Identifying and Computing Nonlinear Normal Modes

Nonlinear Dynamics, Volume 1, 2017

Nonlinear normal modes (NNMs) describe the unforced and undamped periodic responses of nonlinear systems. NNMs have proven to be a valuable tool, and are widely used, for understanding the underlying behaviour of nonlinear systems. They provide insight into the types of behaviour that may be observed when a system is subjected to forcing and damping, which is ultimately of primary concern in many engineering applications. The definition of an NNM has seen a number of evolutions, and the contemporary definition encompasses all periodic responses of a conservative system. Such a broad definition is essential, as it allows for the wide variety of responses that nonlinear systems may exhibit. However, it may also lead to misleading results, as some of the NNMs of a system may represent behaviour that will only be observed under very specific forcing conditions, which may not be realisable in any practical scenario. In this paper, we investigate how the significance of NNMs may differ and how this significance may be quantified. This is achieved using an energy-based method, and is validated using numerical simulations.

2003

Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

Applied Mechanics Reviews, 2010

Two principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer-Rosenberg and Shaw-Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described.

Mechanical Systems and Signal Processing, 2011

Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated structures such as satellites require even the smallest nonlinear behavior to be considered for better performance. Therefore, it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, in most of the cases, test data will be incomplete. These handicaps make most of the methods given in the literature difficult to apply to engineering structures. The aim of this study is to improve a previously developed method considering these practical limitations. The approach proposed can be used for detection, localization, characterization and parametric identification of nonlinear elements by using incomplete FRF data. In order to reduce the effort and avoid the limitations in using footprint graphs for identification of nonlinearity, describing function inversion is used. Thus, it is made possible to identify the restoring force of more than one type of nonlinearity which may co-exist at the same location. The verification of the method is demonstrated with case studies.

Journal of Sound and Vibration, 1999

Mechanical Systems and Signal Processing, 2009

Local nonlinear effects due to micro-slip/slap introduced in boundaries of structures have dominant influence on their lower modal model. This paper studies these effects by experimentally observing the behavior of a clamped-free beam structure with local nonlinearities due to micro-slip at the clamped end. The structure is excited near one of its resonance frequencies and recorded responses are employed to identify the nonlinear effects at the boundary. The nonlinear response of structure is defined using an amplitude-dependent nonlinear normal mode identified from measured responses. A new method for reconstructing nonlinear normal mode is represented in this paper by relating the nonlinear normal mode to the clamped end displacement-dependent stiffness parameters using an eigensensitivity analysis. Solution of obtained equations results equivalent stiffness models at different vibration amplitudes and the corresponding nonlinear normal mode is identified. The approach results nonlinear modes with efficient capabilities in predicting dynamical behavior of the structure at different loading conditions. To evaluate the efficiency of the identified model, the structure is excited at higher excitation load levels than those employed in identification procedures and the observed responses are compared with the predictions of the model at the corresponding input force levels. The predictions are in good agreement with the observed behavior indicating success of identification procedure in capturing the physical merits involve in the boundary local nonlinearities.

Journal of Vibration and Control, 2016

A range of methodologies exist for estimating nonlinear responses of structural systems using numerical simulations. However, efforts in relation to experimental methods in this regard still warrant further investigation. This paper presents an approach for assessing structural nonlinearities using the extremes of dynamic responses of the structural system under consideration. The approach allows revisiting and parameter tuning of theoretical models of structures based on experimental studies. A single degree of freedom system was excited in this study using broadband input excitations and the output dynamic responses were measured using different devices. The type and extent of experimentation required for implementation of the presented technique was investigated along with the effects of the estimates of the measured variables and the effects related to different measurement devices.

Journal of Sound and Vibration, 1992

A theoretical and experimental study of non-linear systems governed by Duffing's equation is presented in this study. The non-linear mode concept is applied to analyze the steady state responses of non-linear systems driven by a harmonic force. In the experimental context, the present modal identification method permits one to obtain the non-linear modal parameters from frequency response tests. By comparison with other methods, the proposed methods appear very interesting in regard to computational time, formulation, and they necessitate fewer computer resources. An experimental work on the steady state response of a beam with local non-linear stiffness is reported to justify the methods. Many aspects of theoretical results, such as the dependence of resonance frequency on the amplitude of vibration and the downward and upward jump phenomena are observed experimentally. The experimental results are in good agreement with theoretical predictions based upon the non-linear modal superposition approach. Recently, the identification of non-linear systems has received considerable attention; parametric and non-parametric techniques have been studied intensively by many authors. 497

Journal of Sound and Vibration, 1986

Experimental modal analysis techniques are based on the theory of linear systems. In this paper the feasibility of their application to non-linear systems is examined. A three degree of freedom non-linear mechanicalsystem was modeled on an analog computer and system responses due to band limited random excitation were examined by using a two channel FFT analyzer. Frequency response functions, natural frequencies and model responses were studied experimentally for a number of example cases. This study shows that the measured transfer function plots can alert one to the fact that the structure is non-linear; measured modal response data base can still give a rough idea about the system behavior. In addition, measured coherence function estimates can be used as the "warning signals" for inherent non-linearities.