PARTIAL FRACTIONS IN SHOCK AND VIBRATION ANALYSIS
Revision K
By Tom Irvine
Email: [email protected]
January 11, 2013
Introduction
Transforming a fraction into the sum of partial fractions is an intermediate step in the
solution of certain shock and vibration problems. The purpose of this tutorial is to
summarize some common cases.
Term

 1 
 
 s 

Appendix



2
s 2  2 n s   n 
 1



 s 2  2 

1



2
s 2  2n s  n 
s  
s   s  
s  

,
A
B
C



1
1



  2  j 2  1   2  j 2  1
D







1  j 2
1  j 2



  1   2   j 2   1   2   j 2



 

E
s2
, ab
s  a s  b 
F
s4
, a  b, a  c, a  d, b  c, b  d, c  d
s  a s  b s  c s  d 
G
1
Term
Appendix
s
H
s  a 2
1
s 2   2  s  2
I
1  1 


s 2  s 2  2 
J
Note that
2
2
s 2  2n s  n  s  n 2  n   2 n 2
(1)
2
2
s 2  2n s  n  s  n 2  n 1   2 


(2)
2
2
s 2  2n s  n  s  n 2  
d
(3)
where
  n 1   2
d
(4)
Thus,
1
2
s 2  2n s  n

1
s  n 2  d2
2
(5)
The inverse Laplace transform f(t) of equation (5) is
f (t) 
1
exp  n t sin d t 
d
(6)
Furthermore, consider the Laplace transform
F̂(s) 
s  n
s  n 2  d2
(7)
The inverse Laplace transform f̂ ( t ) is
f̂ ( t )  exp  n t  cosd t 
3
(8)
APPENDIX A
Example 1

1 
 
 s 




  





2
s  
s 2  2n s  n 




2
s 2  2n s  n 

s  
1
2
1  s 2  2 n s   n    s  s



2
1  s 2  2 n s  n   s 2  s


1     s 2  2n   s  n
2
(A-1)
(A-2)

(A-3)
(A-4)
Equation (A-4) implies three separate equations.
    0
(A-5)
2n     0
(A-6)
2
n  1
(A-7)
Equation (A-7) yields

1
2
n
  
  
1
2
n
  2n 
4
(A-8)
(A-9)
(A-10)
(A-11)
 1 

  2 n 
 2 
 n
 
(A-12)
2
(A-13)
n
Equation (A-1) can thus be expressed as

 1 
 
 s 



2
2
s  2 n s   n 

1
 1
 2

 n
 s



 
 
 

 
 
 





2
2
s  2 n s   n 



 1 
2
s 

 2  
n
 n
(A-14)

1 
 
 s 


1 1  1




2
2
s  2

s 2  2n s  n 

n
n








2
s 2  2n s  n 

(A-15)

 1 
 
 s 


1 1 
1

  2   2
2
s 2  2 n s   n 
n  s  n







s  2n


2
s  n 2  d 
(A-16)
1
1
5
s  2n
The following form is a more convenient format prior to taking the inverse Laplace
transformation,

 1 
 
 s 


1

 
2
2
s 2  2 n s   n 
n

1 
 
s 
1
1

2
n






s  n


2
s  n 2  d 
1

2
n








s  n 2  d2 
n
(A-17)

 1 
 
 s 


1 1 

 
 
2
2 s 
s 2  2 n s   n 

n

1


 1 


  2 
 n 

s  n


s  n 2  d2 


 1  n

  2  d
 n 








s  n 2  d2 
d
(A-18)
The inverse Laplace transform is
f (t) 



1
1
u(t) 
exp  n t  cosd t   n sin d t  , t  0
2
2


d
n
n


where u(t) is the unit step function.
6
(A-19)
APPENDIX B
Example 2
Equation (B-1) effectively represents a number of cases since  and  may each be set
equal to zero.
 1 

 2

2
 s   


 s    






2
2
2
s    
s 2  2ns  n 


s  



2
s 2  2ns  n 

s  
(B-1)
Multiply through by the common denominator.



s    s   s 2  2n s  n  s   s 2  2
2

(B-2)
2
2
s     s3    2n  s 2   2n   n  s   n 

 

  s3   s 2   2s   2
(B-3)
s   
  s3
   2n   s 2
(B-4)
2
 2n   n  2  s




 n  2 


2
Equation (B-4) implies four separate equations.
0
(B-5)
  2n     0
(B-6)
7
2
2n   n  2  
(B-7)
2
(B-8)
 n   2  
Equations (B-5) through (B-8) can be assembled into matrix form.
 1
2
n

 n2

 0
0
0     0 
0
1     0 
  
2 0    
   
0 2       
1
1
2n
2
n
(B-9)
Gaussian elimination is used to simplify the coefficient matrix.
0
1
0
1

