Development of stochastic quadratization for nonlinear systems with application to compliant offshore structures

The exploration and production of oil at offshore locations is extending into ever increasing water depths. To exploit this resource in water depths in excess of 300 meters, a new class of offshore structures called compliant platforms are being developed. The compliant nature of these platforms introduces nonlinear behavior which can not be neglected as in conventional offshore platforms. Consequently, new analytical methods to estimate response statistics are needed. Stochastic linearization is perhaps the most frequently used analytical approximation for analyzing the response of nonlinear systems. Linearization, however, can not predict responses at frequencies outside the excitation frequencies. Therefore, some response statistics may be significantly unconservative. In addition for a gaussian excitation, the linearized solution leads to a gaussian probability distribution, whereas the true response is non-gaussian. In this study, a higher order method termed equivalent stochastic "quadratization" is proposed to circumvent these shortcomings. The nonlinearity is replaced by a polynomial expansion up to quadratic order. The Volterra series method is used to approximate the response of the resulting nonlinear system. The excitation is assumed to be gaussian, however, the response is described by a nongaussian probability distribution. The method is developed for analyzing the stationary response of single and multi-degree-of-freedom systems. A useful application of the proposed method is for analyzing the stochastic response of compliant offshore platforms due to nonlinear drag forces. The method is applied for analyzing a three-degree-of-freedom model of a Tension Leg Platform (TLP) subject to wave and current forces. The proposed method predicts the low frequency response induced by the drag force at the surge natural frequency which linearization can not account for. In addition to nonlinear drag forces, nonlinear potential forces significantly affect the TLP response. These forces are derived in the form of second order Volterra series. A stochastic response analysis of the TLP system due to combined nonlinear drag and nonlinear potential forces is performed to evaluate the relative significance of these forces. The focus of the study is on the nonlinear low frequency and high frequency responses