A Geometric Interpretation of Newtonian Impacts with Global Dissipation Index

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1 A Geometric Interpretation of Newtonian Impacts with Global Dissipation Index Christoph Glocker Institute B of Mechanics, Technical University of Munich, D Garching, Germany (glocker@lbm.mw.tu-muenchen.de) December 31, 2000 Abstract The paper treats the frictionless multi-contact collision problem in scleronomic rigid body dynamics according to Newton s kinematic impact law in terms of inequalities. A geometric construction of the impact law together with related variational formulations is first proposed for the convex case and then extended to the non-convex case of reentrant corners. The resulting impact law is stated as an inclusion and likewise as a stationary point problem. 1 Introduction Newton s classical kinematic impact law provides a method how to calculate the post-impact velocities of two colliding particles. It reverses the sign of the relative velocity γ at the impact and takes into account dissipation by a coefficient of restitution ε such that γ + = ε γ. In this paper we extend the classical approach to frictionless multi-contact collision problems in multibody systems [7] which requires a reformulation of Newton s impact law in terms of inequalities. Only the impact event itself is investigated, where it is assumed in addition that the coefficients of restitution of the different contacts are all equal to each other. Such an event is then called a Newtonian impact with global dissipation index, and it agrees with Moreau s non-smooth dynamical equations [6] when they are evaluated at a single instant of time. In Section 3 we present a geometric interpretation of this class of impacts which is based on the decomposition of the pre-impact velocity with respect to a pair of orthogonal convex cones. It is shown that the impact law can be expressed alternatively by an extended complementarity condition, an inclusion, a variational inequality, an orthogonal projection, or a nearest point problem. Most of these results are already found in [6]. We have put them together to find a natural generalization including also the non-convex case of reentrant corners which is addressed at the end of the paper. The resulting formulation goes far beyond simple unilaterally constrained motion and includes, for example, the situation of two rectangular blocks hitting each other at their corners. The different representations of the impact law are developed by using tools from convex analysis, in particular convex cones. The according definitions and related variational expressions are shortly summarized in Section 2. 1

2 2 Cones in Non-Smooth Analysis In this section some basic definitions, notations and properties of cones used in non-smooth analysis are put together. These cones characterize sets in the neighborhood of a chosen point and indicate whether convexification as the next step available after linearization is possible or not. As a standard tool for inequality systems, these cones will be used throughout the paper; the related representations, variational formulations and decomposition rules as far as needed are presented in this section. The material is mainly taken from [1] and [8] but applied only to finite-dimensional spaces. 2.1 Cones and convexification Let C be a non-empty subset of R f and q 0 C. Following [1] the contingent cone K C and the tangent cone T C to C at q 0 are defined as K C (q 0 ) := ( ) 1 (C q µ 0) + δb, (1) δ>0 α>0 0<µ α T C (q 0 ) := ( ) 1 (C q) + δb (2) µ δ>0 α>0 β>0 0<µ α q B C (q 0,β) where δb denotes the closed ball at center zero with radius δ, and B C (q 0, β) := C (q 0 +βb). We set K =, T = and K C (q 0 ) =, T C (q 0 ) = if q 0 / C. Both cones are closed subsets of R f and, if not empty, they always contain 0. Furthermore, T C (q 0 ) is convex and is always contained in K C (q 0 ). For C being convex we have T C (q 0 ) = K C (q 0 ). If C is a smooth manifold of class C 1, then T C (q 0 ) = K C (q 0 ) and they are identified with the usual tangent vector space to C at q 0 which is a subspace of R f. Note that for q 0 int C both cones become T C (q 0 ) = K C (q 0 ) = R f. A set C is called to be tangentially regular at a point q 0 if T C (q 0 ) = K C (q 0 ). Tangential regularity holds in particular for the three cases mentioned above, i.e. when C is a convex set or a smooth manifold, or when q 0 is not a boundary point of C. For non-convex sets we may have T C (q 0 ) = K C (q 0 ) but also points q 0 C at which this equality does not apply, called the re-entrant corner points of C. We may further introduce an object playing the role of normal vectors to C at q 0. This object is called the normal cone to C at q 0 and is defined via the variational inequality N C (q 0 ) = { r r T v 0 v T C (q 0 ) }, (3) providing the set of all linear maps acting on v with values in R 0. We set N = and N C (q 0 ) = if q 0 / C. The normal cone is a closed, convex cone always containing 0 if not empty. The cones T C (q 0 ) and N C (q 0 ) are mutually polar by (3) which allows to express the orthogonality of spaces in terms of polarity of cones. One observes that N C (q 0 ) = R f T C (q 0 ) = 0 and vice versa. If T C (q 0 ) is a half-space then N C (q 0 ) degenerates to a ray orthogonal to the boundary of C at q 0. If T C (q 0 ) is a subspace of R f, then N C (q 0 ) is its orthogonal complement in the dual (R f ). For C a subset of a differentiable manifold M the cones K C (q 0 ) and T C (q 0 ) at some point q 0 C are contained in the tangent space of M at q 0, whereas N C (q 0 ) is a subset of the cotangent space. We denote by M(q 0 ) the natural isomorphism induced by the metric on 2

