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Dynamics cheat sheet

my dynamics notes - 12000.org

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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 />

27

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