Vibrations of a lumped parameter mass–spring–dashpot system wherein the spring is described by a non-invertible elongation-force constitutive function

Abstract

The standard setting concerning vibrations of lumped parameter systems is based on the assumption that the mechanical response of the elements of the system is given explicitly in terms of kinematical variables. In particular, the force in a spring element is assumed to be given as a function of the displacement from the equilibrium position. However, some simple mechanical systems such as linear springs with limited compressibility/extensibility do not fit into the standard setting. In this case the displacement must be written as a function of the force. In general, the mechanical response of such elements must be described by an implicit relation between the force and kinematical variables. We study the behaviour of a particular lumped parameter system whose mechanical response is given by a non-invertible expression for the displacement in terms of the force, under harmonic external force. We show that a solution to the original system wherein the displacement is given in terms of the force can be obtained as a limit of a sequence of approximate problems. The approximate problems are designed in such a way that they can be solved using standard numerical methods, and one can avoid using concepts such as set valued mappings. Moreover, we show that the “bounce back” behaviour of the system with linear spring with limited compressibility/extensibility is a direct consequence of the assumed constitutive relation. There is no need to a priori supply the rules for the bounce back (impact rules). Further, we show that the advocated approximation procedure is capable of describing the behaviour of the lumped parameter system even in the situations where the governing ordinary differential equation collapses to an algebraic equation. Representative results are demonstrated by a numerical experiment.