3 Figure 1: The cones K C, T C and T C at different points of C. M in use which takes at q 0 elements from the tangent to the cotangent space and introduce, in addition, the image and pre-image of the cones T C (q 0 ) and N C (q 0 ), N C (q 0 ) := M(q 0 ) T C (q 0 ), T C (q 0 ) := M 1 (q 0 ) N C (q 0 ). (4) By (2), (3) and (4) we have thus obtained two pairs of orthogonal cones (T C (q 0 ), T C (q 0)) and (N C (q 0 ), N C (q 0)) characterized by v v 0 v T C (q 0 ), v TC (q 0 ), r r 0 r N C (q 0 ), r NC (q 0 ), (5) where the dots denote the associated inner products on the tangent and the cotangent space, respectively. Figure 1 shows the cones K C (q 0 ), T C (q 0 ) and TC (q 0) for different situations. In (a) (d) one always has T C = K C due to convexity of C. Non-convex sets C are depicted in Figures (e) (l), where (e) (h) shows the cones for points at which C is tangentially regular and (i) (l) configurations at re-entrant corner points of C. 2.2 Representation of the normal cone The cones introduced in the preceding section are, in general, quite hard to get. We will only discuss the case that a Riemannian manifold is restricted to a set C by simple unilateral 3

4 Figure 2: Normal and tangent cone for simple unilateral constraints. smooth constraints g i (q) 0 as depicted in Figure 2, C := { q g i (q) 0, i = 1,..., m }. (6) Denote by w i the differential of g i and by H the indices of the active set of constraints at some point q 0 C, wi T (q 0 ) := g i q q0, H(q 0 ) := { i g i (q 0 ) = 0 }, (7) and assume that w i 0 when i H. The normal cone N C (q 0 ) to C at q 0 is then generated by the vectors w i (q 0 ) of the active set, N C (q 0 ) = { r r = α i w i, α i 0 }, (8) and the elements of the tangent cone are by the inverse of (3) characterized as v T C (q 0 ) α i wi T v 0, α i 0. (9) It should be noted that the normal and the tangent cone associated with simple unilateral constraints are already polyhedral; more general situations occur if the set C of admissible displacements may not be obtained by the intersection of simple smooth inequality constraints, such as the circular cone { λ(e 3 + D 12 ), λ 0 } in R 3, where e 3 denotes the unit vector in the 3-direction and D 12 the closed unit disc centered at 0 in the 1-2-plane. 2.3 Orthogonal decompositions In this section we review briefly the decomposition of a vector with respect to a pair of orthogonal convex cones and discuss some other related formulations. The following statement is contained as a special case in Moreau s Theorem, see e.g. [8] for the full version and the proof, and generalizes the classical orthogonal vector decomposition. 4

