Experimental and numerical characterization of the structural dynamics of flapping beams

The nonlinear structural dynamics of a slender beam in flapping motion is examined both experimentally and computationally. In the experiments the periodic flapping motion is imposed on the clamped edge of the cantilever beam using a 4-bar crank-and-rocker mechanism. Aluminum beams with nominal dimensions of 150 mm×25 mm×0.4 mm are tested in air over a range of flapping frequencies up to 1.3 times the linear first modal frequency at two different flapping amplitudes, 15° and 30°. The response of the beam is characterized experimentally through bending strain and tip displacement data obtained from foil strain gage and high-speed camera, respectively. It was determined that for the particular combination of beam specimen (dimensions, material properties) and forcing parameters investigated, all experimental responses were periodic. The frequency response curves based upon the experimental bending strain data reveal a secondary superharmonic peak in addition to the primary resonance peak. As the flapping frequency is increased, the response of the beam is observed to change from symmetric (with respect to equilibrium position) periodic vibrations with a period equal to the flapping period to asymmetric periodic vibrations with higher harmonic content featuring local oscillations in the time histories. Experimental tip displacement results show that the beam spends more time during stroke reversals when the flapping frequency is near primary and secondary resonance regions. In addition to experiment, numerical simulations are performed using two-node, isoparametric degenerate-continuum based geometrically nonlinear beam elements. The HHT-α version of the Newmark finite difference scheme is used to discretize the problem in time and a linear viscous damping model is assumed. Overall the numerical simulations agree well with the experiments and capture most of the nonlinear dynamical features of the beam response. It is, however, found that in resonance regions the simulation overpredicts response magnitudes, possibly due to the use of the linear damping model and linear elastic constitutive model. Additional numerical simulations of the beam tip response reveal dynamics which include periodic, asymmetric periodic, quasi-periodic and aperiodic motions.