts, gearsand other parts that are subjected to friction and wear.The plain, flat surface is machined using a horizontal-spindle (peripheral) surface grinder. Peripheral grind-ing produces a straight, parallel pattern of grindinggrooves which is expected to cause anisotropic fric-tional behavior. Using confocal microscopy, the grind-ing grooves are found to be parallel with a mean spac-ing of approximately 8µm and the arithmetic averageof the roughness profile orthogonal to the grooves isRa = 0.35µm.

5.1 Experimental analysis of sliding paths

To experimentally confirm that the friction force in thecase of anisotropic friction is not necessarily collinearto the sliding direction, the sliding paths of disks slidingon a surface having anisotropic friction properties areanalyzed. A schematic view of the experimental setupis shown in Fig. 18a.

The anisotropic surface is a horizontal surfaceground steel disk, having a diameter of 30 cm. On top

of the steel plate, a disk with a diameter of 28 mmand a height of 13 mm is accelerated and slides underthe influence of friction until it comes to a stop on theplate. Not the whole bottom surface of the sliding diskis in contact with the steel plate. Instead, three circu-lar contact areas, each having a diameter of 7 mm, areequally distributed on a circle with diameter 21 mm.The material at the contact areas is polyvinyl chloride(PVC) and the weight of the disk is 18 g. The accel-eration unit, consisting of a linear solenoid and a cen-tering device, is used to accelerate the disk at a speci-fied angle ϕ with respect to the direction of the grind-ing grooves. The initial velocity v0 of the sliding diskdepends on the applied voltage at the solenoid. The slid-ing path is recorded with a Sony RX10 II camera with aframe rate of 500 fps. For further image processing, therecorded frames are filtered using a two-dimensionalGaussian filter. A red mark centered on the top of thesliding disk is tracked by evaluation of the RGB chan-nels of the images. The convex hull of the detectedred pixels of the mark is determined using Delaunaytriangulation. To find the center of the mark, a circleis fitted to the convex hull using the method of leastsquares. Figure 18b shows an example of the trackedpoints with a time interval of 2 ms between two pointsand the last frame of the record in the background.From the position of the tracked mark on the top ofthe sliding disk, the displacement of the disk can becalculated.

Figure 19a shows the resulting sliding paths for theconstant angle ϕ = 47◦ and varying initial velocitiesv0. The sliding disk is clearly deflected from a straightline. Higher initial velocities obviously lead to longersliding paths. In addition, the figure shows that the finalsliding direction in all cases is in the direction of thegrinding grooves. The same is seen in Fig. 19b for theinitial velocity v0 = 1.49m/s and different angles ϕ.When the initial sliding velocity is parallel or orthog-

123


Fig. 19 Measured slidingpaths of a PVC disk on ahorizontal surface groundsteel plate. a Results forϕ = 47◦ and different initialvelocities. b Results forv0 = 1.49m/s and differentangles ϕ

(b)(a)

Fig. 20 Experimentalpin-on-disk setup for themeasurement of frictionforces

k

onal to the grinding grooves, the disk is not deflected.Furthermore, it holds that the sliding path is longerwhen the disk is sliding parallel to the grinding groovesthan when it is sliding orthogonal to them.

The experiments show that themagnitude of the fric-tion force depends on the sliding direction. Moreover,the deflection of the sliding disk proves that the direc-tion of the friction force is also dependent on the slidingdirection. In general, anisotropic friction forces are notcollinear to the sliding direction.

5.2 Measurement of friction forces

The previous section is concerned with the qualitativebehavior of anisotropic dry friction. Next, an experi-mental setup tomeasure the friction forces is described.The experimental results can be used to determine theforce reservoir C and direction set D of the extendednormal cone inclusion friction law.

Experimental Setup

Various kinds of experimental setups to measure fric-tion forces are used in the field of tribology. Such exper-iments utilize linear or rotationalmotionwith a constantor oscillating sliding velocity. Point, line and flat con-tact areas are usually studied. Most conventional tri-bometers measure the friction force solely in the direc-tion of sliding, even if anisotropic frictional behavioris considered (e.g., [72]). The findings of Sect. 5.1,however, show that if anisotropic friction forces areexpected, it is not sufficient to measure the force injust one direction, as the friction force is not collinearwith the sliding direction. The frictional behavior canonly be analyzed accurately by measuring the forces intwo directions in the contact plane, for different slidingdirections.

