Fast Convolution Integration–Based Nonstationary Response Analysis of Linear and Nonlinear Structures with Nonproportional Damping

Abstract

The frequency domain approach (FDA) based on the power spectral description of input and output has been widely applied in the random vibration analysis of structures because of its high efficiency and distinct relationship between the input and output. However, for large structures with closely spaced natural frequencies and nonproportional damping, the efficiency of this method for nonstationary response analysis is impacted by the high demand posed by the time-history analysis. In this study, a fast convolution integration method based on Duhamel integration is introduced to enhance the efficiency of the FDA for the nonstationary response analysis of nonproportionally damped structures. First, an efficient impulse excitation approach is proposed to identify the discrete-time impulse response of linear structures. Accordingly, the nonstationary response statistics can be directly evaluated by the convolution with respect to the discrete-time impulse response. Subsequent application of the fast Fourier transform (FFT)–based algorithm accelerates the evaluation of the convolution. Therefore, the proposed scheme is efficient as it eliminates a direct time-history analysis and benefits from the implementation of the FFT. The proposed method is also illustrated to enhance the analysis of nonlinear structures under nonstationary random excitations by invoking an equivalent statistical linearization scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficacy of the proposed method.