Finding the forcing frequency to maximize the amplitude

[Jamin2112]

Homework Statement

Consider the following underdamped oscillator governed by:

u''(t) + ¥u'(t) + w₀²u(t)=Fcos(wt)

(a) Find the ge...
(b) The hom...
(c) What is the forcing frequency w for which the amplitude R in the previous part attains a maximum? Show that it is always less than the natural frequency w₀.

Homework Equations

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The Attempt at a Solution

So, I solved the problem in part (a) and put it in the form Rcos(wt-µ) in part (b). I've looked over my work several times; the answer is messy. You can try it for yourself if you really feel like it. Otherwise, trust me when I say that the long term solution (getting rid of the homogenous equation since it involves crap in the form e-^at, where a is positive) is

[F(w₀2-w²)]/[(w₀²-w²)²+(wy)²]*√[1+(wy)²/(w₀²-w²)²]*cos(wt-tan^-1(wy/(w₀²-w²)).

...so how do I find the right w to maximize the amplitude? I have no idea. Please explain in detail. This assignment is due tomorrow morning.

 

[lanedance]

if you have a function in terms of w for the amplitude, could you differentiate and set to zero to find the maxima?

 

[Jamin2112]

if you have a function in terms of w for the amplitude, could you differentiate and set to zero to find the maxima?

There has to be a better way

 

[lanedance]

you solution looks too complicated - have a look at this
http://en.wikipedia.org/wiki/Harmonic_oscillator#Sinusoidal_driving_force

 

[lanedance]

note there should be no homogenous part in the steady state forced solution anyway, only the driven frequency term

 

[lanedance]

actually maybe you can clean that up a bit and its ok? - notice when you find the amplitude transfer function (divide by Fcoswt), the cos part with the tan inside will disappear as its just the same frequency response shifted by a phase given by the tan term