Identification of the dynamic characteristics of nonlinear structures

0.016
0.014
0.012
0
10
15
25
 
Relative Norm Changes (stiffness)
Calculated Damping Matrix 
iteration)
1.1 
1.0 
0.9
 0.8
 0.7
 
0 2
4
6
10 12 14 16 18
22
26 28 30
 
Relative Norm Changes (damping)
Fig.6.8 The Iteration Results

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Identification of Mathematical Model of Dynamic 
190
During the calculation, it was found that when the stiffness modelling errors were greater
than 
convergence became a problem (for this specific case). It is therefore
considered that if the modelling errors are localised and there are some ways of locating
them, then the number of unknowns involved in the updating process can be reduced and
hopefully the restrictions imposed by the original assumption (the Euclidean norm of the
modification should be of second order) can be relaxed. To illustrate this point, some
100% stiffness modelling errors are introduced to the same basic system mentioned above
and the modelling errors are located in the ways which have been discussed in some detail
in Chapter 5. After the location, only the unknowns which contain modelling errors are
retained in the updating process. Even when measured 
FRF 
data covering only the first
mode are used, convergence of the results was obtained.
From what is shown in Fig.6.7, a complete eigensolution of the analytical system is
necessary during each iteration. For large practical problems, this could lead to a huge
amount of calculation and so, if possible, this complete eigensolution should be avoided.
This will be discussed in some detail later on in this Chapter.
EXTENSION OF THE METHOD TO THE MODELLING
OF NONLINEAR STRUCTURES
The above developed method can be extended to the case where the structure is nonlinear.
As mentioned before, when a structure to be 
is nonlinear, its mathematical
model has to be established on a mode by mode basis because of the inconsistency of
measured data (even when response control is used to linearise the structure, the FRF data
measured around different resonances could be the data from different 
systems
due the different modeshapes). For this reason, a mathematical model of a nonlinear
structure cannot, in general, be established based on the use of measured modal data
because, even when the nonlinearity location information is given, the data for one mode
are, in most cases, insufficient for the problem to be solved. In order to establish a
mathematical model of a nonlinear structure, it becomes essential that measured frequency
response function data should be used instead of modal data.
It is assumed here that the nonlinearity of the structure is localised, as is usually the case
in practice, and that the linearised frequency response functions corresponding to different
response levels are measured. Also available is an analytical model of the structure which
may be in error for the linear part (corresponding to very low response amplitude). What
needs to be established is a mathematical model of the nonlinear structure corresponding

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