Chapter 30 Constructing Backbone Curves from Free-Decay Vibrations Data in Multi-Degrees of Freedom Oscillatory Systems Mattia Cenedese and George Haller Abstract Backbone curves are often the best representation of the nonlinear behavior for the vibrations of mechanical systems. Several approaches for obtaining them are present in literature, either analytical, numerical or experimental ones. However, they often make assumptions that unavoidably limit the range of applicability, such as the dynamics of the underlying conservative system and the modeling of damping terms. Here, we describe a mathematical theory and the corresponding numerical methodology that is able to rigorously extract backbone curves from free-decay vibrations data and that can overcome some of the main limitations of existing methods. We illustrate our findings with synthetic and real experiment vibration measurements. Keywords Backbone curves · Nonlinear vibrations · Damped vibrations · Nonlinear system identification · Spectral submanifolds 30.1 Introduction In the field of nonlinear structural dynamics, backbone curves often represent the most valuable information to be extracted, either from a model or an experimental setup. Indeed, these amplitude-frequency plots are able to properly characterize the dynamics quantifying the nonlinear behavior and also predicting eventual interactions. The original definition of backbone curve was proposed by Nayfeh and Mook in [1] that refer to it as the frequencies of amplitude maxima in frequency responses of different forcing amplitudes. In literature, the same name is also given to the relation between amplitude and frequency of conservative periodic orbits in nonlinear normal modes (NNMs) [2] or Lyapunov subcenter manifolds (LSMs) [3]. Even though fundamentally different, these two different objects are often observed to be close, in some cases even identical. However, rigorous results that establish a relation between forced-damped backbone curves and conservative ones are still missing [4]. In order to compute backbone curves, other than generic methods such as analytical techniques (e.g. method of multiple scales [1] and averaging [5]) or numerical ones like continuation (see the packages AUTO [6], MATCONT [7] and COCO [8]), several ad-hoc developments do exist. For conservative backbone curves, popular analytical techniques rely on the normal form method of [9], while several numerical ones have been established within the framework of nonlinear normal modes [10], like the shooting technique in [11]. The most common experimental method for backbone curve extraction is a two-step procedure [12]. First, one applies the force appropriation method [13] where the main underlying assumption is that introducing a periodic forcing with a 90◦ phase lag in a conservative system with a small linear damping term should preserve exactly a periodic orbit of the conservative part. Once this orbit is detected, the second step exploits the nonlinear resonance decay method [14, 15] in which excitation is turned off. Here, it is assumed that the decaying vibrations are close to conservative ones such that, by properly post-processing the signal, the instantaneous amplitude-frequency plot describes the backbone curve. Other approaches utilize experimental continuation [16], always starting from the assumptions of the force appropriation method. In spite of being considerably less expensive than generic techniques, all these ad-hoc methods make strong assumptions often limiting their range of applicability: small enough oscillations amplitude, accurate knowledge of the nonlinear normal mode manifold, small position independent linear damping and the yet unproven relation between forced-damped backbone curves and conservative ones [4]. In this contribution, we propose a data-driven method for extracting forced-damped backbone curves that, having its foundations on the work of Haller and Ponsioen [17], should overcome most of these limitations. In this work, they define M. Cenedese ( ) · G. Haller Institute for Mechanical Systems, ETH Zürich, Zürich, Switzerland e-mail: mattiac@ethz.ch © Society for Experimental Mechanics, Inc. 2020 G. Kerschen et al. (eds.), Nonlinear Structures and Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12391-8_30 221
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