Responses of a strongly forced Mathieu equation Part 2: constant loading

The present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Nonresonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order ε0, where epsilon is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order epsilon and grows with the strength of parametric excitation squared. An n-th order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances at order n. Subharmonic resonances do not produce a nonzero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enables responses of order epsilon or ε0. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.