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3 Identification of
Nonlinearity Using Higher-order
8 8
harmonic probing method and correlation analysis using random input are the most
commonly referred methods and have been given much discussion in the literature. The
theoretical basis for the measurement of higher-order frequency response functions by the
harmonic probing method has already been presented and what is to be discussed next is
the practical applicability of the method through numerical case study. The theoretical
basis of the correlation technique and the numerically simulated case studies of
order frequency response function (Wiener kernel transform) measurement will be given
later in this Chapter.
As explained earlier, if the input to a nonlinear system is
where the frequency components
are incommensurable, then the
frequency response function
which is the first-order approximation of
the
Volterra kernel transform
can be estimated using (3-27).
On the thus- estimated
with the exception of
which
is the leading term, all the higher kernel transforms
may have a contribution. This, as already pointed out, may make the measured
a better representation of the system than the corresponding Volterra
kernel transforms
since we are only able to deal with the truncated
series. Measurement of the first-order frequency response functions of nonlinear systems
has already been well established
and here the measurement of higher-order (mainly
second-order) frequency response functions is discussed. Although the technique can
theoretically be readily extended to the the measurement of higher- (than the second) order
terms, because of the time and effort involved, it is hardly practical to measure beyond the
third and for most practical nonlinear systems, only the lower few terms are in general
required in order to provide accurate representation of the system.
Based on the above-mentioned theory, numerical simulation of the measurement of the
second-order frequency response functions for a square-law system described by (3-34)-
(3-35) and an SDOF nonlinear system given by (3-3) was carried out. For computational
convenience, the input was set to be
where A is real and equal to
B (in real practical measurements, A and B can be set independently and can be complex
to accommodate the relative phase difference of these two sinusoids). The response of the
system x(t) was calculated using a numerical integration technique and, after the transient
dies away, the signal was sampled and Fourier transformed to find the
and
frequency components of x(t). Suppose the frequency components
of x(t) at
and
are
and
respectively, then
based on
the following four points of
on
plane are given by:
3
Identification of Nonlinearity Using Higher-order
89
(3-56)
(3-57)
=
(3-58)
The measured
and
lie on the diagonal of
of
plane. When
and
are varied, the value of
in any desired region on
plane can be obtained. Considering the mathematical symmetry of
if
the frequency range of interest is
then what needs to be measured is the triangular
region where
and
as shown in
0
Fig.3.12 Illustration of Measurement Frequency Region Required
In this way, the second-order frequency response functions of the square-law system and
the SDOF nonlinear system are measured and they are shown in
and 3.14.
When comparing figure 3.13 with its analytical counterpart figure 3.9, it can be seen that
except for some spurious spikes at both diagonals of
and
the
measurement results are quite acceptable. The spurious spikes appear because in the
numerical simulation,
and were chosen to be integers and so the condition that
and
should be incommensurable was violated and for this specific system, it can be
shown that such violation only causes errors when
and then,
is
overestimated by 100%. This problem can be removed by measuring the diagonal
elements of
at
and
using one single sinusoid input and then
3 Identification of Nonlinearity Using Higher-order
90
measuring the second harmonic component (on the diagonal
and the DC
component (on the diagonal
of the output. On the other hand, in the case of a
nonlinear SDOF system, the situation becomes somewhat complicated. When comparing
figure 3.14 with its analytical counterpart figure 3.5, we see that the measured results are
not very bad except for some small spurious spikes appearing on the plane, again
showing the effects of the violation of the condition that
and
must be
incommensurable.
ig.3.13 Measured Second Order FRF of the Square-law System
Using Harmonic Probing
Method (Modulus Linear Scale, x-axis
260, y-axis
-260 260
Identification of Nonlinearity Using Higher-order
91
‘ig.3.14 Measured Second Order FRF of the SDOF Nonlinear System Using Harmonic
Probing Method (Modulus Linear Scale, x-axis
260, y-axis
-260 260
Of course, it is possible to set and
to be incommensurable in the measurement, but
then there may be a leakage problem in the DFT of response x(t) because, in this case, it
is not possible to make all these major frequency components
contained in the x(t) coincide with frequency lines. Some further research is
needed to investigate how cleanly and consistently the second-order frequency response
functions of a nonlinear system can be measured using the harmonic probing method
based on DFT algorithms. However, one possible way of getting around the leakage
problem, which the author suggests here, is to use the correlation technique. say, the
frequency component
of x(t) at
is of interest, then instead of obtaining
using the
(which can sometimes cause serious leakage errors), the correlation
technique can be used by multiplying
+
to x(t) and then
integrating the product with time as:
T
2
x(t) [
+
+
+
dt
(3-59)
+
T
Based on this correlation technique, together with the diagonal elements of
to
be measured using single sinusoid input, clean and consistent measurement results could
be obtained.
3
Identifkation of Nonlinearity Using Higher-order
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