A generalized method for the stationary probabilistic response of nonlinear dynamical system

Abstract

Statistical properties of random response essentially depend on the system and excitation features. The mathematical expressions for the response statistical properties should include the information on the system and excitation. In order to adequately illustrate the dependence of random response on the system and excitation features, only the stationary response is concerned. The exponential-polynomial-closure (EPC) method is further generalized and explicitly includes this information in the response probability density function (PDF) approximation. The unknown coefficients in the EPC approximate solution, which are constants in the previous stationary expression, are generalized to be functions of the system and excitation parameters. With the consideration of the independence among these parameters, each unknown coefficient is finally expressed as the product of several functions of each parameter. Based on the simulated response PDF results, the function of each parameter was determined by the least-squares method. Such a generalization deeply depicts the essentials of the response statistical properties and uniquely determines the response PDF expression. Three typical nonlinear systems are taken as examples to verify the efficiency of the proposed method. Numerical results show that approximate results obtained by the proposed method agree well with the simulated or exact ones. In addition, the evolution of the response PDF to the system or excitation parameters is obtained, which can be utilized for stochastic sensitivity analysis.