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 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.

arXiv comment39 pages, 9 figures

References in corpus (9)

• Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation
• Unsteady ballistic heat transport in a 1D harmonic crystal due to a source on an isotopic defect
• Non-stationary localized oscillations of an infinite Bernoulli-Euler beam lying on the Winkler foundation with a point elastic inhomogeneity of time-varying stiffness
• Non-stationary localized oscillations of an infinite string, with time-varying tension, lying on the Winkler foundation with a point elastic inhomogeneity
• The anti-localization of non-stationary linear waves and its relation to the localization. The simplest illustrative problem
• Passage through a resonance for a mechanical system, having time-varying parameters and possessing a single trapped mode. The principal term of the resonant solution
• Non-stationary elastic wave scattering and energy transport in a one-dimensional harmonic chain with an isotopic defect
• Non-stationary oscillation of a string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a sub-critical speed
• Comment on WKB series of all orders

Cited by in corpus (1)

• Anti-localization of non-stationary quasi-waves in a strongly nonlinear $\beta$-FPUT chain