Nonsmooth Mechanics

Thank you for opening the second edition of this monograph, which is devoted to the study of a class of nonsmooth dynamical systems of the general form: ::i; = g(x, u) (0. 1) f(x, t) 2: 0 where x E JRn is the system's state vector, u E JRm is the vector of inputs, and the function f (-, . ) represents a unilateral constraint that is imposed on the state. More precisely, we shall restrict ourselves to a subclass of such systems, namely mechanical systems subject to unilateral constraints on the position, whose dynamical equations may be in a first instance written as: ii= g(q, q, u) (0. 2) f(q, t) 2: 0 where q E JRn is the vector of generalized coordinates of the system and u is an in put (or controller) that generally involves a state feedback loop, i. e. u= u(q, q, t, z), with z= Z(z, q, q, t) when the controller is a dynamic state feedback. Mechanical systems composed of rigid bodies interacting fall into this subclass. A general prop erty of systems as in (0. 1) and (0. 2) is that their solutions are nonsmooth (with respect to time): Nonsmoothness arises primarily from the occurence of impacts (or collisions, or percussions) in the dynamical behaviour, when the trajectories attain the surface f(x, t) = O. They are necessary to keep the trajectories within the subspace = {x : f(x, t) 2: O} of the system's state space.

1 Distributional model of impacts. - 1. 1 External percussions. - 1. 2 Measure differential equations. - 1. 2. 1 Some properties. - 1. 2. 2 Additional comments. - 1. 3 Systems subject to unilateral constraints. - 1. 3. 1 General considerations. - 1. 3. 2 Flows with collisions. - 1. 3. 3 A system theoretical geometric approach. - 1. 3. 4 Descriptor variable systems. - 1. 4 Changes of coordinates in MDEs. - 1. 4. 1 From measure to Carathéodory systems. - 1. 4. 2 Decoupling of the impulsive effects (commutativity conditions). - 1. 4. 3 From measure to Filippov's differential equations: the Zhuravlev-Ivanov method. - 2 Approximating problems. - 2. 1 Simple examples. - 2. 1. 1 From elastic to hard impact. - 2. 1. 2 From damped to plastic impact. - 2. 1. 3 The general case. - 2. 2 The method of penalizing functions. - 2. 2. 1 The elastic rebound case. - 2. 2. 2 A more general case. - 2. 2. 3 Uniqueness of solutions. - 3 Variational principles. - 3. 1 Virtual displacements principle. - 3. 2 Gauss' principle. - 3. 2. 1 Additional comments and studies. - 3. 3 Lagrange's equations. - 3. 4 External impulsive forces. - 3. 4. 1 Example: flexible joint manipulators. - 3. 5 Hamilton's principle and unilateral constraints. - 3. 5. 1 Introduction. - 3. 5. 2 Modified set of curves. - 3. 5. 3 Modified Lagrangian function. - 3. 5. 4 Additional comments and studies. - 4 Two bodies colliding. - 4. 1 Dynamical equations of two rigid bodies colliding. - 4. 1. 1 General considerations. - 4. 1. 2 Relationships between real-world and generalized normal di-rections. - 4. 1. 3 Dynamical equations at collision times. - 4. 1. 4 The percussion center. - 4. 2 Percussion laws. - 4. 2. 1 Oblique percussions with friction between two bodies. - 4. 2. 2 Rigid body formulation: Brach's method. - 4. 2. 3 Additional comments and studies. - 4. 2. 4 Rigid body formulation: Frémond's approach. - 4. 2. 5 Dynamical equations during the collision process: Darboux-Keller's shock equations. - 4. 2. 6 Stronge's energetical coefficient. - 4. 2. 7 3 dimensional shocks- Ivanov's energetical coefficient. - 4. 2. 8 A third energetical coefficient. - 4. 2. 9 Additional comments and studies. - 4. 2. 10 Multiple micro-collisions phenomenon: towards a global coef-ficient. - 4. 2. 11 Conclusion. - 4. 2. 12 The Thomson and Tait formula. - 4. 2. 13 Graphical analysis of shock dynamics. - 4. 2. 14 Impacts in flexible structures. - 5 Multiconstraint nonsmooth dynamics. - 5. 1 Introduction. Delassus' problem. - 5. 2 Multiple impacts: the striking balls examples. - 5. 3 Moreau's sweeping process. - 5. 3. 1 General formulation. - 5. 3. 2 Application to mechanical systems. - 5. 3. 3 Existential results. - 5. 3. 4 Shocks with friction. - 5. 4 Complementarity formulations. - 5. 4. 1 General introduction to LCPs and Signorini's conditions. - 5. 4. 2 Linear Complementarity Problems. - 5. 4. 3 Relationships with quadratic problems. - 5. 4. 4 Linear complementarity systems. - 5. 4. 5 Additional comments and studies. - 5. 5 The Painlevé's example. - 5. 5. 1 Lecornu's frictional catastrophes. - 5. 5. 2 Conclusions. - 5. 5. 3 Additional comments and bibliography. - 5. 6 Numerical analysis. - 5. 6. 1 General comments. - 5. 6. 2 Integration of penalized problems. - 5. 6. 3 Specific numerical algorithms. - 6 Generalized impacts. - 6. 1 The frictionless case. - 6. 1. 1 About "complete" Newton's rules. - 6. 2 The use of the kinetic metric. - 6. 2. 1 The kinetic energy loss at impact. - 6. 3 Simple generalized impacts. - 6. 3. 1 2-dimensional lamina striking a plane. - 6. 3. 2 Shock of a particle against a pendulum. - 6. 4 Multiple generalized impacts. - 6. 4. 1 The rocking block problem. - 6. 5 General restitution rules for multiple impacts. - 6. 5. 1 Introduction. - 6. 5. 2 The rocking block example continued. - 6. 5. 3 Additional comments and studies. - 6. 5. 4 3-balls example continued. - 6. 5. 5 2-balls. - 6. 5. 6 Additional comments and studies. - 6. 5. 7 Summary of the main ideas. - 6. 5. 8 Collisions near singularities: additional comments. - 6. 6 Constraints with Amontons-Coulomb friction. - 6. 6. 1 Lamina with friction. - 6. 7 Additional comments and studies. - 7 Stability of nonsmooth dynamical systems. - 7. 1 General stability concepts. - 7. 1. 1 Stabili