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8.2.2 under-damped ζ < 1<br />
The general solution is<br />
u (t) = e −ξωt (A cos ω d t + B sin ω d t) + F k<br />
From initial conditions<br />
Hence the solution is<br />
u (t) = e −ξωt ( (<br />
u (0) − F k<br />
A = u (0) − F k<br />
B = u′ (0) + u (0) ξω − F k ξω<br />
ω d<br />
) (<br />
u ′ (0) + u (0) ξω − F k<br />
cos ω d t +<br />
ξω<br />
) )<br />
sin ω d t + F ω d k<br />
8.2.3 Critical damping ζ = 1<br />
The general solution is<br />
u(t) = (A + Bt)e −ωt + F k<br />
Where from initial conditions<br />
A = u(0) − F k<br />
B = u ′ (0) + u(0)ω − F k ω<br />
8.2.4 Over-damped ζ > 0<br />
The solution is<br />
u (t) = Ae p1t + Be p2t + F k<br />
Where now<br />
B =<br />
F<br />
k p 1 − u 0 p 1 + u ′ (0)<br />
(p 2 − p 1 )<br />
A = u (0) − F k − B<br />
Hence the solution is<br />
u (t) = Ae p 1t + Be p 2t + F k<br />
Where<br />
p 1 = − c<br />
√ ( c<br />
2m + 2m<br />
p 2 = − c<br />
2m − √ ( c<br />
2m<br />
) 2 k −<br />
m = −ωξ + ω √<br />
n ξ 2 − 1<br />
) 2 k −<br />
m = −ωξ − ω √<br />
n ξ 2 − 1<br />
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