# Forced Oscillations and Resonance

Forced Oscillations and Resonance

A person swinging in a swing without anyone pushing it is an example of  oscillation. However if someone pushes the swing periodically the swing forced or driven oscillations Two angular frequencies are associated with a system undergoing driven osciIlations (I) the natural angular frequency w of the system which is the angular frequency at which it would oscillate if It were suddenly turbed and then left to osciIlate freely and (2) the angular  requency external driving force causing the driven oscillations. We can use to represent an idealized forced simple harmonic  lator if we allow the structure marked “rigid support” to move up and down avariable angular frequency Such a forced osciIlator oscillates at the angu frequency of the driving force and its displacement x(t). where x m is the amplitude of the oscillations.

How large the displacement amplitude xmis depends on a complicated function of  The velocity amplitude vnt of the oscillations is easier to describe greatest when a condition called resonance is also approximately the conditioin which the displacement amplitude of the oscillations is greatest Thus if push a swing at its natural angular frequency displacement and velocity  tudes wiII increase to large values, a fact that children learn quickly by trial error If you push at other angular frequencies either higher or lower the displment and velocity amplitudes will be smaller the displacement amplitude of an oscillator on the angular frequencyof the driving force, for three values of the  b Note that for all three the amplitude is approximately greatest -that is hen the resonance condition of satisfied. curves of that less damping gives a taller and  resonance peak All mechanical structures have one or more natural angular frequencies a structure is subjected to a strong external driving force hat matches one The displacement amplitude of a forced oscillator varies as the angular frequency  of the driving force is varied The amplitude is greatest approximately at  the resonance condition The curves here correspond  to three values of the damping constant b.