# Finding the forcing frequency to maximize the amplitude

1. May 2, 2010

### Jamin2112

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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3. 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.

2. May 2, 2010

### lanedance

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

Last edited: May 2, 2010
3. May 2, 2010

### Jamin2112

There has to be a better way

4. May 2, 2010

5. May 2, 2010

### lanedance

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

6. May 2, 2010

### 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