In this work, an experimental setup consisting ofa turning anisotropic surface and a stationary pin (pin-on-disk tribometer) is utilized [71]. In contrast to many

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Fig. 21 Schematic diagramof the experimental setup

tribometers, friction forces aremeasured in two orthog-onal directions and the rotation angle of the disk isrecorded. During one full rotation of the disk, thefriction forces for all possible sliding directions areobserved. The pin-on-disk setup is shown in Fig. 20.The rotating disk has a diameter of 30 cm. It is drivenwith a synchronous servomotor with an integratedresolver and a harmonic drive gear such that the max-imum speed of the disk is 200 rpm. The normal forceis applied using weights that are placed on a linear ballspline shaft. The friction forces in tangential and radialdirection of the disk, as well as the normal force aremeasured using a 3D force sensor. A schematic dia-gram of the experimental setup is shown in Fig. 21.

The tested pins have hemispherical or cylindricalshapes (diameter dP ) with a maximum diameter of 7mm, resulting in a maximum apparent contact area of38mm2. The position of the pin in radial direction ofthe rotating disk, denoted by rP , is variable in the rangefrom 60 to 149 mm and is constant during a measure-ment. The ratio between the diameter and radial posi-tion of the pin is chosen such that dP/rP � 1. Thus, theinfluence of varying sliding velocities along the radiusof the disk,which leads to combined sliding anddrillingfriction, is minimized. For a given position, the relativesliding velocity is adjusted by the rotation speed of thedisk.

During a measurement, the time signals of the fric-tion forces in radial and tangential direction, the normalforce, and the rotation angle of the disk are recorded.

Signal Processing

The output signals F = [Fx Fy Fz

]T of the force sen-sor do not directly correspond to the components of thecontact force λ = [λr λt λN ]T in radial, tangential, andnormal direction. Crosstalk in multi-axis sensors pro-duces an output signal in channels that do not correlatewith the direction of the applied load. To minimize theeffect, a compensation matrix T is used to transform

the channel output into the reference frame of the sen-sor, i.e., λ = T F. The matrix is determined by apply-ing known loads using weights in each of the referencedirections and calculating the inverse of thematrix con-sisting of the measured values. For the sensor used inthiswork, thematrixT is close to the identitymatrix buthas nonzero off-diagonal terms. The measured signalsare filtered using a zero-phase low-passButterworth fil-ter. With the obtained data, it is possible to determinethe shapes of the force reservoir C and direction setD that define the extended normal cone inclusion fric-tion law (see Definition 16). The friction forces λr andλt are measured in the I -frame of the sensor in radialand tangential direction of the disk (see Fig. 22a). Withthe measured current rotation angle ϕ of the disk, theforces can be transformed into the co-rotating frame Kof the disk such that

[−λT1−λT2

]=

[cosϕ − sin ϕ

sin ϕ cosϕ

] [λtλr

], (119)

Plotting the transformed forces in a diagram gives dis-crete points on the boundary of the force reservoir C .The friction lawdefines the sliding direction by normal-ity to the setD . In the described experimental setup, therelative sliding velocity is known to point in tangentialdirection of the rotating disk. In the K -frame the vectorγ T can therefore be expressed as

γ T = ‖γ T ‖[cosϕ

sin ϕ

]. (120)

We address points on the boundary ofD with the vectorrD as shown in Fig. 22b. With the magnitude rD andtan θ = λT2/λT1 , the vector rD can be expressed in theK -frame as a function of θ ,

rD (θ) = rD (θ)

[cos θ

sin θ

]. (121)

123


Fig. 22 a Top view of therotating disk withcoordinate frames. bCalculated sets C and D

(a) (b)

Fig. 23 Experimentalresults for steel pin onsandblasted steel disk. aMeasured friction forces intangential and radialdirection. b Calculated setsC and D

(a) (b)

Its derivative with respect to θ ,

drD (θ)

dθ:= r ′

D (θ) =[r ′D (θ) cos θ − rD (θ) sin θ

r ′D (θ) sin θ + rD (θ) cos θ

],

(122)

is tangential to the boundary of the set D . Since γ T isnormal to D , the condition

r ′D (θ)⊥γ T ⇐⇒ r ′

D (θ) · γ T = 0 (123)

has to be met. Substitution of Eqs. (120) and (122) intothe scalar product for ‖γ T ‖ �= 0 yields the first-orderhomogeneous differential equation

r ′D (θ) − tan(θ − ϕ)rD (θ) = 0. (124)

for the magnitude of rD . From Fig. 22a, the relationtan(θ −ϕ) = λr/λt is obtained. A solution of Eq. (124)is found by separation of variables and integration giv-ing

rD (θ) = rD (0)e∫ θ0

λrλtdθ

. (125)

As described in Sect. 3.4, the setD must be scaled suchthat D ⊆ C . This is achieved by choosing an appro-priate initial value rD (0). With the measured values ofone full rotation of the disk, the sets C and D can beexperimentally determined.

Experimental Results

The experimental setup can be used tomeasure the fric-tion forces for various material combinations and sur-face conditions. All experiments are conducted underroom temperature (20 − 22◦) and in the unlubricatedstate. Before anisotropy is examined, a sample with africtional behavior that is expected to be isotropic isanalyzed.

