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8.1.3 critically damped free vibration ξ = c<br />
c r<br />
= 1<br />
The solution is<br />
u (t) = (A + Bt) e −( cr<br />
2m<br />
)<br />
t<br />
= (A + Bt) e −ωt<br />
where A, B are found from initial conditions A = u (0),B = u ′ (0) + u (0) ω, hence<br />
8.1.4 over-damped free vibration ξ = c<br />
c r<br />
> 1<br />
The solution is<br />
where A, B are found from initial conditions.<br />
u (t) = ( u (0) + ( u ′ (0) + u (0) ω ) t ) e −ωt<br />
u (t) = Ae λ 1t + Be λ 2t<br />
A = u′ (0) − u (0) λ 2<br />
2ω √ ξ 2 − 1<br />
B = −u′ (0) + u (0) λ 1<br />
2ω √ ξ 2 − 1<br />
where λ 1 and λ 2 are the roots of the characteristic equation<br />
8.2 Constant input F<br />
8.2.1 Undamped case ζ = 0<br />
λ 1 = − c<br />
√ ( c<br />
2m + 2m<br />
λ 2 = − c<br />
2m − √ ( c<br />
2m<br />
) 2 k −<br />
m = −ξω + ω√ ξ 2 − 1<br />
) 2 k −<br />
m = −ξω − ω√ ξ 2 − 1<br />
mu ′′ + ku = F<br />
u ′′ + ω 2 u = F<br />
u (t) = u h + u p<br />
Where u h = A cos ωt + B sin ωt and u p = F k<br />
, the solution is<br />
Applying initial conditions gives<br />
u (t) = A cos ωt + B sin ωt + F k<br />
And complete solution is<br />
A = u (0) − F k<br />
B = u′ (0)<br />
ω<br />
u (t) = F (<br />
k + u (0) − F )<br />
cos ωt + u′ (0)<br />
sin ωt<br />
k<br />
ω<br />
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