5 Figure 3: Orthogonal cones R, R and the orthogonal tangent cone of R at ( ), T R ( ). Theorem 2.1. Let U be a finite-dimensional real inner product space, R a closed convex cone in U and R the closed convex cone in U orthogonal to R defined by R = { v v v 0 v R }, (10) where denotes the inner product on U. Any u U can then be expressed uniquely as a sum u = v + v such that v R, v R, v v = 0. (11) This decomposition is depicted in the left part of Figure 3. For u / R R one obtains for the decomposition v 0, v 0, whereas v = 0 or v = 0 as soon as u R or u R. Apparently is v the nearest point to u in the set R, which is usually denoted by v = prox R (u). The corresponding map u prox R (u) is called a proximation which is, in fact, a projection because prox 2 R (u) = prox R(u). In addition, this projection is orthogonal since prox R (u) (u prox R (u)) = 0. Of course, the same properties apply for the second term v in the decomposition due to symmetry, i.e. v = prox R (u). Proposition 2.2. Let U, R and R be defined as in Theorem 2.1. Then the following conditions are equivalent to each other: v R, v R, v v = 0 (12) v R, v T R (v) (13) v R, v T R (v ) (14) Here, T R (v) and T R (v ) denote the cones orthogonal to the tangent cones to R at v and R at v, respectively. The cone TR (v) is depicted in the right part of Figure 3 for some elements v R. Proof. In order to show the equivalence of (12) and (13) we first recall that the normal cone to a convex set R at a point v R is N R (v) = { r r T (v v) 0, v R, v R }, (15) 5

6 see e.g. [8]. It consists of all vectors r in the dual U of U which do not make an acute angle with any line segment in R with v as endpoint. The corresponding cone in U is thus by (4) the set T R (v) = M 1 N R (v), i.e. T R (v) = { u u (v v) 0, v R, v R }. (16) with u = M 1 r. Further, we note that (i) R = { u u v 0, v R } (ii) TR (v) = { u u z 0, v R, z R v } (iii) TR (v) R with TR (0) = R where (i) is the definition of R from (10), (ii) is obtained from (16) when setting z := v v, and (iii) holds because (ii) is more restrictive on u than (i): The set R v is the translate of the closed convex cone R in the direction v with v R, hence R v R. Suppose now that (12) holds. The second condition, v R, means by (i) that v v 0 v R. Subtracting from this expression the third condition v v = 0 in (12) gives v (v v) 0 v R. This is by (16) together with v R already v TR (v) in (13). To show the converse we rewrite v TR (v) in (13) with the help of (16) as a variational inequality, v (v v) 0, which has to hold for any v R when v R. Choose now v = 2v R and v = 0 R, and evaluate the variational expression for both cases. This gives v v 0 and v v 0, hence v v = 0 which is the third condition in (12). It remains to show that v R which has already been done in (iii). To prove the equivalence of (12) and (14) one proceeds exactly in the same manner. 3 Newtonian Impacts with Global Dissipation Index We consider a second order scleronomic dynamical system with f degrees of freedom and denote by t time, by q(t) R f the generalized coordinates, and by u(t) R f the generalized velocities as functions of bounded variation with q(t) = u(t) almost everywhere. Further, M(q) R f,f is the symmetric and positive definite mass matrix of the system, and h(q, u) the vector of finite gyroscopical accelerations and applied forces. A total of m simple perfect holonomic scleronomic unilateral (frictionless) constraints g i (q) are imposed. We denote by γ i (t) the associated relative velocities, i.e. the bounded variation velocities induced by u(t) for which ġ i (q(t)) = γ i (t) almost everywhere. The reactions caused by the unilateral constraints are taken into account by Lagrange s equations of first kind, such that the overall problem stated in terms of an equality of measures is M(q) du h(q, u) dt m w i (q) dλ i = 0, g i (q) 0, w T i (q) := g i q i=1, i = 1,..., m, γ i (t) = w T i (q) u(t), i = 1,..., m, contact-impact law. Note that a particular contact-impact law relating the contact percussion-force measure dλ i with the kinematic terms g i and γ i has not yet been specified. (17) 6

