Direct detection of nonlinear modal interactions from time series measurements

Abstract

Abstract This work describes a new method for the detection and characterization of strongly nonlinear modal interactions that occur in dynamical systems with an underlying linear structure and smooth, local nonlinearities directly from transient response data. The method relies on proper orthogonal decomposition (POD) to extract energy-dependent proper orthogonal mode shapes, which are used as trial vectors in the Rayleigh quotient to estimate the system’s frequency content at varying energy levels. By plotting the estimated frequencies as functions of energy, an approximate frequency-energy plot (FEP) is created, which reveals the rich and interesting nonlinear dynamical interactions directly from the measured time series. The method is first applied to the simulated response of a cantilever beam with local, smooth nonlinearity and then to the experimentally measured response of a comparable cantilever beam. In both applications, the approximate FEP reveals the presence of nonsmooth perturbations in the frequency estimates emanating from the curves that connect different nonlinear normal modes (NNMs) of the system. Using wavelet-bounded empirical mode decomposition and slow-flow analysis, the nonsmooth perturbations are shown to be the result of internal resonances between two or more interacting NNMs. Ultimately, the method provides significant insight into the nonlinear physics governing dynamical systems with local, smooth nonlinearities while remaining conceptually and computationally simple compared to traditional methods.