# Forced Vibrations : Resonance

#### Master J

I am looking at forced vibrations and I have come across this:

(w_max)^2 = (w_0)^2 - (1/2)y^2

Now I am not entirely sure of what the (w_max) is. ANd where does this equation come from? It was simply stated without a derivation.

THanks guys!

#### HallsofIvy

There is simply a blank page with that formula on it?

Surely there is some sort of context. As you write it I not only do not know what w_max is, I also don't know what w_0 is or what y is!

#### Master J

W_O is the natural/resonance frequency for the system being driven by a force with frequency w.

y is the width, or the damping constant divided by the mass of the oscillator.

The equation comes up in resonance. I think it has to do with the maximum amplitude of the system ?

#### Andy Resnick

That expression comes up a lot in damped driven oscillations- not just masses and springs, but any sort of linear oscillator.

W_max is the exact resonance frequency, w_0 is the "classical resonance frequency" (i.e. the resonance without damping present).

This expression is strightforward to derive beginning with the 2nd order ODE for a damped, driven oscillator.

http://en.wikipedia.org/wiki/Harmonic_oscillator

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