7 In order to obtain the impact equations the first line in (17) is integrated over a singleton {t 0 }. Let C be as defined in (6) and q(t 0 ) =: q 0 C. Integration yields M(q 0 )(u + (t 0 ) u (t 0 )) = w i (q 0 ) Λ i (t 0 ), γ ± i (t 0) = w T i (q 0 ) u ± (t 0 ), i H(q 0 ), impact law with Λ i the contact percussion at t 0, u ± and γ ± i the right and left limits of u and γ i at t 0, and H(q 0 ) denoting the active set (7) already incorporating the contact part of the contactimpact law in (18): For an open contact (i / H) there is no reaction, in particular no contact percussion (Λ i (t 0 ) = 0). In the following Newton s impact law is investigated. It relates the relative velocities γ ± i (t 0) with the percussions Λ i (t 0 ) of the closed contacts (i H(q 0 )). For brevity we suppress the dependencies on q 0 and t Newton s Impact Law When stated in terms of inequalities, Newton s impact law is a complementarity condition on the kinematics of the form (18) (γ + i + ε i γ i ) 0, Λ i 0, (γ + i + ε i γ i ) Λ i = 0. (i H) (19) It completes the impact equations (18) and incorporates the following physical behavior: The contact percussion Λ i acts only as a compressive magnitude (Λ i 0). When positive, the relative velocity is reversed according to the classical rule γ + i = ε i γ i with 0 ε i 1 the restitution coefficient. If Λ i = 0 the associated contact does not participate in the impact, thus even bigger post-impact velocities (γ + i ε i γ i ) may be allowed. Suppose now that ε := ε i = ε j i, j H, i.e. a Newtonian impact with global dissipation index. By setting ξ i := 1 1+ε (γ+ i + εγ i ) (20) the impact law (19) becomes ξ i 0, Λ i 0, ξ i Λ i = 0 and may thus be written for all contacts i H together as a variational inequality Λ i (ξi ξ i ) 0 (21) with ξ i 0 and for all ξi 0. These remaining two sets of inequalities are, again, formulated in variational form, ξ i such that α j ξ j 0 α j 0, j H ξi such that α j ξj (22) 0 α j 0, j H which clearly implies ξ i 0 and ξi 0. The Newtonian impact law (19) for all contacts i H is thus equivalently expressed by the conditions (20) (22) and completes the impact equations (18). 7

8 3.2 A Variational Formulation of the Impact Equations In this section a variational form of the Newtonian impact (18), (20) (22) in the configuration space is presented by eliminating the local contact-impact terms Λ i, γ ± i, ξ i. We start with eliminating γ ± i by substituting the second equation of (18) into (20). The complete set of impact equations is then M(u + u ) = w i Λ i, ξ i = wi T ( 1 1+ε u+ + ε 1+ε u ), Λ i (ξi ξ i ) 0, ξ i such that α j ξ j 0 α j 0, (23) j H ξi such that α j ξj 0 α j 0. j H Denote now by v the term 1 1+ε u+ + ε 1+ε u which yields ξ i = wi T v, and regard the variations ξi as linear functions of some velocity variations v, i.e. ξi = wi T v, in order to eliminate ξ i and ξi from (23). This gives M(u + u ) = w i Λ i, v = 1 1+ε u+ + ε 1+ε u, Λ i wi T (v v) 0, v such that α j w T (24) j v 0 α j 0, j H v such that α j wj T v 0 α j 0. j H Next, we introduce the overall contact percussion R := w i Λ i in R f and take into account the representation of the tangent cone (9) for simple unilateral constraints to rewrite the last two equations in (24), M(u + u ) = R, v = 1 1+ε u+ + ε 1+ε u, R T (v v) 0, v T C (q 0 ), v T C (q 0 ). (25) After elimination of R the Newtonian impact problem may thus be stated in the following variational form: For and given u find u + such that v = 1 1+ε u+ + ε 1+ε u (26) (v v) T M(u + u ) 0, v T C (q 0 ), v T C (q 0 ). (27) These are by (15) exactly the conditions for an element M(u + u ) to belong to the normal cone to the convex set T C (q 0 ) at the point v, M(u + u ) N TC (q 0 )(v), v T C (q 0 ), (28) which expresses Newton s impact law as an inclusion [6]. We have thus proven the following: 8

