Resonance

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.

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