Synchronization of coupled friction-induced oscillators - A Dynamic Mode Decomposition Perspective

Abstract

The synchronization phenomenon in two linearly coupled friction-induced oscillators is analyzed in this paper using a data-driven approach based on dynamic mode decomposition (DMD). The tip mass at the end of a cantilever beam in each of the oscillators is in frictional contact with a rigid rotating disc. The cantilever beams are subjected to base excitation and a linear spring between the tip masses provides the coupling between the two oscillators. The partial differential equation governing the motion of the system is reduced to a set of ordinary differential equations employing the method of modal projection. The qualitative nature of the coupled oscillations is determined by analyzing the time displacement response, Fast Fourier Transform (FFT), Poincaré map, and the phase plane diagrams. DMD approximates the dynamical system in terms of coherent structures known as spatiotemporal modes. The frequency information is captured in the corresponding spatiotemporal modes. The influence of each frequency component on the whole dynamics of the system is studied by reconstructing the motion of each subsystem using the corresponding spatiotemporal mode. The contribution of a single dynamic mode towards the overall synchronized motion of the coupled system is analyzed by evaluating the linear correlation between those modes. Evaluation of the similarity measures helps to unearth how far each spatial and temporal mode behaves similarly in time.