0 2n

2
0 n
1
 2n
2
2  n
0
0     0 
1     0 
  
0      
   
2       
(B-10)
Equation (B-10) can be reduced to a 3 x 3 matrix.
 1

2n
 2
 n
 2n
2
2  n
0
1     0 

0     
2       
(B-11)
Complete the solution using Cramer’s rule.
 1

det 2n
 2
 n
 2n
2
2  n
0
1 

2
2
2
0   2  2  n   2 2n 2  n  2  n 





2 
8
(B-12)
 1

det 2n
 2
 n
 2n
2
2  n
0
1 

2 2
0    2  n   2  n 2

 
2 
(B-13)
1
 0  2n


2
2

det    n
0
2
 2   2   2   2

0
2 
n
n



(B-14)
1
2
2  2n    2  n  



2
 2   2   2   2
n
n


(B-15)
 1


det 2n
2
 2   2   2   2
 2
n
n
 n


1

1

 0
 2 
0
(B-16)
2
2  2n  n 
(B-17a)
2
 2   2   2   2
n
n


 2   2    2 
n
n

 
2
 2   2   2   2
n
n


(B-17b)
Recall equation (B-5).
  
(B-18)
9
 2   2    2 
n
n

 
2
 2   2   2   2
n
n


(B-19)
 1


det 2n
2
 2   2   2   2
 2
n
n
 n


1
 2n
2
2  n
0
0


 
 2   2   2 2   2 3
n
n
n


2
 2   2   2   2
n
n



 2   2   2 2   2 3
n
n 
n



2
 2   2   2   2
n
n

(B-20)
(B-21a)
(B-21b)

The coefficients are summarized in equation (B-22).
2


  2  n    2n 




 


2
 
2  2n    2  n  


1


 


2   2    2 

   2


2 2

2

n
n
     n   2  n  




 


3
2


2
2

 2n      n   2n   
(B-22a)
10


 

 
1

 

2
   2
2
2

     n   2  n  



 


  2   2  
 2n

n 
 


2


  2  n 
2

2




n   

2n
  2   2  

n 


  2   2   2 2 
n
n 

3 


 
  2n 
















(B-22b)
Equation (B-1) can thus be rewritten as


1


 s 2   2 



 
2
2
s  2 n s   n 
s  
  2


2
2 
 2
2
     n    2n  s  2   n       n  




2


2
  2   n   2   n 2   s 2   2 



 
 2
 
 2
2
3
2
2 
    n    2n s   2n       n   2n   

 

 

2


2
2
  2   n   2   n 2  s 2  2 n s   n 




 
(B-23)
11
An alternate form is
 1 
 2

 s   2 


2
s 2  2 n s   n 
s  


    2   2   s  2  2 
n 
n
 


 2
2 2
2
2 2
    n   2   n   s  






 2n s    2  n2  
 


 2
2 2
2
2 2
    n   2   n   s  




  2   2  s   2 3 
n 
n

  


2
 2
2
2
2 2
    n   2   n   s  2 n s   n 



 

2n s   2  n2   2n 2 



 2
2 2
2
2 2




n   2   n   s  2 n s   n 


 

(B-24)
12
 1 
 2

 s   2 


2
s 2  2 n s   n 
s  


    2   2   s  2  2 
n 
n
 


 2
2 2
2
2 2




n   2   n   s  






 2n s    2  n2  
 


 2
2 2
2
2 2




n   2   n   s  




  2   2  s   2 3 
n 
n

  


 2
2 2
2
2
2





n   2   n   s   n   d 







2n s   2  n2   2n 2 

 2
2 2





n





2
 2   n 2  s   n 2  d 




(B-25)
13
The inverse Laplace transforms for the four terms on the left-hand-side are found as
follows
  2
2
2


     n   s  2   n 


F1 (s)  
2


2
  2   n   2   n 2   s 2   2 



 
f1 ( t ) 



2



2
 2  n   2  n 2 



(B-26)

  2
2
    n   cost   2  n sin t 

(B-27)
F2 (s)  
f 2 (t) 
 2n s    2  n2  
 

2


2
  2   n   2   n 2   s 2   2 



 
(B-28)



 1 2
2
  2n cost       n   sin t 
2



 

