Nonlinear response of flapping beams to resonant excitations under nonlinear damping

Abstract

The effect of excitation and damping parameters on the superharmonic and primary resonance responses of a slender cantilever beam undergoing flapping motion is investigated. The problem is cast into mathematical form using a nonlinear inextensible beam model which is subjected to time-dependent boundary conditions and linear or nonlinear damping forces. The flapping excitation is assumed to be non-harmonic, composed of two sine waves with different amplitudes. We employ a combination of Galerkin and perturbation methods to arrive at the frequency–response relationships associated with the second- and third-order superharmonic and primary resonances. The resonance solutions of the spatially discretized equation of motion, which involves both quadratic and cubic nonlinear terms, are constructed as first-order uniform asymptotic expansions via the method of multiple timescales. The effect of excitation and damping parameters on the steady-state resonance responses and their stability is described quantitatively using approximate analytical expressions. The critical excitation amplitudes leading to bistable solutions are identified. For the second-order superharmonic resonance, the critical excitation amplitude is determined to be dependent on the first-harmonic amplitude in the case of nonlinear damping. The third-order superharmonic resonance is determined to be independent of the second-harmonic excitation amplitude regardless of the damping types considered. The perturbation solutions are compared with numerical time-spectral solutions for different flapping amplitudes. The first-order perturbation solution is determined to be in very good agreement with the numerical solution up to 5° while above this amplitude differences in the two solutions develop, which are attributed to phase estimation accuracy.