Nonlinear Rocking Motions. II: Overturning under Random Excitations

Abstract

Rocking responses of rigid objects under combined deterministic and stochastic excitations of arbitrary relative intensities are examined from a fully probabilistic perspective. The associated Fokker-Planck equation is derived and numerically solved by a path-integral solution procedure to obtain the joint probability density functions (JPDFs). The evolutions and the steady states of the JPDFs are employed to elucidate the global behavior of the rocking responses. As found in the companion paper, numerical results confirm that the presence of stochastic excitation bridges the domains of attraction of coexisting responses, and that overturning attractors are of the greatest relative stability. Thus, all rocking response trajectories that visit near the heteroclinic orbit will eventually lead to overturning under the influence of stochastic excitation. A rapid leakage of the probability (mass) out of the “safe” (bounded, chaotic) domain to the overturning regime implies weak stability of the chaotic attractor. Using mean first-passage time as a performance index, sensitivity of rocking responses to system parameters and (non)stationarity of the stochastic excitation is also investigated.