2
 2  n   2  n 2 



(B-29)
14
 2
2 
3
   n  s   2n 
F3 (s)  
2


2
2
 2  n   2  n 2  s  n 2   
d 


 
F3 (s)   

F3 (s)  
2

2
  2   n 


2

2
  2   n 


2

2
  2   n 


(B-30)
 2
2
    n  s

2
 2   n 2  s   n 2   
d

 
 2 3 
n



2
 2   n 2  s   n 2   
d

 
(B-31)
 2
2
    n  s



2
 2   n 2  s   n 2   
d

 
  2 3 
d

n
 

2

 d   2

     2   2   2  s   2   2 
n
n
n
d 


 
(B-32)
15
f 3 (t) 
 exp  n t 


2 2
 2  n   2  n 2 





3

 2   2  cos t     2n  sin  t 
d
d 
n
 



d




(B-33)
F4 (s)  
2n s   2  n2   2n 2 




2
2
 2  n   2  n 2  s  n 2   


d


 
F4 (s)   2n

2

1
  2   2   2 2 

n
n 

2n  

2
 2
2
2
2
2
    n   2   n   s   n   d 






(B-34)
s
16
(B-35)


 1  2

2
  n   2n 2   n 






 2n exp  n t 
2n  








f4 (t) 
cos

t

sin

t

d
d 
2
 2


d
2

2 
   n   2  n   










f 4 (t) 
f4 (t) 
 exp  n t 
 2
2 2
2
   n   2  n  



 exp  n t 
 2
2 2
2
   n   2  n  




  2
2
2

    n   2n    n 2n
2n cosd t   
d






 sin d t 







  2
2
2 2
2


    n   2n    2 n 




2

cos

t

sin

t



n
d
d 
d








17
(B-36)
(B-37)
(B-38)


  2
2
2






2



n
n






 exp  n t 






f 4 (t) 
2

cos

t

sin

t



n
d
d 
 2
 



d
2 2
2
    n   2   n   







18
(B-39)
APPENDIX C
Example 3
Equation (C-1) effectively represents a number of cases since  and  may each be set
equal to zero.




s  
 p   m 

 
s   s    s     s   
s  
,

 p 
 m 

s   s    
s   s  
s   
s   
(C-1)
(C-2)
s  
 p s     m s   
(C-3)
s  
  p s  p   m s  m
(C-4)
s  
 p  ms  p  m
(C-5)
Equation (C-5) implies equations (C-6) and (C-7).
  p  m
(C-6)
  p  m
(C-7)
Equation (C-6) yields
m p
(C-8)
Substitute equation (C-8) into (C-7).
  p    p
(C-9)
    p  
(C-10)
      p
(C-11)
19
  p    
(C-12)
  

(C-13)
p
Substitute equation (C-13) into (C-8).
    
m  

  
m
(C-14)
       

m
(C-15)
      
m
(C-16)

      

m
(C-17)
  

(C-18)
Substitute equations (C-18) and (C-13) into (C-1).




s  
 1            

 
s   s         s     s    
(C-19)
The inverse Laplace transform is
1
 1  1

f (t )  
   sin t       sin  t  

     

20
(C-20)
APPENDIX D
Example 4
The transfer function H times its complex conjugate is



1
1


HH *   
  2  j 2  1   2  j 2  1
(D-1)
Solve for the roots R1 and R2 of the first denominator.
R1, R 2 
R1, R 2 
j2 
 j22  4(1)
2
j2 
 42  4
2
(D-2)
(D-3)
R1, R 2  j  1   2
(D-4)
Solve for the roots R3 and R4 of the second denominator.
R 3, R 4 
R 3, R 4 
 j 
 j22  4(1)
2
 j2 
 42  4
2
(D-5)
(D-6)
R 3, R 4   j  1   2
(D-7)
(D-8)
Summary,
R1   j  1   2
(D-9)
21
R 2   j  1   2
(D-10)
R 3   j  1   2
(D-11)
R 4   j  1   2
(D-12)
Note
R2 = -R1*
(D-13)
R3 = R1*
(D-14)
R4 = -R1*
(D-15)
Now substitute into the denominators.




1


HH *  








2
2
2
2 
    j  1      j  1      j  1      j  1    





(D-16)


1
H H *   

   R1  R 2  R 3  R 4
(D-17)


1
H H *   

   R1  R1 *  R1 *  R1
(D-18)
22
Expand into partial fractions.


1
   R1  R1 *  R1 *  R1 




  R1

  R1*


  R1*


  R1

(D-19)
Multiply through by the denominator on the left-hand side of equation (D-19).

