| A frequency domain treatment of the probabilistic dynamic response of systems which may contain nonlinearities in their loading or in their structural properties is given. Particular applications are made to offshore structures subjected to nonlinear environmental (wind, wave and current) loads. A third-order Volterra series approach is used for analysis. As a Taylor series with memory, it is a useful tool for analyses of a class of nonlinear dynamic systems. It is particularly suitable for systems like offshore structures whose excitation is a function of their relative motion with respect to waves or wind. At the third order, the Volterra series approach can preserve effects of both statistically asymmetric and statistically symmetric nonlinearities. Nonlinearities in the loading processes are attributable to one or a combination of the mathematical forms of aerodynamic and hydrodynamic drag; a stretched model of wave kinematics; a second-order random Stokes perturbation model of wave kinematics; or intermittence of forces due to the fluctuation of the ocean free surface. In polynomial or perturbation series forms, these nonlinearities are readily handled within the Volterra approach. If they do not have such forms, a mean square error minimization or a moment-based Hermite polynomialization is used to cast them as polynomials. The approximate polynomial generated via mean square error minimization is constrained to match the second-order statistics of the original force while the moment-based Hermite polynomialization additionally applies constraints to the skewness and coefficient of excess of the approximation. The Volterra series analysis approach yields the power spectral density and cumulants of the response. The cumulants may be used to develop an appropriate non-Gaussian probability density function model as well as to quantify extreme response via peak factors. |
