# 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

## TL;DR

This paper analyzes the passage through resonance in an infinite string system with time-varying parameters and a trapped mode, deriving an asymptotic solution and validating it with numerical simulations.

## Contribution

It provides the first asymptotic expression for the resonant response of a string with a trapped mode and time-varying parameters, verified by numerical methods.

## Key findings

- Asymptotic principal term of the resonant solution derived.
- Good agreement between analytical and numerical results near resonance.
- Validated the theoretical results with finite difference numerical simulations.

## Abstract

We consider a forced oscillation and passage through resonance for an infinite-length system, having time-varying parameters and possessing a single trapped mode. The system is a string, lying on the Winkler foundation and equipped with a discrete linear mass-spring oscillator of time-varying stiffness. We obtain the principal term of the asymptotic expansion for the resonant solution describing the motion of the inclusion (i.e., the mass-spring oscillator). The obtained result was verified by independent numerical calculations based on solution of the corresponding partial differential equation by means of the method of finite differences. The comparison demonstrates a good agreement in a neighbourhood of the instant of resonance.