Nonsmooth analysis of threedimensional slipping and rolling in the presence of dry friction
 112 Downloads
Abstract
In this paper, the nonsmooth dynamics of two contacting rigid bodies is analysed in the presence of dry friction. In three dimensions, slipping can occur in continuously many directions. Then, the Coulomb friction model leads to a system of differential equations, which has a codimension2 discontinuity set in the phase space. The new theory of extended Filippov systems is applied to analyse the dynamics of a rigid body moving on a fixed rigid plane to explore the possible transitions between the slipping and rolling behaviour. The paper focuses on finding the socalled limit directions of the slipping equations at the discontinuity. This leads to a complete qualitative description of the possible scenarios of the dynamics in the vicinity of the discontinuity. It is shown that the new approach consistently extends the information provided from the static friction force of the rolling behaviour. The methods are demonstrated on an application example.
Keywords
Nonsmooth dynamics Filippov systems Coulomb friction Slipping1 Introduction
If dry friction is assumed between the surfaces of rigid bodies, the dynamical model of the bodies leads to discontinuous behaviour. By considering the simple Coulomb model in the twodimensional (2D) contact problems, the friction force changes sign at zero relative velocity of the surfaces. The situation is similar but more complicated in the threedimensional (3D) contact dynamics. Then, for infinitesimally small relative velocities, the Coulomb friction model provides continuously many directions of the friction force with a constant finite amplitude.
The direct substitution of the discontinuous friction models into the dynamical equations leads to discontinuous systems of differential equations. In the 2D case, the Coulomb friction leads to Filippov systems (for an overview and examples, see [7]). Considering the friction as a setvalued force law leads to differential inclusions, which is a completely different point of view of modelling (see [13] for an overview). A further approach can be found in [6, 16, 17].
The generalization of the Filippov systems to codimension2 discontinuity sets in the phase space leads to the concept of extended Filippov systems (see [1] and [4]). This type of differential equation can be used for modelling and analysis of 3D mechanical systems with dry friction, which was demonstrated in specific mechanical examples in [4]. The early results about two general contacting bodies have been presented by the authors in [3].
In this paper, we analyse the dynamics of a single rigid body in contact with a fixed rigid plane. During the motion of the body, rolling or slipping can occur, and the slipping case reveals to be described by an extended Filippov system. We focus on the transitions between the slipping and rolling dynamics when applying the theory of extended Filippov systems. The socalled limit directions can be determined, which are strongly connected to the possible slipping–rolling transitions. One of our main motivations is to provide a deeper understanding of the qualitative dynamics in the neighbourhood of the discontinuity. But the results makes the possibility for new numerical methods for simulating these mechanical systems, as well.
The paper is organized as follows: In Sect. 2, the dynamic equations of the moving body are derived by appropriate choice of the state variables for the subsequent analysis. In Sect. 3, the basic concepts and definitions of extended Filippov systems are presented. The main part of the paper is Sect. 4, where the theory of extended Filippov systems is applied to the dynamics of the moving body. From the analysis, we get four typical cases of the limit directions. In Sect. 5, the mechanical consequence of the four cases is explained, and the relation with the static friction force is presented. In Sect. 6, the results are demonstrated on a mechanical example. In Sect. 7, an overview can be found about the possible extension of the results to more complicated contact models.
This paper is a significantly extended version of the conference paper [3]. The content of Sects. 2–5 has been rearranged and improved, and most importantly, the former conjectures have been developed into a series of proved statements about the possible slipping–rolling transitions, as can be found in the current paper. Sections 6 and 7 are completely new.
2 Dynamics of a rigid body on a flat surface
Important notation of the mechanical system
Notation  Quantity 

C  Centre of gravity of the rigid body 
P  Contact point 
\(\mathbf {r}_{PC}\)  Position vector between P and C 
m  Mass of the body 
\(\mathbf {J}\)  Mass moment of inertia of the body 
\(\mu \)  Friction coefficient between the surfaces 
\(\mathbf {F}_C,\mathbf {T}_C\)  Resultant force and torque of external forces computed at C 
\(\mathbf {n}\)  Normal unit vector at P 
\(\mathbf {t}_1,\mathbf {t}_2\)  Tangential unit vectors at P 
\(N\mathbf {n}\)  Normal force at P 
\(\mathbf {F}_f\)  Friction force at P 
\(\mathbf {F}_P\)  Total contact force at P 
\(\rho _n\)  Reciprocal of the normal curvature of the body in the direction of motion 
\(\mathbf {v}_C,\mathbf {v}_P\)  Velocities of C and P 
\(u_1,u_2\)  Components of the velocity \(\mathbf {v}_P\) (slipping) 
\({{\varvec{\omega }}}\)  Angular velocity of the body 
\(\omega _1,\omega _2,\omega _3\)  Components of \({{\varvec{\omega }}}\) 
q  Vector of generalized coordinates 
s  Vector of quasivelocities 
2.1 Kinematics
During the motion of the body, the contact point P corresponds to different material points of the rigid body. We can take the time derivatives of (1) by using two different approaches: either by following the material point of the body currently located at P, or, following the motion of the instantaneous geometric contact point P.
