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Homework Statement
A Force F(t) = F0(1 - e-at), where both F0 and a are constants, acts over a damped oscillator. In t = 0, the oscillator is in it's equilibrium position. The mass of the oscillator is m, the spring constant is k = 2ma2 and the damping constant is b = 2ma.
Find x(t)
Homework Equations
Well, the differential equation is: d2x / dt2 + 2adx/dt + 2a2x = F0/m - F0/m * e-at
Also, x(0) = 0; x'(0) = 0.
The Attempt at a Solution
First, I tried to find the solution to the homogeneous equation associated.
So I tried a solution of the kind x = ept, and the I found: p2 + 2ap + 2 = 0
p = (-2a [tex]\pm[/tex] [tex]\sqrt{-4a^2}[/tex]) / 2
p = -a [tex]\pm[/tex] ia
So, it's an under-dampeing case, which the general solution is A*e-at*cos(at + [tex]\theta)[/tex].
Then I have to find the particular solutions, where xp1 = F0/m and xp2 = -F0/m * e-at.
xp1 is rather easy. C*xp1 = F0/m ; 2a2*xp1 = F0/m ; xp1 = F0/2ma2
Now I'm stuck at the other one. I tried solutions like x = C*ept and x = C*t2*ept (C = constant) and haven't got it right.
The answer is: x(t) = F0/ma2 * [[tex]\sqrt{2}[/tex]*e-at*cos(at + P/4) + 1 - 2e-at]
How do I find the other particular solution? What am I missing?
(By the way, should this be here or in the Calculus sub-forum?)
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