Page 335 - Mechanical Engineers' Handbook (Volume 2)

326 Mathematical Models of Dynamic Physical Systems

Bs B 2
1
F(s)
(s
j )(s
j )
Bs B 2
1

2
s 2
s
2
2
Bs B 2
1

2
(s
) 2
From the transform tables the Laplace inverse is
ƒ(t) e
t [B cos t B sin t]
3
1
Ke
t cos( t )
where B (1/ )(B aB )
2
1
3
2
K B B 2 3
1
1
tan (B /B )
1
3
Proper and Improper Rational Functions
If F(s) is not a strictly proper rational function, then N(s) must be divided by D(s) using
synthetic division. The result is
N(s) N*(s)
F(s) P(s)
D(s) D(s)
where P(s) is a polynomial of degree m n and N*(s) is a polynomial of degree n 1.
Each term of P(s) may be inverted directly using the transform tables. The strictly proper
rational function N*(s)/D(s) may be inverted using partial-fraction expansion.
Initial-Value and Final-Value Theorems
The limits of ƒ(t) as time approaches zero or infinity frequently can be determined directly
from the transform F(s) without inverting. The initial-value theorem states that
ƒ(0 ) lim sF(s)

s→
where the limit exists. If the limit does not exist (i.e., is infinite), the value of ƒ(0 )is

undefined. The final-value theorem states that
ƒ( ) lim sF(s)
s→0
provided that (with the possible exception of a single pole at s 0) F(s) has no poles with
nonnegative real parts.
Transfer Functions
The Laplace transform of the system I/O equation may be written in terms of the transform
Y(s) of the system response y(t)as
G(s)N(s) F(s)D(s)
Y(s)
P(s)D(s)

F(s)
G(s)
N(s)
P(s) D(s) P(s)