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Oscillation / Phase Space Question

  1. Feb 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Thornton and Marion, chapter 3, problem 21:

    Use a computer to produces a phase space diagram similar to Figure 3-11 for the case of critical damping. Show analytically that the equation of the line that the phase paths approach asymptotically is [tex] \dot{x}=-\beta x[/tex]. Show the phase paths for at least three initial positions above and below the line.

    [tex]\beta>0[/tex] is the usual damping parameter.

    2. Relevant equations

    Equation of motion for critically damped oscillator:

    [tex] x = A\exp \left(-\beta t\right) + Bt\exp \left(-\beta t\right)[/tex].

    And,

    [tex] \dot{x} =-A\beta\exp\left(-\beta t \right) +B\exp\left(-\beta t \right)-B\beta t \exp\left(-\beta t \right)[/tex].

    3. The attempt at a solution

    The phase diagram is done and correct. My problem is in showing the equation of the asymptote. My first inclination was to examine the limits of [tex]x[/tex] and [tex]\dot{x}[/tex] as [tex]t \to \infty[/tex]. But they both go to zero, correct?

    But I took at peak at the solution and they have

    [tex] \lim_{t \to \infty} x = Bt\exp \left(-\beta t\right) [/tex]

    and

    [tex] \lim_{t\to \infty} \dot{x} = -B\beta t \exp\left(-\beta t \right)[/tex].

    Therefore, [tex] \dot{x} = -\beta x [/tex] as [tex]t \to \infty[/tex].

    What?! Aren't those limits zero?! Am I so sleep deprived that I can't even take limits anymore? What am I doing wrong?

    Thanks!
     
  2. jcsd
  3. Feb 18, 2010 #2

    ideasrule

    User Avatar
    Homework Helper

    Those limits are indeed zero, but the limit of x-dot divided by the limit of x is -beta. That's exactly what the question wanted you to prove.
     
  4. Feb 18, 2010 #3
    Oh, OK. So you're taking the limit of the ratio [tex]\dot{x}/x[/tex]. Why would be interested in that limit? I'm having a hard time finding motivation for all these things that the problems want you to do.
     
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