Global Parametrization of the Invariant Manifold Defining Nonlinear Normal Modes Using the Koopman Operator

2015 ◽  
Author(s):  
Giuseppe I. Cirillo ◽  
Alexandre Mauroy ◽  
Ludovic Renson ◽  
Gaëtan Kerschen ◽  
Rodolphe Sepulchre
Keyword(s):  
Invariant Manifold ◽  
Normal Modes ◽  
Geometric Approach ◽  
Nonlinear Normal Modes ◽  
Koopman Operator ◽  
The Past ◽  
Modes Of Vibration ◽  
Nonlinear Normal ◽  
Damped Systems

Nonlinear normal modes of vibration have been the focus of many studies during the past years and different characterizations of them have been proposed. The present work focuses on damped systems, and considers nonlinear normal mode motions as trajectories lying on an invariant manifold, following the geometric approach of Shaw and Pierre. We provide a novel characterization of the invariant manifold, that rests on the spectral theory of the Koopman operator. A main advantage of the proposed approach is a global parametrization of the manifold, which avoids folding issues arising with the use of displacement-velocity coordinates.

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

Characterizing the Dynamics of 3-D Elastic Nonlinear Continua Using the Method of Proper Orthogonal Decompositions

2005 ◽  
Author(s):  
Ioannis Georgiou ◽  
Dimitris Servis
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Normal Modes ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Modes Of Vibration ◽  
Nonlinear Normal ◽  
Proper Orthogonal Decompositions ◽  
Nonlinear Finite Element Model ◽  
Proper Orthogonal

A novel and systematic way is presented to characterize the modal structure of the free dynamics of three-dimensional elastic continua. In particular, the method of Proper Orthogonal Decomposition (POD) for multi-field dynamics is applied to analyze the dynamics of prisms and moderately thick beams. A nonlinear finite element model is used to compute accurate approximations to free motions which in turn are processed by POD. The extension of POD to analyze the dynamics of three-dimensional elastic continua, which are multi-field coupled dynamical system, is carried out by vector and matrix quantization of the finite element dynamics. An important outcome of this study is the fact that POD provides the means to systematically identify the shapes of nonlinear normal modes of vibration of three-dimensional structures from high resolution finite element simulations.

scholarly journals Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

2004 ◽  
Vol 10 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Mathias Legrand ◽  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotating Shaft ◽  
Nonlinear Mode ◽  
Large Amplitude Motions ◽  
Nonlinear Normal ◽  
The Individual

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig–Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig–Bampton model.

scholarly journals Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

2004 ◽  
Vol 10 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Mathias Legrand ◽  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven Shaw
Keyword(s):  
Invariant Manifold ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotating Shaft ◽  
Nonlinear Normal

The Construction of Nonlinear Normal Modes for Systems With Internal Resonance: Application to Rotating Beams

2002 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Internal Resonance ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Reduced Order Model ◽  
Degree Of Freedom ◽  
Beam Model ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Normal

A numerical method for constructing nonlinear normal modes for systems with internal resonances is presented based on the invariant manifold approach. In order to parameterize the nonlinear normal modes, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are constrained to these ‘seed’ variables, resulting in a system of nonlinear partial differential equations governing the constraint relationships, which must be solved numerically. The solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two nonlinear normal modes is constructed, resulting in a reduced-order model that accurately captures the system dynamics. The methodology is then applied to a more large system, namely an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the nonlinear two-mode reduced-order model is verified by time-domain simulations.

scholarly journals A spectral characterization of nonlinear normal modes

2016 ◽  
Vol 377 ◽  
pp. 284-301 ◽  
Author(s):  
G.I. Cirillo ◽  
A. Mauroy ◽  
L. Renson ◽  
G. Kerschen ◽  
R. Sepulchre
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Spectral Characterization ◽  
Nonlinear Normal

Instantaneous Center Manifolds and Nonlinear Modes of Vibration

2012 ◽  
Author(s):  
Hamid A. Ardeh ◽  
Matthew S. Allen
Keyword(s):  
Nonlinear System ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Oscillatory System ◽  
Nonlinear Normal Modes ◽  
Center Manifolds ◽  
Invariance Properties ◽  
Modes Of Vibration ◽  
Instantaneous Center ◽  
Nonlinear Normal

Nonlinear Normal Modes (NNM) have been defined in various ways; first by Rosenberg as a subset of periodic solutions of a nonlinear system and then by Shaw and Pierre as invariant manifolds tangent to the vector field of a nonlinear system at its equilibrium point. This work presents an alternative approach, namely Instantaneous Center Manifold (ICM), that extends the concept of modes of vibration to nonlinear systems, using both periodicity and invariance properties. Instantaneous Center Manifolds are invariant manifolds that contain all of the periodic invariant solutions of the nonlinear oscillatory system. The ICM approach is explained through three simple analytical examples, and is shown to be capable of finding solutions that have been remaining latent using the aforementioned approaches. New branches of nonlinear normal modes, separate from the main branches that are a continuation of linear modes, are illustrated. It is shown that these new branches connect the main branches of Rosenberg’s NNMs, and make it possible to travel from one main branch to another. Some natural extensions and applications of the ICM approach are briefly discussed in the conclusion.

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