Identification of the dynamic characteristics of nonlinear structures

3.4 CORRELATION ANALYSIS USING RANDOM INPUT
The correlation method for the measurement of frequency response functions using
random input has been widely used in structural modal testing because of its convenience.
In the study of nonlinear structures, as discussed in Chapter 2, first-order frequency
response functions (first-order Wiener kernel transforms) can be measured using random
excitation. Corresponding to different excitation levels (or input power spectra), the thus
measured first-order frequency response functions of a nonlinear system are in general
different and, therefore, the existence of nonlinearity can be detected by comparing FRF
data measured at different excitation levels. The theoretical aspects of this first-order
frequency response function analysis based on random input are given in refs. 
Nevertheless, anything beyond the detection in the identification of nonlinear systems by
application of the thus measured first-order frequency response functions is difficult.
However, in addition to the auto- and cross-correlation analysis which is used in the
calculation of first-order FRF, if we can do higher-order correlation analysis, then as in
the case of the Volterra kernel measurement, there is a systematic way of 
a
nonlinear system by measuring its higher-order Wiener kernels using random input. The
theory behind this practice is the Wiener series of nonlinear systems.
3.4.1 THE WIENER SERIES
In the Wiener theory of nonlinear systems, if the input f(t) is a white Gaussian time series
with autocorrelation function 
then the output x(t) of a nonlinear system can
be expressed by the orthogonal expression:
x(t) =
f(t) 
(3-60)
n = l
in which
is the set of Wiener kernels of the nonlinear system which,
like the set of Volterra kernels
serve to describe the system and 
is a complete set of orthogonal 
For a linear system, all the higher-order
kernels except 
and 
are zero. Unlike the 
Volterra functional, which is
homogeneous and defined as

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Identification of Nonlinearity Using Higher-order