What is the point of writing \(\mathbf {a}_C\) in the form of (3)? In the subsequent calculations, we are using the components of \(\mathbf {v}_P\) as phase variables. As the Coulomb friction model is discontinuous exactly at \(\mathbf {v}_P=\mathbf {0}\), this choice of variables, the discontinuity set of the resulting differential equation can be treated easily.
2.2 Dynamics
Equations (6)–(9) lead to a system of differential equations in the case of rolling or slipping. In this paper, we focus on the slipping equations with special attention to their behaviour close to the rolling state.
2.3 Differential equations for the slipping case
Important notation of extended Filippov systems
Notation  Quantity 

\(\mathcal {D}\)  Phase space of the system (subset of \(\mathbb {R}^m\)) 
x  Vector of phase variables (element in \(\mathcal {D}\)) 
F(x)  Vector field of the system 
\(\Sigma \)  Discontinuity manifold of F(x) (\(m2\)dimensional subset of \(\mathcal {D}\)) 
\(\mathcal {T}_{x_0}\Sigma \)  Tangent space of \(\Sigma \) at \(x_0\) 
\(\mathcal {O}_{x_0}\Sigma \)  Orthogonal space of \(\Sigma \) at \(x_0\) 
\(n_1,n_2\)  Orthogonal basis vectors of \(\mathcal {O}_{x_0}\Sigma \) at \(x_0\) 
\(\phi \)  Aangle parametrizing the directions of \(\mathcal {O}_{x_0}\Sigma \) around \(x_0\) 
\(n(\phi )\)  Set of unit normal vectors to \(\Sigma \) at \(x_0\) 
\(F^*(\phi )(x_0)\)  Limit vector field (directional limit of F at \(x_0\) from the different directions \(\phi \)) 
\(R(\phi )\)  Radial component of \(F^*(\phi )\) 
\(V(\phi )\)  Circumferential component of \(F^*(\phi )\) 
Consequently, Eqs. (15), (10), and (11) form a set of six differentialalgebraic equations in the generalized coordinates (12), the quasivelocities (14), and the normal force N.
Equations (15) and (18) form a system of ten firstorder ODEs for the variables (12) and (14). Due to the discontinuity of the contact force (17), the system is not defined at \(u_1=u_2=0\), which corresponds to the rolling behaviour. For the rolling states, a different set of differential equations can be derived by excluding the slipping Coulomb law (9) but including the rolling constraint (8). To obtain a deeper insight to the switches between rolling and slipping, this paper focuses on the analysis of slipping system (15) and (18) in the vicinity of the discontinuity \(u_1=u_2=0\).
Note that this discontinuity is located at the states where two variables (\(u_1\) and \(u_2\)) are zero at the same time. For the analysis of differential equations with such discontinuity, we can use effectively the theory of extended Filippov systems, which is presented briefly in the next section.
3 Overview of extended Filippov systems
The concept of extended Filippov systems was introduced by the authors in [1] and [4]. Roughly speaking, these dynamical systems are vector fields containing \(m2\)dimensional discontinuities in the mdimensional phase space. We will show in Sect. 4 that the contact problem of the rigid body presented in Sect. 2 leads to an extended Filippov system.
In this section, only the most important concepts and definitions of these dynamical systems are presented, which are utilized in the subsequent analysis of the mechanical system. For a more detailed presentation of the theory of extended Filippov systems, see [1] and [4]. The notation of the important corresponding quantities can be found in Table 2.
Definition 1
 (a)
The vector field F is smooth on \(\mathcal {D}{\setminus }\Sigma \).
 (b)The limitexists for all \(x_0\in \Sigma \) and for all \(\phi \in [0,2\pi )\).$$\begin{aligned} F^*(\phi )(x_0):=\lim _{\epsilon \rightarrow 0^+} F\left( x_0+\epsilon n(\phi )(x_0)\right) \end{aligned}$$(23)
 (c)
For all \(x_0\), there exist \(\phi _1,\phi _2\in [0,2\pi )\) for that \(F^*(\phi _1)\ne F^*(\phi _2)\).
In the sense of Definition 1, \(\Sigma \) is called a codimension2 discontinuity manifold of\({{\varvec{F}}}\). At a chosen point \(x_0\in \Sigma \), the function \(F^*(\phi )\) is called the limit vector field of F (see Fig. 2), which contains the directional limits of F from the different directions parameterized by \(\phi \).
The three conditions of Definition 1 formally express that there is no discontinuity outside \(\Sigma \) (see (a)), there is indeed discontinuity at any point of \(\Sigma \) (see (c)), and the directional limit does not diverge from any direction (see (b)).
Let us now define the concept of limit directions, which are strongly connected to the behaviour of the trajectories at the discontinuity.
Definition 2
Consider an extended Filippov system \(\dot{x}=F(x)\) and a point \(x_0\in \Sigma \) of the discontinuity manifold. The roots of the equation \(V(\phi )=0\) are called the limit directions of \(x_0\) with respect to F.
Definition 3
A limit direction \(\phi _1\) with \(V(\phi _1)=0\) is called attracting if \(R(\phi _1)<0,\) and it is called repelling if \(R(\phi _1)>0\).
It can be proved (see [4]) that if \(x_0\in \Sigma \) possesses at least one limit direction, then all trajectories tending to \(x_0\) (either forward or backward time) approach \(x_0\) along the limit directions.
In this sense, the limit directions are somewhat analogous to the eigenvectors of equilibrium points of smooth systems, but there are fundamental differences. Firstly, an eigenvector of a saddle or node is bidirectional (corresponding to a line), while a limit direction is unidirectional (corresponding to a halfline). Secondly, the eigenvectors of equilibria correspond to infinitetime (exponential) convergence of the solutions in forward or backward time, while trajectories reach \(x_0\) in finite time in forward (attracting) or backward (repelling) direction of time.
By continuing the analogy with the equilibria, we can separate the nodelike (sliding) and saddlelike (crossing) behaviour.
Definition 4
Consider a point \(x_0\in \Sigma \) which possesses at least one limit direction. If all the limit directions are either attracting or repelling, then we say that \(x_0\) is located in the sliding region of \(\Sigma \). If there is at least one attracting and one repelling limit direction, then we say that \(x_0\) is located in the crossing region of \(\Sigma \).
The terminology of crossing and sliding was introduced in [1] and [4] by generalizing of the crossing and sliding region of classical Filippov systems with codimension1 discontinuities (see [7]). In the crossing case, there is at least one incoming and one leaving halftrajectory at \(x_0\), which can be concatenated to a trajectory crossing through \(\Sigma \) at \(x_0\). In the sliding case, there are either only incoming or only leaving trajectories and the dynamics of F gets stuck into \(\Sigma \) in forward or backward time, respectively. Then, the socalled sliding dynamics generated inside the discontinuity manifold \(\Sigma \). For the derivations and a more detailed explanation, see [4].
The introduction of the extended Filippov systems was originally motivated by 3D contact problems of rigid bodies. In these mechanical problems, the slipping of the bodies in the presence of Coulomb friction leads to extended Filippov systems, and the rolling or sticking of the bodies corresponds to the sliding dynamics inside the discontinuity manifold. In the following central part of the paper, the analysis of the limit directions is applied to explore the transitions between slipping and rolling between the bodies.
4 Analysis of limit directions at the rigid body
4.1 The resulting extended Filippov system
Theorem 1
The system (29) is an extended Filippov system.
Proof
4.2 Analysis of the limit directions of the system
The discontinuity set \(\Sigma \) of (29) is defined by \(u_1=u_2=0\) (see (30)), which corresponds to the rolling of the body on the plane. In this subsection, we categorize the points of \(\Sigma \) according to the number and type of limit directions, which are strongly connected to the transitions between rolling and slipping.
4.2.1 Possible formal simplifications
Proposition 1
In (33), the coefficients satisfy \(B_1=A_2\).
Proof
Proposition 2
The coefficients \(A_1\) and \(B_2\) in (38) are strictly negative for the physically relevant parameters.
Proof
Proposition 3
Proof
4.2.2 Possible number of limit directions
In the expression of (35) and (44), \(V(\phi )\) is a truncated Fourier series containing terms up to the second harmonics. According to [10], determining the zeroes of such function leads to the eigenvalue problem of a 4by4 complex matrix. Alternatively, finding the zeroes of \(V(\phi )\) is equivalent to solving the following fourthorder polynomial equation.
Proposition 4
Proof
Equation (45) can be derived from (44) by direct calculation using basic trigonometric identities. \(\square \)
Proposition 5
The function \(V(\phi )\) in (44) has maximum four zeroes except if \(V(x_0)(\phi )\) is identically zero.
