Chapter 3 Direct Detection of Nonlinear Modal Interactions and Model Updating Using Measured Time Series Keegan Moore, Mehmet Kurt, Melih Eriten, D. Michael McFarland, Lawrence A. Bergman, and Alexander F. Vakakis Abstract We describe a new method for identifying mechanical systems with strongly nonlinear attachments using measured transient response data. The procedure is motivated by the desire to quantify the degree of nonlinearity of a system, with the ultimate goal of updating a finite-element or other mathematical model to capture the nonlinear effects accurately. Our method relies on the proper orthogonal decomposition to extract proper orthogonal mode shapes (POMs), which are inherently energy dependent, directly from the measured transient response. Using known linear properties, the system’s frequencies are estimated using the Rayleigh quotient and an estimated frequency-energy plot (FEP) is created by them as functions of the system’s mechanical energy. The estimated FEP reveals distinct linear and nonlinear regimes which h are characterized by constant frequency (horizontal lines) and large frequency changes, respectively. The nonlinear regimes also contain spikes that connect different modes and indicate strongly nonlinear modal interactions. The nonlinearity is identified by plotting the estimated frequencies as functions of characteristic displacement and fitting a frequency equation based on the model of the nonlinearity. We demonstrate the method on the response of a cantilevered, model airplane wing with a nonlinear energy sink attached at its free end. Keywords Proper orthogonal decomposition · Nonlinear system identification · Frequency-energy plot · Nonlinear normal modes · Nonlinear energy sink 3.1 Method While the techniques to be presented are applicable to either discrete or continuous structures, it is most convenient to present them in the context of a discrete model. This is hardly a limitation in practice, considering that experimental data acquisition is inherently limited to discrete points in space, and that finite element analysis can produce a viable discrete model for most distributed systems of interest [1]. A large number of degrees of freedom can be accommodated, but it is often desirable to reduce the model dimension, either for computational efficiency or to avoid the need to infer quantities that are difficult to measure directly. An example of the latter is rotation about an in-plane axis at a point on a beam or plate undergoing bending. Such quantities can often be systematically eliminated from the equations of motion by an established technique such as Guyan reduction [2], with little effect on the behavior of the dynamical model. Given the mass and linear stiffness properties of a system, represented by mass and stiffness matrices, we turn to the processing of time series obtained by measuring the response of the system following an input of known magnitude (typically an impulse applied to one degree of freedom). Using proper orthogonal decomposition [3], we construct a set of basis vectors which can be used to represent the response of the system over an interval when its total energy is nearly constant. Thus, the method applies to a damped system provided sufficient response is available at the energy of interest. K.Moore ( ) · A. F. Vakakis Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL, USA e-mail: kmoore14@illinois.edu M. Kurt Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ, USA M. Eriten Department of Mechanical Engineering, University of Wisconsin, Madison, WI, USA D. M. McFarland · L. A. Bergman Department of Aerospace Engineering, University of Illinois, Urbana, IL, USA © The Society for Experimental Mechanics, Inc. 2019 G. Kerschen (ed.), Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74280-9_3 23
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