Normal vibrations of a general class of conservative oscillators

Mikhlin, Yuri V.

doi:10.1007/bf00045048

Cited by 18 publications

(26 citation statements)

References 24 publications

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“…The nonlinear normal modes (NNMs), which are the generalization of the normal modes of linear systems, are considered in [27,28]. The method of NNMs is an effective tool to study engineering problems.…”

Section: Nonlinear Normal Modes Of Shellsmentioning

confidence: 99%

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Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

2006

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Donnell equations are used to simulate free nonlinear oscillations of cylindrical shells with imperfections. The expansion, which consists of two conjugate modes and axisymmetric one, is used to analyze shell oscillations. Amplitudes of the axisymmetric motions are assumed significantly smaller, than the conjugate modes amplitudes. Nonlinear normal vibrations mode, which is determined by shell imperfections, is analyzed. The stability and bifurcations of this mode are studied by the multiple scales method. It is discovered that stable quasiperiodic motions appear at the bifurcations points.The forced oscillations of circular cylindrical shells in the case of two internal resonances and the principle resonance are analyzed too. The multiple scales method is used to obtain the system of six modulation equations. The method for stability analysis of standing waves is suggested. The continuation algorithm is used to analyze fixed points of the system of the modulation equations.

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Section: Nonlinear Normal Modes Of Shellsmentioning

confidence: 99%

“…In this section the stability and bifurcations of the NNM (28,19) is studied. It is important, that if ε = 0, the system (9) is essentially nonlinear.…”

Section: Bifurcations and Stability Of Nonlinear Normal Modementioning

confidence: 99%

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

2006

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“…a family of periodic orbits, that persist in the vicinity of fixed points with the presence of nonlinearity. This definition has been used by Rosenberg and other subsequent investigators who retained this framework [25,26,27,28]. However, a more general definition has emerged since the work by Shaw and Pierre [29], who used the framework provided by the center manifold theorem [30] in order to define a NNM as an invariant manifold in phase space, tangent at origin to the corresponding eigenmode.…”

Section: Introductionmentioning

confidence: 99%

On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Blanc

Touzé

Mercier

et al. 2013

Mechanical Systems and Signal Processing

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Numerical computation of Nonlinear Normal Modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three different mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the center manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve.

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“…Since then, a large body of literature has addressed, with notable success, the qualitative and quantitative analysis of nonlinear phenomena using NNMs [4]. NNMs are a useful mathematical tool to obtain minimal descriptions in structural dynamics and also to identify the underlying structure of the system nonlinear response [5]. They have the potential to handle strong structural nonlinearity and address the individualistic nature of nonlinear systems [6,7].…”

Section: Introductionmentioning

confidence: 99%

Energy-based formulation for nonlinear normal modes in cable-stayed beam

Wang

Luo

2015

Applied Mathematics and Computation

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Normal vibrations of a general class of conservative oscillators

Abstract: Abstract. This paper considers normal vibrations with curvilinear trajectories in a configuration space of systems which are close to systems permitting rectilinear normal modes of vibration. Analysis of trajectories of normal vibrations in the configuration space is used.

Cited by 18 publications

References 24 publications

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Energy-based formulation for nonlinear normal modes in cable-stayed beam

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