Nonlinear response of flapping beams to resonant excitations under nonlinear damping

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 5A degrees while above this amplitude differences in the two solutions develop, which are attributed to phase estimation accuracy.