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
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