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Damped oscillator given odd initial conditions

  1. Oct 26, 2013 #1
    1. The problem statement, all variables and given/known data
    (A) The damped oscillator is described by the equation mx''+bx'+kx=0. What is the condition for critical damping expressed in terms of m,b,k. Assume this is satisfied.

    (B) For t<0 the mass is at rest (x=0). This mass is set in motion at t=0 by a sharp impulsive force so that the velocity is v0 at time t=0. Determin the position x(t) for t>0.

    (C) Suppose k/m=(2*pi rad/s)2 and v0=10 m/s. Plot an accurate graph of x(t) using an appropriate range for t.


    2. Relevant equations

    For critical damping; B=W0

    General solution for critically damped oscillator:

    x(t)=(C1+C2t)e-βt

    3. The attempt at a solution
    I am running into issues at (B) where I'm not entirely sure how to find the values of the two constants in the general solution.

    (A) I have shown through the given relations that b2=4km which I believe is the correct answer.

    (B) Given the general solution for a critically damped oscillator I can take its derivative to find v(t):

    v(t)=e-βt(C2-C2βt-C1β)

    I know that at t=0 v(0)=v0 so I can solve the solution for v(t) for a constant:

    C2=v0+C1β

    I'm just not sure about how to solve for x(t). Can I consider x(0)=0 since it is set in motion at t=0 but since no time has elapsed it has not moved?

    If not, how do I proceed in solving for the constants? I believe the rest of the problem will fall into place without much issue after that point.

    Thank you for your time and help!
     
  2. jcsd
  3. Oct 26, 2013 #2

    ehild

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    The initial conditions are x(0)=0 and v(0)=vo. Impulsive force means that the mass is set into motion in an infinitesimally short time, so the displacement during that time is negligible.

    ehild
     
  4. Oct 26, 2013 #3
    Ahh, fantastic. That definitely makes sense, thank you.

    So, C1=0 and C2=v0 making the general solution:

    x(t)=v0te-βt

    (C) Since k/m=(2*π rad/s)2 and w02=k/m and w0

    β=2π rad/s

    x(t)=10te-2πt

    The plot:
    http://www.wolframalpha.com/input/?i=Plot(x(t)=10*t*e^(-2*pi*t)),+(t,+0,+1)

    Thanks again!
     
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