On the Relation of the Principle of Maximum Dissipation to the Principle of Gauss

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1 On the Relation of the Principle of Maximum Dissipation to the Principle of Gauss Kerim Yunt P.O Box 7 Zurich, Switzerland 82 kerimyunt@web.de is about conditions which specify when sticking and sliding take place, which are related to the tangential relative contact velocity v T and acceleration a T. The possibility to formulate friction models on acceleration level is shown systematically by Glocker in his monograph [9] using nonsmooth analysis though already in [2] by Pozharitskii the spatial friction force model of tresca-type on acceleration level as well as friction acceleration potentials are used without definition, in order to extend the principle of Gauss to systems with dry friction. The classical PMD for frictional contacts states that given a sliding velocity, the forces are adjusted so as to cause maximum dissipation. This statement is moulded into an optimization problem as follows: γ Ti = µ i γ Ni max γ T i sliding γ Ti, v Ti v Ti v Ti, i I sliding. The aim of this work is to establish the relation between two well-known principles of dynamics for finite-dimensional Lagrangian systems subject to non-impulsive dissipative force laws. These two principles are the principle of maximum dissipation PMD and the principle of Gauss. For dynamics, where the evolution requires the deteration of the accelerations of the system, it is shown that in the presence of dissipative force laws a similar principle holds, which requires the augmentation of the optimization problem of least constraints by the time rate of total dissipation. The dual problem of least constraints is derived from the maximization of the total time derivative of the total dissipation with respect to passive dissipative forces. Introduction The principle of Gauss has been published in [5] by Gauss and provided a variational criteria on admissible accelerations of bilaterally holonomically constraint mechanical systems. The principle of maximal dissipation was introduced by Moreau in [9] solely for single point frictional contacts. The principle considers the friction as maximizing dissipation over all feasible friction forces at a contact. The considered frictional contact law in this work, is the dry friction law of tresca type. There are two main features of dry friction. The first feature is a set C T to which frictional contact force γ T belongs as admissible forces. This set depends on the magnitude of the normal contact force γ N. The friction coefficient µ serves as a transmissive multiplier of the normal force on the tangential direction. The second feature Address all correspondence related to ASME style format and figures to this author. The radius of application of the PMD is extended by making use of the definition of a general dissipative force in this work. The PMD is a formulation on velocity level because it deals with forces that are velocity dependent and in dynamics they are related to the principle of Jourdain. However, the principle of Jourdain states only the condition on the admissible velocities without exactly specifying the evolution of the system, whereas depending on the uniqueness of the solution of the dissipative forces might be specified and the evolution of the system is thereby specified on acceleration level. The principle of Jourdain provides tools to integrate forces that derive from velocity potentials. The principle of Gauss that bears the the principle of least constraints enables in addition to the above cited principles to detere the evolution of a non-impulsive mechanical process. The evolution of the mechanical system is detered by its acceleration and the principle of least constraints is about an optimization problem to detere accelerations for bilaterally constraint holonomic mechanical systems. The principle of maximal dissipation and Gauss s principle of least constraints are reigning in two distinct domains of mechanics. The principle of maximal dissipation is the means of evaluating dissipative mechanical problems in continuum mechanics in the absence of accelerations. In the works of Klarbring [] and [2] the application of the principle of maximal dissipation to quasi-static and static problems with unilateral contacts and friction is elucidated and for all types the complementarity problems developed. The need to distinguish among different contact states in a multicontact problem gives rise to complementarity problems which are well studied. Starting with the works of Lötstedt [3] and [4], the literature on the complementarity modelling and simulation of contact systems with friction is vast and good overviews are provided in [], [2], [6], [23], [25], [26]. Klarbring formulates a nonconvex optimization problem starting from the PMD for quasistatic problems in structural mechanics. The nonconvexity of the optimization problem arises due to the Coulomb-type friction for which

