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Steady State Solution of Forced, Damped Harmonic Oscillator

  1. Sep 21, 2016 #1
    1. The problem statement, all variables and given/known data
    A damped harmonic oscillator is driven by an external force of the form $$F_{ext}=F_0sin(\omega t)$$
    Show that the steady state solution is given by $$x(t)=A(\omega)sin(\omega t-\phi)$$
    where $$ A(\omega)=\frac{F_0/m}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^{1/2}} $$
    and $$tan\phi=\frac{2\gamma\omega}{\omega_0^2-\omega^2}$$

    2. Relevant equations
    Equation of motion: $$\ddot{x}+2\gamma\dot{x}+\omega_0^2x=\frac{F_0}{m}sin(\omega t)$$

    3. The attempt at a solution
    Using a characteristic equation ##\lambda^2+2\gamma\lambda+\omega_0^2## I found the solution to the equation of motion to be $$x(t)=Acos(\omega t)+Bsin(\omega t)$$ with another term that goes to zero at steady state, which we are considering only steady state. ##A## and ##B## are both arbitrary constants.

    Then using the steady state equation found, I plugged this back into the equation of motion to find the coefficients.
    $$\dot{x} = -A\omega sin(\omega t)+B\omega cos(\omega t)$$
    $$\ddot{x} = -A\omega^2cos(\omega t)-B\omega^2sin(\omega t)$$
    so
    $$-A\omega^2cos(\omega t)-B\omega^2sin(\omega t)-2\gamma A\omega sin(\omega t)+2\gamma B\omega cos(\omega t)+\omega_0^2Acos(\omega t)+\omega_0^2Bsin(\omega t) = \frac{F_0}{m}sin(\omega t)$$
    grouping sines and cosines:
    $$(-A\omega^2+2\gamma B\omega+\omega_0^2)cos(\omega t)+(-B\omega^2-2\gamma A\omega+\omega_0^2B)sin(\omega t) = \frac{F_0}{m}sin(\omega t)$$

    Since the coefficients for cosines on the left have to match the ones on the right, similarly for the sines, I got
    $$-A\omega^2+2\gamma B\omega+\omega_0^2A = 0$$
    $$-B\omega^2-2\gamma A\omega+\omega_0^2B=\frac{F_0}{m}$$

    Then solving for the coefficients A and B, I got
    $$A=\frac{-2F_0\gamma\omega}{m(\omega_0^2-\omega^2)^2+4m\gamma^2\omega^2}$$
    $$B=\frac{F_0(\omega_0^2-\omega^2)}{m(\omega_0^2-\omega^2)^2+4m\gamma^2\omega^2}$$

    Which when combining the sine and cosine term into ##A(\omega)cos(\omega t-\phi)## where ##A(\omega)=\sqrt{A^2+B^2}## and ##tan(\phi)=\frac{B}{A}##,

    I receive exactly what I was supposed to get with the exception that I get $$tan(\phi)=\frac{-2\gamma\omega}{\omega_0^2-\omega^2}$$ rather than the positive version of the argument in the trig function.

    I got told that "the method is wrong" and the " 'A' you received is not the same as the one we want". I don't see how the method is wrong, seeing as how it does in fact give a steady-state solution to the equation of motion (I did verify this to be sure). When using complex roots you get the correct signs on all the terms, but where did I go wrong with solving the differential equation using this method? If this method actually does not work, why does it not work when it was how we were taught in an Elementary Differential Equations course?

    I can post the finding of the coefficients if need be, but since it was a few fairly ugly matrices, I left it out of the initial post.
     
    Last edited: Sep 21, 2016
  2. jcsd
  3. Sep 21, 2016 #2

    TSny

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    The problem statement asks for the solution in terms of a sine function with an argument that has ##\phi## subtracted. You found a solution in terms of a cosine function with an argument that has ##\phi## added.

    I don't see why your method of solution would be considered incorrect.
     
  4. Sep 21, 2016 #3
    When I mentioned the combination of sine and cosine, the expression should actually be ##A(\omega)cos(]omega t-\phi)## rather than with ##+##. I've changed it in the original post. But because of that minus, I have a different phase shift, so it's not the EXACT answer the book was looking for. The solution the book gives has ##\phi## being positive, so it would shift the solution to the right, whereas since mine is negative, it is shifted to the left, thus not being the same solution to the equation of motion. This is why they call it incorrect. That was a typing mistake on my part.
     
  5. Sep 21, 2016 #4

    TSny

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    I'm not understanding why you are putting your answer in terms of the cosine function while the problem asks for the answer in terms of the sine function.
     
  6. Sep 21, 2016 #5
    I honestly have no idea. I just noticed that shortly before you posted this. I think maybe in our notes it's what we did so I just did that without thinking of the problem statement. I can't find verification other than one .com site, but combining into a sine term would be ##Rsin(\omega t +\phi)## correct? If so this would yield the correct answer since tangent and arctangent are odd, so the negative pulls out making it minus the positive ##\phi##.
     
  7. Sep 21, 2016 #6

    TSny

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    You can convert ##A \cos \omega t + B \sin \omega t## into any of the following forms:

    ##A(\omega) \sin \left (\omega t + \phi \right)##
    ##A(\omega) \sin \left (\omega t - \phi \right)##
    ##A(\omega) \cos \left (\omega t + \phi \right)##
    ##A(\omega) \cos \left (\omega t - \phi \right)##

    where, of course, the function ##A(\omega)## is different from the constant ##A##, and ##\phi## will be different for each of the four forms listed above.

    Each form is fine. But your problem stated that it wants the form ##A(\omega) \sin \left (\omega t - \phi \right)##. If you express the answer in this form, then I think you should get expressions for ##A(\omega)## and ##\phi## as given in the problem statement.
     
  8. Sep 21, 2016 #7
    If I used ##A(\omega)sin(\omega t+\phi)##, yeah. I'll have to mention how my answer is equivalent. I got a 10% on the problem with the work I have here. Thanks for helping spot the mistake I had missed.
     
  9. Sep 21, 2016 #8

    TSny

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    OK. Note that the answer for ##\phi## will differ in sign for the two cases ##A(\omega)sin(\omega t+\phi)## and ##A(\omega)sin(\omega t - \phi)##.

    So, it makes a difference that they asked for ##A(\omega)sin(\omega t - \phi)## rather than ##A(\omega)sin(\omega t + \phi)##

    I definitely think you deserved more than 40%. :frown:
     
  10. Sep 21, 2016 #9
    I actually forgot to look at the back of the page, it turned out to be a 10%. Its all good though. Each homework assignment is just a tad over 1% of our overall grade so losing some points doesn't mean a whole lot. I will talk to my professor and see about getting a few back though. But yeah thanks again for helping see what was up with this problem.
     
  11. Sep 21, 2016 #10

    TSny

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    OK. Good luck with retrieving some points.
     
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