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

Abstract

We consider a quite general problem concerning a linear free oscillation of a discrete mass-spring-damper system. This discrete subsystem is embedded into a one-dimensional continuum medium described by the linear telegraph equation. In a particular case, the discrete subsystem can move along the continuum one at a sub-critical speed. Provided that the dissipation in both discrete and continuum subsystems is absent, if parameters of the subsystems are constants, under certain conditions (the localization conditions), a non-vanishing oscillation localized near the discrete subsystem 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 subsystem 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 subsystem. 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. In previous studies, several non-dissipative particular cases of the problem under consideration are investigated using a similar approach, provided that only one parameter of the discrete-continuum system is a slowly time-varying non-constant quantity, whereas all other parameters are constants. We show that asymptotics obtained in previous studies are particular cases of the universal asymptotics. The existence of a universal asymptotics in the form of a function is a non-trivial fact, which does not follow from the summarization of the results obtained for particular cases. Finally, we have justified the universal asymptotics by numerical calculations.