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5.3 no loading (free) mu ′′ + cu ′ + ku = 0<br />
⎧<br />
⎨ −iω<br />
roots<br />
⎩<br />
+iω<br />
ζ = 0<br />
u ′′ + ω 2 u = 0<br />
ζ < 1<br />
ζ = 1<br />
u (t) = u(0) cos ωt + u′ (0)<br />
ω<br />
sin ωt<br />
⎧<br />
u (t) = A cos (ωt − φ)<br />
⎪⎨ √<br />
( )<br />
A = u(0) + u ′ 2<br />
(0)<br />
ω<br />
( )<br />
⎪⎩ φ = tan −1 u ′ (0)/ω<br />
u(0)<br />
⎧<br />
⎨ −ξω + iω √ 1 − ξ 2<br />
roots<br />
⎩<br />
−ξω − iω √ 1 − ξ 2<br />
(<br />
u (t) = e −ξωt u(0) cos ω d t + u′ (0)+u(0)ξω<br />
ω<br />
⎧<br />
d<br />
⎨ −ω<br />
roots<br />
⎩<br />
−ω<br />
)<br />
sin ω d t<br />
ζ > 1<br />
u (t) = (u(0)(1 + ωt) + u ′ (0)t) e −ωt<br />
⎧<br />
⎨ λ 1 = −ωξ + ω √ ξ 2 − 1<br />
roots<br />
⎩<br />
λ 2 = −ωξ − ω √ ξ 2 − 1<br />
⎧<br />
u (t) = Ae<br />
⎪⎨<br />
λ1ωt + Be λ 2ωt<br />
A = u′ (0)−u(0)λ 2<br />
2ω √ ξ 2 −1<br />
⎪⎩<br />
B = −u′ (0)+u(0)λ 1<br />
2ω √ ξ 2 −1<br />
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