Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification Mehmet Kurt a,n , Melih Eriten b , D. Michael McFarland c , Lawrence A. Bergman c , Alexander F. Vakakis a a Department of Mechanical Science and Engineering, University of Illinois at Urbana – Champaign, Urbana, IL 61801, USA b Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA c Department of Aerospace Engineering, University of Illinois at Urbana – Champaign, Urbana, IL 61801, USA article info Article history: Received 13 June 2012 Received in revised form 8 November 2013 Accepted 8 November 2013 Handling Editor: W. Lacarbonara Available online 23 December 2013 abstract We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed- modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end. & 2013 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jsvi Journal of Sound and Vibration 0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.11.021 Abbreviations: DOF, degree-of-freedom; EMD, empirical mode decomposition; FEP, frequency–energy plot; FT, Fourier transform; WT, wavelet transform; IMF, intrinsic mode function; IMO, intrinsic modal oscillator; NIM, nonlinear interaction model; NSI, nonlinear system identification; POD, Proper Orthogonal Decomposition; NNM, nonlinear normal mode; ROM, reduced-order model; RGM, reduced Guyan model n Corresponding author. E-mail address: kurt2@illinois.edu (M. Kurt). Journal of Sound and Vibration 333 (2014) 2054–2072