Responses of a Forced Mathieu Equation With Quasiperiodic Excitation

Resonances of the externally forced Mathieu equation under quasiperiodic excitation are studied. The external forcing frequency is assumed to be independent of the parametric-stiffness frequency and the system’s natural frequency. The response is analyzed by using a second-order multiple-scale approach. The system has secondary resonances at O(ϵ) and O(ϵ²) due to quasiperiodic forcing. These resonances occur when the forcing frequency matches the Mathieu equation’s natural response frequencies, which themselves are functions of the parametric frequency and the natural frequency without excitation. In addition, at specific frequencies where the unforced Mathieu equation exhibits instabilities, resonances are observed at O(ϵ) and O(ϵ²) simultaneously. For a few selected resonances, the steady-state amplitude and phase are determined, and the multiple-scale solutions are compared to numerical simulations for verification. The effects of system parameters, such as the damping ratio and the parametric stiffness, on the response near the resonances are evaluated.