Proof
Equation (45) is a fourthorder polynomial in \(\cos \phi \), which leads to maximum four different roots for \(\phi \) on the interval \(\phi \in [0,2\pi )\). In the degenerate case when \(V(\phi )\) is identically zero, all \(\phi \in [0,2\pi )\) are limit directions. \(\square \)
Proposition 6
The function \(V(\phi )\) in (44) has minimum two zeroes.
Proof
In the form (44) of \(V(\phi )\), the constant term vanishes. Therefore, \(V(\phi )\) is a periodic continuous function with zero mean value. Thus, it needs to have at least two zeroes on \(\phi \in [0,2\pi )\). \(\square \)
Proposition 7
Proof
Theorem 2
 1.
If \(C_1^{2/3}+C_2^{2/3}>(B_2A_1)^{2/3},\) then \(x_0\) has 2 limit directions.
 2.
If \(C_1^{2/3}+C_2^{2/3}<(B_2A_1)^{2/3},\) then \(x_0\) has 4 limit directions.
 3.
If \(C_1^{2/3}+C_2^{2/3}=(B_2A_1)^{2/3}\ne 0,\) then \(x_0\) has 3 limit directions.
 4.
If \(C_1=C_2=B_2A_1=0,\) then \(x_0\) has continuously many limit directions.
Proof
Point 3 of the Theorem is contained by Proposition 7. The condition (46) separates the space of the coefficient \(A_1,B_2,C_1,C_2\) into two regions, where there can be 2 or 4 limit directions (see Propositions 5 and 6). In the case \(C_1^{2/3}+C_1^{2/3}\gg (B_2A_1)^{2/3}\), the last term in (44) is negligible, that is, there are two roots of \(V(\phi )\) on \([0,2\pi )\), resulting in Point 1 of the Theorem. Point 2 can be proved similarly by checking the extreme case \(C_1^{2/3}+C_1^{2/3}\ll (B_2A_1)^{2/3}\), when there are four roots of (44). In Point 4, \(V(\phi )\) is identically zero and all \(\phi \in [0,2\pi )\) are limit directions. \(\square \)
4.2.3 Attracting and repelling limit directions
Proposition 8
Proof
The condition \(V(\phi _1)=R(\phi _1)=0\) is equivalent to \(F^*_1(\phi _1)=F^*_2(\phi _1)=0\) in (42). By direct calculation, we get \(\tan \phi _1=A_1C_2/(B_2C_1)\) and the condition (48) of Proposition. \(\square \)
Theorem 3
 1.
If \(C_1^2/A_1^2+C_2^2/B_2^2<1,\) then \(x_0\) has only attracting limit directions and no repelling limit directions.
 2.
If \(C_1^2/A_1^2+C_2^2/B_2^2>1,\) then \(x_0\) has at least one attracting limit direction and exactly one repelling limit direction.
 3.
If \(C_1^2/A_1^2+C_2^2/B_2^2=1,\) then \(x_0\) has attracting limit directions and a limit directions on the boundary of being attracting and repelling.
Proof
In Point 3 of the Theorem, we can find the condition of Proposition 8 which ensures the existence of a limit directions between being repelling and attracting (see Definition 3). This condition (48) separates the space of the coefficient \(A_1,B_2,C_1,C_2\) into two regions, and the number of the repelling (or attracting) direction changes by one when crossing the boundary (48).
4.2.4 The four generic cases
The boundary (46) in Proposition 7 divides the space of the coefficients \(A_1,B_2,C_1,C_2\) into two typical regions (see Theorem 2). Similarly, the boundary (48) in Proposition 8 creates two typical regions (see Theorem 3). This creates four generic regions in the space of \(A_1,B_2,C_1, and C_2\). For fixed values of \(A_1\) and \(B_2\), these regions can be visualized in the plane of \(C_1\) and \(C_2\).
 I.
2 attracting limit directions,
 II.
4 attracting limit directions,
 III.
1 attracting and 1 repelling limit directions,
 IV.
3 attracting and 1 repelling limit directions.
The typical structure of the vector field in the four cases can be seen in Figs. 6–9. The figures show the projection of the vector field into the orthogonal space \(\mathcal {O}_{x_0}\Sigma \). The origin of the diagram corresponds to the given point \(x_0\) of the discontinuity \(u_1=u_2=0\) of the rolling behaviour. The direction of the vector field induces that the trajectories approach the discontinuity along the attracting limit directions (denoted by solid lines), and they leave the discontinuity along the repelling limit directions (denoted by dashed lines).
4.2.5 Angularly stable and unstable limit directions
Definition 5
The limit direction \(\phi _1\) of (29) is called angularly stable if \(\mathrm {d}V(\phi )/\mathrm {d}\phi \) is negative at \(\phi =\phi _1,\) and it is called angularly unstable if \(\mathrm {d}V(\phi )/\mathrm {d}\phi \) is positive at \(\phi =\phi _1\).