  R1  R1 *  R1 *  R1
  R1

  R1  R1*  R1*  R1

  R1 *

  R1  R1*  R1*  R1

  R1 *

  R1  R1*  R1 *  R1

  R1
1

(D-20)
1
    R1*  R1*  R1
    R1  R1*  R1
    R1  R1*  R1
    R1  R1*  R1*
(D-21)
23


   2   R1  R1*  R1R1*  R1
   2   R1  R1*  R1R1*  R1
   2   R1  R1*  R1R1*  R1*
   2  R1*2   R1
1
(D-22)
1

  3  R1 2  R1 *2   R1R1 *2





   3  R1  R1  R1 * 2  R1R1 * R12  R1R1 *  R12 R1 *
   3  R1 * R1  R1 * 2   R1R1 * R1R1 * R1 *2   R1R1 *2 
  3  R1  R1  R1 * 2   R1R1 * R12  R1R1 *   R12 R1 *
(D-23)
1


   3  R1 *  2  R12   R12 R1 *
   3  R1 *  2  R12   R12 R1 *
   3  R1 2  R1 *2   R1R1 *2 
  3  R1 2  R1 *2   R1R1 *2
(D-24)
1
          3
  R1  R1*   R1*   R1 2


  R1 *2   R12   R12   R1*2  

  R1R1 *2   R12 R1 *   R12 R1 *   R1R1 *2 

(D-25)
24
1
        3
 R1  R1*   R1*   R1 2


  R1*2   R12   R12   R1*2  
   R1*   R1  R1  R1*  R1R1*
(D-26)
Equation (D-26) can be broken up into four separate equations,
  0
(D-27)
 R1  R1*   R1*   R1  0
(D-28)
  R1 *2   R12   R12   R1 *2   0


(D-29)
  R1*   R1  R1  R1*  R1R1*
(D-30)
1
The four equations are assembled into matrix form.
 1
 R1

 R1*2

  R1*
1
R1*
 R12
 R1
0
   






 R1*
 R1     
0



 R12  R1*2     
0
  

 R1
 R1*    1 / R1R1*
1
1
(D-31)
Recall
R1   j  1   2
(D-32)



R1R1*   j  1  2   j  1  2 



(D-33)
25
R1 R1* =1
 1
 R1

 R1*2

  R1*
1
R1*
 R12
 R1
(D-34)
   0 
 R1*
 R1     0

 R12  R1*2     0
   
 R1  R1*    1
1
1
(D-35)
Multiply the first row by R1 *2 and add to the third row.
1
 1
 R1
R1*

 0
 R12  R1*2

 R1
 R1*
1
 R1*
 R12  R1*2
 R1
1    0 
 R1     0

0     0 
   
 R1*   1
(D-36)
Scale the third row.
1
1
1    0 
 1
 R1
R1*  R1*  R1     0


 0
1
1
0     0 

   
 R1*  R1  R1  R1*   1
(D-37)
Multiply the third row by -1 and add to the first row.
0
0
1    0 
 1
 R1
R1*  R1*  R1     0


 0
1
1
0     0 

   
 R1*  R1  R1  R1*   1
(D-38)
26
Multiply the first row by -R1 and add to the second row. Also multiply the first row by
R1* and add to the fourth row.
0
0
1     0 
1
0 R1*  R1*  2 R1     0

  
0
1
1
0     0 

   
0  R1  R1  2 R1*   1
(D-39)
Multiply the third row by -R1* and add to the second row. Also, multiply the third row
by R1 and add to the fourth row.
    0 
0  2R1*  2 R1     0

1
1
0     0 
   
0  2R1  2 R1*    1
1
0

0

0
0
0
1
(D-40)
The first row equation yields
  
(D-41)
The third row equation yields
  
(D-42)
Equation (D-40) thus reduces to
 2R1 *  2R1    0
  2R1  2R1 *   1

   
(D-43)
 2R1 *   2R1   0
(D-44)
 R1 *   R1
(D-45)

 R1

R1 *
(D-46)
2R1  2R1*   1
(D-47)
27
  R1
2R1 
 R1*

  2R1*   1

1
   R1 

 R1* 
R1 

2
  R1* 


1
1
2    R1 


R
1
*
R1 


  R1* 


1
R1*
2 [  R12  R1*2 ]


(D-48)
(D-49)
(D-50)
(D-51)
 R1

R1 *
(D-52)
R1
1
2 [  R12  R1*2 ]
(D-53)
Recall,
  
(D-54)
  
