Multiwave non-linear couplings in elastic structures

Kovriguine, D. A.; Maugin, Gérard A.; Potapov, A. I.

doi:10.1016/s0020-7683(02)00365-7

Cited by 5 publications

(4 citation statements)

References 3 publications

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“…This yields the concept of cascade wave processes following along ideas of Richardson and Landau. We emphasize the necessarily sketchy nature of this paper as cumbersome formulas of repetitive form are only to be found in a long report [10] and also some of our recent papers [8,9]. This does not exhaust the subject matter since direct numerical solutions of the true threedimensional equations would be most instructive.…”

Section: Two-dimensional Examplementioning

confidence: 92%

“…For these we refer the reader to recent synthesis works [1, 2, 4-12, 14, 15, 17, 18]. The one-dimensional structural examples briefly examined in this contribution have revealed the essential properties of three-wave and four-wave resonance couplings [8]. Although the emphasis has been placed on mechanical consequences of these couplings, the analogy with nonlinear optical systems (of which the jargon is often used) is more than obvious.…”

Section: Two-dimensional Examplementioning

confidence: 96%

“…Then the linear coupling between modes in (3.2) with vanishing right-hand sides can be said to be weak. On discarding μ-terms, (3.5) yield the following separate dispersion relations whenever we implement the inextensibility condition of the middle line of the ring V = 0; see [4]: 8) for bending and azimuthal waves, respectively. In these conditions it is shown that the amplitudes of the azimuthal, A k,n , and bending, B k,n , components of the normal waves are linearly related by B k,n = −ip k,n A k,n (3.9) with coefficients p k,n (k = 1,2), given approximately by The linear dynamical solution (3.8)-(3.10) is usually considered as quite satisfactory.…”

Section: Waves Traveling Around a Closed Circular Ringmentioning

confidence: 99%

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Multiwave nonlinear couplings in elastic structures

Kovriguine

Maugin

Potapov

2006

Mathematical Problems in Engineering

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This short contribution considers the essentials of nonlinear wave properties in typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate. It is found that nonlinear resonance is experienced in all the systems exhibiting continuous and discrete spectra, respectively. Multiwave interactions and the stability of coupled modes with respect to small perturbations are discussed. The emphasis is placed on mechanical phenomena, for example, stress amplification, although some analogies with some nonlinear optical systems are also obvious. The nonlinear resonance coupling in a plate within the Kirchhoff-Love approximation is selected as a two-dimensional example exhibiting a rich range of resonant wave phenomena. This is originally examined by use of Whitham's averaged Lagrangian method. In particular, the existence of three basic resonant triads between longitudinal, shear, and bending modes is shown. Some of these necessarily enter cascade wave processes related to the instability of some mode components of the triad under small perturbations.

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Section: Two-dimensional Examplementioning

confidence: 92%

Section: Two-dimensional Examplementioning

confidence: 96%

Section: Waves Traveling Around a Closed Circular Ringmentioning

confidence: 99%

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Multiwave nonlinear couplings in elastic structures

Kovriguine

Maugin

Potapov

2006

Mathematical Problems in Engineering

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“…A general form of non-linearity requires special analysis. In this case the relation of dispersion and additional requirements for solutions should be studied (for example, in [19] a coupled Klein -Gordon system is investigated, in [12], non-linear waves in typical mechanical systems are analyzed; in [22], an analysis of the four-wave resonant interactions in shallow water is presented).…”

Section: Introductionmentioning

confidence: 99%

Asymptotical Analysis of Some Coupled Nonlinear Wave Equations

Krylovas

Kriauzienė

2011

Mathematical Modelling and Analysis

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We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein -Gordon, Hirota -Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro -differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro -differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

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