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7
Possibilities and Limitations of Analytical Model Improvement
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extensional stiffness
Y
<
X
bending stiffness
Fig.7.26 Exact Stiffness Error Matrix
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Possibilities and Limitations of Analytical Model Improvement
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-122
l
- 1 4 6 ,
-106
409
410
FREQUENCY HZ
point receptance
161
266 216
408
FREOUENCY HZ
transfer receptance
Fig.7.27
of Analytical and ‘Experimental’ Models
analytical, ‘experimental’
7
and Limitations of Analytical Model Improvement
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Y
X
Y
<
X
extensional stiffness
bending stiffness
Fig.7.28 First Iteration Estimation
7
Possibilities and Limitations of Analytical Model Improvement
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extensional stiffness
Y
<
X
bending stiffness
Fig.7.29 4th Iteration Estimation
7 Possibilities and Limitations of Analytical Model Improvement
246
extensional stiffness
bending stiffness
Fig.7.30 7th Iteration Estimation
7
Possibilities and Limitations of Analytical Model Improvement
247
extensional stiffness
bending stiffness
Fig.7.31 12th Iteration Estimation
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Possibilities and Limitations of Analytical Model Improvement
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7.6 CONCLUSIONS
In this Chapter, the limitations and difficulties of some of the recently-developed methods
based on full matrix updating such as Berman’s method and the Error Matrix method
have been discussed. The mathematical underdetenninacy associated with these methods
is explained. Then, the possibility of updating a condensed (Guyan-reduced) model with
error location based on Kidder’s expansion method is examined. It has been
demonstrated that when measured modes and/or coordinates are incomplete, as they are
in practice, updating of the analytical model based on full matrix updating or the
reduced model with error location is truly difficult, if not impossible. Such a target as to
update an analytical model by considering every element of mass and stiffness matrices to
be (potentially) in error is overambitious and not necessarily appropriate when the
inevitable limitations in measured data are considered.
order to solve the updating
problem, it becomes clear that the physical connectivity of the analytical model should be
respected during the updating process so that the total number of unknowns involved can
be reduced and the limited measured data can have the possibility of solving the problem.
It has been illustrated that the modelling process that is, to obtain a constant coefficient
model of a continuous structure is mathematically a dynamic condensation process and
since the exact model is frequency-dependent (the mass matrix is a function of
frequency), it is apparent that such a constant coefficient model can only be an
approximation with limits on the frequency range of applicability. On the other hand, as
has been demonstrated in the numerical example, it is quite possible for a constant
coefficient model with well preserved physical connectivity (heavily banded) to represent
the structure accurately as far as the lower modes of a structure are of interest (this is a
limitation of any constant coefficient models whether they are fully populated or heavily
banded).
When the physical connectivity of the analytical model is applied, the measured data
required in order to update an analytical model are usually within the capability of
practical measurements. Hence, it is possible to establish the criteria on the minimum
measured data required to solve the updating problem. Such criteria are important not
only because modal testing is costly, but also because they enable the analyst to judge
whether a set of measured data will have the potential to solve the updating problem so
that ‘blind attempts can be avoided.
Possibilities and Limitations of Analytical Model Improvement
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For those cases where measured coordinates are complete, the Eigendynamic Constraint
method has been developed and employed in this Chapter to establish the criterion
concerning the number of modes required to solve the updating problem. As shown in
numerical case studies, this method requires less modes as compared with other existing
model updating methods based on the use of
data. When the measured coordinates
are incomplete, a direct solution of the updating problem is, in general, not possible and
some kind of iterative perturbation or sensitivity analysis has to be devised. In this case,
the Inverse Eigensensitivity Analysis method has been employed to establish the criterion
on the minimum data (measured modes and coordinates) required to update an analytical
model. Furthermore, it has been shown that these criteria can be generalised when
measured FRF data are used based on the method developed in Chapter 6.
The criteria developed have been verified numerically and the methods presented have
been applied to the analytical model updating exercise called ‘GARTEUR’ which is
intended to represent the realistic practical problem in terms of the incompleteness of both
measured modes and coordinates.
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