9 Proposition 3.1. The impact laws (26), (27) and (26), (28) are equivalent to the formulation (18), (19) of Newton s multi-contact impact law for simple unilaterally constrained systems with global dissipation index in the sense that for any u + satisfying (26), (27) or (26), (28) values of Λ i and γ i can be found such that the local impact laws in (19) hold. Note, however, that (26) (28) can be regarded as a generalization of (18), (19) because these formulations hold beyond simple unilateral constraints. They are still valid for (tangentially regular) sets C which can not be expressed by the intersection of a finite number of inequalities g i (q 0 ) 0 with non-vanishing gradients g i (q 0 ) = M 1 (q 0 ) w i (q 0 ) at their boundaries. Along with the impact equations, the impact law R N TC (q 0 )(v) for systems with global dissipation index singles out one particular element R of N C (q 0 ). This element is unique as we will see in the next section, and is always contained in N C (q 0 ), because N TC (q 0 )(v) N C (q 0 ) for v T C (q 0 ) by (iii) in the proof of Proposition 2.2 and (4) which connects T ( ) and N ( ). There is nothing axiomatic behind this impact rule, and one can not expect this law to hold for any impact event in a physical system. The only prediction which may be taken for granted is that R lies in the normal cone N C (q 0 ), see in particular the paper [3] for a general framework on impact laws. We will finally state equation (28) in the tangent space at point q 0. With T T C (q 0 ) (v) = M 1 N TC (q 0 )(v) the cone orthogonal to T TC (q 0 )(v) as a subset of the tangent space we obtain (u + u ) T T C (q 0 )(v), v T C (q 0 ), (29) which can be rewritten according to Proposition 2.2, (12), (13) as (u + u ) T C (q 0 ), v T C (q 0 ), (u + u ) v = 0, (30) where the dot denotes the inner product with respect to the kinetic metric. This formulation will further be used to investigate the geometry of the impact. 3.3 A Geometric Impact Law With u being the pre-impact velocity of the system, we propose the following geometric construction of the impact law: According to Theorem 2.1, perform an orthogonal decomposition of u into such that the vectors v and v satisfy v T C (q 0 ), u = v + v (31) v T C (q 0 ), v v = 0. (32) This decomposition is unique. The term v plays the role of the tangential component of u which remains unchanged by the impact, whereas the normal component v is inverted by the impact rule v := εv (0 ε 1), (33) where ε is the coefficient of restitution as before. The post-impact velocity is then set to be u + := v + v (34) according to the construction shown in Figure 4, see e.g. also [2] for comments on Moreau s sweeping process with soft constraints (ε = 0) in the frictionless case. 9

