Lab 8 - Claymore - Grand Valley State University
Lab 8 - Claymore - Grand Valley State University
Lab 8 - Claymore - Grand Valley State University
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Homogeneous:<br />
Therefore:<br />
Particular:<br />
Therefore:<br />
2 ⎛ K ⎞<br />
ω& + ω ⋅ ⎜<br />
⎟ = 0<br />
(2)<br />
⎝ J m ⎠<br />
t = e<br />
α<br />
ω (3)<br />
⋅t<br />
Let: ( )<br />
Homogeneous<br />
α⋅t<br />
( t)<br />
= α ⋅ e<br />
ω& (4)<br />
Homogeneous<br />
Substituting (3) and (4) into (2) yields:<br />
( t)<br />
⎛ K<br />
α + ⎜<br />
⎝<br />
2<br />
J m<br />
⎛<br />
∴α<br />
= − ⎜<br />
⎝<br />
Homogeneous<br />
⎞<br />
⎟ = 0<br />
⎠<br />
K 2<br />
J m<br />
⎟ ⎞<br />
⎠<br />
⎛ 2<br />
K ⎞<br />
−⎜<br />
⎟⋅t<br />
⎜ J ⎟<br />
⎝ m ⎠<br />
ω = C ⋅ e<br />
(5)<br />
Let: ( ) 2 C<br />
t Particular =<br />
1<br />
ω (6)<br />
( t)<br />
= 0<br />
ω& (7)<br />
Particular<br />
Substituting (6) and (7) into (1) yields:<br />
0 +<br />
∴ C<br />
⎛ K<br />
⎞<br />
K<br />
2<br />
s<br />
C ⋅ ⎜ Vs<br />
−<br />
J ⎟ = ⋅ ⎜<br />
m J m R ⎟<br />
2<br />
⋅<br />
2<br />
⎝<br />
⎠<br />
⎛<br />
⎝<br />
Vs<br />
Tload<br />
⋅ R<br />
= − 2<br />
K K<br />
Vs<br />
Tload<br />
⋅ R<br />
ω ( t)<br />
Particular = −<br />
(8)<br />
2<br />
K K<br />
Combining (5) and (8) yields:<br />
⎛ 2<br />
K ⎞<br />
−⎜<br />
⎟⋅t<br />
⎜ J ⎟ Vs<br />
Tload<br />
⋅ R<br />
⎝ m ⎠<br />
ω ( t)<br />
= C1<br />
⋅ e + −<br />
(9)<br />
2<br />
K K<br />
⎞<br />
⎠<br />
T<br />
load<br />
J<br />
m<br />
4