2 also the normal contact force is a variable of the optimization. Originally, Gauss formulated his celebrated principle for Lagrangian systems which were constrained bilaterally. Moreau showed in [6] and [7], how to generalize the principle of Gauss and the principle of least constraints to finitedimensional Lagrangian systems with frictionless unilateral constraints and derived the quadratic programg procedure for detering the accelerations for such mechanical systems. Pozharitskii formulated in [2] the extension of the principle of least constraints for mechanical systems with planar isotropic dry friction with known normal forces by a function he called the work of all forces applied to the system with a virtual acceleration. Glocker in [8] proposed the most general optimization problem by putting the whole constraints in to an acceleration potential Ψ C q in the form: f q = 2 q, M q h, q Ψ C q, 2 which depending on the structure of the domain C is a strictly convex function, with M being the symmetric and positivedefinite mass matrix and h a continuous function of the generalized positions and velocities of the mechanical system. In [7] Glocker considered nonconvex potentials which are regular. The dual principle of least constraints, which is an optimization problem in constraint forces, is also mentioned in [9]. Pozharitskii considers following form of extension to the principle of least constraints in [2] for problems with tresca-type dry friction: q, M q h, q Ψ. 3 q 2 Pozharitskii defines the quantity: Ψ = n i= Q i a i i I NP µ i k ni a Ti 4 as the work of all the forces applied to the system with a virtual acceleration, where Q is a generalized force and the distance function is in the Euclidean norm; a Ti is the relative tangential acceleration at a sticking contact; k Ni is the given normal force at a frictional contact, and formulates following principle: If in a system with contacts one knows all maxima of the frictional forces, then the actual motion which agrees with the law of friction will differ from all neighboring conceivable motions by the fact that for a real motion the difference between the energy of acceleration and the work of all froces on the real acceleration will be smaller than that same difference is for any virtual acceleration that is near a real one; furthermore, there will always exist at least one motion which satisfies this condition. The distance of dissipation potentials are formulated in the Euclidean norm, for which norm he shows that the function is positive-definite. In this work, for mechanical systems with convex dissipation potentials such as tresca-type of friction, signorini type normal contact law, viscous friction etc. following extended problem of least constraints is proposed: q, M q h, q Ṗ D, 5 q 2 where Ṗ D is the total time rate of change of the total dissipation function. The proposed Gauss principle associated with 5 is presented as well. Further, the relation of the PMD to the principle of Gauss as well as to the dual principle of least constraints is established, by considering the conditions under which the time-rate of the total dissipation function P D with respect to passive dissipative forces is maximized. The relation of the PMD to the principle of Gauss and the principle of least constraints is accomplished by the generalization of a principle of Pozharitskii to general dissipative forces and showing that the function Pozharitskii added to the "energy of acceleration" is the total time derivative of the total dissipation function. This is achieved by the techniques of nonsmooth analysis and by using the most general definition of dissipative forces indeed more general classes of forces then only friction forces are covered. Pozharitskii provided in [2] an extensive analysis for the existence and uniqueness of the solution in generalized accelerations for the imization of his proposed augmented function, which is handled in this work by the properties of