(D-55)
The complete solution set is thus
 
 
1
 1
   2 [  R12  R1*2
 
 
 R1 *
 R1 





R
1
]


 R1 * 
(D-56)
28
Note that


 R12  R1 *2   j 4 1   2 


(D-57)

 


 


1
  1
 
  2  
2  
 
  j 4 1     

 
 


1   2  j 

1   2  j 
1   2  j 
1   2  j 
(D-58)

 

 

j
 

 


8 1   2   
 



 


1   2  j 

1   2  j 
1   2  j 
1   2  j 
(D-59)
 
 
 
 
 
 

   j

1
   j

2     j
8 1    


 j

1  2 

1  2 
1   2 
1   2 
(D-60)
Let


(D-61)
1  2
29
 
    j
 


   1     j
   8     j 
 


 
    j
(D-62)
Recall
R1   j  1  2
(D-63)


1
   R1  R1 *  R1 *  R1 






1
 

 8
2
   1    j   


 j
1
 

2  j   8 


1






 j
1
 

  8 
2
   1    j 


 j
1
 

 8
2
   1    j   


 j
(D-64)
30
APPENDIX E
Example 5
A transfer function times its complex conjugate is






1  j 2
1  j 2
HH *   


  1   2   j 2   1   2   j 2



 

(E-1)
Rearrange equation (E-1) as
 1  4  2 2  

1
HH *   


 2  j 2  1  2  j 2  1
(E-2)
Solve for the roots R1 and R2 of the first denominator.
R1, R 2 
R1, R 2 
 j22  4(1)
j2 
2
 42  4
2
j2 
R1, R 2  j 
(E-3)
(E-4)
1  2
(E-5)
Solve for the roots R3 and R4 of the second denominator.
R 3, R 4 
R 3, R 4 
 j2 
 j22  4(1)
2
 j2 
R3, R 4   j 
 42  4
2
(E-6)
(E-7)
1  2
(E-8)
31
Summary,
R1   j 
1  2
(E-9)
R 2   j 
1  2
(E-10)
R3   j 
1  2
(E-11)
R 4   j 
1  2
(E-12)
Note
R2 = -R1*
(E-13)
R3 = R1*
(E-14)
R4 = -R1*
(E-15)
Now substitute into the denominators.




2
2
1 4 

HH *   





2
2
2
2
    j  1      j  1      j  1      j  1    





(E-16)


1  4  2 2
HH *   

   R1  R 2  R 3  R 4
(E-17)
By substitution,


1  4  2 2
HH *   

   R1  R1 *  R1 *  R1
32
(E-18)
Expand into partial fractions.


1  4  2 2


   R1  R1 *  R1 *  R1 

  R1


  R1 *


  R1 *


  R1

(E-19)
Multiply through by the denominator on the left-hand side of equation (E-19).
1  4 2 2  

  R1  R1 *  R1 *  R1
  R1

  R1  R1 *  R1 *  R1

  R1 *

  R1  R1 *  R1 *  R1

  R1 *

  R1  R1 *  R1 *  R1

  R1

(E-20)
33
1  4 2 2  
    R1 *  R1 *  R1
    R1  R1 *  R1
    R1  R1 *  R1
    R1  R1 *  R1 *
(E-21)
1  4 2 2  


  2   R1  R1 *  R1R1 *  R1
  2   R1  R1 *  R1R1 *  R1
  2   R1  R1 *  R1R1 *  R1 *
  2  R1 *2   R1
(E-22)
1  4 2 2  


   3  R1  R1  R1 *2   R1R1 * R12  R1R1 *  R12 R1 *
   3  R1  R1  R1 *2  R1R1 * R12  R1R1 *  R12 R1 *
   3  R1 * R1  R1 *2   R1R1 * R1R1 * R1 *2   R1R1 *2 
  3  R12  R1 *2   R1R1 *2
(E-23)
34
1  4 2 2  


   3  R1 * 2  R12   R12 R1 *
   3  R1 * 2  R12   R12 R1 *
   3  R12  R1 *2   R1R1 *2 
  3  R12  R1 *2   R1R1 *2

(E-24)