10 Figure 4: The geometry of impacts with global dissipation index. Proposition 3.2. For u T C (q 0 ) the impact law (31) (34) yields kinematically admissible post-impact velocities, i.e. u + T C (q 0 ). Proof. Denote by L(q 0 ) a two-dimensional subspace containing u, v and v. This L(q 0 ) is uniquely generated as soon as two out of the above three vectors are linearly independent. Let further be F C (q 0 ) := L(q 0 ) T C (q 0 ) and FC (q 0) := L(q 0 ) TC (q 0) which are two closed convex cones orthogonal to each other such that one of the two configurations in Figure 5 is met. Consider first the case ϕ 90 for which FC (q 0) F C (q 0 ). In this case it holds for any ε 0 and any v F C (q 0 ), v FC (q 0) that u + = v εv F C (q 0 ), i.e. admissible post-impact velocities are assured independent of the value of u. Consider now ϕ < 90. We have F C (q 0 ) FC (q 0). For u F C (q 0 ) by assumption and a proper choice of L(q 0 ) if any, we have thus u FC (q 0), and the orthogonal decomposition yields v = 0, v = u with v F C (q 0 ). This means by (33), (34) that u + = εv F C (q 0 ) T C (q 0 ) and completes the proof, because this or the other situation applies for any possible L(q 0 ). Note that the impact law (31) (34) may produce non-compatible post-impact velocities u + when the pre-impact velocities u are not from T C (q 0 ), for instance when initial conditions are arbitrarily chosen or as a result of numerical round-offs. For ϕ < 90 as in the right diagram of Figure 5 one has in general u + FC (q 0) which might, of course, violate u + F C (q 0 ). For completely inelastic impacts (ε = 0), however, one has always compatible post-impact velocities. From Figure 4 one also recognizes that the impact law is always energetically consistent: The kinetic energy T satisfies 2 T + = u + 2 u 2 = 2 T, where equality holds for ε = 1 and maximal dissipation is achieved for ε = 0. A completely elastic impact (ε = 1) can be interpreted as a reflection on a hyperplane H with normal v, whereas a completely inelastic impact (ε = 0) corresponds to an orthogonal projection of u on H to give v. In terms of a minimization problem are v and v the nearest points to u in the sets T C (q 0 ) and T C (q 0), respectively. The corresponding maps are called proximations 10

11 Figure 5: On the kinematic compatibility of Newton s impact law. and are denoted by v = prox TC (q 0 )(u ), v = prox T C (q 0 )(u ). (35) For example, (31) (34) might equivalently be stated in terms of proximations as u + = (1 + ε) prox TC (q 0 )(u ) ε u (36) when the first equation in (35) is used. Further, we recognize that the proximation in (36) becomes the identity whenever u T C (q 0 ). In this case u + u, thus no impact occurs. Proposition 3.3. The construction given by equations (31) (34) and in Figure 4 is the geometric version of Newton s impact law (30), (26) for systems with global dissipation index. Proof. We take (31), (33), (34) and solve them for v and v, u = v + v, u + = v εv, v = 1 1+ε (u+ + εu ), v = 1 1+ε (u+ u ), (37) in order to substitute v in the orthogonal decomposition (32), v T C (q 0 ), (u + u ) 1 1+ε T C (q 0), v (u + u ) 1 1+ε This is exactly (30), because we may cancel 1 1+ε 3.4 On Impacts at Re-entrant Corners = 0. (38) > 0 from (38) since T C (q 0) is a cone. We finally consider the case that the boundary of the admissible displacements is not tangentially regular at the point of impact q 0, but can be approximated well by the (non-convex) contingent cone K C (q 0 ). Note that such a configuration can not be generated by the intersection of smooth simple unilateral constraints, but occurs in practice when, for example, two rectangular blocks hit each other at their corners. We try to find a reasonable extension of Moreau s impact law which we do by just replacing the tangent cone with the contingent cone in the appropriate equations. The set of admissible virtual velocities is now defined by 11

