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Find the eigenfrequencies for systems

  1. Mar 10, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex] (M + m) \ddot x + m l \ddot\theta - ml \dot\theta ^2\theta = 0[/tex]

    [tex]\ddot\theta + \frac{\ddot x}{l} + \frac{g}{l}\theta = 0[/tex]

    I am not quite sure how to get a x(t) and theta(t) that actually fit for these equations...
    For the second equation, I was thinking something like
    x(t) = At^2 + Bt+C
    theta(t) = Dt^2 + Et + C, but of course that doesn't work.

    I know how to find the eigenfrequencies for systems that say have a Ae^(iwt) term in them, but for something like this, I have no idea... If I could get the correct forms of x(t) and theta(t) I think I could probably find them, but I am a bit lost in how to get the forms for something like this.
     
    Last edited: Mar 10, 2009
  2. jcsd
  3. Mar 10, 2009 #2
    Re: Eigenfrequency

    I would think in terms of trying to cast this as a matrix eigenproblem, but that last term in the first equation somewhat messes things up with the theta-dot^2 factor. Are you sure you have this written correctly?
     
  4. Mar 10, 2009 #3
    Re: Eigenfrequency

    Thanks for your reply.
    It is written correctly; however, I *may* (not positive but upon thinking about it..) be able to say that the theta-dot^2 term is small so that it is effectively zero...
    [tex]
    (M + m) \ddot x + m l \ddot\theta = 0
    [/tex]
    [tex]
    \ddot\theta + \frac{\ddot x}{l} + \frac{g}{l}\theta = 0
    [/tex]

    I'm afraid I'm not quite sure how to proceed even if this were the case. I am not familiar exactly with matrix eigenproblems.

    If this were a pure math problem I suppose I would do something like
    (A-I*lambda)x = 0 but since I don't have the system of equations in terms of x(t) and theta(t) I'm not quite sure what to do.
     
    Last edited: Mar 10, 2009
  5. Mar 10, 2009 #4
    Re: Eigenfrequency

    "...since I don't have the system of equations in terms of x(t) and theta(t) I'm not quite sure what to do." But if you were to substitute x = X*sin(omega*t) and theta = Theta*sin(omega*t), then you would have such a system.

    But dropping that theta-dot^2 term seems shaky unless you have some physical argument for it.
     
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