convexity in nonsmooth analysis. It is shown, that the extended problem of least constraints as in 5 possesses always a imum due to the strict convexity of the goal functional. This relatively simple argumentation on the uniqueness of the imum is one of the advantages one derives from nonsmooth analysis. The strict convexity is of the extended problem of least constraints requires that the dissipation potentials involved on velocity and acceleration level are convex. Though dissipation potentials due to viscous friction and nonconvex dissipation potentials are not addressed in this work, viscous friction can be treated straight forward with the results of this work, whereas the consideration of nonconvex dissipation potentials which often are encountered in continuum mechanics, of course makes the optimality conditions less sharp with regards to sufficiency. The dissipation function was introduced by Rayleigh in his classical treatise for resisting forces which are linear in the velocities. Lurie extended this idea to dissipative forces which were higher powers of the velocity in his monograph [5]. He obtained by setting the power to zero the dissipation function for the dry friction forces and by setting to one the Rayleigh dissipation function. In modern mechanics, which is equipped with potential theory and nonsmooth analysis the dissipation functions of Rayleigh and Lurie are velocity potentials and need not to be differentiable. A superpotential is a lower-semicontinuous proper convex function Φ such that the inclusions x Φy and y Φ x hold between

3 two dual variables x any y and expresses some physical law. The second inclusion expresses the so-called inverse law and Φ x is the conjugate of Φy. In the sequel, superpotentials for normal and friction forces are introduced, which are related to dissipation functions in the classical sense. Further, these superpotential laws for frictional and normal unilateral contact are extended to the acceleration level in the sense of Glocker s work. The class of superpotentials is defined by Moreau in [8]. It is known as stated in [] that in the case of Tresca type friction in which the normal force does not depend on the dynamics, the friction force is modelled by using nonsmooth superpotentials which is not the case for Coulomb type dry friction. The research that has been conducted on systems with friction and contact mechanics is vast. A broad overview on finite-dimensional contact mechanics is provided in [3], [4] and [9] and in their cited references. The excellent reviews of Stewart in [23] and [24] provide overviews on rigid body dynamics with friction and impact. The mathematical tools used for friction of the modelling and the derivation of the equivalence heavily relies on nonsmooth analysis and its application to the theory of optimization. The concept of variational inequalities has been an outcome of the nonsmooth analysis which, plays an important role in applied mathematics since its introduction by Hartman and Stampacchia []. The proofs for the main theorems and the detailed reading on various definitions in convex analysis can be verified in [22]. 2 Preliaries Let f be an extended-valued convex function from R n to [, ] and let x be a point where f is finite. The one-sided directional derivative of f at x with respect to a vector y is defined as: f fx γy fx x; y = lim. 6 γ γ If it exists and are allowed. A vector x is a subgradient of a convex function f at a point x if fz fx x, z x, z. 7 The set of all subgradients of f at x constitute the subdifferential of f at x and is denoted by fx and is a closed convex set. The tangent cone T C x to a set C at a point x C is a nonempty convex closed cone and is polar to a certain nonempty closed convex cone N C x: N C x = {z y,z, y T C x}, 8 T C x = { y y,z, z N C x }. 9 The set N C x is defined to be the normal cone to C at x. the function Ψ C is the indicator function of the set C : Ψ C x =, x C, Ψ C x =, x / C. Fig.. In one dimension: a The distance function to the origin, b Conjugate of the distance function to the origin, c Subdifferential of the distance, d Subdifferential of the conjugate. An important special case in the theory of subgradients is the case where f is the indicator of a non-empty convex set C, then its subdifferential x Ψx is given by the normal cone at x to the set C, x Ψx = N C x. Let f be any closed convex function on R n, then the conjugate of fx is defined as: f x = sup { x, x fx}. 2 x The conjugate of f is f. The conjugate of f together with f fulfill following equality: f x fx = x, x. 3 The distance function d C which measures the distance to a closed set and is given by: d C x = inf{ x s s C }. 4 The conjugate of the distance function to a convex set d C is given by: d C x = sup{ x, y d C y} = Ψ C x Ψ B. 5 y The function Ψ C is the conjugate of the indicator function of the set C. Let f be a proper convex function. The domain of f, dom f is defined as the region where f is finite. For x / domf, fx is empty. For x in the relative interior of dom f, fx

