How to find formula for resonant frequency of a forced oscillator.

alimon.cioro

In a damped forced harmonic oscillator the amplitude is determined by a series of paramenters according to :

A = (Fo/m)/ (sqrt( (wo^2-w^2)^2+(wy)^2) ).

where

Fo= driving force,
m=mass of spring
wo=natural frequency of system.
w=driving frequency
y=damping constant.

Now my question is how do you find the driving angular frequency w at which A is maximum, which should be the resonante frequency (that is not exactly Wo).

The resonant frequency formula is :

Wres = sqrt(Wo^2-(y^2)/2)) .

I though that differentiating the formula for A in terms of dA/dw and equating it to zero should give me an answer but the maths look to convoluted for such a simple answer. Any ideas of how to get the resonante frequency?

nrqed

That's precisely what you must do. The equation you get is actually easy to solve.
Note that you get something over ( (wo^2-w^2)^2+(wy)^2)^(3/2). You may multiply both sides of the equation by ( (wo^2-w^2)^2+(wy)^2)^(3/2) and you are left with a simple expression equal to zero, which is then easy to solve.

AlephZero

The easy way is to see that if A is a maximum, 1/A is a mimimum, and 1/A² is also a minimum.

Differentiating d (1/A²) / dw is easy.

BTW the formula you gave in the OP for the resonant frequency is wrong. If can't possibly be right to subtract a frequency squared and a damping corefficient squared, they don't have the same units!

bgc

The damping coefficient is s^-1 Rather convoluted to show. OTOH, note the equation for the position. X = A*exp(-dc*t)*cos(etc.) for the simple harmonic damped oscillator.

bc