Primary Parametric Amplification in a Weakly Forced Mathieu Equation

The present study deals with the response of a forced Mathieu equation with damping, with weak harmonic direct excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds parametric amplification at primary resonance. The parametric effect on the primary resonance behavior occurs with a slow time scale of second-order, although the effect on the steady-state response is of order 1. As the parametric excitation level increases, the response at primary resonance stretches before becoming unbounded and unstable. Analytical expressions for predicting the response amplitudes are presented and compared with numerical results for a specific set of system parameters. Dependence of the amplification behavior, and indeed possible deamplification, on parameters is examined. The effect of parametric excitation on the response phase behavior is also presented.

Automated summary

The present study investigates the response of a damped Mathieu equation under weak harmonic direct excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales reveals parametric amplification at primary resonance. The parametric effect occurs with a slow time scale of second-order, and the response at primary resonance stretches and becomes unbounded and unstable as the parametric excitation level increases. Analytical expressions for predicting the response amplitudes are presented and compared with numerical results for a specific set of system parameters.