View source for Resonance

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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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.P. Strelkov,   "Introduction to oscillation theory" , Moscow  (1951)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR></table>