Short-time dynamic mode decomposition: a data-driven forced response analysis of nonlinear systems under swept-sine excitation

Abstract Recently, data-driven approaches to predict the dynamical properties of nonlinear dynamical systems have become popular in the field of structural dynamics. Dynamic mode decomposition with control (DMDc) is an application of dynamic mode decomposition (DMD) to dynamical systems with control terms, which can now be applied to systems that receive external input. In this paper, we propose short-time dynamic mode decomposition (STDMD), which represents the spatio-temporal characteristics of a system by means of operators constructed for each short-time domain with a time window for swept external input and their state snapshot pairs. Based on the physical properties obtained, the time response and the frequency response of the system during steady-state vibration are clarified when the system is time evolved by external input data under conditions different from those used to construct the DMDc and STDMD models. As a numerical analysis, a forced vibration system by mass-spring-damper system that can be expressed by the Duffing equation is used. First, as in general studies, time evolution of the system was attempted using the same external force data as the external input used for model construction. Next, snapshots of the system under harmonic excitation were used for DMD model construction, and the time history and frequency responses were investigated using external input data under harmonic excitation at different frequencies for time evolution. Finally, data from the system undergoing swept sine excitation were used for model construction, and harmonic excitation data at each frequency within the sweep range were used for time evolution. Prediction of the time and frequency responses by ordinary DMDc was a linear approach that was able to represent nonlinear characteristics, but prediction became difficult when the nonlinearity was strong. On the other hand, the frequency response obtained by STDMD has been shown to be able to represent nonlinear characteristics such as jump phenomenon and hysteresis, even for systems with strong nonlinearities.