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Heavily Damped Simple Harmonic System - How To Start?

  1. Mar 23, 2009 #1
    1. The problem statement, all variables and given/known data

    A heavily damped simple harmonic system is displaced a distance F from its equilibrium positio and released from rest. Show that in the expression for the displacement

    [tex]x=e^{-pt}(F\cosh qt + G\sinh qt)[/tex]

    where

    [tex]p=\frac{r}{2m}[/tex]

    and

    [tex]q=(\frac{r^2}{4m^2}-\frac{s}{m})^{\frac{1}{2}}[/tex]

    that the ratio

    [tex]\frac{G}{F}=\frac{r}{(r^2-4ms)^{\frac{1}{2}}}[/tex]


    2. The attempt at a solution

    I've been thinking and thinking and thinking, but no luck. I'd really appreciate it if someone could just tell me where to start.

    Thanks!
    phyz
     
  2. jcsd
  3. Mar 23, 2009 #2

    lanedance

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    Homework Helper

    hi phyz,

    the differential equation of the damped harmonic oscillator is probably a good place to start...
     
  4. Mar 24, 2009 #3
    Hi lanedance!

    You mean I should simply calculate xdot and xddot and plug them into

    [tex]m\ddot{x}+r\dot{x}+sx=0[/tex]

    ???

    Let me give it a go :smile:
     
    Last edited: Mar 24, 2009
  5. Mar 24, 2009 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    [tex]x'(t)= -pe^{-pt}(F\cosh qt+ G\sinh qt)+ e^{-pt}(qF\sinh qt+ qG\cosh qt)[/tex]

    Since it is released from rest, we have
    [tex]x'(0)=-pF+ qG= 0[/tex]
    That should be enough.
     
  6. Mar 26, 2009 #5
    Thank you! That did the trick :smile:
     
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