4 is non-empty, f x; y is closed and proper as a function of y, and f x; y = sup{ x, y x fx} = Ψ x fx y 6 is valid. Further, fx is a nonempty bounded set if and only if x is an element of the interior of the domain of f, in which case f x; y is finite for every direction y. At the imum of a convex function f following condition needs to hold: x fx. 7 3 Dissipative Force Laws and Dissipative Force Potentials A force law characterizes the relation between the kinematics and forces. This characterization encompasses the specification of the force direction in relation to the relative kinematics and/or the magnitude of the force. A force γ is called a dissipative force, if the power of the force element P γ v fulfills: P γ v := γv,v. 8 The nonpositivity of the power of the force element is visualized best if one considers a typical friction force law. One immediately notices that during sticking of the contact the power dissipation is zero, and is negative only during sliding. A dissipative force is passive if the velocity is zero. A dissipative force is strictly dissipative if P γ v < whenever the velocity is nonzero. If there exists dissipative superpotentials in the sense of Moreau s work [9], then there exists a scalar potential Φ, which is a lower-semicontinuous convex function with domφ /, such that the dissipative force is given by: γ v Φv. 9 The conjugate of the scalar potential Φ relates the force and the velocity in the following form: v γ Φ γ. 2 Let q, q, q represent the position, velocity and acceleration in the generalized coordinates of the basic configuration of a finite-dimensional Lagrangian system with n maximal degrees of freedom DOF, respectively. A contact between two rigid bodies is seen as a point-to-point contact so that line-to-surface, line-to-line and surface-to-surface contacts are not considered within the modelling framework of this work. Let the scalar function d N q denote the shortest normal distance between two rigid bodies in the system as depicted in figure 3. The normal contact distances are always nonnegative due to the impenetrability at rigid body contacts. Further, there exist no attracting forces at the rigid body contacts. A contact is considered as closed if the normal contact distance d N, normal contact velocity v N are both zero. The relation between these kinematic entities can be given as follows: ḋ N = v N = d N q q = JT N q. 2 The normal contact acceleration is analogously given by: v N = J T N q J T N q = J T N q p N. 22 The tangential relative contact velocity and acceleration are defined by: v T = J T T q and v T = JT T q J T T q = JT T q p T, 23 respectively. Though anisotropic friction on any convex set can be modelled within this framework,without loss of generality the friction force is assumed to exist on a disk with normal contact force dependent radius in R 2. The friction force is limited to the following set: γ T µγ N B 2 = C T, 24 where B 2 is the unit ball in two dimensions. Let for a given positive γ N and µ, let f T be defined as: f T = γ T µγ N. 25 It is obvious that f T B 2. By the resemblance of the dissipativity condition γ T,v T to the definition of the normal cone in 8, following inclusion is proposed: v T N B2 f T. 26 The subdifferential of the indicator function of a convex set is given by its normal cone at that location. By making use of this relation, one has: v T Ψ B2 f T. 27 The inclusion of the 3-D friction in 27 stems from Moreau as first described in [2]. According to inclusion the conjugate of the superpotential Φ is Ψ B2 f T. The superpotential itself is obtained by conjugacy: Ψ B 2 v T = sup f T { f T, v T Ψ B2 f T }. 28