1  4  2 2 
         3
  R1  R1 *   R1 *   R1 2


  R1 *2   R12   R12   R1 *2  

  R1R1 *2   R12 R1 *   R12 R1 *   R1R1 *2 

(E-25)
1  4 2 2  
        3
 R1  R1 *   R1 *   R1 2


  R1 *2   R12   R12   R1 *2  
   R1 *   R1  R1  R1 *  R1R1 *
(E-26)
35
Equation (E-26) can be broken up into four separate equations.
  0
(E-27)
 R1  R1*   R1*   R1  42
(E-28)
  R1*2   R12  R12   R1*2   0
(E-29)
  R1*   R1  R1  R1*  R1R1*  1
(E-30)
The four equations are assembled into matrix form.
 1
 R1

 R1 *2

  R1 *
1

 R1 *  R1 
 R12  R1 *2 

 R1  R1 * 
1
R1 *
 R12
 R1
1
0
  

    42 
 

  

0
  

  1 / R1R1 *
(E-31)
Recall
R1   j  1   2
(E-32)



R1R1*   j  1  2   j  1  2 



(E-33)
R1 R1* =1
(E-34)
By substitution,
 1
 R1

 R1 *2

  R1 *
1
R1 *
 R12
 R1

 R1 *  R1 
 R12  R1 *2 

 R1  R1 * 
1
1
   0 
   42 
 

   0 
  

   1 
(E-35)
36
Multiply the first row by R1 *2 and add to the third row.
1
 1
 R1
R1 *

2
 0
 R1  R1 *2

 R1
 R1 *
1 
 R1 
0 

 R1 *
1
 R1 *
 R12  R1 *2
 R1
   0 
   42 
 

   0 
  

   1 
(E-36)
Scale the third row.
1
1
1 
 1
 R1
R1 *  R1 *  R1 

 0
1
1
0 


 R1 *  R1  R1  R1 *
   0 
   42 
 

   0 
  

   1 
(E-37)
Multiply the third row by -1 and add to the first row.
0
0
1 
 1
 R1
R1 *  R1 *  R1 

 0
1
1
0 


 R1 *  R1  R1  R1 *
   0 
   42 
 

   0 
  

   1 
(E-38)
Multiply the first row by -R1 and add to the second row. Also multiply the first row by
R1* and add to the fourth row.
0
0
1 
1
0 R1 *  R1 *  2 R1 


0
1
1
0 


0  R1  R1  2 R1 *
   0 
   42 
 

   0 
  

   1 
(E-39)
Multiply the third row by -R1* and add to the second row. Also, multiply the third row
by R1 and add to the fourth row.
37
1
0

0

0

0  2R1 *  2 R1 
1
1
0 

0  2R1  2 R1 * 
0
0
1
   0 
   42 
 

   0 
  

   1 
(E-40)
The first row equation yields
  
(E-41)
  
(E-42)
The third row equation yields
Equation (E-40) thus reduces to
 2R1 *  2R1    4 2 

  2R1  2R1 *   

    1 
(E-43)
Complete the solution using Cramer's rule.
 2R1 *  2R1   
det er min ant 
 4  R12  R1 *2 





  2R1  2R1 * 
(E-44)
Recall
R1   j  1   2
(E-45)



R12   j  1  2   j  1  2 



(E-46)




R12   2  1   2  j 2 1   2 






R12  1  22  j 2 1  2 


(E-47)
(E-48)
R1*   j  1  2
(E-49)
38



R1 *2   j  1  2   j  1  2 







R1 *2  2  1  2  j 2 1  2 


(E-51)


(E-50)


R1 *2  1  22  j 2 1  2 


(E-52)
Thus,


R12  R1 *2  j 4 1  2 



(E-53)



4 R12  R1 *2  j 16  1  2 


 2R1 *  2R1   
determinan t 
  

2
R
1

2
R
1
*

 

(E-54)


j 16  1  2 


42
determinan t 


 1
j 16  1  2 


1
(E-55)
 2R1 

 2R1 *
(E-56)
 4 2  2R1 *  2R1




j 16  1   2 


(E-57)
 4 2  R1 *  R1




j 8  1   2 


(E-58)
Recall,
R1   j  1   2
(E-59)
39

 4 2    j  1   2   j  1   2





j 8  1   2 



 1  4 2   1   2   j  1  4 2 








   1  4 2   j  1  4 2   1   2 

 


(E-61)


j 8  1   2 



2
8  1   



 2R1 * 42 
det er min ant 


2
2
R
1
1



j 16  1  



(E-60)
1
(E-62)
(E-63)
  2R1* 4 2R1
(E-64)
  R1* 4 R1
(E-65)
2
j 16  1  2 



2
j 8  1  2 


R1   j  1  2
(E-66)
 
 
2
2 
2 
    j  1     4   j  1   

j 8  1   2 


40
(E-67)
  j  1  2  j43  42 1  2 


 
j 8  1  2 





(E-68)

