Damped and Unforced Motion

Asked May 21 '17 at 17:37

Active May 21 '17 at 20:08

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Find the general solution x(t) to the equation for damped harmonic motion

$$x''+ 2kx'+ω_{0}^2x=0$$ in the following cases: (i) $k < ω_{0}$, (ii) $k = ω_{0}$, (iii)$ k > ω_{0}$.

Assume that, at time t = 0, the body is released from rest at x = 1.

I believe the 3 general cases for each case is:

i) $x(t) = e^{-kt}(Acos(ωt)+Bsin(ωt)),\,\,\,ω = (k^2 - ω_{0}^2)^{1/2}$

ii)$x(t) = (A + Bt)e^{-kt}$

iii)$x(t) = Ae^{(-k+q)t} + Be^{(-k-q)t},\,\,\,q = (k^2 - ω_{0}^2)^{1/2}$

I would just like an explanation as to what conditions I should apply to the equations, and a small run through the first example would be nice.

edited May 21 '17 at 20:08

asked May 21 '17 at 17:37

James

is there meant to be an $x$ after $\omega_0^2$ given the context of SHM? – PhysicsMathsLove May 21 '17 at 17:45
are you sure that you have made no typo? – Dr. Sonnhard Graubner May 21 '17 at 17:48
you're right. My apologies. – James May 21 '17 at 17:49
i have got $$x(t)=c_1 e^{t \left(-\sqrt{k^2-\text{w0}^2}-k\right)}+c_2 e^{t \left(\sqrt{k^2-\text{w0}^2}-k\right)}$$ – Dr. Sonnhard Graubner May 21 '17 at 18:02
Could you tell me what initial conditions you would use to find c1 and c2 – James May 21 '17 at 18:09
@James At $t=0$, the body is released from rest at $x=1$. You know an $(x,t)$ of the body and a $(v,t)$ of the body. Use the initial conditions, but you will need to find a way to get an expression for velocity from $x(t)$. – PhysicsMathsLove May 21 '17 at 18:24
In case (i), $ω$ should read $(ω_0^2-k^2)^{1/2}$ and $e^{kt}$ should read $e^{-kt}$, for reasons coming either from the physics of the problem, or from its maths. – Did May 21 '17 at 19:14

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