Page 458 - Mechanical Engineers' Handbook (Volume 2)

2 Laplace Transforms 449

By definition
1
L[u (t)] U e 1 dt
ts
s
s
0 s
Example 2 Determine the Laplace transform of ƒ(t):
0 t 0
ƒ(t) t
e t 0
1
ts t
L[ƒ(t)] F(s) ee dt
0 s
Example 3 Determine the Laplace transform of the function ƒ(t) given by
0 t 0
ƒ(t) t 0 t T
T T t
By definition

F(s) e ƒ(t) dt
ts
0
T
ts
ts
et dt eT dt
0 T
1 e Ts 1
(1 e Ts )
s 2 s 2 s 2
In transforming differential equations, entire equations need to be transformed. Several
theorems useful in such transformations are given next without proof.

T1. Linearity Theorem

L[ ƒ(t) g(t)] L[ƒ(t)] L[g(t)] (3)
T2. Differentiation Theorem
L
dƒ
dt sF(s) ƒ(0) (4)
L 2 dƒ
d ƒ
2
dt 2 sF(s) sƒ(0) dt (0) (5)
L n n 1 n 2 dƒ d n 1
d ƒ
n
dt n sF(s) s ƒ(0) s dt (0) dt n 1 (0) (6)
T3. Translated Function (Fig. 6)
L[ƒ(t )u (t )] e s F(s ) (7)
s