Laplace+TransferFunctions
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M.D. Bryant ME 344 notes 03/25/08 5<br />
∞<br />
• Define X(s) = ⌡ ⌠ x(t) e- st dt<br />
t= 0<br />
• Integrate by parts<br />
∞<br />
dx<br />
dt<br />
e- st<br />
∞<br />
| t=0 +s dx<br />
⌡ ⎮⎮⌠ dt e - st dt +2ζω n {x(t) e- st<br />
∞<br />
| t=0<br />
t= 0<br />
∞<br />
+s ⌡ ⌠<br />
x(t) e- st dt } + ω 2 n X(s) = H<br />
O<br />
t= 0<br />
("s) e - st<br />
∞<br />
| t=0<br />
• Rearrange, note<br />
- x ˙ (0) + s (- x(0) + s<br />
!<br />
X(s) ) + 2ζωn {- x(0) +<br />
s X(s) } + ω 2 n X(s) = H o<br />
s<br />
!<br />
lim<br />
t "#<br />
e $ st = 0<br />
Note: L {dx/dt} = - x(0) + s X(s)<br />
L {d 2 x/dt 2 } = L {d x ˙ /dt} =- x ˙ (0) + sL {dx/dt}<br />
• Result: algebraic equation in s<br />
H<br />
(s 2 2<br />
+ 2"# ! !<br />
n<br />
s +#<br />
n<br />
)X(s) =<br />
O<br />
Note: x ˙ (0) = x1 ,<br />
s + x 1 + (s + 2"# n )x 0<br />
x(0) = xo
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