Autoresonance energy transfer versus localization in weakly coupled oscillators

TL;DR

This paper explores how energy is distributed in weakly coupled oscillators under autoresonance conditions, revealing different energy transfer behaviors depending on the excitation method and providing explicit asymptotic formulas.

Contribution

It analytically and numerically compares autoresonance phenomena in linear and nonlinear oscillators with different excitation protocols, offering new insights into energy transfer mechanisms.

Findings

Autoresonance occurs in both oscillators with decreasing stiffness excitation.

Only the nonlinear oscillator exhibits autoresonance with increasing frequency excitation.

Explicit asymptotic formulas closely match numerical results.

Abstract

In this paper we investigate the distribution of energy between weakly coupled linear and nonlinear oscillators in a two-degree-of-freedom (2D) system. Two classes of problems are studied analytically and numerically: (1) a periodic force with constant frequency is applied to the nonlinear (Duffing) oscillator with slowly time-decreasing linear stiffness; (2) the time-independent nonlinear oscillator is excited by a force with slowly increasing frequency. In both cases, stiffness of the attached linear oscillator and linear coupling remain constant, and the system is initially engaged in resonance. This paper demonstrates that in the systems of the first type autoresonance (AR) occurs in both oscillators while in systems of the second type AR occurs only in the excited nonlinear oscillator but the coupled linear oscillator exhibits small bounded oscillations. Considering slow detuning, we obtain explicit asymptotic approximations for the amplitudes and the phases of oscillations close to exact (numerical) results.