Time-Frequency Method for Nonlinear System Identification and Damage Detection

This paper presents a method for extracting system nonlinearities and time-localized transient response to impulsive loading by processing stationary/transient responses using the Hilbert—Huang transform (HHT) and a sliding-window fitting (SWF) technique. Time-dependent dynamic characteristics of nonlinear systems are derived using perturbation analysis. The SWF is introduced mainly to show the mathematical implications of HHT and the differences between HHT and the discrete Fourier transform. Similar to the wavelet transform the SWF uses windowed predetermined regular harmonics and function orthogonality to extract local harmonic components. It simultaneously decomposes a signal into just a few regular/distorted harmonics, and the obtained time-varying amplitudes and frequencies of the harmonics can reveal system nonlinearities. On the other hand the HHT uses the apparent time scales revealed by the signal's local maxima and minima and cubic splines of the extrema to sequentially sift components of different time scales, starting from high-frequency to low-frequency ones. Because HHT does not use predetermined basis functions and function orthogonality for component extraction, components are extracted without distortion and hence their time-varying amplitudes and frequencies can be accurately computed using the Hilbert transform to reveal system characteristics and nonlinearities. Moreover, because the first component extracted from HHT contains all discontinuities of the original signal, its time-varying frequency and amplitude are excellent indicators for pinpointing the time instants of impulsive loads. However, the discontinuity-induced Gibbs' phenomenon makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis suffers less from Gibbs' phenomenon at the two data ends, but it cannot extract accurate time-varying frequencies and amplitudes because the use of predetermined basis functions and function orthogonality in the sliding-window fitting process distorts the extracted components. Numerical and experimental results show that the proposed method with the use of HHT can provide accurate extraction of intrawave amplitude and phase modulations, distorted harmonic response under a single-frequency harmonic excitation, softening and hardening effects, different orders of nonlinearity, interwave amplitude and phase modulations, multiple-mode vibrations caused by internal/ external resonances, and time instants of impact loading on a structure. These are key phenomena for performing dynamics-based system identification and damage detection.