10.1 Introduction

Many nonlinear mechanical systems have been extensively studied since thirty years: many of these systems possess a finite number of degrees of freedom and almost all of them correspond to a smooth mathematical model ([Ueda (1979)], [Lorenz (1963)], [Sparrow (1982)], [Lamarque and Malasoma (1992)]) except in the case of systems with plasticity, impacts or friction forces. Smooth nonlinearities are generally due to large displacements (geometrical nonlinearities) and nonlinear elasticity (constitutive laws of materials).

Impacts and friction have been introduced because of their many engineering applications: Study of joints [Paoli et. al. (1992)], impacts in gear-boxes [Pfeiffer (1988a)], models of beams [Dowell and Schwartz (1983a)], [Dowell and Schwartz (1983b)], [Whiteman and Ferri (1996)], rotor-casing [Li and Yorke (1975)], etc.

Many numerical works (for example, let us cite the works of [Whiston (1987)] (numerical study of a single degree of freedom vibro-impact-system), [Shaw and Shaw (1989)], [Shaw and Rand (1989)], [Awrejcewicz and Delfs (1990a)], [Awrejcewicz and Delfs (1990b)], [Foale and Bishop (1994)], [Popp and Stelter (1990)], [Cone and Zadoks (1995)]) are available in the case of one or two degrees of freedom systems.

Let us notice that in each case the problem of existence and uniqueness is solved by a direct quadrature of equations so that the mathematical difficulty disappears. Exact equations have to be solved in order to obtain impact-time and/or stick or slip time.

In general cases it is not always possible to obtain a piecewise-first-integral corresponding to the model. So two problems are considered:

• Find existence...