Identification of the dynamic characteristics of nonlinear structures

 
 
 
 
n = l
k = O
(3-30)
(3-3 1)
=
n!
k! (n k)!
n)
(3-32)
n = l
k = O
=
n!
(n k)!
(2k
(3-33)
n = l
k = O
The first few terms of 
and 
are
= 
+ 2 H,(o) 
+ 
(3-34)
= 
+ 3 
+ 3 
(3-35)
Substitute 
into (3-30) and set the coefficients of 
at both sides to be
equal:
1
k
+ 
(3-36)
The first-order Volterra kernel transform is independent of the nonlinear terms present in
the equation of motion and represents the dynamic characteristics of the linear part of the
nonlinear system.

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Identification of Nonlinearity Using Higher-order 
8 0
Now, let 
and substitute into 
then
 
N = O
.H
(3-37)
Differentiate x(t) to get x(t) and 
and then substitute into (3-30) in similar way as for
the 
calculation, and let the coefficients of 
on both sides be equal:
(3-38)
From (3-38) it can be seen that 
has all the poles which 
has and is
proportional to the coefficient of the quadratic nonlinearity term 
Similarly, if we let 
then
+ 
+ 
(M+k)! k! (N+j)! j!
(3-39)
Upon substituting into (3-30) and letting the coefficients of 
both sides
be equal:
+ 
+ 
+ 
+
+ 
+ 
(3-40)
In fact, it has been established in 
that for a physically realisable system specified by
the nonlinear differential equation as
 y + 
 
= 
(3-41)

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3
Identification of Nonlinearity Using Higher-order 
8 1
where F(x) is a function of the differential operator 
the 
Volterra kernel
transform is given by
n
1
a ,