Identification of the dynamic characteristics of nonlinear structures

 
 
 
 
n = l
(3-20)
where, 
denotes the nh-order Volterra kernel transform 
with
the first k of the values equal to 
and the remaining (n-k) values equal to 
From 
any frequency component which is present in x(t) due to input 
can be calculated. For example, the 
component of x(t) is
m = O
m! 
(3-21)
If the input 
then in 
should be correspondingly replaced by
The same type of argument shows that when 
the 
component of x(t) is ( N, M 0)
= 0
(N+k)! k!
2
(3-22)
The four subscripts of 
mean that 
and the first 
values of 
are equal to (or), the next 1 values equal to 
the next N+k values equal
t o
a n d t h e l a s t k v a l u e s e q u a l t o
S i m i l a r l y ,
w h e n
the 
frequency component in x(t) is
(L, M 
0)
= O j = 0
(M+k)! k! (N+j)! j!
(2-23)

---

3
Identification of Nonlinearity Using Higher-order 
7 7
From 
the leading terms for the frequency component are:
 
 
H,(o) + 
+ . ..] + CC
(3-24)
where CC means complex conjugate since the response component must be real.
The leading term for the frequency component 
in (3-22) if 
and 
are
incommensurable 
and 
are said to be incommensurable if 
cannot be
expressed as 
where and are integers), is:
+ . . . . . . + cc
(3-25)
Similarly, the leading term for the frequency component 
in (3-23) if 
and 
are incommensurable, is:
 + 
2
+ . . . . . . + CC
(3-26)
On the other hand, 
frequency response function 
which 
is
experimentally measurable, is defined as the output component 
of x(t) at
frequency 
due to the input 
(here 
can be complex to accommodate the different phase shifts) divided by the input
spectra, that is
 
(3-27)
Comparing 
(3-25) and (3-26) with the definition of the higher-order frequency
response function of equation 
it can be seen that the measured 
frequency
response function 
is the first-order approximation of the 
Volterra kernel transform 
To illustrate this point, take the 
order frequency response function as an example. If only the leading term is considered
in equation (3-25) and the contribution of other kernels (even-ordered kernels after the
second) at frequency 
can be neglected, then it becomes clear that the measured
second-order frequency response function 
based on (3-27) will be the same as
the second-order Volterra kernel transform 
In general, however, there will be

---

3
Identification of Nonlinearity Using Higher-order 
7 8
some contribution from the higher even-ordered Volterra kernels and the estimated
second-order frequency response function is an approximate of the uniquely defined
second-order Volterra kernel transform. The same argument holds for other higher-order
frequency response functions. Based on this observation, the Volterra kernel
and its transform 
have direct physical meaning and
interpretation.
It is worth pointing out here that the Volterra kernel transforms
mathematically unique. However, the 
-order frequency response functions
are usually input-output dependent like the classical first-order
frequency response function H,(o) measured using a sine wave excitation. Since we are
only able to deal with truncated series, these measured frequency response functions will,
in some cases, give more accurate representation than the equivalent
transforms, which are by no means measurable.
Volterra kernel
3.3.2 ANALYTICAL CALCULATION OF FREQUENCY
RESPONSE FUNCTIONS
So far, it has been shown how the output x(t) and input f(t) of a nonlinear system are
related through the system’s frequency response functions (or, more strictly, the Volterra
kernel transforms), and it is appropriate here to investigate what forms and what
characteristics the higher-order frequency response functions of typical nonlinear
mechanical systems possess. There are some different methods for analytically calculating
the frequency response functions of a known nonlinear system and what is discussed
here is the harmonic probing method 
Suppose that the input f(t) is
 
A, 
(3-28)
where the 
values are incommensurable and, for simplicity, let 
since the
analytical 
FRF, i.e. the 
Volterra kernel transform (we define
as the analytical
order FRF), is unique. Substituting into 
then 
is given 
by

---

3 Identification of Nonlinearity Using Higher-order 
7 9
= coefficient of 
term in the expression of x(t) 
(3-29)
Based on 
it is possible to compute 
of a nonlinear
mechanical system successively. To illustrate this, first consider an SDOF system given
+ 
+ kx + 
+ 
= f(t)
and substitute into 
then