12 Figure 6: Impact at a re-entrant corner. the contingent cone, whereas the impulsive forces at the impact are still taken from the normal cone with the corresponding set TC (q 0) in the tangent space. We thus rewrite Newton s impact law (29) now as (u + u ) T K C (q 0 )(v), v K C (q 0 ) (39) with v as defined in (26) and note that the inclusion T K C (q 0 ) (v) T C (q 0) still holds. Of course, the orthogonal decomposition performed in (30) does not longer apply, because convexity of the two participating cones is required in Theorem 2.1. By using the same notation for the decomposition of u and u + as in (37), the impact law (39) becomes u =: v + v, u + =: v εv, (40) v T K C (q 0 )(v), v K C (q 0 ) (41) and is the version of (32) extended to re-entrant corners with the geometrical meaning as sketched in Figure 6. A representation in terms of proximal points also fails, because due to the lack of convexity of K C (q 0 ) one is no longer able to express (41) as a minimization problem. The extended statement of (35) is: Find the stationary points of the function Φ(v) = 1 2 v u 2 + I KC (q 0 )(v), (42) where I N (v) denotes the value of the indicator function of N at v, i.e. I N (v) = 0 for v N and I N (v) = + for v / N. The solution set of this problem contains, among others, the proximal points to u in K C (q 0 ) and is set up by the zeros of the generalized gradient of (42). These are the values of v which satisfy 0 Φ(v) = M(v u ) + I KC (q 0 )(v) = M(v u ) + N KC (q 0 )(v). (43) 12

13 Figure 7: Non-uniqueness of the impact law at re-entrant corners. Or, with v = u v and by taking the terms in (43) from the cotangent space to the tangent space, we obtain v T K C (q 0 )(v), (44) which is together with the restriction v K C (q 0 ) again the impact law (41). Due to the non-convexity of K C (q 0 ) one can not expect the solution set of (41) to consist of one element only. Non-unique post-impact velocities must be accepted to occur for certain configurations, such as for the symmetric case depicted in Figure 7 for ε = 1 and ε = 0, where three different solutions are met. An impact law for re-entrant corners was also introduced in [5] in the sense that (42) is taken as a minimization problem leading, however, to a reduced set of solutions. 4 Conclusion In this paper we have presented several versions of Newton s kinematic impact law including the classical one-point collision problem, but also Moreau s multi-contact formulation for frictionless constraints, and even the case of re-entrant corners in the feasible domain of displacements. Although clearly originating in the collision problem of rigid bodies under simple unilateral constraints, the presented methods must be looked upon as a theoretical framework useful for many impact problems in general finite-degree-of-freedom dynamics, with even a clear geometric meaning relating to projections and reflections. There is a second classical approach to impacts based on an impulse ratio which requires a decomposition of the impact into two succeeding phases, known as Poisson s law. A comparison of both concepts for multi-contact impacts with global dissipation index may be found in [4], where the theory is also extended to moving sets reflecting rheonomic unilateral constraints. References [1] Aubin, J.; Ekeland, I.: Applied nonlinear analysis, John Wiley & Sons, New York [2] Brogliato, B.: Nonsmooth mechanics, 2nd edition, communications and control engineering series, Springer, London [3] Frémond, M.: Rigid Bodies Collisions, Physics Letters A 204, 33 41,

14 [4] Glocker, Ch.: On frictionless impact models in rigid body systems, Phil. Trans. Royal. Soc., to appear. [5] Kane, C.; Repetto, E.A.; Ortiz, M.; Marsden, J.E.: Finite element analysis of nonsmooth contact, Comp. Methods Appl. Mech. Eng. 180, 1 26, [6] Moreau, J. J.: Unilateral contact and dry friction in finite freedom dynamics, In: Non- Smooth mechanics and applications, CISM Courses and Lectures Vol. 302 (eds J. J. Moreau & P. D. Panagiotopoulos), 1 82, Springer, Wien [7] Pfeiffer, F.; Glocker, Ch.: Multibody dynamics with unilateral contacts, John Wiley & Sons, New York [8] Rockafellar, R. T.: Convex analysis, Princeton Univ. Press, Princeton, New Jersey