Wavelet-bounded empirical mode decomposition for measured time series analysis

• Published in: Mechanical systems and signal processing 2018-01, Vol.99, p.14-29
• Main Authors: Moore, Keegan J., Kurt, Mehmet, Eriten, Melih, McFarland, D. Michael, Bergman, Lawrence A., Vakakis, Alexander F.
• Format: Article
• Language: English
• Subjects: Cantilever beams; Computer simulation; Decomposition; Empirical analysis; Empirical mode decomposition; Masking; Nonlinear analysis; Nonlinear systems; Optimization; Time series; Transient response; Wavelet analysis; Wavelet transform; Wavelet transforms
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2017.06.005

•We describe a new method for optimizing EMD using wavelet transforms.•The method drastically reduces the effect of mode-mixing.•The method’s success and robustness is studied using a two-component signal.•We extract IMFs that representative of the NNMs for a beam with local nonlinearity.•The IMFs are used to prove the existence of a 3:1 internal resonance.•Energy-dependent periodic solutions are constructed using the IMFs. Empirical mode decomposition (EMD) is a powerful technique for separating the transient responses of nonlinear and nonstationary systems into finite sets of nearly orthogonal components, called intrinsic mode functions (IMFs), which represent the dynamics on different characteristic time scales. However, a deficiency of EMD is the mixing of two or more components in a single IMF, which can drastically affect the physical meaning of the empirical decomposition results. In this paper, we present a new approached based on EMD, designated as wavelet-bounded empirical mode decomposition (WBEMD), which is a closed-loop, optimization-based solution to the problem of mode mixing. The optimization routine relies on maximizing the isolation of an IMF around a characteristic frequency. This isolation is measured by fitting a bounding function around the IMF in the frequency domain and computing the area under this function. It follows that a large (small) area corresponds to a poorly (well) separated IMF. An optimization routine is developed based on this result with the objective of minimizing the bounding-function area and with the masking signal parameters serving as free parameters, such that a well-separated IMF is extracted. As examples of application of WBEMD we apply the proposed method, first to a stationary, two-component signal, and then to the numerically simulated response of a cantilever beam with an essentially nonlinear end attachment. We find that WBEMD vastly improves upon EMD and that the extracted sets of IMFs provide insight into the underlying physics of the response of each system.