Multiwave nonlinear couplings in elastic structures

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

doi:10.1155/mpe/2006/76041

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…all possessing nonlinear resonances [2]. Nonlinear resonances appear in a great amount of typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate [3]. The so-called "nonlinear resonance jump", important for the analysis of a turbine governor positioning system of hydroelectric power plants, can cause severe damage to the mechanical, hydraulic and electrical systems [4].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of nonlinear resonances in Hamiltonian systems

Bustamante

Kartashova

2009

Europhys. Lett.

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It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We show that 1) most frequently met clusters are described by integrable dynamical systems, and 2) construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for system with cubic Hamiltonian. PACS numbers: 47.10.Df, 47.10.Fg, 02.70.Dh 1. Introduction. A notion of resonance runs through all our life. Without resonance we wouldn't have radio, television, music, etc. The general properties of linear resonances are quite well-known; their nonlinear counterpart is substantially less studied though interest in understanding nonlinear resonances is enormous. Famous experiments of Tesla show how disastrous resonances can be: he studied experimentally vibrations of an iron column which ran downward into the foundation of the building, and caused sort of a small earthquake in Manhattan, with smashed windows and swayed buildings [1]. Another example is Tacoma Narrows Bridge which tore itself apart and collapsed (in 1940) under a wind of only 42 mph, though designed for winds of 120 mph. Nonlinear resonances are ubiquitous in physics. Euler equations, regarded with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.) all possessing nonlinear resonances [2]. Nonlinear resonances appear in a great amount of typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate [3].The so-called "nonlinear resonance jump", important for the analysis of a turbine governor positioning system of hydroelectric power plants, can cause severe damage to the mechanical, hydraulic and electrical systems [4]. It was recently established that nonlinear resonance is the dominant mechanism behind outer ionization and energy absorption in near infrared laser-driven rare-gas or metal clusters [5]. The characteristic resonant frequencies observed in accretion disks allow astronomers to determine whether the object is a black hole, a neutron star, or a quark star [6]. Thermally induced variations of the helium dielectric permittivity in superconductors are due to microwave nonlinear resonances [7]. Temporal processing in the central auditory nervous system analyzes sounds using networks of nonlinear neural resonators [8]. The non-linear resonant response of biological tissue to the action of an electromagnetic field is used to investigate cases of suspected disease or cancer [9].

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Section: Introductionmentioning

confidence: 99%

Dynamics of nonlinear resonances in Hamiltonian systems

Bustamante

Kartashova

2009

Europhys. Lett.

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“…Nonlinear resonances are ubiquitous in physics. They appear in a great amount of typical mechanical systems [22], in engineering [23], astronomy [21], biology [9], etc. etc.…”

Section: Introductionmentioning

confidence: 99%

Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

Bustamante

Kartashova

2011

Commun. comput. phys.

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Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in k-space, and 2) existence of "gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: kinetic, described by wave kinetic equations and power energy spectra; discrete, characterized by resonance clustering; and mesoscopic, where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others -only for particular initial conditions. Integrable clusters yield pattern formation in kinetic and mesoscopic wave turbulent regimes.

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“…Nonlinear resonances are ubiquitous in physics. They appear in a great amount of typical mechanical systems [13,41], in engineering [8,17,42,57], astronomy [40], biology [15], etc. etc.…”

Section: Introductionmentioning

confidence: 99%

Resonance clustering in wave turbulent regimes: Integrable dynamics

Kartashova¹,

Bustamante²

2010

Preprint

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Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in k-space, and 2) existence of "gaps" in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: kinetic, described by wave kinetic equations and power energy spectra; discrete, characterized by resonance clustering; and mesoscopic, where both types of wave field time evolution coexist. In this paper we study integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we establish that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others -only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.

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Multiwave nonlinear couplings in elastic structures

Cited by 5 publications

References 12 publications

Dynamics of nonlinear resonances in Hamiltonian systems

Dynamics of nonlinear resonances in Hamiltonian systems

Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

Resonance clustering in wave turbulent regimes: Integrable dynamics

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