Identification of the dynamic characteristics of nonlinear structures

 
+ 
+
 
(6-19)

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Identification of Mathematical Model of 
Structures
198
Here 
is a constant residual term representing the contribution of higher modes which
can be determined as follows by using the 
which has been calculated before
= 
= 
+ 
(6-20)
where 
is the receptance matrix corresponding to the contribution of the calculated
modes.
It has been shown that during model updating process, the problem of an eigensolution
being involved in each iteration is actually reduced to the problem of dynamic reanalysis.
Hence, performing the complete eigensolution 
 
is unnecessary. On the other
hand, due to the fact that the contribution of the higher modes to the receptances in the
lower frequency range decreases quadratically in terms of frequency separation, only a
partial eigensolution is necessary in order to calculate accurately the receptance data
needed in the model updating process. These two observations make the Inverse Iteration
method as discussed above the most appropriate method for the specific problem
addressed.
6.5 
GENERALISATION OF MODEL UPDATING METHODS
A model updating method based on the correlation between analytical model and
measured frequency response functions has been developed and it will be advantageous to
examine the relationship between the present method and the existing methods which are
based on the correlation between analytical and measured modal data. It can be 
that model updating methods can be generalised if measured FRF data are used instead of
the modal data. This argument is self-evident because modal data are effectively derived
from measured FRF data at the resonance frequencies but a brief mathematical proof is
given here.
When the measured coordinates are complete, the present method of updating is
formulated based on equation (6-6). Since model updating methods based on the use of
modal data rely only on the data points where 
(at the natural frequencies), it is
interesting to know to what equation (6-6) will degenerate when 
Unfortunately,
in (6-6) is not defined when 
(the damping of the experimental model is
made to be zero in order to permit a comparison with the methods based on modal data
which are supposed to be from an undamped system), therefore what needs to be

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6
Identification of Mathematical Model of Dynamic Structures
199
discussed is what will equation (6-6) degenerate to as 
and the damping matrix
It is not difficult to see that when 
can be expressed 
as
(6-21)
where 
is due to the influence of the other modes and can be considered as a constant
vector when there is slight change in frequency and change in damping. However, when
the damping matrix 
the coefficient 
Therefore, when 
equation (6-6) becomes
(6-22a)
 
(6-22b)
From equation 
it can be seen that equation (6-6) degenerates to the simple
eigendynamic equation when 
and since all the model updating methods using
modal data are derived from the basic equation 
it can be concluded that, in a
broad sense, model updating methods based on modal data are discrete versions of the
present generalised method based on frequency response data.
When the measured coordinates are incomplete, as mentioned earlier, the present method
is derived from some sort of perturbation analysis and again, this perturbation analysis in
the domain of frequency response functions can, in a broad sense, be regarded as a
generalised version of the perturbation analysis in the modal domain on which some of
the methods, such as the inverse sensitivity analysis as will be discussed in detail in
Chapter 7, are based.
As discussed above, model updating methods based on modal data are discrete versions
of the present generalised method because modal data are virtually the data of frequency
response functions at resonance frequencies. Furthermore, in the case of model updating
using measured FRF data, each data point contains information from