5 According to relation 5, we have Ψ B 2 v T = d {} v T, which is the distance function of the tangential velocity to the origin. By this relation following is valid: γ T µγ N d {} v T. 29 The function µγ N d {} v T is a nonsmooth function, which is the dissipation function for spatial isotropic friction. It encompasses also the behaviour of the frictional contact at sticking which is enabled by its nonsmoothness at the origin. However, it serves as a superpotential in the sense of Moreau only in case of tresca-type dry friction where the normal force is known. The relation between the kinematics in normal direction and normal contact force γ N for a single contact is expressed as Signorini type complementarity relations: d N > γ N =, 3 d N =, v N > γ N =, 3 d N =, v N =, v N = γ N, 32 d N =, v N =, v N > γ N =. 33 It is easily verified that the Signorini type unilateral contact law defines a dissipative force law if one notices that v N, γ N = is valid. The dissipation condition requires v N, γ N to hold, which is in accordance with 33 moulded into a normal cone inclusions: v N N R γ N, γ N N R v N 34 for contacts having d N =. The dissipation function of the normal contact force law is: Φv N = Ψ R v N. 35 This definition of dissipativity encompasses the normal force potential formulated on velocity level. 3. The Local Behaviour of the Dissipation Functions The nonsmooth dissipation potentials capture the behaviour of the dissipative force laws also at the origin. The relative velocity is approximated by the Taylor series for small enough τ: vt τ vt vtτ. 36 The generalized directional derivative of the function d {} v T at the origin in the direction v T t is given by: d {} vt ; v T = limsup v T τ d {} vt v T τ d {} v T v T τ =. 37 This result shows that locally around v T =, one has d {} v T = d {} v T. By the relation 6 following is valid: d {} vt = ; v T = Ψ B v T = d {} v T. 38 The upper subderivative of the normal force velocity potential Ψ R v N at the origin in any direction v N t given by is given Ψ R vn = ; v N = {, if v N / T R v N =,, if v N T R v N =. 39 The tangent cone T R at is given again given by R. So the upper subderivative in is given as: Ψ R vn = ; v N According to 6 relations hold: Ψ R v N = sup γ N { d {} v T = sup f T = ΨR v N = Ψ v R N. 4 γ N, v N }, γn Ψ R v N,4 { ft, v T, ft d {} v T }. 42 By conjugacy the equality in 3 applies: Ψ R v N ΨR γ N = v N, γ N, vn =, 43 d {} v T ΨR γ T = v T, f T, vt =. 44 The time-rate of change of the friction force dissipation function except at the origin is given by: vt µγ N ḋ {} v T = µγ N v T, a T = γ T v T, a T. 45 At the origin it is given by: d dt µγn d {} v T vt = = µγ N sup η, a T, η B η which amounts to: d µγn d dt {} v T vt = = µγ N d {} a T. 47 The derivative of the normal contact velocity potential with respect to time is given by: Ψ R v N = a, η, η Ψ R v N. 48

6 Fig. 2. The spatial frictional disc C T γ N with normal force dependent radius and various normal cones red. Fig. 3. The geometry of a single spatial rigid body contact between two rigid bodies. 4 Mathematical Modelling of Mechanics In order to formulate the index sets that account for the force laws properly following index sets are introduced: I S = {i d Ni = }, I N = {i d Ni =, v Ni = }. The set I S denotes the set of all contacts that are closed on position level of the system. The set I N, with a number of k elements, denotes the set of all contacts at which normal contact velocity and normal contact distance equal to zero. In this case the active dissipative friction forces are the contacts at which sliding takes place. The set of passive friction forces consist of those which stick v T =. The index sets of sliding contacts and sticking contacts are named I NA and I NP, respectively. The normal contact force is passive if the associated normal contact velocity is zero. The sticking and the sliding contact forces are incorporated by the appropriate generalized force directions in the equations of motion, which are the columns of the linear operators J TA and J TP, respectively. For a finite-dimensional Lagrangian system with n degrees of freedom and l active sliding contacts and m passive sticking contacts, which have spatial friction, J TA q is a n by 2l and J TP q is a n by 2m linear operators of generalized friction force directions. The linear operator J N q is a n by k linear operator of generalized normal force directions. Given this setting the differential inclusion of a mechanical system with tresca-type friction at the contacts is stated as: M q h J TA γ TA J TP γ TP J N γ N =, a.e. 49 γ Ni Ψ R a Ni, i I N, 5 v Ti γ TAi = µ i k Ni v Ti, i I NA, 5 γ TPi µ i k Ni atpi d {} a TPi, i I NP. 52 Mq is the symmetric positive-definite PD mass matrix and hq, q represents the