Universal formal asymptotics for localized oscillation of a discrete mass-spring-damper system of time-varying properties, embedded into a one-dimensional medium described by the telegraph equation with variable coefficients

TL;DR

This paper develops a universal asymptotic description for localized oscillations in a discrete mass-spring-damper system embedded in a variable medium, accounting for small dissipation and slowly changing parameters.

Contribution

It derives a general asymptotic formula for localized oscillations in a discrete-continuum system with time-varying properties, extending previous static or constant-parameter models.

Findings

Asymptotic amplitude expressions depend on instantaneous parameters in non-dissipative cases.

In dissipative cases, amplitude involves integrals over system history.

Numerical validation confirms the accuracy of the derived asymptotics.

Abstract

We consider a quite general problem concerning a linear free oscillation of a discrete mass-spring-damper system. This discrete sub-system is embedded into a one-dimensional continuum medium described by the linear telegraph equation. In a particular case, the discrete sub-system can move along the continuum one at a sub-critical speed. Provided that the dissipation in both discrete and continuum sub-systems is absent, if parameters of the sub-systems are constants, under certain conditions (the localization conditions), a non-vanishing oscillation localized near the discrete sub-system can be possible. In the paper we assume that the dissipation in the damper and the medium is small, and all discrete-continuum system parameters are slowly varying functions in time and in space (when applicable), such that the localization condition is fulfilled for the instantaneous values of the parameters in a certain neighbourhood of the discrete sub-system position. This general statement can describe a number of mechanical systems of various nature. We derive the expression for the leading-order term of a universal asymptotics, which describes a localized oscillation of the discrete sub-system. In the non-dissipative case, the leading-order term of the expansion for the amplitude is found in the form of an algebraic expression, which involves the instantaneous values of the system parameters. In the dissipative case, the leading-order term for the amplitude, generally, is found in quadratures in the form of a functional, which depends on the history of the system parameters, though in some exceptional cases the result can be obtained as a function of time and the instantaneous limiting values of the system parameters. Finally, we have justified the universal asymptotics by numerical calculations for some particular cases.