Chapter 6 Subspace and Nonlinear-Normal-Modes-Based Identification of a Beam with Softening-Hardening Behaviour C. Grappasonni, J.P. Noël, and G. Kerschen Abstract The capability to reproduce and predict with high accuracy the behaviour of a real system is a fundamental task of numerical models. In nonlinear structural dynamics, additional parameters compared to classical linear modelling, which include the nonlinear coefficient and the mathematical form of the nonlinearity, need to be identified to bring the numerical predictions in good agreement with the experimental observations. In this context, the present paper presents a method for the identification of an experimental cantilever beam with a geometrically nonlinear thin beam clamped with a prestress, hence giving rise to a softening-hardening nonlinearity. A novel nonlinear subspace identification method formulated in the frequency domain is first exploited to estimate the nonlinear parameters of the real structure together with the underlying linear system directly from the experimental tests. Then a finite element model, built from the estimated parameters, is used to compute the backbone of the first nonlinear normal mode motion. These numerical evaluations are compared to a nonlinear normal modes-based identification of the structure using system responses to stepped sine excitation at different forcing levels. Keywords Nonlinear system identification • Subspace identification • Experimental test • Softening hardening behaviour • Nonlinear normal modes 6.1 Introduction The physical behaviour of a structure undergoing high energy vibrational regimes can represent a challenging problem even in the case of simple one-dimensional flexible structures. When investigated using linear system identification techniques, the dynamical phenomena can be erroneously interpreted and lead to an inaccurate model. This results in the inapplicability of traditional, well-established linear techniques that needs to be reformulated in order to include the (predominant) nonlinearities in the measuring, identification and modelling processes. Several techniques are available today for the experimental identification of the linear structure dynamics and they can be classified as Phase Resonance or Phase Separation methods, depending on the sequential or simultaneous excitation of the normal modes, respectively. In the first case an accurate estimate of each mode is achieved by means of resonance excitation and concurrent response measuring. On the other hand, when a band-limited signal is used to vibrate the structure in the whole band of interest, the Frequency Response Functions (FRFs) between the system outputs and inputs can be evaluated and one of the many available modal analysis algorithms can be applied to assess the modal parameters. Starting from the sophisticated and advanced subspacebased algorithms [1, 2] the frequency-domain nonlinear subspace identification (FNSI) method has been developed and successfully applied to nonlinear structures [3]. The FRFs of the underlying linear structure and the nonlinear restoring force law can be estimated by this approach starting from the measurements of both the system responses and excitations. Therefore, the modal parameters, such as the natural frequencies, the damping ratios and the mode shapes, and the nonlinear coefficients can be estimated and used to implement an accurate model of the structure under investigation. The robustness C. Grappasonni ( ) • J.P. Noël • G. Kerschen Space Structures and Systems Laboratory (S3L), Department of Aerospace and Mechanical Engineering, University of Liège, Liège, Belgium e-mail: chiara.grappasonni@ulg.ac.be; jp.noel@ulg.ac.be; g.kerschen@ulg.ac.be G. Kerschen (ed.), Nonlinear Dynamics, Volume 2: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04522-1__6, © The Society for Experimental Mechanics, Inc. 2014 55
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