Alain Leger (Marseille, France)
Dynamics of discrete systems
with nonregularized impact and friction
Abstract: We are studying the dynamics of discrete systems made of particles or rigid bodies possibly
connected to each others or to a rigid frame by some elastic devices, and such that their
trajectories may involve shocks and friction with some obstacles. The simplest example of such
systems is given by a single particle attached by springs to a rigid frame, moving in the plane
and constrained to remain in a half-plane. We shall keep strictly non-regularized contact and
friction conditions, but for the sake of simplicity we shall concentrate on a simple example and
just comment on the points which either can be extended to any finite dimensional systems or
remain difficult open problems in general.
The lecture will concentrate on three points.
1. The first point is the foundation of the analysis: we give the convenient mathematical
statement of the problem, in which the reaction of the obstacle and the acceleration are
both measures so that the velocity of the particle is a function of bounded variation. The
result of this part is that, given initial data, a trajectory exists as soon as the external forces
are integrable functions but in general the Cauchy problem is ill-posed and uniqueness of
the trajectory is recovered only if the external forces are extremely smooth.
2. The second point deals with computational aspects, describing an algorithm of the time
stepping type, which is shown to converge no matter the solution involves isolated shocks
or infinitely many shocks in finite time.
3. The last point is closer to classical analyses of dynamical systems, but the results are very
different. Choosing external data such that the initial value problem is well-posed, we
study the dynamics under oscillating excitation. The {frequency - amplitude} plane of
the excitation appears to be divided into parts where there exist only equilibrium points,
and other parts where there exist only periodic solutions among which some are of small
or very small amplitude while others are of large amplitude. Moreover, in some transition
ranges there exist inifinitely many equilibrium points or infinitely many periodic
solutions.
pdf of the abstract.