  1  42 1  2  j 1  42 


 
j 8  1  2 



 
(E-69)

  1  42  j 1  42 1  2 


 
j 8  1  2 


(E-70)
Recall,
  
(E-71)
  
(E-72)
The complete solution set is thus
 
  

 
 
 
1
 
 
 
2  
  8 1    
 

 
 
1  4 2   j 
 
2

1  4   j 
 
2
1  4   j 
 
2
1  4   j 
 
1  4 2 

2
1  4 

2
1  4 

2
1  4 


1  2 

1  2 


2
1  

1  2 

(E-73)
41
The partial fraction expansion is thus
H H *  

  R1


  R1 *


  R1 *


  R1

(E-74)
42
APPENDIX F
s2




s  a s  b 
sa sb
s 2  s  a s  b  
(F-1)

s  a s  b   s  a s  b
s  a 
s  b
(F-2)
s 2    s 2  (a  b) s  ab    s  b    s  a 


(F-3)
s 2  s 2  (a  b) s  ab  s  b  s  a
(F-4)
s 2  s 2  (a  b)     s  ab  b  a
(F-5)
Equation (F-5) yields three individual equations.
 1
(F-6)
(a  b)     0
(F-7)
ab  b  a  0
(F-8)
(a  b)     0
(F-9)
ab  b  a  0
(F-10)
    (a  b )
(F-11)
b  a  ab
(F-12)
43
The equations are assembled into matrix form.
1 1    (a  b)
b a       ab 

  

(F-13)
Apply Cramer’s rule.

 (a  b) 1  
1
det 
a   
ab
  ab
(F-14)

 a 2  ab  ab
ab
(F-15)

 a2
ab
(F-16)

 1  (a  b ) 
1
det 
 ab 
ab
b
(F-17)

 ab  b(a  b)
ab
(F-18)
 ab  ab  b 2

ab
(F-19)
b2

ab
(F-20)
The partial fraction expansion is thus
  a 2  1   b2    
s2


 1 

, a  b
s  a s  b 
 a  b   s  a   a  b   s  b 
44
(F-21)
APPENDIX G
s4
s  a s  b s  c s  d 
  a2  1


2
2
2


  b   1     c   1   d   1  







 1  

1




 




  a  b   s  a   a  b   s  b    c  d   s  c   c  d   s  d  
(G-1)
Let
 a2 

A
 a  b 
(G-2)
 b2 

B
 a  b 
(G-3)
  c2 

C
 c  d 
(G-4)
 d2 

D
 c  d 
(G-5)
45
s4
s  a s  b s  c s  d 

 1 
 1  
 1 
 1 
 1  A 
 B
 D
1  C 




s  a 
 s  b  
s  c 
 s  d 

(G-6)
s4
s  a s  b s  c s  d 

 1 
 1 
 1  C 
 D


s  c 
 s  d 

 1 
 1 
 1 
A
 D
1  C 



 s  a 
s  c 
 s  d 
 1 
 1 
 1 
 B
 D
1  C 



 s  b 
s  c 
 s  d 
(G-7)
s4
s  a s  b s  c s  d 

 1 
 1 
 1  C 
 D


s  c 
 s  d 

 1 
 1  1 
 1   1 
 A 
 C
 D






s  a  s  c 
 s  a   s  d 
 s  a 
 1 
 1  1 
 1   1 
 B 
 C
 D






s  b  s  c 
 s  b   s  d 
 s  b 
(G-8)
Recall
1
s  a s  b
1 
 1   1
 

, ab


 a  b   s  a s  b 
46
(G-9)
s4
s  a s  b s  c s  d 

 1 
 1 
 1  C 
 D


s  c 
 s  d 

 1 
1 
1 
 1   1
 1   1
 A 
 C

 D








a  c  s  a s  c 
 a  d   s  a s  d 
 s  a 
 1 
1 
1 
 1   1
 1   1
 B 
 C

 D








 b  c  s  b s  c 
 b  d   s  b s  d 
 s  b 
(G-10)
s4
s  a s  b s  c s  d 

 1 
 1 
 1  C 
 D


s  c 
 s  d 

 1 
1 
1 
 1   1
 1   1
 A 
 C

 D








a  c  s  a s  c 
 a  d   s  a s  d 
 s  a 
 1 
1 
1 
 1   1
 1   1
 B 
 C

 D








 b  c  s  b s  c 
 b  d   s  b s  d 
 s  b 
(G-11)
s4
s  a s  b s  cs  d 

 1 
 1 
 1  C 
 D


s  c 
 s  d 

 A   AC    1   AC   1   AD    1   AD   1  
 









 s  a   a  c   s  a   a  c   s  c   a  d   s  a   a  d   s  d  
 B   BC    1   BC   1   BD    1   BD   1  
 









