A novel probabilistic analysis method for long-term dynamical response analysis

Uncertainty propagation and quantification analysis in nonlinear systems are among the most challenging issues in engineering practice. Probabilistic analysis methods, based on the statistical information (i.e., mean and variance) of random variables, can account for uncertainties in the dynamical analysis of nonlinear systems. The statistical information of responses obtained by the Polynomial chaos expansion (PCE) method for nonlinear systems with random uncertainties deteriorates as the time history increases. Thus, the significant difficulty arises in analyzing the stochastic responses and long-term uncertainty propagation of nonlinear dynamical systems. To solve this problem, this paper proposes the PCE-HHT method by embedding a classical signal decomposition technique named Hilbert–Huang transform (HHT) in the PCE. Firstly, the HHT technique decomposes the multi-component response of a nonlinear system into a sum of several single vibration components and a trend component. Secondly, the PCE employs Hermite polynomials to approximate the instantaneous amplitudes and phases of each vibration component and the trend component, thereby establishing a coupled model of the system response, which can be used to determine the mean and variance of the dynamical response. Finally, considering parameter uncertainties in the Duffing–Van der Pol oscillator, the rigid double pendulum, and the spatially rigid-flexible crank-slider mechanism, the effectiveness of the PCE-HHT method is validated. Numerical results demonstrate that the PCE-HHT method exhibits desirable computational accuracy in the long-term random dynamical analysis of nonlinear systems.