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The phenomenon of increasing amplitudes of [[Forced oscillations|forced oscillations]] when the frequency of the external action approximates one of the frequencies of the eigenoscillations (cf. [[Eigen oscillation|Eigen oscillation]]) of a dynamical system. Resonance is simplest in a linear dynamical system. The differential equation of motion of a linear system with one degree of freedom in an environment with viscous friction and with harmonic external action takes the form:
 
The phenomenon of increasing amplitudes of [[Forced oscillations|forced oscillations]] when the frequency of the external action approximates one of the frequencies of the eigenoscillations (cf. [[Eigen oscillation|Eigen oscillation]]) of a dynamical system. Resonance is simplest in a linear dynamical system. The differential equation of motion of a linear system with one degree of freedom in an environment with viscous friction and with harmonic external action takes the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816201.png" /></td> </tr></table>
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$$a\ddot q+b\dot q+cq=H\sin(pt+\delta),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816202.png" /> is a generalized coordinate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816203.png" /> are constant parameters characterizing the system, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816206.png" /> are the amplitude, the frequency and the initial phase of the external action, respectively. The stationary forced oscillations occur in accordance with the harmonic law with frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816207.png" /> and amplitude
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where $q$ is a generalized coordinate, $a,b,c$ are constant parameters characterizing the system, and $H$, $p$, $\delta$ are the amplitude, the frequency and the initial phase of the external action, respectively. The stationary forced oscillations occur in accordance with the harmonic law with frequency $p$ and amplitude
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816208.png" /></td> </tr></table>
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$$D=\frac{H}{a\sqrt{(k^2-p^2)^2+b^2p^2/a^2}}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r0816209.png" /> is the frequency of the eigenoscillations in the absence of energy dissipation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162010.png" />. The amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162011.png" /> has a maximum value when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162012.png" />, and with low energy dissipation it is close to this value when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162013.png" />. Sometimes by resonance is meant that case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162015.png" /> then, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162016.png" />, the amplitude of the forced oscillations increases proportional to time. If a linear system has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081620/r08162017.png" /> degrees of freedom, then resonance begins when the frequency of the external force coincides with one of the eigenfrequencies of the system. With non-harmonic action, resonance may occur only when the frequencies of its harmonic spectrum coincide with the frequencies of eigenoscillations.
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where $k=\sqrt{c/a}$ is the frequency of the eigenoscillations in the absence of energy dissipation $(b=0)$. The amplitude $D$ has a maximum value when $p/k=\sqrt{1-b^2/2ac}$, and with low energy dissipation it is close to this value when $p=k$. Sometimes by resonance is meant that case where $p=k$. If $b=0$ then, when $p=k$, the amplitude of the forced oscillations increases proportional to time. If a linear system has $n$ degrees of freedom, then resonance begins when the frequency of the external force coincides with one of the eigenfrequencies of the system. With non-harmonic action, resonance may occur only when the frequencies of its harmonic spectrum coincide with the frequencies of eigenoscillations.
  
 
====References====
 
====References====

Revision as of 12:24, 13 August 2014

The phenomenon of increasing amplitudes of forced oscillations when the frequency of the external action approximates one of the frequencies of the eigenoscillations (cf. Eigen oscillation) of a dynamical system. Resonance is simplest in a linear dynamical system. The differential equation of motion of a linear system with one degree of freedom in an environment with viscous friction and with harmonic external action takes the form:

$$a\ddot q+b\dot q+cq=H\sin(pt+\delta),$$

where $q$ is a generalized coordinate, $a,b,c$ are constant parameters characterizing the system, and $H$, $p$, $\delta$ are the amplitude, the frequency and the initial phase of the external action, respectively. The stationary forced oscillations occur in accordance with the harmonic law with frequency $p$ and amplitude

$$D=\frac{H}{a\sqrt{(k^2-p^2)^2+b^2p^2/a^2}}$$

where $k=\sqrt{c/a}$ is the frequency of the eigenoscillations in the absence of energy dissipation $(b=0)$. The amplitude $D$ has a maximum value when $p/k=\sqrt{1-b^2/2ac}$, and with low energy dissipation it is close to this value when $p=k$. Sometimes by resonance is meant that case where $p=k$. If $b=0$ then, when $p=k$, the amplitude of the forced oscillations increases proportional to time. If a linear system has $n$ degrees of freedom, then resonance begins when the frequency of the external force coincides with one of the eigenfrequencies of the system. With non-harmonic action, resonance may occur only when the frequencies of its harmonic spectrum coincide with the frequencies of eigenoscillations.

References

[1] S.P. Strelkov, "Introduction to oscillation theory" , Moscow (1951) (In Russian)


Comments

References

[a1] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
How to Cite This Entry:
Resonance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resonance&oldid=15377
This article was adapted from an original article by N.V. Butenin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article