Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

Arquier, Rémi; Bellizzi, Sergio; Bouc, R.; Cochelin, Bruno

doi:10.1016/j.compstruc.2006.01.011

Cited by 52 publications

(36 citation statements)

References 28 publications

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“…For instance, the simplest one consists in setting one component of the initial conditions vector to zero, as in [17]. This is illustrated in Fig.…”

Section: Shooting Methodsmentioning

confidence: 99%

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Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Peeters

Viguié

Sérandour

et al. 2009

Mechanical Systems and Signal Processing

433

369

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The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part I. One reason of the still limited use of NNMs in structural dynamics is that their computation is often regarded as impractical. However, when resorting to numerical algorithms, we show that the NNM computation is possible with limited implementation effort, which paves the way to a practical method for determining the NNMs of nonlinear mechanical systems. The proposed algorithm relies on two main techniques, namely a shooting procedure and a method for the continuation of NNM motions. The algorithm is demonstrated using four different mechanical systems, a weakly and a strongly nonlinear two-degree-of-freedom system, a simplified discrete model of a nonlinear bladed disk and a nonlinear cantilever beam discretized by the finite element method.

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“…For instance, the simplest one consists in setting one component of the initial conditions vector to zero, as in [17]. This is illustrated in Fig.…”

Section: Shooting Methodsmentioning

confidence: 99%

“…It is a semi-analytical technique that is based on a power series expansion of the unknowns parameterized by a control parameter. It is utilized in [17] to follow the NNM branches in conjunction with finite difference methods, following a framework similar to that of [12].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Peeters

Viguié

Sérandour

et al. 2009

Mechanical Systems and Signal Processing

433

369

View full text Add to dashboard Cite

show abstract

“…For conservative systems, different reference algorithms to compute NNMs exist [11,13,32] . For instance, the "exact" manifolds can be computed using the technique developed in [13] , which combines shooting and pseudo-arclength continuation.…”

Section: Algorithm Demonstration On Conservative Examplesmentioning

confidence: 99%

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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“…Alternatively, the harmonic balance and asymptotic numerical methods are used in Ref. [9]. The main difficulty associated with NNMs is that their calculation for large-scale models is expensive.…”

Section: Introductionmentioning

confidence: 99%

Bridging the Gap Between Nonlinear Normal Modes and Modal Derivatives

Sombroek

Renson

Tiso

et al. 2016

Nonlinear Dynamics, Volume 1

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. AbstractNonlinear Normal Modes (NNMs) have a clear conceptual relation to the classical linear normal modes (LNMs), yet they offer a solid theoretical framework for interpreting a wide class of non-linear dynamical phenomena with no linear counterpart. The main difficulty associated with NNMs is that their calculation for largescale models is expensive, particularly for distributed nonlinearities. Repeated direct time integrations need to be carried out together with extensive sensitivity analysis to reproduce the frequency-energy dependence of the modes of interest.In the present paper, NNMs are computed from a reduced model obtained using a quadratic transformation comprising LNMs and Modal Derivatives (MDs). Previous studies have shown that MDs can capture the essential dynamics of geometrically nonlinear structures and can greatly reduce the computational cost of time integration.A direct comparison with the NNMs computed from another standard reduction technique highlights the capability of the proposed reduction method to capture the essential nonlinear phenomena. The methodology is demonstrated using simple examples with 2 and 4 degrees of freedom.

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Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

Cited by 52 publications

References 28 publications

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

Nonlinear Normal Modes of Nonconservative Systems

Bridging the Gap Between Nonlinear Normal Modes and Modal Derivatives

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