In the angularly stable case, the limit direction is attracting the adjacent trajectories (see \(\phi _1\) in Fig. 6). In the angularly unstable case, the adjacent trajectories get far from the limit directions in the sense of the angle \(\phi \) (see \(\phi _2\) in Fig. 6).
The special case \(\mathrm {d}V(\phi )/\mathrm {d}\phi =0\) is a fold bifurcation of (52), which corresponds to the fold of limit directions in (29). This condition was already discussed in Proposition 7 (see (47)); thus, the fold of directions coincides with the condition (46). The next Proposition completes our analysis of the limit directions of (29).
Proposition 9
If a limit direction of (29) is repelling, then it is an angularly stable limit direction.
Proof
Number and type of the limit directions in the four generic cases of the system
Case  I  II  III  IV 

Total number of limit directions  2  4  2  4 
Attracting, angularly stable  1  2  0  1 
Attracting, angularly unstable  1  2  1  2 
Repelling, angularly stable  0  0  1  1 
Repelling, angularly unstable  0  0  0  0 
To summarize our results, the number and properties of the limit trajectories can be found in Table 3.
5 Slipping–rolling transitions
5.1 Mechanical consequence of the limit directions
The system F(x) in (28) was introduced to describe the slipping behaviour of the body. The discontinuity manifold \(\Sigma \) is the set \(u_1=u_2=0\), which coincides with the condition of the rolling constraint (8).
In this subsection, the slipping–rolling transitions are analysed by considering purely the limit directions of the slipping equations determined in Sect. 4. The relation to the dynamical conditions of the rolling equations is presented in the next subsection.
5.1.1 Case I: 2 attracting directions
In this case, all trajectories in the vicinity of \(x_0\) tend to the discontinuity at \(u_1=u_2=0\) (see Fig. 6). That is, the behaviour of the body turns from slipping to rolling. It is proved in [4] that the trajectories reach \(u_1=u_2=0\) in finite time. In some sense, the rolling motion is stable with respect to slipping perturbations, because the effect of a small perturbation in \(u_1\) and \(u_2\) is eliminated by the dynamics in finite time.
Note that almost all solutions reach the rolling state along the angularly stable limit direction (\(\phi _1\) in Fig. 6). The angle \(\phi \) can be imagined not only as an angle in the phase space but as an angle of the slipping velocity \(\mathbf {v}_P\), as well. Therefore, the dominant behaviour of the limit direction \(\phi _1\) causes that the slipping velocity points typically into the direction \(\phi _1\) when the motion changes from slipping into rolling. There is only a single trajectory which approaches the state \(u_1=u_2=0\) from the direction \(\phi _2\). The trajectories close to \(\phi _2\) contain a highcurvature turning when reaching \(x_0\). Hence, the direction of the friction force changes rapidly just before the transition from slipping to rolling.
5.1.2 Case II: 4 attracting directions
This case has a behaviour similar to Case I: all surrounding trajectories tend to the discontinuity (\(u_1=u_2=0\)) in finite time (see Fig. 7. From mechanical point of view, this means that the rolling motion is realizable, because small perturbations causing slipping are eliminated by the dynamics and the body starts rolling again.
5.1.3 Case III: 1 repelling and 1 attracting direction
When an attracting limit direction turns into repelling, the structure of the phase changes significantly. In Case III, we can find an attracting and a repelling limit direction (see Fig. 8). The attracting direction is angularly unstable and the repelling direction is angularly stable (according to Proposition 9). That is, the typical behaviour of the system is slipping, and almost all trajectories avoid the discontinuity at \(u_1=u_2=0\).
In the vicinity of the discontinuity set (rolling behaviour), the trajectories tend to the repelling limit direction \(\phi _1\) and they diverge from the rolling state. That is, a slipping motion is generated with a typical direction \(\phi _1\) of the slipping velocity. There exists one single trajectory which reaches the discontinuity, and this happens along the limit direction \(\phi _2\). But the system reaches the rolling state just for a moment, and it starts slipping immediately in the direction of \(\phi _2\).
5.1.4 Case IV: 1 repelling and 3 attracting directions
However, in the regions \(\phi _2<\phi <\phi _3\) and \(\phi _3<\phi <\phi _4\), the trajectories tend to the angularly stable attracting direction \(\phi _3\), and they all reach the discontinuity at \(u_1=u_2=0\). There is rolling only for a moment, and the system starts slipping with a slipping velocity described by the direction \(\phi _1\). In contrast to Case III, not just a single trajectory is connected to the discontinuity, but a large portion of the phase plane tends to \(u_1=u_2=0\). That is, the typical longtime behaviour is slipping, but for many initial conditions, rolling can occur for a moment.
