The nonlinear structural dynamics of slender cantilever beams in flapping motion is studied through experiments, numerical simulations, and perturbation analyses.

A flapping mechanism which imparts a periodic flapping motion of certain amplitude and frequency on the clamped boundary of the appended cantilever beam is constructed. Centimeter-size thin aluminum beams are tested at two amplitudes and frequencies up to, and slightly above, the first bending mode to collect beam tip displacement and surface bending strain data. Experimental data analyzed in time and frequency domains reveal a planar, single stable (for a given flapping amplitude-frequency combination) periodic beam response with superharmonic resonance peaks. Numerical simulations performed with a nonlinear beam finite element corroborate the experiments in general with the exception of the resonance regions where they overpredict the experiments. The discrepancy is mainly attributed to the use of a linear viscous damping model in the simulations. Nonlinear response dynamics predicted by the simulations include symmetric periodic, asymmetric periodic, quasi-periodic, and aperiodic motions.

To investigate the above-mentioned discrepancy between experiment and simulation, linear and nonlinear damping force models of different functional forms are incorporated into a nonlinear inextensible beam theory. The mathematical model is solved for periodic response by using a combination of Galerkin and a time-spectral numerical scheme; two reduced order methods which, along with the choice of the inextensible beam model, facilitate parametric study and analytical analysis. Additional experiments are conducted in reduced air pressure to isolate the air damping from the material damping. The frequency response curves obtained with different damping models reveal that, when compared to the linear viscous damping, the nonlinear external damping models better represent the experimental damping forces in the regions of superharmonic and primary resonances. The effect of different damping models on the stability of the periodic solutions are investigated using the Floquet theory. The mathematical models with nonlinear damping yield stable periodic solutions which is in accord with the experimental observation.

The effect of excitation and damping parameters on the steady-state superharmonic and primary resonance responses of the flapping beam is further investigated through perturbation analyses. The resonance solutions of the spatially-discretized equation of motion (via 1-mode Galerkin approximation of the inextensible beam model), which involves both quadratic and cubic nonlinear terms, are constructed as first-order uniform asymptotic expansions via the method of multiple time scales. The critical excitation amplitudes leading to bistable solutions are identified and are found to be consistent with the experimental and numerical results. The approximate analytical results indicate that a second harmonic is required in the boundary actuation spectra in order for a second order superharmonic response to exist. 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 angle differences in the two solutions develop, which are attributed to phase estimation accuracy.