Nonlinear Mathieu equation and coupled resonance mechanism
The method of multiple scales is used to determine a third-order solution for cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-recurrent case yield the standard Landau equation. Several types of parametric Landau equation are derived in the neighbourhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term, or a parametric second-order complex conjugate term, or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are given in each case. Stability criteria for non-trivial steady-state response are discussed, and transition curves are given in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions where the induced stability tongues within stability tongues are observed. The amplitude of periodic coefficients of Mathieu-type equations plays a dual role in the stability criteria for nonlinear Mathieu equations.
scientific article; zbMATH DE number 1884577
scientific article