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Homework Help: Finding the forcing frequency to maximize the amplitude

  1. May 2, 2010 #1
    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

    ?

    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. jcsd
  3. May 2, 2010 #2

    lanedance

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    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
  4. May 2, 2010 #3
    There has to be a better way
     
  5. May 2, 2010 #4

    lanedance

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  6. May 2, 2010 #5

    lanedance

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    note there should be no homogenous part in the steady state forced solution anyway, only the driven frequency term
     
  7. May 2, 2010 #6

    lanedance

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    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
     
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