Wavelet-bounded empirical mode decomposition for vibro-impact analysis

We consider the response of a single-degree-of-freedom, linear oscillator (LO) coupled to a vibro-impact (VI) nonlinear energy sink (NES), and subject to an impulsive load. A previous study of this system used the empirical mode decomposition (EMD) technique to decompose the LO’s and NES’s responses into smooth components (the first harmonic) and nonsmooth components (all other harmonics). The smooth components were used to demonstrate the existence of a 1:1 transient resonance capture (TRC) between the first harmonics of the LO and the NES. When faced with the possibility of a 3:1 TRC between the first and third harmonic, previous studies were unable to extract the third harmonic from the nonsmooth component and, thus, did not provide an analysis of its nonlinear interaction with the first harmonic. Consequently, it was conjectured that the higher-order TRCs that appeared in the response were nonphysical and the result of numerical artifacts produced by the VIs. In this work, we argue that this conjecture is incorrect, in the sense that higher-order TRCs do populate the dynamics and, moreover, that the higher harmonics and higher-order TRCs contribute significantly to the VI response. To show this, we employ a new and more powerful version of EMD, namely the recently developed wavelet-bounded EMD method, to decompose the responses into multiple smooth components, each physically representative of a single harmonic contained in the original response. The components reveal the relative contributions of each harmonic and are used to demonstrate the existence of higher-order TRCs, including the 3:1 TRC between the first and third harmonics. Hence, the present work highlights the efficacy of wavelet-bounded EMD to extract the nonlinear interactions from VI responses, a result which, to the authors’ knowledge, appears for the first time in the literature.