Archived Oscillation with Green's Function

Menteith

1. Homework Statement
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. Homework 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.

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Couchyam

The final position can be found by balancing the limiting force with Hook's law.

Homework Helper
Gold Member
2018 Award
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.

"Oscillation with Green's Function"

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