vector of gyroscopic, centripetal and coriolis forces and encompasses also smooth force elements such as springs and dampers. Here k Ni are the normal contact forces at the tresca-type frictional contacts for which the normal force is not dependent on the future generalized accelerations at a given state. Solving 49 for the generalized accelerations yields: q = M h J TA qγ TA J TP qγ TP J N qγ N. 53 Insertion of 53 into relations 22 and 23 reveals the relative contact normal and tangential accelerations in relation to the dynamics and contact forces: a TP = J T TPM h J TA γ TA J TP γ TP J N γ N p TP, a N = J T N M h J TA γ TA J TP γ TP J N γ N p N, a TA = J T TA M h J TA γ TA J TP γ TP J N γ N p TA. The relations above are rearranged as: [ an a TP ] = [ GNN G NP G PN where various entries are given by: G PP ][ γn γ TP ] [ κn G NN = J T N M J N, G NP = J T N M J TP, G PP = J T TP M J TP, G PN = J T TP M J N, κ N = J T NM h J T NM J TA γ TA p N, κ TP = J T TP M h J T TP M J TA γ TA p TP. The equation 54 is abbreviated in the sequel by: κ TP ], 54 a = Gγ κ. 55

7 Having set the stage, the total dissipation function is given as: P D v T,v N = Ψ R v ni µ i k ni d {} v Ti. i I S i I NP I NA 56 The time rate of change Ṗ D is evaluated by making use of 45, 46, 48 as follows: Ṗ D = i I N Ψ R a ni i I NP µ i k ni d {} a Ti 5 Principle of Least Constraints for Dissipative Mechanical Systems The generalized acceleration of the mechanical system is related to passive dissipative forces by the set of relations given by 49 to 52 for given state specified by the position q and the generalized velocity q. Any optimization in the passive dissipative forces can thereby related to a search of the generalized acceleration in the next moment. Following extended problem of least constraints is proposed in this work: q, M q h, q Ṗ D. 58 q 2 The unconstrained imization problem 58 is principle of least constraint for the considered type of mechanical systems. The necessary condition of optimality of 58 takes following form: 6 The Equivalence of the Dual Principle of Least Constraints and the Maximization of the Time-Rate of Dissipation The PMD states that given the position q and the velocity q of the mechanical system, that the dissipative forces arrange themselves such that the dissipation is maximized, however, it does not specify what dissipative forces shall do if the relative velocity vanishes. This maximization of the dissipativity happens recurrently at every position and velocity of the mechanical system. However, the dissipative γ Ti v Ti, a Ti. i I NA forces, influence the evolution of the system by detering 57 the acceleration q in general by entering the equations of motion. In the sequel, the maximization of Ṗ D is investigated, for which no criteria in relation to velocities exist. In this section it is shown that the maximization of Ṗ D with respect to passive dissipative forces: max γ N,γ TP Ṗ D 63 reveals the dual problem of least constraints. This approach is analogous to the PMD on velocity level where dissipation velocity potentials constitute the dissipation function. The dissipation time-rate Ṗ D is equivalently expressed by making use of 43 and 44 as: Ṗ D = i I N γ Ni a Ni, a Ni Ψ R γ Ni 64 i I NP γ Ti a Ti, a Ti Ψ CTi γ Ti i I NA γ Ti v Ti, a Ti Mq q hq, q q Ṗ D 59 The directional derivative of the unconstrained goal functional requires for a imum following condition: Ṗ D ; q q M q h, q q 6 The vector q denotes the arbitrary unconstrained virtual accelerations. The requirement of 6 represents the extended Gauss Principle proposed by the author. The dependence of the relative accelerations as in Ṗ D are given by relations 2 and 22. The directional derivative of Ṗ D becomes by making use of 2 and 22: Ṗ D ; q q = J N γ N J TP γ TP J TA γ TA, q q 6 By combinining 6 and 6 following optimality condition is obtained: M q h J N γ N J TP γ TP J TA γ TA, q q, 62 which for arbitrary unconstrained virtual accelerations q is only fulfilled if the equations of motion given by 49 hold. The dissipation time-rate 64 is strictly concave in the passive dissipative forces. The directional derivative of Ṗ D with respect to feasible directions of passive contact forces must fulfill following condition at a maximum: Ṗ D ; γ T ã T γ T a T, γ N ã N γ N a N. 65 The directional derivative in 65 is explicitly given by: Ṗ D ; γ γ = γ Ni γ Ni, a Ni Ψ R ; γ Ni γ Ni i I N γ Ti γ Ti, a Ti Ψ C Ti ; γ Ti γ Ti. 66 i I NP The necessary condition in 66 is given in compact form by: Ṗ D ; γ γ = a, γ γ Ψ C γ ; γ γ, 67 where the set is C γ defined as: C γ = R... R C T... C Ts. 68 }{{} ktimes

