DETERMINISTIC AND STOCHASTIC RESPONSE OF NONLINEAR COUPLED BENDING-TORSION MODES IN A CANTILEVER BEAM

The purpose of this study is to understand the main differences betwee n the deterministic and random response characteristics of an inextens ible cantilever beam (with a tip mass) in the neighborhood of combinat ion parametric resonance. The excitation is applied in the plane of la rgest rigidity such that the bending and torsion modes are cross-coupl ed through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parame tric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the fir st bending and torsion natural frequencies. The dependence of the resp onse amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the a bsence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimental ly. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to exp erience a single response, two possible responses, or non-stationary r esponses, depending on excitation level. Experimentally, it is possibl e to obtain two different responses for the same excitation level by p roviding a small perturbation to the beam during the test.