On the use of the auto-bispectral density for detecting quadratic nonlinearity in structural systems

Higher-order spectra appear often in the analysis and identification of nonlinear systems. The auto-bispectral density is one example of a higher-order spectrum and may be used in the analysis of stationary structural response data to detect the presence of certain types of structural nonlinearities. In this work a closed-form expression for the auto-bispectral density, derived previously by the authors, is used to find the bispectral frequency most sensitive to the nonlinearity. The properties of nonlinearity detectors based on estimates of the magnitude of the auto-bispectral density at this frequency are then explored. Estimates of the auto-bispectral density are obtained using the direct method based on the discrete Fourier transform. The bias associated with this estimator is derived here and combined with previously derived expressions for the estimator variance to give both Type-I and Type-II errors for the detector. Detector performance is quantified using a receiver operating characteristic (ROC) curve illustrating the trade-off between false positives (Type-I error) and power of detection (1.0-Type-II error). Theoretically derived ROC curves are compared to those obtained via numerical simulation and show excellent agreement. Results are presented for different levels of nonlinearity in both the stiffness and damping terms for a spring–mass system. Possible consequences are discussed with regard to the detection of damage-induced nonlinearities in structures.