8 This variational inequality is equivalently expressed as an inclusion: Ψ C γ ; γ γ a, γ γ Gγκ Ψ Cγ γ. 69 The variational inequality and the inclusion as given in 69 are recognized as the necessary and sufficient conditions for the following convex quadratic programg problem: γa 2 [ γn [ κn γ TP ] T [ GNN κ TP ], ][ ] G NP γn G PN G PP γ TP [ ] γn Ψ γ Cγ γ. 7 TP In [9] an alternate method in obtaining the QP in 7 is presented by inserting the relation 55 into: a Ψ Cγ γ 7 for general superpotential force laws without obtaining first a variational inequality as a necessary condition for an optimization problem. This insertion reveals a normal cone inclusion which is the optimality condition for the dual problem of least constraints of the type: γ 2 γt Gγ κ, γ. References [] V. Acary and B. Brogliato. Numerical Methods for Nonsmooth Dynamical Systems-Applications in Mechanics and Electronics, volume 35 of Lecture Notes in Applied and Computational Mechanics. Springer, 28. [2] M. Anitescu and A. P. Florian. A time-stepping method for stiff multibody dynamics with contact and friction. Int. Journal for Num. Meth. in Engr., 55: , 2. [3] B. Brogliato. Nonsmooth Mechanics: Models, Dynamics and Control. Communications and Control Engineering. Springer, 2nd edition, 999. [4] B. Brogliato. Dissipative Systems Analysis and Control: Theory and Applications. Communications and Control Engineering. Springer, 2nd edition, 27. [5] C. F. Gauss. Über ein neues allgemeines grundgesetz der mechanik. J. Reine Angew. Math., 4: , 829. [6] B. I. Gavrea, M. Anitescu, and F. A. Potra. Convergence of a class of semi-implicit time-stepping schemes for nonsmooth rigid multibody dynamics. SIAM J. Optimization, 9:969, 28. [7] Ch. Glocker. Formulation of rigid body systems with nonsmooth and multivalued interactions. Nonlinear Analysis, Theory, Methods and Appl., 38: , 997. [8] Ch. Glocker. The principles of d alembert, jourdain, and gauss in nonsmooth dynamics part : Sclerenomic multibody systems. Zeitschrift für Angewandte Mechanik und Mathematik, 78:2 37, 998. [9] Ch. Glocker. Set-Valued Force Laws Dynamics of Non-Smooth Systems, volume of Lecture Notes in Applied Mechanics. Springer Verlag, 2. [] P. Hartman and G. Stampacchia. On some nonlinear elliptic differential equations. Acta Math., 5:53 88, 966. [] A. Klarbring. Contact, friction and discrete mechanical structures and mathematical programg. In Lecture Notes for the CISM Course-Contact Problems: Theory, Methods, Applications, pages 55. Springer, 999. [2] A. Klarbring. Contact, friction and discrete mechanical structures: Analogies and dynamic problems. In Lecture Notes for the CISM Course-Multibody Dynamics with Unilateral Contact, pages Springer, 2. [3] P. Lötstedt. Coulomb friction in two-dimensional rigid body systems. ZAMM, 6:65 65, 98. [4] P. Lötstedt. Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math., 42:28 296, 982. [5] Lurie. Analytical Mechanics. Springer, 22. [6] J. J. Moreau. Les liaisons unilatérales et le principe de gauss. Ibid., 256:87 874, 963. [7] J. J. Moreau. Quadratic programg in mechanics: Dynamics of one-sided constraints. J. SIAM. Control, 4:53 58, 966. [8] J. J. Moreau. La notion de sur-potentiel et les liasions unilatérales en élastostatique. C. R. Acad. Sci. Paris, A27: , 968. [9] J. J. Moreau. New variational techniques in Mathematical Physics, chapter On unilateral constraints, friction and plasticity, pages [2] J. J. Moreau. Unilateral Contact and dry friction in finite freedom dynamics, chapter On unilateral constraints, friction and plasticity, pages [2] G. K. Pozharitskii. Extension of the principle of gauss to systems with dry coulomb friction. Journal of Appl. Math. and Mech., 253:586 67, 96. [22] R. T. Rockafellar. Convex Analysis. Princeton Landmarks in Mathematics and Physics. Princeton University Press, 97. [23] D. E. Stewart. Rigid-body dynamics with friction and impact. SIAM Review, 42:3 39, 2. [24] D. E. Stewart. Finite-dimensional contact mechanics. Phil. Trans. R. Soc. Lond. A, 359: , 2. [25] D. E. Stewart and J. C. Trinkle. An implicit timestepping scheme for rigid-body dynamics with inelastic collisions and coulomb friction. Int. Journal for Num. Meth. in Engr., 39: , 996. [26] C. Studer and Ch. Glocker. Representation of normal cone inclusion problems in dynamics via nonlinear equations. Archive of Applied Mechanics, 765-6: , 26.