Nonlinear Random Response of Ocean Structures Using First- and Second-Order Stochastic Averaging

This paper deals with the dynamic response of nonlinear elastic structure subjected to random hydrodynamic forces and parametric excitation using a first- and second-order stochastic averaging method. The governing equation of motion is derived by using Hamilton's principle, taking into account inertia and curvature nonlinearities and work done due to hydrodynamic forces. Within the framework of first-order stochastic averaging, the system response statistics and stability boundaries are obtained. Unfortunately, the effects of nonlinear inertia and curvature are not reflected in the final results, since the contribution of these nonlinearities is lost during the averaging process. In the absence of hydrodynamic forces, the method fails to give bounded response statistics, and the analysis yields stability conditions. It is the second-order stochastic averaging which can capture the influence of stiffness and inertia nonlinearities that were lost in the first-order averaging process. The results of the second-order averaging are compared with those predicted by Gaussian and non-Gaussian closures and by Monte Carlo simulation. In the absence of parametric excitation, the non-Gaussian closure solutions are in good agreement with Monte Carlo simulation. On the other hand, in the absence of hydrodynamic forces, second-order averaging gives more reliable results in the neighborhood of stochastic bifurcation. However, under pure parametric random excitation, the stochastic averaging and Monte Carlo simulation predict the on-off intermittency phenomenon near bifurcation point, in addition to stochastic bifurcation in probability.     

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 between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric 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 first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence 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 experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.