Identification of the dynamic characteristics of nonlinear structures

3
Identification of Nonlinearity Using Higher-order 
101
r
Measured Second Order Wiener kernel of the SDOF Nonlinear System
(Modulus Linear Scale, x-axis 
0 
1, y-axis 
0 
3.5.2 
REMOVAL OF LINEAR CONTRIBUTION
It has been demonstrated in the numerical case studies that in order to calculate 
order (second-order) Wiener kernel transforms of a nonlinear system efficiently, removal
of the linear contribution in the response signal x(t) becomes necessary. This is due to the
fact that for nonlinear systems like the SDOF and 3DOF systems in the numerical case
studies the nonlinearities are such that the response component due to the nonlinear terms
(mainly quadratic term in the estimation of second-order Wiener kernels) is usually of
second order when compared with that of the linear contribution in the response x(t). As a
result, although the term expressed in equation (3-76) (due to the linear contribution) for
the estimation of second-order Wiener kernel should mathematically go to zero as
averaging time increases, the time required for this to become valid could be very long
indeed and so to improve the calculation efficiency, it is necessary to remove the linear
contribution 
before the correlation process takes place.
It is suggested here that this removal of the linear contribution from the system response
x(t) can be achieved by performing the averaging process in the frequency domain rather
than in the time domain. The whole procedure is discussed next. Suppose the response
component due to the quadratic and higher even terms of nonlinearity 
be expressed

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3 Identification of Nonlinearity Using Higher-order 
102
as 
(where 
is the DC component which is supposed to be removed
and 
is the linear contribution), then, by replacing 
with 
equation (3-71) can be rewritten as