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Archived Oscillation with Green's Function

  1. Mar 26, 2013 #1
    1. The problem statement, all variables and given/known data
    A force Fext(t) = F0[ 1−e(−αt) ] acts, for time t > 0, on an oscillator which is at rest at x=0 at time 0.
    The mass is m; the spring constant is k; and the damping force is −b x′. The parameters satisfy these relations:
    b = m q , k = 4 m q2 where q is a constant with units of inverse time.
    Find the motion. Determine x(t); and hand in a qualitatively correct graph of x(t).
    (B) Determine the final position.


    2. Relevant equations
    Green's Function:
    x(t) = ∫-∞tF(t')G(t,t')dt'
    where:
    G(t,t') = (1/(m*ω1))*e-β(t-t')*sinω1(t-t') for t≥t'
    = 0 for t<t'



    3. The attempt at a solution
    I have solved using green's function to obtain this mess:
    (F0/m)*(e-βt/(β212))*((β/ω1)*sinω1t - cosω1t) - (F0/m)/((α-β)212)*[e-αt-e-βt*(cosω1t-((α-β)/ω1)sinω1t)]

    From here, however, I am unsure of how to find the final position without the final time.
     
    Last edited by a moderator: Oct 24, 2016
  2. jcsd
  3. Oct 24, 2016 #2
    The final position can be found by balancing the limiting force with Hook's law.
     
  4. Oct 26, 2016 #3

    Charles Link

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    Homework Helper

    I would suggest you work the problem with a simple differential equation. The Green function approach for this one is rather difficult. If your instructor specified he requires a Green's function type solution, then it is the route you need to go, but for this one, the Green's function looks like a difficult approach. editing... I'm looking at the date on the OP. This one appears to be a couple of years old.
     
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