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5.2 constant loading mu ′′ + cu ′ + ku = F<br />
⎧<br />
⎨ −iω<br />
roots<br />
ζ = 0 ⎩<br />
+iω<br />
ζ < 1<br />
ζ = 1<br />
u (t) = ( u 0 − F )<br />
k cos ωt +<br />
v 0<br />
ω<br />
sin ωt + F k<br />
⎧<br />
√<br />
⎨ −ξω + iω n 1 − ξ 2<br />
roots<br />
⎩<br />
√<br />
−ξω − iω n 1 − ξ 2<br />
( (u0<br />
u (t) = e −ξωt − F )<br />
k cos ωd t +<br />
⎧<br />
⎨ −ω<br />
roots<br />
⎩<br />
−ω<br />
(<br />
) )<br />
v 0 +u 0 ξω− F k ξω<br />
ω d<br />
sin ω d t + F k<br />
u (t) = (( u 0 − F ) (<br />
k + v0 + u 0 ω − F k ω) t ) e −ωt + F k<br />
⎧<br />
√<br />
⎨ −ω n ξ + ω n ξ 2 − 1<br />
roots<br />
⎩<br />
√<br />
−ω n ξ − ω n ξ 2 − 1<br />
ζ > 1<br />
B = F k p 1−u 0 p 1 +v 0<br />
(p 2 −p 1 )<br />
A = u 0 − F k − B<br />
p 1 = − c<br />
2m + √ ( c<br />
2m<br />
p 2 = − c<br />
2m − √ ( c<br />
2m<br />
u (t) = Ae p 1t + Be p 2t + F k<br />
) 2 −<br />
k<br />
m = −ω √<br />
nξ + ω n ξ 2 − 1<br />
) 2 −<br />
k<br />
m = −ω √<br />
nξ − ω n ξ 2 − 1<br />
24