5.1.5 Summary of the typical types of behaviour
After the detailed survey of the possible types of solution, let us summarize the typical four cases of behaviour from the mechanical point of view.
Corollary 1

In Cases I and II, the perturbed body returns to rolling in finite time and then it maintains the rolling state.

In Case I, the slipping velocity vanishes from a certain direction for almost all perturbations (see Fig. 6).

In Case II, the slipping velocity vanishes from two certain directions for almost all perturbations (see Fig. 7).

In Cases III and IV, the perturbed body is unable to maintain a lasting rolling state and it continues slipping.

In Case III, the slipping velocity remains finite for almost all perturbations, thus, pure slipping continues (see Fig. 8).

In Case IV, two types of behaviour occur according to the direction of the perturbation. Either, the body continues pure slipping like in Case III. Or, the slipping velocity vanishes in finite time, rolling motion occurs for a single moment, and then, the body continues slipping again (see Fig. 9).
5.2 Comparison with the rolling condition of the friction law
Up to this point, we analysed the rolling–slipping transitions based on purely the phase space of the slipping system (29). It can be seen that a detailed, consistent structure of the behaviour can be obtained from this analysis. But what is the relation between these results and the ones from the rolling condition of the friction law with the static friction force?
The equations of the rolling vector field can be derived either from the Newton–Euler equations (7) with the rolling constraint (8), or directly from the limit vector field (32) of the slipping case.
In the latter case, we consider the dynamics on \(\Sigma \) generated by \(F^*(x)\) by a convex combination, which is called sliding dynamics in the terminology of Filippov systems and extended Filippov systems (see [1, 7] and [4]). In mechanical problems, we have to be careful with this terminology because sliding dynamics correspond to the mechanical rolling and not to the mechanical slipping.
When the static and dynamic friction coefficient is equal (\(\mu _0=\mu \)) then we can state the following theorem.
Theorem 4
 1.
If the rolling is strictly admitted by (58), that is, \(\mathbf {F}_f<\mu _0 N\), then \(x_0\) possesses no repelling limit directions.
 2.
If the rolling is not admitted by (58), that is, \(\mathbf {F}_f>\mu _0 N\), then \(x_0\) possesses a repelling limit direction.
 3.
In the special case \(\mathbf {F}_f=\mu _0 N\), \(x_0\) possesses attracting limit directions and a limit direction on the boundary of being attracting and repelling.
Proof
The rolling dynamics ensures \(u_1=u_2=0\) permanently, that is, the derivatives \(\dot{u}_1\) and \(\dot{u}_2\) have to be zero, as well. If \(\mu _0=\mu \) then third statement of the theorem with \(\mathbf {F}_f=\mu _0 N\) gives back the condition (9) of the dynamic friction. That is, the rolling and slipping dynamics is valid at the same time. In the slipping dynamics, the condition \(\dot{u}_1=\dot{u}_2=0\) is equivalent to the condition (48) (see the proof of Proposition 8), which decides whether there exists a repelling limit direction or not (see Theorem 3). The magnitude \(\mathbf {F}_f\) of the friction force tends to zero when \(A_1\rightarrow 0\) and \(B_2\rightarrow 0\) [compare (6), (11), (17) and (29)]. Consequently, the three cases of Theorems 3 and 4 are pairwise equivalent. \(\square \)
Proposition 10
By considering the improved slipping friction model (60)–(61) instead of (9), Theorem 4 remains valid for different values of the static and dynamic friction coefficients.
Proof
By the first limit of (61), the dynamic friction coefficient tends to \(\mu _0\) when \(\mathbf {v}_P\rightarrow 0\). Then, \(\mu \) can be replaced by \(\mu _0\) in \(F^*\) and the related quantities all along the analysis of the paper. Then, the proof of Theorem 4 can be repeated. \(\square \)
Theorem 4 and Proposition 10 state that the analysis of the limit directions in Sect. 4 is consistent with the checking of the maximal admissible friction force. In Cases I and II with only attracting limit directions (Figs. 6–7), the condition (58) is satisfied (rolling is realizable). In Cases III and IV with a repelling limit direction (Figs. 8–9), the condition (58) is violated (rolling is not realizable).
6 Application example
Consider a wheel moving on a plane (see Fig. 11), where the symmetry axis of the wheel remains parallel with the plane (no tilting of the wheel). The wheel is modelled by a rigid disc with a radius \(\rho \) and a negligible thickness. The external forces acting on the wheel are the gravity force (mg), the steering moment (\(M_s\)), the driving moment (\(M_d\)), and the balancing moment (\(M_b\)), ensuring the horizontal orientation of the symmetry axis. All the other notations are the same as in Sects. 2–4 (see Tables 1). It is shown in [9] that the rolling motion of this model is equivalent to the motion of the Chaplyginsleigh, which is an important benchmark problem of nonholonomic mechanics.
