Transient Response Sensitivity Analysis of Localized Nonlinear Structure Using Direct Differentiation Method

Abstract

Based on the direct differentiation method, sensitivity analysis of transient responses with respect to local nonlinearity is developed in this paper. Solutions of nonlinear equations and time-domain integration are combined to compute the response sensitivities, which consist of three steps: firstly, the nonlinear differential equations of motion are solved using Newton–Raphson iteration to obtain the transient response; secondly, the algebraic equations of the sensitivity are obtained by differentiating the incremental equation of motion with respect to nonlinear coefficients; thirdly, the nonlinear transient response sensitivities are determined using the Newmark-β integration in the interested time range. Three validation studies, including a Duffing oscillator, a nonlinear multiple-degrees-of-freedom (MDOF) system, and a cantilever beam with local nonlinearity, are adopted to illustrate the application of the proposed method. The comparisons among the finite difference method (FDM), the Poincaré method (PCM), the Lindstedt–Poincaré method (LPM), and the proposed method are conducted. The key factors, such as the parameter perturbation step size, the secular term, and the time step, are discussed to verify the accuracy and efficiency. Results show that parameter perturbation selection in the FDM sensitivity analysis is related to the nonlinear features depending on the initial condition; the consistency of the transient response sensitivity can be improved based on the accurate nonlinear response when a small time step is adopted in the proposed method.