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3
Identification of Nonlinearity Using Higher-order
103
In the ‘state-space analysis’, a-priori information about the
number of degrees of
freedom and the physical connectivity of the system to be analysed should be given.
Since, mathematically, measured first- and second-order FRF data are functions of
the
physical parameters (mass, nonlinear stiffness and damping elements), given known
measured
and second-order
these parameters can, in theory, be calculated
provided enough data have been measured. The physical parameter identification problem
of a nonlinear system can therefore be formulated mathematically as the solution of the
following linear algebraic equation:
=
(3-8 1)
where (p) is the unknown vector of physical parameters, and [A] and
are the
coefficient matix and vector formed using the measured first- and second-order
frequency response functions. To illustrate this process, take the nonlinear SDOF system
described by (3-3) as an example. The system mass m, linear stiffness k and damping c
can be calculated based on the familiar analysis of the measured first-order FRF
while the coefficient of the second-order nonlinear term can be obtained from equation
(3-38) using the measured second-order FRF
In fact, one data point on the
plane is enough to determine
although more data points are recommended in
practice in order to have a reliable averaged estimation.
The ‘modal-space analysis’ is based on the mathematical observation that, in general, the
second-order frequency response function (second-order Volterra kernel transform) can
be decomposed as
=
(3-82)
where N is the number of degrees of freedom of the system and 2K represents the
number of poles corresponding to “nonlinear coupling modes” which are the
combinational resonances of the system
When one of the variables
or
is
fixed, then equation (3-82) reduces to the following
form:
=
(3-83)
3
of Nonlinearity Using Higher-or&r
104
Curve-fitting
of this polynomial function can be made using the well-developed
polynomial curve-fitting algorithms
used in linear modal analysis, then all the and
which are the natural frequencies of the system can be obtained and the analytical
model of
in its polynormial expression can be established. Such analytical
models can be used for further applications such as response prediction, as shown in
Fig.3.21 for the system described by (3-3) with
and with input being a pure
sinusoid of frequency half of the natural frequency of the system. The improvement of
prediction accuracy by including second-order FRF is clearly demonstrated.
Prediction Using First Order
Prediction Using First and Second Order FRF
____
response,
Fig.3.21 Response Prediction Using Second Order Frequency Response Function
3.7 CONCLUSIONS
In this Chapter, the basic theory of Volterra and Wiener series of nonlinear systems has
been introduced and the measurement of higher-order frequency response functions has
been discussed. The relationships between the Volterra and Wiener kernels and their
corresponding measured frequency response functions have been demonstrated. By
extending the classical fast-order frequency response function analysis to higher-order
frequency response function analysis, it can be seen that the linear system theory is
extended in a logical way to cover nonlinear systems.
From the system identification point of view, the measured higher-order frequency
response functions provide considerable information about the nature of the nonlinearity
of the system which the classical first-order frequency response functions cannot provide.
Among them are the following:
(a) since for certain nonsymmetric nonlinear systems, such as the quadratic and bilinear
systems mentioned in this Chapter, the measured first-order frequency response functions
are effectively linear and therefore, cannot be used to detect existence of nonlinearity, the
3 Identification of Nonlinearity Using Higher-order
105
measured higher- (second-) order frequency response functions give an indication of the
nonlinearity in the system;
(b) the different characteristics of higher-order frequency functions may give
categorization and so identification of common mechanical nonlinear systems by
comparing the measured higher-order frequency response functions with those analytical
ones of known nonlinear systems;
(c) from a system response prediction point of view, the higher-order frequency
response functions together with the first-order frequency response functions give more
accurate response prediction to any input than just using the first-order frequency
response functions and;
(d) the measured higher-order frequency response functions can be analysed in a
similar way to the case of first-order frequency response functions in order to identify
either the physical parameters or the modal parameters of a nonlinear system so that its
mathematical model can be established.
The existing numerical difficulties concerning the successful measurement of
order frequency response functions have been discussed and possible ways of
overcoming these difficulties suggested, both in the case of measurement using harmonic
probing technique and correlation analysis with random input. In the harmonic probing
method, the main problem involved is leakage in the
of the response signal. This
leakage problem can be overcome using the correlation technique as suggested. For the
correlation analysis with random input, the main problem involved is the removal of the
linear contribution from the total response so that the computational efficiency can be
improved. For this purpose, averaging in the frequency domain instead of the time
domain as discussed is recommended.
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