Identification of Nonlinear Normal Modes of Engineering Structures under Broadband Forcing

The objective of the present paper is to develop a two-step methodology integrating system identification and numerical continuation for the experimental extraction of nonlinear normal modes (NNMs) under broadband forcing. The first step processes acquired input and output data to derive an experimental state-space model of the structure. The second step converts this state-space model into a model in modal space from which NNMs are computed using shooting and pseudo-arclength continuation. The method is demonstrated using noisy synthetic data simulated on a cantilever beam with a hardening-softening nonlinearity at its free end.Comment: Journal pape

Numerical computation of nonlinear normal modes in mechanical engineering

This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear systems to strongly nonlinear industrial structures. © 2015 Elsevier Ltd

Mathematical Model Identification of Self-Excited Systems Using Experimental Bifurcation Analysis Data

Self-excited vibrations can be found in many engineering applications such as flutter of aerofoils, stick-slip vibrations in drill strings, and wheel shimmy. These self-excited vibrations are generally unwanted since they can cause serious damage to the system. To avoid such phenomena, an accurate mathematical model of the system is crucial. Self-excited systems are typically modelled as dynamical systems with Hopf bifurcations. The identification of such non-linear dynamical system from data is much more challenging compared to linear systems. In this research, we propose two different mathematical model identification methods for self-excited systems that use experimental bifurcation analysis data. The first method considers an empirical mathematical model whose coefficients are identified to fit the measured bifurcation diagram. The second approach considers a fundamental Hopf normal form model and learns a data-driven coordinate transformation mapping the normal form state-space to physical coordinates. The approaches developed are applied to bifurcation data collected on a two degree-of-freedom flutter rig and the two methods show promising results. The advantages and disadvantages of the methods are discussed

Control-based continuation of nonlinear structures using adaptive filtering

Control-Based Continuation uses feedback control to follow stable and unstable branches of periodic orbits of a nonlinear system without the need for advanced post-processing of experimental data. CBC relies on an iterative scheme to modify the harmonic content of the control reference and obtain a non-invasive control signal. This scheme currently requires to wait for the experiment to settle down to steady-state and hence runs offline (i.e. at a much lower frequency than the feedback controller). This paper proposes to replace this conventional iterative scheme by adaptive filters. Adaptive filters can directly synthesize either the excitation or the control reference adequately and can operate online (i.e. at the same frequency as the feedback controller). This novel approach is found to significantly accelerate convergence to non-invasive steady-state responses to the extend that the structure response can be characterized in a nearly-continuous amplitude sweep. Furthermore, the stability of the controller does not appear to be affected

A spectral characterization of nonlinear normal modes

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new, global parametrization of the invariant manifolds is established. Unlike the classical parametrization using a pair of state-space variables, this parametrization remains valid whenever the invariant manifold undergoes folding, which extends the computation of NNMs to regimes of greater energy. The proposed ideas are illustrated using a two-degree-of-freedom system with cubic nonlinearity.Belgian Network DYSCO (Dynamical Systems, Control, and Optimization) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy OfficeThis is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2016.05.01

Numerical continuation in nonlinear experiments using local Gaussian process regression

Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of geometric features of the response surface such as folds. The method uses Gaussian process regression to create a local model of the response surface on which standard numerical continuation algorithms can be applied. The local model evolves as continuation explores the experimental parameter space, exploiting previously captured data to actively select the next data points to collect such that they maximise the potential information gain about the feature of interest. The method is demonstrated experimentally on a nonlinear structure featuring harmonically coupled modes. Fold points present in the response surface of the system are followed and reveal the presence of an isola, i.e. a branch of periodic responses detached from the main resonance peak