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A numerical scheme for impact problems. I: The one-dimensional case. (English) Zbl 1021.65065
The numerical solution of initial value problems for mechanical systems with one degree of freedom subject to impact constraints are considered. More precisely if \( u(t) \in \operatorname{Re} \) is the function that describes the position of the system at time \(t\) it is assumed that \( u(t) \geq 0 \) for all \(t\) and any impact time \( t^* \) in which \( u(t^*)=0\) the velocity is reversed and multiplied by some given restitution coefficient \( e \in [0,1]\) so that \( \dot{u} ( t^* + 0) = - e \dot{u} ( t^* - 0)\).
Hence if \( \ddot{u} = f ( \cdot , u , \dot{u}) \) is the differential equation that describes the dynamics of the free system the authors take \( \ddot{u} = \mu + f ( \cdot , u , \dot{u}) \) to describe the constrained system where \( \mu \) is a suitable nonnegative measure. In this context a numerical scheme is proposed and the existence of a discrete solution and some estimates for the velocity and acceleration are given. Further some additional properties are studied. The paper ends presenting the results of some numerical experiments carried out with the linear equation: \( \ddot{u} + 2 \alpha \dot{u} + u = a \cos ( \omega t) \).

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
65L05 Numerical methods for initial value problems
70F25 Nonholonomic systems related to the dynamics of a system of particles
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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