7 Overview of more complex contact models
 1.
the assumption of planar geometry of the fixed body,
 2.
the assumption of rigid bodies,
 3.
assuming that the contact state is either slipping or rolling,
 4.
the assumption of the Coulomb friction model,
 5.
the assumption of a welldefined, single contact point of the unloaded bodies.
If we replace the fixed plane by a rigid body with an arbitrary curved fixed surface, then the normal plane of the contacting surfaces is changing during the motion. It would make the dynamic equations more complicated, but it would not be a structural modification of the model. Thus, the different scenarios of the limit directions are expected to be preserved in this case.
If we do not neglect the deformation of the bodies, then we have to consider the formation of the contact area around the theoretical contact point. In the literature, it is usual to separate the motion into two parts: the rigid body motion of the whole body and the local deformations in the vicinity of the contact area. The Hertz theory (see, e.g. [25], p. 55 or [19] p. 84) assumes elliptical contact area and a parabolic distribution of the normal pressure between the bodies in the frictionless case. Similar but higher order theories exist for the normal pressures (see, e.g. [20]). Combining these models with friction leads to theoretical and computational challenges [19, 20, 25]. For the purpose of dynamical applications, analytical and semianalytical models of the contact forces can be derived from these theories. When we consider the combined effect of the slipping and drilling motion, the contact laws are determined by the CoulombContensou friction model (see, e.g. [21]). The combined effect of slipping and rolling motion leads to the contact laws of creep models (see, e.g. [18]). By improving our analysis by these models, highercodimensional (35) discontinuities are expected to appear. This can be the topic of the further research work. The concept of limit directions probably remain important in these cases, as well, to find the possible transitions between rolling and slipping.
We assumed that the normal contact force N is strictly positive, and the surfaces remain in permanent contact at the contact point. Then, depending on the state of the system, the friction model decides whether slipping or rolling behaviour occurs. However, when the contact force N decreases to zero, the bodies can separate from each other (liftoff), and the nonsmooth behaviour of the dynamics with the discontinuity set vanishes. However, the switching of the contact and the nocontact states introduces a further discontinuity, containing the impacts of the bodies, as well. The generalization of the results of the paper to these cases would need additional extensive research work. The Painlevé paradox of the contact states [11, 15] causes further complications.
In the analysis, we first considered the simple Coulomb law for modelling dry friction. Then, in Proposition 10, the results are generalized for a class of friction models similar to the Stribeck friction law. However, several different friction models can be found in the literature (see [22] and [23] for an overview). It is an open question how further effects like hysteresis (e.g. the Karnopp model) or internal variables (e.g. the Dahl model) modify the qualitative structure of the dynamics at the discontinuity.
A further complication can be the coexistence of multiple contact points between the contacting bodies. The results in [1] show that the concept of limit directions is applicable to two contact points, but still, a throughout analysis would be necessary. An even further case is the contact of conforming bodies, where there is a finite contact area even with the rigid body assumption. (The simplest example is a block moving on a plane.) Then, information is needed about the normal pressure distribution, and the integration on the contact area is expected to lead to highercodimensional discontinuities as we expected in the deformable models, too.
8 Conclusion
The dynamical equations of a rigid body were derived, in which body is in 3D slipping or rolling contact with a rigid plane in the presence of dry friction. It was shown that by assuming Coulomb friction model, the differential equations of the dynamics of the body lead to an extended Filippov system. That is, the phase space of the system contains a codimension2 discontinuity set.
The nonsmooth differential equations of slipping were analysed by the methods of new theory of extended Filippov systems. The possible number and type of limit directions of the points of the discontinuity were determined, where the transitions between slipping and rolling occur. We got four structurally different cases of limit directions; there can be two or four limit directions, from which maximum one can be repelling. The effect of these scenarios of the mechanical behaviour was discussed in detail. It was shown that in case of simple Coulomb model and the Stribeck model, the limit directions lead to such conditions of rolling, which are consistent with the condition of maximum admissible friction force. Furthermore, the results of the new approach provides more information about the qualitative behaviour of these mechanical systems near the discontinuity. The result was demonstrated on an example of a wheel. A part of the further work would be to apply the results to other wellknown systems such as the rolling disc (see [5, 8]) and the classical skate problem (see, e.g. [9]).
The considered contact model is clearly the simplest mechanical model which is capable to describe the problem of the different possible directions of transitions between rolling and slipping in three dimensions. However, several possibilities were presented in the last section to improve the contact model in different ways.
