Editing Oscillation (section)

== Driven oscillations ==
In addition, an oscillating system may be subject to some external force, as when an AC [[Electronic circuit|circuit]] is connected to an outside power source.  In this case the oscillation is said to be ''[[driven oscillations|driven]]''. 

The simplest example of this is a spring-mass system with a [[Sine wave|sinusoidal]] driving force.
<math display="block">\ddot{x} + 2 \beta\dot{x} + \omega_0^2 x = f(t),</math>where <math>f(t) = f_0 \cos(\omega t + \delta).</math>

This gives the solution: <math display="block">x(t) = A \cos(\omega t - \delta) + A_{tr} \cos(\omega_1 t - \delta_{tr}),</math>
where <math>A = \sqrt{\frac {f_0^2} {(\omega_0^2 - \omega ^2)^2 + 4 \beta^2 \omega^2}}</math> and <math>\delta = \tan^{-1}\left(\frac {2 \beta \omega} {\omega_0^2 - \omega^2}\right)</math>

The second term of {{math|''x''(''t'')}} is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system.

Some systems can be excited by energy transfer from the environment.  This transfer typically occurs where systems are embedded in some [[fluid]] flow.  For example, the phenomenon of [[Aeroelastic flutter|flutter]] in [[aerodynamics]] occurs when an arbitrarily small displacement of an [[aircraft]] [[wing]] (from its equilibrium) results in an increase in the [[angle of attack]] of the wing on the air flow and a consequential increase in [[coefficient of lift|lift coefficient]], leading to a still greater displacement.  At sufficiently large displacements, the [[stiffness]] of the wing dominates to provide the restoring force that enables an oscillation.

=== Resonance ===
[[Resonance]] occurs in a damped driven oscillator when ω = ω<sub>0</sub>, that is, when the driving frequency is equal to the natural frequency of the system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations.