A Global-local Approach to Nonlinear System Identification Young S. Lee(1), Alexander Vakakis(2), D. Michael McFarland(3), Lawrence Bergman(3) (1) Department of Mechanical and Aerospace Engineering, New Mexico State University, 1040 S. Horseshoe St., Las Cruces, NM 88003, U.S.A.; younglee@nmsu.edu (2) Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green St., Urbana, IL 61801, U.S.A.; avakakis@illinois.edu (3) Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 104 S. Wright St., Urbana, IL 61801, U.S.A.; dmmcf, lbergman@illinois.edu ABSTRACT We present the basic components of a time-domain nonlinear system identification (NSI) method with promise of applicability to a broad class of smooth and non-smooth dynamical systems. The proposed NSI method is based on the close correspondence between analytical and empirical slow-flow dynamics, and relies on direct analysis of measured time series without any a priori assumptions on the system dynamics. The central assumption is that the measured time series can be decomposed in terms of a finite number of oscillating components that are in the form of fast monochromatic oscillations modulated by slow amplitudes. The empirical slow-flow model of the dynamics is obtained from empirical mode decomposition, and its correspondence to the analytical slow-flow model establishes a local nonlinear interaction model (NIM). A NIM consists of a set of intrinsic modal oscillators (IMOs) that can reproduce the measured time series over different time scales and can account for (even strongly) nonlinear modal interactions. Hence, it represents a local model of the dynamics, identifying specific nonlinear transitions. By collecting energy-dependent frequency behaviors from all identified IMOs, a frequency-energy plot (in the modal space) can be constructed, which depicts global features of the dynamical system. 1. INTRODUCTION The need for system identification and reduced-order modeling in dynamical systems arises from the fact that, presented with sensor data, the analyst is often unaware of details of the underlying system from which they originated. The straightforward approach to this dilemma is to assume that the dynamical system is linear and that the measured responses are stationary in time. This facilitates the use, for example, of the numerical Fourier transform (FT) followed by experimental modal analysis (EMA [1]) to extract natural frequencies, mode shapes and modal damping ratios, from which the parameters of the assumed linear model can be determined if the mass distribution is known. This approach, which is fully nonparametric, has served the dynamics and controls community well, even in the presence of weakly nonlinear system behavior. Clearly, though, as systems become more complex, incorporating not only electrical and mechanical components but also biological and biomimetic elements, the likelihood exists that the observed data will reflect strong nonlinearity and nonstationarity. Such behaviors can result from, for example, local buckling, clearance and backlash, friction, hysteresis, local damage, large displacements and/or strains, and so forth (see, for example, [2]). And as we think more in terms of multi-physics problems, one must also include nonlinearities due to interfacial effects such as shear lag between actuator and structure, fluid-structure interactions, sensor-tissue interactions, and others. Thus, a physically-based parametric model of the system will not, in general, be known a priori. However, given a sufficiently dense set of sensors, measured time series recorded throughout the system will contain all of the information reflecting both nonlinearity and nonstationarity. Methods based upon the FT are not able to properly isolate and extract this information and, in fact, may lead the less experienced analyst to mistake phenomena such as internal and combination resonances for natural frequencies [3], to mistake time dependence for damping, and to fail to T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 513 DOI 10.1007/978-1-4419-9299-4_43, © The Society for Experimental Mechanics, Inc. 2011
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