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5.4 impulse F 0 δ (t) loading<br />
⎧<br />
⎨ −iω<br />
roots<br />
ζ = 0<br />
⎩<br />
+iω<br />
u ′′ + ω 2 u = 0<br />
ζ < 1<br />
ζ = 1<br />
ζ > 1<br />
transient<br />
{ }} {<br />
u (t) = u(0) cos ωt + u′ (0)<br />
⎧<br />
ω<br />
⎨ −ξω + iω √ 1 − ξ 2<br />
roots<br />
⎩<br />
−ξω − iω √ 1 − ξ 2<br />
steady state<br />
{ }} {<br />
sin ωt + F 0<br />
sin ωt<br />
mω<br />
transient<br />
steady state<br />
{ (<br />
}} ){<br />
{ ( }} ){<br />
u (t) = e −ξωt u(0) cos ω d t + u′ (0) + u(0)ξω<br />
sin ω d t + e −ξωt F0<br />
sin ω d t<br />
⎧<br />
ω d<br />
mω d<br />
⎨ −ω<br />
roots<br />
⎩<br />
−ω<br />
u (t) = (u (0) (1 + ωt) + u ′ (0) t) e −ωt + F 0t<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 λ2ωt +<br />
⎪⎩<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 />
m e−ωt<br />
F 0<br />
m<br />
2ω √ ξ 2 −1 eλ 1ωt −<br />
The impulse response can be implemented in Mathematica as<br />
parms = {m -> 10, c -> 1.2, k -> 4.3, a -> 1};<br />
tf = TransferFunctionModel[a/(m s^2 + c s + k) /. parms, s]<br />
sol = OutputResponse[tf, DiracDelta[t], t];<br />
F 0<br />
m<br />
2ω √ ξ 2 −1 eλ 2ωt<br />
Plot[sol, {t, 0, 60}, PlotRange -> All, Frame -> True,<br />
FrameLabel -> {{z[t], None}, {Row[{t, " (sec)"}], eq}},GridLines -> Automatic]<br />
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