Asymptotic Approach for Non-Linear Periodical Vibrations of Continuous Structures

Andrianov, Igor V.; Danishevskyy, Vladyslav V.

doi:10.1006/jsvi.2001.3878

Cited by 24 publications

(20 citation statements)

References 37 publications

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“…The homotopy perturbation technique can only be satisfactorily applied in connection with an effective method of summation. It has already been shown [6,7,[35][36][37][38][39][40] that accurate results can be obtained using the Padé approximants (PA).…”

Section: Modified Bubnov-galerkin Methodsmentioning

confidence: 99%

“…Me and V. Danishevs'kyy [35,36] proposed to introduce homotopy parameter near nondiagonal terms of infinite system (see also Refs. [6,7,[37][38][39][40]).…”

Section: Modified Bubnov-galerkin Methodsmentioning

confidence: 99%

“…Many modifications of Bubnov-Galerkin approach were proposed [35][36][37][38][39][40][41][42][43][44][45][46][47]. I propose to treat various variants of Bubnov-Galerkin approach as variants of homotopy perturbation method [48].…”

Section: Modified Bubnov-galerkin Methodsmentioning

confidence: 99%

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Asymptotic construction of nonlinear normal modes for continuous systems

Andrianov

2007

Nonlinear Dyn

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This paper provides a review of some asymptotic methods for construction of nonlinear normal modes for continuous system (NNMCS). Asymptotic methods of solving problems relating to NNMCS have been developed by many authors. The main features of this paper are that (i) it is devoted to the basic principles of asymptotic approaches for constructing of NNMCS; (ii) it deals with both traditional approaches and, less widely used, new approaches; and (iii) it pays a lot of attention to the analysis of widely used simplified mechanical models for the analysis of NNMCS. The author has paid special attention to examples and discussion of results. Keywords Nonlinear oscillations . Normal mode . Asymptotic approach . Continuous system 1 Definition of nonlinear normal modes for continuous system First definition of nonlinear normal modes for continuous system (NNMCS) is based on exact separation of spatial and time variables [1]. Exact separation of spatial and time variables is possible for some simplified models of continuous systems (Kirchhoff model of nonlinear beam [2], Berger model of nonlinear plate [3,4], Berger-like model of shallow shell [5-7]) for simply supported ends. Naturally, solutions of this type were constructed earlier [2], but Wah was the first who pointed to links between these solutions and those proposed by R.M. Rosenberg for discrete system concepts of NNM [8][9][10]. Naturally, area of applicability of Wah definition is very restricted.Shaw and Pierre proposed the following definition of NNMCS [11]: "If the displacement u 0 (t) = u(s 0 , t) and velocity v 0 (t) = v(s 0 , t) of a single point, s = s 0 , are known, then the entire displacement and velocity field can be determined by the dynamics of that single point u(s, t) = U (u 0 (t), v 0 (t), s, s 0 ), v(s, t) = V (u 0 (t), v 0 (t), s, s 0 )."

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Section: Modified Bubnov-galerkin Methodsmentioning

confidence: 99%

“…Me and V. Danishevs'kyy [35,36] proposed to introduce homotopy parameter near nondiagonal terms of infinite system (see also Refs. [6,7,[37][38][39][40]).…”

Section: Modified Bubnov-galerkin Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic construction of nonlinear normal modes for continuous systems

Andrianov

2007

Nonlinear Dyn

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“…Maccari [36] proposed an asymptotic method for a bar with nonlinear boundary conditions. Andrianov and Danishevs'kyy developed the asymptotic approach to locate the periodic response of a beam with a cubic nonlinearity [37]. Chen [38] used an asymptotic perturbation technique to derive a new family of two-dimensional nonlinear dispersive equation for a pre-stressed hyperelastic compressible plate.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Chen

Chen²

2009

J Eng Math

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Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic behaviors. A coordinate transformation is introduced to derive Mote's model of transverse motion from the governing equation of the stationary string. Mote's model leads to Kirchhoff's model by replacing the tension with the averaged tension over the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models. The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses.

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“…The backbone curves were drawn and internal resonance between modes of vibration was investigated. The longitudinal vibration of a rod and lateral vibration of a beam subjected to nonlinear response of the surrounding medium, posing weakly nonlinear equations, were studied by Andrianov et al [3]. In all of these works, structures sustained large deformation but infinitesimal strain.…”

Section: Introductionmentioning

confidence: 99%

Free vibration of a rod undergoing finite strain

Mousavi

Fariborz

2012

J. Phys.: Conf. Ser.

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The finite strain longitudinal free vibration of a rod is studied. Utilizing second Piola-Kirchhoff stress and Green strain tensors, the equation of motion is written in terms of displacement in reference configuration. Three different types of homogenous boundary conditions may be considered for the rod, leading to three nonlinear eigenvalue problems. The series solutions with three terms satisfying the boundary conditions are utilized and the relationships between amplitudes of vibration are obtained by means of the Galerkin method. The backbone curves are drawn and the internal resonance between different modes of vibration is analyzed.

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Asymptotic Approach for Non-Linear Periodical Vibrations of Continuous Structures

Cited by 24 publications

References 37 publications

Asymptotic construction of nonlinear normal modes for continuous systems

Asymptotic construction of nonlinear normal modes for continuous systems

Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Free vibration of a rod undergoing finite strain

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