Identification of the dynamic characteristics of nonlinear structures

partial differential operator given as:
 
. . . 
 
Under this operator, it can be seen that
 
 
r = l
s = l
Upon substitution, (3-4) becomes
. . .
n = l
) 
s = l
(3-9)
(3-10)
(3-11)
After some further mathematical manipulation, (3-11) can be written as:
 
s = l
. . .
n = 1
) 
n!
12)
On the other hand, under the differential operator defined in 
the 
Volterra
kernel transform can be rewritten as:
 
. . .
) 
r = l
(3-13)
where

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3
Identification of Nonlinearity Using Higher-order 
7 5
n
14)
s = l
To illustrate the validity of equation 
consider the derivation of 
In this
case, 
and so 
becomes
 
 + 
 
(3-15)
and upon substitution,
= 
[ (a, 
+ 
)(a, 
+ 
r = l
= e
+ 
+ 
+ 
(3-16)
Substituting (3-16) into (3-13) and considering the symmetry property of 
equation 
13) becomes
 
(3-17)
Using this preliminary mathematics, it is now possible to establish an input-output
relationship of a general nonlinear system when the input to the system is in the form of
sinusoid.
When 
then
 
 
s = l
where 
is given by (3.14). According to the binormial theorem,
I”
n
n!
2”
_
(3-19)
k = O
(3-18)

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3
Identification of Nonlinearity Using Higher-order 
7 6
Substituting (3-19) into 
and using (3-13) for 
gives