Non-linear normal modes and invariant manifolds

Shaw, Steven W.; Pierre, Christophe

doi:10.1016/0022-460x(91)90412-d

Cited by 293 publications

(225 citation statements)

References 4 publications

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“…To provide a rigorous extension of the NNM concept to damped systems, Shaw and Pierre defined an NNM as a two-dimensional invariant manifold in phase space [3] . Such a manifold is invariant under the flow (i.e., orbits that start out in the manifold remain in it for all time), which generalizes the invariance property of LNMs to nonlinear systems.…”

Section: Review Of Normal Modes For Nonlinear Systemsmentioning

confidence: 99%

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Nonlinear Normal Modes of Nonconservative Systems

Renson¹,

Kerschen²

2013

Topics in Nonlinear Dynamics, Volume 1

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Linear modal analysis is a mature tool enjoying various applications ranging from bridges to satellites. Nevertheless, modal analysis fails in the presence of nonlinear dynamical phenomena and the development of a practical nonlinear analog of modal analysis is a current research topic. Recently, numerical techniques (e.g., harmonic balance, continuation of periodic solutions) were developed for the computation of nonlinear normal modes (NNMs). Because these methods are limited to conservative systems, the present study targets the computation of NNMs for nonconservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, a new finite element technique is proposed to solve the set of partial differential equations governing the manifold geometry.

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Section: Review Of Normal Modes For Nonlinear Systemsmentioning

confidence: 99%

“…Shaw and Pierre proposed a generalization of Rosenberg's definition that provides an elegant extension of the NNM concept to damped systems. Based on geometric arguments and inspired by the center manifold theory, they defined an NNM as a two-dimensional invariant manifold in phase space [3,4] .…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Normal Modes of Nonconservative Systems

Renson¹,

Kerschen²

2013

Topics in Nonlinear Dynamics, Volume 1

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“…To conclude this analysis, a more thorough comparison of real normal form with NNMs computation using the center manifold technique [11][12][13], as proposed by Shaw and Pierre [32,33], is here given. For the sake of clarity, the original dynamical equations (1) are restricted to a two dofs problem, so that…”

Section: Normal Forms and Nonlinear Normal Modesmentioning

confidence: 99%

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

2012

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Abstract Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form com-

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“…The most diffused ones are probably the nonlinear normal modes (under this name are classified the techniques based on the centre manifold theorem, the normal form theory and the inertial manifold) [1][2][3][4][5][6][7][8][9], including the most diffused version with asymptotic approach, the discretization of the equations of motion by using global (i.e. defined on the whole structure) admissible functions [10][11][12][13], the proper orthogonal decomposition method [14][15][16][17][18] and the natural mode discretization [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

Amabili

2013

Phil. Trans. R. Soc. A.

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Reduced-order models are essential to study nonlinear vibrations of structures and structural components. The natural mode discretization is based on a twostep analysis. In the first step, the natural modes of the structure are obtained. Because this is a linear analysis, the structure can be discretized with a very large number of degrees of freedom. Then, in the second step, a small number of these natural modes are used to discretize the nonlinear vibration problem with a huge reduction in the number of degrees of freedom. This study finds a recipe to select the natural modes that must be retained to study nonlinear vibrations of an angle-ply laminated circular cylindrical shell that the author has previously studied by using admissible functions defined on the whole structure, so that an accuracy analysis is performed. The higher-order shear deformation theory developed by Amabili and Reddy is used to model the shell.

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Non-linear normal modes and invariant manifolds

Cited by 293 publications

References 4 publications

Nonlinear Normal Modes of Nonconservative Systems

Nonlinear Normal Modes of Nonconservative Systems

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Reduced-order models for nonlinear vibrations, based on natural modes: the case of the circular cylindrical shell

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