A Re-Examination of Various Resonances in Parametrically Excited Systems1

Abstract

The dynamics of parametrically excited systems are characterized by distinct types of resonances including parametric, combination, and internal. Existing resonance conditions for these instability phenomena involve natural frequencies and thus are valid when the amplitude of the parametric excitation term is zero or close to zero. In this paper, various types of resonances in parametrically excited systems are revisited and new resonance conditions are developed such that the new conditions are valid in the entire parametric space, unlike existing conditions. This is achieved by expressing resonance conditions in terms of “true characteristic exponents” which are defined using characteristic exponents and their non-uniqueness property. Since different types of resonances may arise depending upon the class of parametrically excited systems, the present study has categorized such systems into four classes: linear systems with parametric excitation, linear systems with combined parametric and external excitations, nonlinear systems with parametric excitation, and nonlinear systems with combined parametric and external excitations. Each class is investigated separately for different types of resonances, and examples are provided to establish the proof of concept. Resonances in linear systems with parametric excitation are examined using the Lyapunov–Poincaré theorem, whereas Lyapunov–Floquet transformation is utilized to generate a resonance condition for linear systems with combined excitations. In the case of nonlinear parametrically excited systems, nonlinear techniques such as “time-dependent normal forms” and “order reduction using invariant manifolds” are employed to express various resonance conditions. It is found that the forms of new resonance conditions obtained in terms of ‘true characteristic exponents’ are similar to the forms of existing resonance conditions that involve natural frequencies.