 Nonlinear resonance

In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude.
Contents
Description
Generically two types of resonances have to be distinguished – linear and nonlinear. From the physical point of view, they are defined by the fact whether or not external force coincides with the eigenfrequency of the system or not (linear and nonlinear resonance correspondingly). The frequency condition of nonlinear resonance reads
with possibly different being eigenfrequencies of the linear part of some nonlinear partial differential equation. Here is a vector with the integer subscripts i being indexes into Fourier harmonics – or eigenmodes – see Fourier series. Accordingly, the frequency resonance condition is equivalent to a Diophantine equation with many unknowns. The problem of finding their solutions is equivalent to the Hilbert's tenth problem that is proven to be algorithmically unsolvable.
Main notions and results of the theory of nonlinear resonances are:^{[1]}
 The use of the special form of dispersion functions appearing in various physical applications allows to find the solutions of frequency resonance condition.
 The set of resonances for given dispersion function and the form of resonance conditions is partitioned into nonintersecting resonance clusters; dynamics of each cluster can be studied independently (at the appropriate timescale).
 Each resonance cluster can be represented by its NRdiagram which is a plane graph of the special structure. This representation allows to reconstruct uniquely 3a) dynamical system describing timedependent behavior of the cluster, and 3b) the set of its polynomial conservation laws which are generalization of Manley–Rowe constants of motion for the simplest clusters (triads and quartets)
 Dynamical systems describing some types of the clusters can be solved analytically.
 These theoretical results can be used directly for describing reallife physical phenomena (e.g. intraseasonal oscillations in the Earth's atmosphere) or various wave turbulent regimes in the theory of wave turbulence.
Nonlinear resonance shift
Nonlinear effects may significantly modify the shape of the resonance curves of harmonic oscillators. First of all, the resonance frequency ω is shifted from its "natural" value ω_{0} according to the formula
 ω = ω_{0} + κA^{2},
where A is the oscillation amplitude and κ is a constant defined by the anharmonic coefficients. Second, the shape of the resonance curve is distorted (foldover effect). When the amplitude of the (sinusoidal) external force F reaches a critical value F_{crit} instabilities appear. The critical value is given by the formula
where m is the oscillator mass and γ is the damping coefficient. Furthermore, new resonances appear in which oscillations of frequency close to ω_{0} are excited by an external force with frequency quite different from ω_{0}.
See also
Notes and references
Notes
References
 Landau, L. D.; Lifshitz, E. M. (1976), Mechanics (3rd ed.), Pergamon Press, ISBN 0080210228 (hardcover) and 0080291414 (softcover)
External links
 Elmer, FranzJosef (July 20, 1998), Nonlinear Resonance], University of Basel, http://monet.physik.unibas.ch/~elmer/pendulum/nonres.htm, retrieved 27 October 2010
Categories: Mechanical vibrations
Wikimedia Foundation. 2010.
Look at other dictionaries:
Resonance — This article is about resonance in physics. For other uses, see Resonance (disambiguation). Resonant redirects here. For the phonological term, see Sonorant. Increase of amplitude as damping decreases and frequency approaches resonant frequency… … Wikipedia
Nonlinear metamaterials — A nonlinear metamaterial is an artificially constructed material that can exhibit properties not found in nature. Its response to electromagnetic radiation can be characterized by its permittivity and material permeability. The product of the… … Wikipedia
Suprathreshold stochastic resonance — is a particular form of stochastic resonance, that is, the phenomenon where random fluctuations, or noise, provides a signal processing benefit in a nonlinear system. Unlike most of the nonlinear systems where stochastic resonance occurs,… … Wikipedia
Stochastic resonance — (also known as SR) is observed when noise added to a system improves the systems performance in some fashion. More technically, SR occurs if the signal to noise ratio of a nonlinear system or device increases for moderate values of noise… … Wikipedia
Secular resonance — A secular resonance is a type of orbital resonance. Secular resonances occur when the precession of two orbits is synchronised (a precession of the perihelion, of the ascending node, or both). A small body in secular resonance with a much larger… … Wikipedia
Chirikov criterion — The Chirikov criterion or Chirikov resonance overlap criterion was established by the Russian physicist Boris Chirikov. Back in 1959, he published a seminal article (Atom. Energ. 6: 630 (1959)), where he introduced the very first physical… … Wikipedia
Boris Chirikov — Boris Walerianowitsch Tschirikow (russisch Борис Валерианович Чириков, wiss. Transliteration Boris Valerianovič Čirikov, englische Transliteration Boris Chirikov; * 6. Juni 1928 in Orjol, Russland; † 12. Februar 2008 in Akademgorodok in… … Deutsch Wikipedia
Boris Tschirikow — Boris Walerianowitsch Tschirikow (russisch Борис Валерианович Чириков, wiss. Transliteration Boris Valerianovič Čirikov, englische Transliteration Boris Chirikov; * 6. Juni 1928 in Orjol, Russland; † 12. Februar 2008 in Akademgorodok in… … Deutsch Wikipedia
Boris Walerianowitsch Tschirikow — (russisch Борис Валерианович Чириков, wiss. Transliteration Boris Valerianovič Čirikov, englische Transliteration Boris Chirikov; * 6. Juni 1928 in Orjol, Russland; † 12. Februar 2008 in Akademgorodok in Nowosibirsk) war ein russischer… … Deutsch Wikipedia
Dinitromethan — Strukturformel Allgemeines Name Dinitromethan Summenformel … Deutsch Wikipedia