# Nonlinear springs with dynamic frictionless contact

Article

## Abstract

This paper focuses on mathematical and numerical approaches to dynamic frictionless contact of nonlinear viscoelastic springs. This contact model is formulated by a nonlinear ordinary differential equation system and a pair of complementarity conditions. We propose three different numerical schemes in which each of them consists of several numerical methods. As a result, three groups of time-discrete numerical formulations are established. We use the coefficient of restitution to prove convergence of numerical trajectories, passing to the limit in the time step size. The Banach-fixed point theorem is applied to show the existence of global solutions satisfying all conditions. A new form of energy balance is derived, which is verified theoretically and numerically. All of the three schemes are implemented and their numerical results are compared with each other.

## Keywords

Nonlinear springs Signorini contact conditions Complementarity conditions Euler methods Runge–Kutta method

## Mathematics Subject Classification

Primary 65L20 Secondary 74H20 74H15 74M20

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