Page 104 - Fluid Power Engineering

78 Cha pte r T h ree

3. L [e − at f ( )] = ( F s + ) a (3A.5)
t
+
(
(
0
4. Initial value problem f() = Lim f t) = Lim sF s) (3A.6)
t→ 0 s→∞
5. Final value problem f()∞= Lim f t ( ) = Lim sF s ( ) (3A.7)
t→∞ s→0
⎡ df t()⎤ df
n
+ +
n
6. L ⎢ n ⎥ = sF s() − s n−1 f(0 + ) − s n−2 (0 )
⎣ dt ⎦ dt
2
n −3 df + − d n −1 f +
− s (0 ) − L (0 ) (3A.8)
dt 2 dt n −1
⎡ df t ()⎤
+
0
L ⎢ ⎥ = sF s() − f( ) (3A.9)
⎣ dt ⎦
⎡ df t ()⎤ •
2
+
+
2
L ⎢ 2 ⎥ = sF s ( ) − sf 0() − f 0() (3A.10)
⎣ dt ⎦
7. L ⎡ ft dt ⎤ = 1 ⎡ Fs) + ∫ f t dt ⎤ (3A.11)
()
()
(
∫ ⎣ ⎦ s ⎣ 0 + ⎦
Laplace Transform Tables
The pairs of Laplace transforms are given in math textbook tables. The
following are the pairs of Laplace transforms of some key functions:
F(s) f(t)
1 δ(t)
k
k
s
kn(!)
kt n
s n+1
k ke − at
+
sa
k kt e − at
+
( sa) 2
kn(!) kt e − at
n
+
( sa) n+1
k ω k sinω
s + ω 2 t
2
ks k cosω t
s +ω 2
2