Perturbation Analysis of the Combination Resonances of a Thin Cantilever Beam under Vertical Excitations

Abstract

It is useful to consider a thin cantilever beam to investigate the stability of nonlinear oscillations. When the thin cantilever beam is subjected to harmonic base excitation in vertical direction, it reveals many interesting behavior caused by nonlinear effect. In order to analyze the nonlinear oscillation phenomena, we derived two partial differential governing equations under combined parametric and external excitations and converted into two-degree-of-freedom ordinary differential Mathieu equations by using the Galerkin method. Among many perturbation techniques, we employed the method of multiple scales in order to analyze one-to-one combination resonance. We could obtain the eigenvalue problem and analyze the stability of the system. By representing the eigenvalues in nondimensional form, we could obtain the relationship between the eigenvalue and the amplitude ratio, which could explain the physical meaning of the eigenvalues. The real and imaginary plots with respect to the frequency detuning parameter could provide the stability criterion based on the eigenvalues.