Asymptotic Analysis of Autoresonant Oscillator Chains

This paper investigates the emergence of autoresonance (AR) with growing energy in a chain of time-invariant linear oscillators weakly coupled to a nonlinear actuator (the Duffing oscillator) driven by an external periodic force. Two types of forcing are studied: (1) harmonic forcing with constant frequency is applied to an actuator with slowly-varying parameters; (2) harmonic forcing with a slowly increasing frequency is applied to an actuator with constant parameters. In both cases, the linear attachment is time-invariant, and the system is initially engaged in resonance. It is shown that in case (1) AR the nonlinear oscillator generates oscillations with growing amplitudes in the attached chain, while in case (2) energy transfer from the nonlinear oscillator is insufficient to excite high-energy motion in the attachment. The difference in the dynamical behavior is explained by different resonance properties of the systems. It is also shown that a slow change of stiffness may enhance the response of the nonlinear oscillator and make it sufficient to support oscillations with growing energy in the linear attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained.