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5.5 Impulse sin function<br />
Input is given by F (t) = F 0 sin (ϖt) where ϖ = 2π<br />
2t 1<br />
= π t 1<br />
ζ = 0<br />
ζ < 1<br />
ζ = 1<br />
⎧<br />
⎨ −iω<br />
roots<br />
⎩<br />
+iω<br />
⎧<br />
⎧<br />
⎪⎨<br />
ϖ = ω →<br />
(<br />
⎪⎨<br />
⎪⎩ −u (0) +<br />
F 0 π<br />
k 2<br />
⎧<br />
(<br />
u (t)<br />
⎪⎨ u (0) cos ωt +<br />
ϖ ≠ ω →<br />
⎪⎩ ⎪⎩<br />
⎧<br />
√<br />
⎨ −ξω + iω n 1 − ξ 2<br />
roots<br />
⎩<br />
√<br />
−ξω − iω n 1 − ξ 2<br />
⎧<br />
⎪⎨<br />
u (t) =<br />
⎪⎩<br />
u (0) cos ωt + u′ (0)<br />
ω sin ωt − F 0<br />
k<br />
ωt<br />
)<br />
cos ωt +<br />
−u ′ (0)+ F 0<br />
k<br />
u ′ (0)<br />
ϖ<br />
( −<br />
(F π/t1<br />
0/k) ω<br />
π/t1<br />
1−(<br />
ω<br />
2 cos (ωt) 0 ≤ t ≤ t 1<br />
π<br />
2t 1<br />
))<br />
) 2<br />
ω<br />
sin ωt t > t 1<br />
sin ωt + F 0/k<br />
1−( π/t1<br />
ω<br />
( )<br />
) 2 sin πt<br />
t 1<br />
0 ≤ t ≤ t 1<br />
u (t 1 ) cos ωt + u′ (t 1 )<br />
ω<br />
sin ωt t > t 1<br />
time constant τ = 1<br />
ζω n<br />
e −ξωt (A cos ω d t + B sin ω d t) + F 0<br />
√ 1<br />
k<br />
(<br />
e −ξωt u (t 1 ) cos ω d t + u′ (t 1 )+u(t 1 )ξω<br />
ω d<br />
sin ω d t<br />
A = u (0) + F 0<br />
k<br />
1 √(1−r 2 ) 2 +(2ξr) 2 sin θ<br />
B = u′ (0)<br />
ω d<br />
⎧<br />
⎨<br />
roots<br />
+ u(0)ξω<br />
ω d<br />
+ F 0<br />
−ω<br />
√ 1<br />
k<br />
ω d (1−r 2 ) 2 +(2ξr)<br />
(<br />
sin<br />
(1−r 2 ) 2 +(2ξr)<br />
)<br />
2<br />
2<br />
(ξω sin θ − ϖ cos θ)<br />
)<br />
t 1<br />
− θ<br />
⎩<br />
−ω<br />
⎧<br />
( )<br />
⎪⎨ (A + Bt) e −ωt + F 0 √ 1<br />
sin k<br />
π t<br />
u (t) =<br />
(1−r 2 ) 2 +(2r) 2 t 1<br />
− θ 0 ≤ t ≤ t 1<br />
⎪⎩<br />
u (t) = (u (t 1 ) + (u ′ (t 1 ) + u (t 1 ) ω) t) e −ωt t > t 1<br />
π t<br />
0 ≤ t ≤ t 1<br />
t > t 1<br />
A = u (0) + F 0<br />
k<br />
1 √(1−r 2 ) 2 +(2r) 2 sin θ<br />
B = u ′ (0) + u (0) ω + F 0<br />
(<br />
√ 1<br />
k<br />
(1−r 2 ) 2 +(2r) 2<br />
)<br />
ω sin θ − π t 1<br />
cos θ<br />
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