Nonlinear system identification of the dynamics of a vibro-impact beam: numerical results

We study the dynamics of a cantilever beam with two rigid stops of certain clearances by performing nonlinear system identification (NSI) based on the correspondence between analytical and empirical slow-flow dynamics. The NSI method in this work can proceed in two directions: One for the numerical data obtained from a reduced-order model by means of the assumed-mode method, and the other for the experimental data measured at the same positions as in numerical simulations. This paper focuses on the analysis of the numerical data, providing qualitative comparison with some experimental results; the latter task will be discussed in detail in a companion paper. First, we perform empirical mode decomposition (EMD) on the acceleration responses measured at ten, almost evenly-spaced, spanwise positions along the beam leading to sets of intrinsic modal oscillators governing the vibro-impact dynamics at different time scales. In particular, the EMD analysis can separate any nonsmooth effects caused by vibro-impacts of the beam and the rigid stops from the smooth (elastodynamic) response, so that nonlinear modal interactions caused by vibro-impacts can be explored only with the remaining smooth components. Then, we establish nonlinear interaction models (NIMs) for the respective intrinsic modal oscillators, where the NIMs invoke slowly-varying forcing amplitudes that can be computed from empirical slow-flows. By comparing the spatio-temporal variations of the nonlinear modal interactions for the vibro-impact beam and those of the underlying linear model (i.e., the beam with no rigid constraints), we demonstrate that vibro-impacts significantly influence the lower-frequency modes, introducing spatial modal distortions, whereas the higher frequency modes tend to retain their linear dynamics between impacts. We introduce a linear correlation coefficient as a measure for studying the linear dependency between the slowly-varying complex forcing amplitudes for the linear and vibro-impact beams and demonstrate that only a set of lower-frequency modes are strongly influenced by vibro-impacts, capturing most of the essential nonlinear dynamics. These results demonstrate the efficacy of the proposed approach to analyze strongly nonlinear measured time series and provide physical insight for strong nonlinear dynamical interactions.