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Damped Driven Harmonic Oscillator.

  1. Jan 14, 2015 #1
    1. The problem statement, all variables and given/known data
    An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the displacement and speed as a function of the driving frequency and find the frequencies for which the phase angle for which the angle is 45 deg.


    2. Relevant equations


    3. The attempt at a solution
    Using the general form of the solution:
    x(t) = A(w) sin(wt-Φ)
    where Φ=atan(2wp/(w_0^2-w^2))
    A(w) = (F/m)/((w_0^2-w^2)^2 + (2wp)^2)^0.5

    I am positive the above equations are correct and come from the differential equation for this case.

    Now, u(t) [speed] = d x(t)/dt.
    = w*A(w)*cos(wt-Φ)
    =w*A(w)*sin(wt-Φ+pi/2)

    My question: Now the speed amplitude, I believe, is wA(w). Won't the phase angle between the displacement and velocity always be pi/2 irrespective of w?


     
  2. jcsd
  3. Jan 14, 2015 #2

    Orodruin

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    Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.
     
  4. Jan 14, 2015 #3
    I agree but the question says the phase between the speed and the displacement. Am I interpreting this wrong?
     
  5. Jan 14, 2015 #4

    Orodruin

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    It is very likely a misprint.
     
  6. Jan 14, 2015 #5
    Thanks! I'm going to quote this as a misprint in my solution with the explanation of my rationale. Hopefully, that is good enough for my professor
     
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