One further important direction of the subsequent research would be to utilize these results to develop effective and reliable numerical methods for simulation of these systems. The information of the structure of the trajectories and limit directions could help to find appropriate eventdriven strategies similar to those of classical Filippov systems [24].
Notes
Acknowledgements
Open access funding provided by Budapest University of Technology and Economics (BME). The research leading to these results has been supported by the Hungarian Academy of Sciences in the Premium Postdoctoral Fellowship Programme under the grant number PPD2018014/2018. The authors wish to thank Professor Peter Varkonyi from the Budapest University of Technology and Economics for the useful discussion about the higher order discontinuities.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
 1.Antali, M.: Dynamics of dualpoint rolling bodies. Ph.D. thesis, Budapest University of Technology and Economics (2017). (Supervisor: G. Stépán)Google Scholar
 2.Antali, M., Stepan, G.: Discontinuityinduced bifurcations of a dualpoint contact ball. Nonlinear Dyn. 83(1), 685–702 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Antali, M., Stepan, G.: Modelling coulomb friction by extended filippov systems. In: Awrejcewitz, J. et. al. (ed.) Proceedings of DSTA 2017, Engineering Dynamics and Life Sciences, vol. 3, pp. 21–32 (2017). (ISBN:9788393531240)Google Scholar
 4.Antali, M., Stepan, G.: Sliding and crossing dynamics in extended filippov systems. SIAM J. Appl. Dyn. Syst. 17(1), 823–858 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Baranyai, T., Varkonyi, P.: Imperfections, impacts, and the singularity of euler’s disk. Phys. Rev. E 96(3), (2017). https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.033005
 6.Batlle, J.A.: The sliding velocity flow of rough collisions in multibody systems. J. Appl. Mech. 63(3), 804–809 (1996)CrossRefzbMATHGoogle Scholar
 7.di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: PiecewiseSmooth Dynamical Systems. Springer, London (2008)zbMATHGoogle Scholar
 8.Borisov, A.V., Kilin, A.A., Karavaev, Y.L.: Retrograde motion of a rolling disk. PhysicsUspekhi 60(9), 931–934 (2017)CrossRefGoogle Scholar
 9.Borisov, A.V., Kilin, A.A., Mamaev, I.S.: On the Hadamard–Hamel problem and the dynamics of wheeled vehicles. Regul. Chaotic Dyn. 20(6), 752–766 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Boyd, J.P.: Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: solving transcendental equations by spectral interpolation and polynomial rootfinding. J. Eng. Math. 56(3), 203–219 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Champneys, A.R., Varkonyi, P.R.: The painlevé paradox in contact mechanics. IMA J. Appl. Math. 81(3), 538–588 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Frankel, T.: The Geometry of Physics. Cambridge University Press, Cambridge (2012)Google Scholar
 13.Glocker, C.: SetValued Force Laws. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
 14.Greenwood, D.T.: Advanced Dynamics. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
 15.Hogan, S.J., Kristiansen, K.U.: On the regularization of impact without collision: the Painlevé paradox and compliance. Proc. R. Soc. A 473(2202), 1–18 (2017)CrossRefzbMATHGoogle Scholar
 16.Ivanov, A.P.: The conditions for the unique solvability of the equations of the dynamics of systems with friction. J. Appl. Math. Mech. 72(4), 372–382 (2008)MathSciNetCrossRefGoogle Scholar
 17.Ivanov, A.P.: On singular points of equations of mechanics. Dokl. Math. 97(2), 167–169 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Iwnicki, S.: Simulation of wheelrail contact forces. Fatigue Fract. Eng. Mater. 26, 887–900 (2003)CrossRefGoogle Scholar
 19.Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
 20.Kalker, J.J.: ThreeDimensional Elastic Bodies in Rolling Contact. Kluwer, Dordrecht (1990)CrossRefzbMATHGoogle Scholar
 21.Leine, R.I., Glocker, C.: A setvalued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. A Solid 22(2), 193–216 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Marques, F., Flores, P., Claro, J.C.P., Lankarani, H.M.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn. 86, 1407–1443 (2016)MathSciNetCrossRefGoogle Scholar
 23.Pennestri, E., Rossi, V., Salvini, P., Valentini, P.: Review and comparison of dry friction models. Nonlinear Dyn. 83, 1785–1801 (2016)CrossRefzbMATHGoogle Scholar
 24.Piiroinen, P.T., Kuznetzov, Y.A.: An eventdriven method to simulate Filippov systems with accurate computing of sliding motions. ACM T. Math. Softw. 34(3), 1–24 (2008)MathSciNetCrossRefGoogle Scholar
 25.Popov, V.L.: Contact Mechanics and Friction. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.