Choosing a material and surface condition to repre-sent isotropic frictional behavior is challenging sincemost surfaces show a surface texture that has an influ-ence on the frictional properties. In this work, a sand-blasted steel disk is considered. A hemispherical steelpin is pressed against the disk with a normal force ofλN = 14.0N. The relative sliding velocity dependson the rotation speed of the disk and the pin position inradial direction of the disk. It is set to ‖γ T ‖ = 50mm/s.The measured friction forces in tangential and radialdirection, λt and λr , are shown in Fig. 23a. The tangen-

123


Fig. 24 Experimentalresults for PVC pin onsurface ground steel disk. aMeasured friction forces intangential and radialdirection. b Calculated setsC and D

(a) (b)

Fig. 25 Experimentalresults for steel pin onsurface ground steel disk. aMeasured friction forces intangential and radialdirection. b Calculated setsC and D

(a) (b)

tial force is constant, whereas the radial force is approx-imately zero. Small deviations are caused by irregular-ities of the surface. As expected for an isotropic mate-rial, the calculated sets C and D shown in Fig. 23bare circular. Therefore, isotropic frictional behavior canbe described with the associated Coulomb friction lawwith a circular force reservoir (see Eq. 33). Since themagnitude of the friction force is independent of thesliding direction, a single value for the friction coeffi-cient can be determined as μ ≈ 0.14.

An example of an anisotropic surface is given by asurface ground steel disk. Experiments with two differ-ent contact partners are conducted. Each measurementstarts when the sliding direction is orthogonal to thegrinding grooves. In Fig. 24a, the results for a cylin-drical PVC pin with a diameter of 7 mm are shown.The experiment is performed with a normal force ofλN = 9.3N and a sliding velocity of ‖γ T ‖ = 50mm/s.The friction forces exhibit a periodicity with two max-ima per full rotation of the disk. This indicates anorthotropic frictional behavior which is expected forthe surface ground disk due to symmetry. The tangen-tial force component oscillates around a constant off-set, while the radial force component shows a sawtooth

oscillation around zero. A similar behavior is found by[22] for a different material combination. Maximumand minimum values of the tangential force occur atzero crossings of the radial force. For ϕ = kπ , wherek ∈ N, the pin is sliding orthogonal to the grindinggrooves and the tangential force is maximal. When thepin is sliding along the grooves, the tangential force isminimal. This behavior can be explained by the factthat PVC is softer than steel and adapts to the grindinggrooves which hinders movement across the grooves.The resulting sets are shown in Fig. 24b. The forcecomponent orthogonal to the grooves is λT1 , whereasλT2 points in the direction of the grooves. The sets sig-nificantly differ from the circular shape in the isotropiccase. Both sets are similar to rounded rectangles. Astandard ellipse, as is often assumed for the force reser-voir in the case of anisotropic friction in the literature,does not accurately describe the experimentally deter-mined sets. The friction coefficients for sliding direc-tions orthogonal and parallel to the grinding groovesare μ1 ≈ 0.54 and μ2 ≈ 0.48, respectively.

In addition to the measurements with a cylindri-cal PVC pin, experiments using a hemispherical steelpin are conducted. The results obtained using a nor-

123


mal force of λN = 18.5N and a sliding velocity of‖γ T ‖ = 100mm/s are shown in Fig. 25a. Again, thetangential component oscillates around a nonzeromeanvalue, while the radial component shows a sawtoothbehavior. However, note that for the steel pin, the max-imum values of the tangential force occur when thepin is sliding along the grinding grooves. This behav-ior agrees with results found in the literature. Mea-surements performed by [73] show a higher frictioncoefficient along the grinding grooves than in orthog-onal direction for a steel ball on a surface ground steelplate under lubricated conditions. Similar results areobtained by [74] under dry conditions for a steel platewith narrow groove-shaped textures. A physical expla-nation might be a higher stiffness of the ridges betweenthe grooves in the longitudinal direction. The higherstiffness hinders deformation and motion of the pin inthe direction parallel to the grinding grooves.

While in the literature usually only sliding alongor across the grooves is regarded, the present studyconsiders all sliding directions. The calculated sets areshown in Fig. 25b. The shapes of C and D substan-tially differ from each other. This behavior can not berepresented by an associated friction law. The extendednormal cone inclusion friction law, however, gives anaccurate description of the frictional behavior.

