Identification of the dynamic characteristics of nonlinear structures

3
 of Nonlinearity Using Higher-order 
69
I
true 
- - - - - - -
excitation frequency = 
excitation 
frequency 
Fig.3.2 Response Prediction Accuracy Using First Order FRF
All this means that the first-order frequency response function analysis is inadequate and
even sometimes inappropriate for some nonlinear systems and more accurate
representation of their dynamic characteristics becomes necessary. For this purpose,
research work on the higher-order frequency response function analysis has been carried
out and is described in this Chapter.
The mathematical basis of higher-order frequency response function analysis lies in the
Volterra series theory which, as the functional series representation of nonlinear systems
and ‘with its rigorous mathematical base, has been found to be quite effective in the
of general nonlinear systems. The theory was first introduced into
nonlinear 
circuit analysis in 1942 by Wiener who later extended the theory and applied
it in a general way to a number of problems. Since Wiener’s early work, many papers
have been published dealing with this subject in system and communication engineering
However, it was not until recently that the theory has been applied to the
identification of nonlinear mechanical structures 
and found to be quite useful.
There is some literature available now on the identification of nonlinear mechanical
structures based on the Volterra series theory, such as references 
However, most of the studies to date are still at the stage of numerical simulation of
certain nonlinear systems and the difficulties in applying this theory to the identification of
practical nonlinear mechanical structures have not been fully investigated although some
experimental work based on specifically designed nonlinear structure has been carried out
The research work presented in this Chapter introduces the basic theory of Volterra
series and of their relation to the higher-order frequency response functions and how the
higher-order frequency response functions generalise linear system theory to cover
nonlinear systems. The harmonic probing method for the Volterra kernel measurement
using multi-tone input 
and correlation technique for the Wiener kernel measurement
using random input 
are investigated and the relationship between the Volterra and

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3
Identification of Nonlinearity Using Higher-order 
7 0
Wiener 
kernels is studied. Possible ways of curve-fitting or surface-fitting the measured
higher-order frequency response functions so that parametric or nonparametric model of
the nonlinear structure can be established are discussed. Considerable attention is given to
the practical assessment of the measurement of higher-order frequency response functions
of realistic nonlinear mechanical systems, both in the case of sinusoidal and random
inputs, by numerically simulating the measurement processes. The existing difficulties
concerning the successful measurement of higher-order frequency response functions are
discussed and possible ways of improving measurement results are suggested. The
applications of higher-order frequency response function analysis in the identification of
nonlinear mechanical systems are also discussed.
3.2 VOLTERRA SERIES REPRESENTATION OF
NONLINEAR SYSTEMS
A nonlinear function f(x) can in general be represented as a Taylor series at a certain point
(e.g. 
and this series approaches f(x) when the variable x is not far from that point.
Similarly, a nonlinear system can in general be 
by a Volterra series which
converges when the nonlinearity of the system satisfies certain general conditions 
Before presenting the theory of Volterra series, it is necessary to examine some of the
basic characteristics of nonlinear systems.
3.2.1 BASIC CHARACTERISTICS OF NONLINEAR SYSTEMS
Since a linear system must satisfy the principle of superposition (as discussed in Chapter
a sinusoid can be regarded as an eigenfunction of the system. For a sinusoid applied
to a linear system, the system only changes its amplitude and phase angle without
distorting its wave form. A nonlinear system however, is 
by the transfer of
energy between frequencies. For a sinusoidal input 
to the nonlinear system
governed by equation
+ 
+ kx + 
+ 
= f(t)
(3-3)
the system will generate harmonic frequency components response in addition to the
fundamental frequency component, as shown in Fig.3.3 (the background curve is due to
numerical inaccuracy). If a multi-tone input 
is applied (the input
signal has two or more frequency components where A, B can be complex numbers to
accommodate the different phase shifts of these two waveforms), then in addition to the

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Identification of Nonlinearity Using Higher-order 
71
fundamental frequencies 
and their harmonics 
there will also be
combinational frequency components 
etc.) as shown in Fig.3.4. In fact,
for this specific system described by equation 
there will be frequency components
present in the response x(t) for all integer values of and 
In order to
establish an input/output model of a nonlinear system which can not only predict the
fundamental frequency, but also the harmonics and combinational frequencies as well, the
Volterra series theory of nonlinear systems was developed.