Second-order spectra for quadratic nonlinear systems by Volterra functional series: Analytical description and numerical simulation

Higher-order spectral analysis techniques are often used to identify nonlinearities in complex dynamical systems. More specifically, the auto- and cross-bispectrum have proven to be useful tools in testing for the presence of quadratic nonlinearities based on knowledge of a system's input and output. In this paper, analytical expressions for the auto- and cross-bispectrum are developed using a Volterra functional approach under the assumption of a zero-mean, stationary Gaussian input; proper simplifications are presented when the whiteness of the input signal is also imposed. These formulae show the contributions of the bispectrum in terms of the system frequency response function and elementary physical properties of the system. Simulations based on a stochastic numerical integration technique accompany the analytical solutions for a mechanical mass–spring–damper system possessing quadratic damping and stiffness coefficients and subjected to Gaussian white noise excitation. Subsequent estimates of the bispectrum based on the simulated signals show excellent agreement with theory. These results show how modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. The presence and extent of nonlinear interactions between frequency components are identified. Advantages of using higher-order spectra techniques will be revealed and pertinent conclusions will be outlined.