Toward a fundamental understanding of the Hilbert-Huang transform in nonlinear structural dynamics

The Hilbert-Huang transform (HHT) has been shown to be effective for characterizing a wide range of nonstationary signals in terms of elemental components through what has been called the empirical mode decomposition (EMD). The HHT has been utilized extensively despite the absence of a serious analytical foundation, as it provides a concise basis for the analysis of strongly nonlinear systems. In this paper, an attempt is made to provide the missing theoretical link, showing the relationship between the EMD and the slow-flow equations of a system. The slow-flow reduced-order model is established by performing a partition between slow and fast dynamics using the complexification-averaging technique in order to derive a dynamical system described by slowly-varying amplitudes and phases. These slow-flow variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a novel nonlinear system identification method, termed the slow-flow model identification (SFMI) method. Through numerical and experimental application examples, we demonstrate that the proposed method is effective for characterization and parameter estimation of multi-degree-of-freedom nonlinear systems.