 s  b   b  c   s  b   b  c   s  c   b  d   s  b   b  d   s  d  
(G-12)
47
s4

s  a s  b s  c s  d 
  C   D   1 
1  A 1  

 

  a  c   a  d   s  a 
  C   D   1 
 B1  

 

  b  c   b  d   s  b 
  A   B   1 
 C 1  

 

  a  c   b  c   s  c 
  A   B   1 
 D 1  

 

  a  d   b  d   s  d 
(G-13)
s4

s  a s  b s  cs  d 
  a 2  
 
  1 
 c2
d2
 1  

 
1 

 a  b    a  c c  d    a  d c  d     s  a 


 b2  
 
 1 
 c2
d2
 1  




 a  b    b  c c  d   c  d b  d    s  b 


  c2  
 
  1 
 a2
b2
 1  

 


 c  d    a  b a  c   a  b b  c    s  c 


 d 2  
 
  1 
 a2
b2






 

1


 c  d    a  b a  d   a  b b  d     s  d 


(G-14)
48
s4

s  a s  b s  cs  d 
  a 2  
 
  1 
c2
d2

 1 

 
1 

 a  b    a  c c  d   a  d c  d    s  a 


 b2  
 
  1 
c2
d2

 1 

 


 a  b    b  c c  d    c  d b  d    s  b 


  c2  
 
  1 
a2
b2
 1  

 








c

d
a

b
a

c
a

b
b

c




 
   s  c 
 

 d 2  
 
  1 
a2
b2
 1  

 








c

d
a

b
a

d
a

b
b

d




 
   s  d 
 

(G-15)
49
APPENDIX H
s

s  a 2
s
As  B
C

s  a  s  a 2
(H-1)
As  Bs  a 2  C
s  a 
(H-2)
s  As  Bs  a   C
(H-3)

(H-4)
s  As 2  aA  Bs  aB  C
(H-5)

s  As 2  Bs  a As  B  C
A=0
(H-6)
s  Bs  aB  C
(H-7)
B 1
(H-8)
C  a
(H-9)
s
s  a 2

1
a

s  a  s  a 2
(H-10)
50
APPENDIX I
1



s 2   2 s  2
1
As  B
s2  2
s2  2 s  2  2 


Cs  D
(I-1)
s  2
As  B
s2  2

Cs  D
(I-2)
s  2
The expansion can be performed using the derivation in Appendix B.
 1 
 2

 s   2 




 
2
2
s  2 s   
1

 2 s    2  2 



 2
2 2  2   2  s 2   2
  





2 s   2  2  22




 2
2 2  2   2  s 2  2 s  2








(I-3)
51
 1 
 2

 s   2 


 
s 2  2s   2 
1


 4  4  s 2   2 
 2  s    2   2



2 s   2  3 2
 4   4 s  2


(I-4)
 1

 2

 s   2 

 
2
2
s  2 s   
1




 s   2   2 /[ 2]
s   2  3 2 /[ 2] 





s  2
( 4   4 ) 
[s 2   2 ]
2
(I-5)
The inverse Fourier transform is
f (t) 





 1 2
2 
  2 cost        sin t 
 2
 


2 2  2 2  
  



1



   sin t 

  2  2  2 2


2

cos

t



2

 2
 

2
2







2







 exp  t 



(I-6)
52
f (t) 





 1


 2  cos t     2   2  sin t 

 


[ 4   4 ] 
1

  2  2 
 exp  t  




2

cos

t

sin

t




 

[ 4   4 ] 

(I-7)
53
APPENDIX J
1  1  s   s  

 2  2
s 2  s 2  2 
s
s  2


1  s   s 2  2  s 3  s 2
(J-1)
(J-2)
1  s 3  2s  s 2  2  s 3  s 2
1    s 3     s 2  2s  2
(J-3)
(J-4)
    0
(J-5)
    0
(J-6)
0
(J-7)
0
(J-8)
  1 / 2
(J-9)
  1 /  2
(J-10)
 1 / 2  1 / 2
1  1

 2  2
s 2  s 2  2 
s
s  2
(J-11)
The inverse Laplace transform is
f (t) 
1
2
t
1
3
sin( t )
(J-12)
54