Detection and identification of nonlinearities by amplitude and frequency modulation analysis

Abstract

This paper presents an amplitude and frequency modulation method (AFMM) for extracting characteristics of nonlinear systems and intermittent transient responses by processing stationary/transient responses using the empirical mode decomposition, Hilbert–Huang transform (HHT), and nonlinear dynamic characteristics derived from perturbation analysis. A sliding-window fitting (SWF) method is derived and used to show the physical implications of the proposed method and other methods for data processing and transformation. Similar to the wavelet transform, the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics, and then the amplitude and frequency modulations of the harmonics are used to identify 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, it provides more accurate instant amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Moreover, because the first component extracted by HHT contains all original discontinuities, its time-varying amplitude and frequency are excellent indicators for pinpointing times and locations of impulsive external loads. However, the discontinuity-induced Gibbs’ phenomenon makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis is not affected by Gibbs’ phenomenon, but it cannot extract accurate time-varying frequencies and amplitudes because of the use of predetermined basis functions, function orthogonality, and windowed curve fitting for component extraction. Numerical results show that the proposed AFMM can provide accurate estimation of softening and hardening effects, different orders of nonlinearity, linear and nonlinear system parameters, and time instants of intermittent transient responses.