5.3 Experimental validation of the friction law

The experiments described in the previous sectionsallow for an experimental validation of the extendednormal cone inclusion friction law. From the measuredfriction forces of the contact pair PVC—surface groundsteel, the force reservoir C and direction set D can be

Table 1 Semi-axes parameters and roundness factors of fittedsets

Set Semi-major Semi-minor Roundness

Ellipsoidal C 5.00 N 4.49 N 2

Superellipsoidal C 4.99 N 4.25 N 2.94

Superellipsoidal D 1.84 1.19 2.51

derived. The experimentally determined sets in com-bination with the friction law are used for a numericalsimulation of a disk sliding on an anisotropic surface.The result can be directly compared to the experimen-tally obtained sliding path for the same material com-bination and initial condition.

Often only the friction coefficients in the directionsorthogonal and parallel to the grinding grooves areknown and an ellipsoidal force reservoirC is assumed.Figure 26a shows such a set for the measured values ofthe PVC pin on the surface ground disk (see Fig. 24b)).A better fit of the measured values is obtained withthe method of least squares for a superellipsoidal setas shown in Fig. 26b. In addition, a fit for the set D isplotted in Fig. 26c. The parameters of the sets are givenin Table 1.

The experimentally determined sets are used fornumerical simulations with different friction laws.The associated friction law is considered with theellipsoidal and superellipsoidal force reservoir. Thecollinear friction law is formulated in combinationwiththe superellipsoidal set. Finally, the extended normalcone inclusion friction law with two superellipsoidalsets is considered. The graphical representation of thefriction laws is shown in Fig. 26. Numerical simula-tions of a sliding disk are performed with the time-

(a) (b) (c)

Fig. 26 Graphical representation of friction laws with fittedforce reservoirs C and direction set D for measured frictionforces of PVC on surface ground steel. a Associated friction law

with ellipsoidal C . b Associated and collinear friction law withsuperellipsoidal C . c Extended normal cone inclusion frictionlaw with superellipsoidal C and D

123


Fig. 27 Measured slidingpath of PVC disk on surfaceground steel andnumerically obtained slidingpaths using different frictionlaws with parameters fromfriction force measurement

stepping method explained in Sects. 2.3 and 3.4. Theinitial sliding velocity v0 = 1.52m/s and the orienta-tion of the semi-axes with respect to the initial slidingdirection ϕ = 58◦ agree with the parameters of a mea-sured sliding path of a PVC disk. The results are plottedin Fig. 27. Note that for reasons of clarity, only everyfifth data point of the measured sliding path is plotted.The time between two plotted data points is 10ms.

The results show a very good agreement of themeasured sliding path and the path obtained with theextended law. Both length and deflection of the slidingpath are similar. For the other friction laws, the deflec-tion of the sliding path is much smaller. The collinearfriction law obviously causes no deflection at all.

6 Conclusions

This work contributes to three research fields: model-ing and simulation of nonsmooth mechanical systems,friction-induced instability and experimental analysisof anisotropic frictional behavior. It has been shownthat all associated friction laws, i.e., friction laws char-acterized by normality of the sliding direction to the setof admissible friction forces are dissipative and that fora given force reservoir the associated flow rule maxi-mizes the rate of dissipation. Different frictional behav-ior can be described by various shapes of the constantforce reservoir. However, associated friction laws arelimited to convex force reservoirs. Motivated by thestudy of non-associated frictional behavior found inthe literature, in this work an extended normal coneinclusion friction law has been formulated using toolsof convex analysis. The force law enables the descrip-tion of set-valued anisotropic friction laws with pos-sibly non-convex but star-shaped force reservoirs. Themathematical framework allows for the direct imple-

mentation of the force law in efficient numerical time-stepping schemes.

Furthermore, the stability of systemswith anisotropicfriction has been analyzed. From the maximal mono-tonicity of the associated Coulomb friction law, it fol-lows that the law is unable to cause instability ofan equilibrium. However, an eigenvalue analysis hasproven that the extended normal cone inclusion lawcan lead to anisotropic friction-induced instability if thesliding direction is in the tangent cone of the force reser-voir.This is impossible for collinear force laws but can occureven for convex force reservoirs. Anisotropic friction-induced instability is a novel instability phenomenonthat potentially occurs in systems with constant nor-mal force and no dependence on the magnitude of thesliding velocity.

Finally, two experimental setups for the study of an-isotropic frictional behavior have been proposed.Usinga high-speed camera, it has been shown that, in general,a disk sliding on an anisotropic surface is deflected.This is caused by anisotropic friction forces that arenot acting parallel to the sliding direction. In addition,friction forces have been measured in two orthogonaldirections with a pin-on-disk tribometer. The describedexperimental procedure and data processing methodallow for the visualization of the friction force reser-voir for different contact pairs and surface conditions.Complex shapes of the force reservoir and no normal-ity of the sliding direction to the force reservoir havebeen found. The experimental results have confirmedthe benefit of the extended normal cone inclusion fric-tion law.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

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· Nonlinear Dyn ORIGINAL PAPER Set-valued anisotropic dry friction laws: formulation, experimental veriﬁcation and