Autoresonant dynamics of weakly coupled oscillators

Abstract

This paper studies the emergence of autoresonance (AR) in a two-degrees-of-freedom system consisting of an excited nonlinear actuator (the Duffing oscillator) linearly coupled to a passive linear oscillator. Two classes of system are considered: (1) In the system of the first type, a periodic force with constant (resonance) frequency is applied to the nonlinear (Duffing) oscillator with slowly time-decreasing linear stiffness; (2) in the system of the second type, the nonlinear oscillator with constant parameters is excited by a force with slowly increasing frequency. In both cases, the linear oscillator and the linear coupling are time-invariant, and the system is initially engaged in resonance. In the first step, the dynamics of the undamped system is analyzed. It is shown here that in the system of the first type, AR may occur in both oscillators, but in the system of the second type, the amount of energy transferred from the nonlinear oscillator is insufficient to excite high-energy motion in the attached oscillator. Different dynamical behavior arises due to different resonance properties of the systems. In the next step, an effect of viscous damping in the oscillators and the coupling is investigated. As this paper demonstrates, the response enhancement may be observed in the system of the first type on a bounded time interval provided that dissipation is weak enough. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is proved.