Identification of the dynamic characteristics of nonlinear structures


7.3.4 THE UPDATING OF AN ANALYTICAL MODEL



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Dynamic characteristics of non-linear system.

221
7.3.4 THE UPDATING OF AN ANALYTICAL MODEL
Now, suppose the 
modelling errors are successfully located, what to do next?
Obviously, the next step will be to update the analytical model since this is the ultimate
target which is sought, but how? As mentioned in Chapter 6, the only way of updating an
analytical model when the measured coordinates are incomplete, as in this case, is by
some kind of perturbation or sensitivity analysis based on an iterative strategy. However,
in this case, since the physical connectivities have been destroyed during the
condensation process, every element of the mass (if mass modelling errors are
considered) and stiffness matrices of the condensed analytical model in those located
areas should be considered as an independent unknown in the updating process. Again,
from the linear algebraic equation point of view, even for this simple numerical case,
such a task as to update the analytical model will be beyond the capability of practical
measurement (at least, as in the GARTEUR exercise, in which only 5 measured modes
and a third coordinates (half the coordinates in 
of the condensed analytical model)
are given, it is impossible to solve the problem addressed). On the other hand, although
the major errors have been located and so considered in the updating process, the
influence of those ‘small’ errors (which are not considered in the updating process) on
the solution process probably cannot be simply overlooked because they are spread over
the whole matrix.
7.4 
STRUCTURAL CONNECTIVITY IN AN ANALYTICAL MODEL
So far, the mathematical difficulties involved in the full matrix updating process have
been demonstrated and the limitations of updating an analytical model when it is in the
form have been discussed. It becomes clear that in order to update an
analytical model using limited measured data, it is essential to preserve the structural
connectivity of the analytical model. As far as the stiffness properties are concerned, this
suggests that any stiffness modelling errors which exist can only occur in the non-zero
elements of the analytical stiffness matrix [K,]. The mathematical validity of such
physical constraints imposed in the updating process, which are essential, is discussed
below.
It is well known that practical structures are continuous and possess an infinite number of
degrees of freedom while the mathematical models sought to represent their dynamic
characteristics have a finite number of degrees of freedom. As shown before, this
discretisation modelling process is mathematically a dynamic condensation process.


7
Possibilities and Limitations of Analytical Model Improvement
2 2 2
Suppose there exist unique mass and stiffness matrices 
where 
which exactly describe the dynamic characteristics of the test structure, then according to
dynamic condensation theory, the mathematical model 
and 
with
with finite degrees of freedom which can exactly represent the test structure in
terms of all these retained coordinates is not unique (the mass matrix is a function of
frequency as can be seen from (7-13a)). On the other hand, during the condensation
process, the physical connectivity of [M]
(which could themselves be
fully populated matrices) has been destroyed, and in general, 
and 
become fully populated. In the problem under consideration, 
are
unknown and it is desired to identify constant 
and 
which can best
represent the test structure. It is apparent therefore that such a constant coefficient model
can only be an approximation with limits on the frequency range of applicability.
Although the exact model of the structure to be identified 
and 
is
fully populated and the mass matrix is, in theory, a function of frequency, experience of
Finite Element Analysis shows that as far as the lower modes of the structure, which are
usually of practical interests, are concerned, a model with constant mass and stiffness
matrices which preserves the physical connectivity can, in general, accurately describe the
test structure. Also, as shown in the following numerical calculation, a fully populated
model is not necessarily a better representation of the structure than a model with physical
connectivity preserved.
To illustrate the above arguments, a free-free beam structure as shown in Fig.7.15 is
considered. Suppose the ‘true’ model of the structure is the FE model formulated by
discretising the beam into 50 elements and consider 2 
at each node: this yields
 
A reduced model of 
corresponding to 52
‘measured’ coordinates (hatched nodes) is to be identified which can best represent the
structure. One choice of such a model is, naturally, the Guyan-reduced model with these
measured coordinates considered as master degrees of freedom. A thus-obtained model is
fully populated. Another choice of a suitable model is the FE representation of the
structure composed of 25 elements with all the ‘measured’ coordinates being considered.
This model preserves the physical connectivity (and is heavily banded). The question to
be answered is: which model represents the structure better? In order to make this
comparison, the first 10 natural frequencies of the ‘true’ model, the Guyan-reduced
model and the model with 25 elements are tabulated in Table 7.1 from which it can be
seen that although the Guyan-reduced model and the model with 25 elements are quite
different in terms of their spatial forms (one is fully populated and another is heavily
banded), the first 10 modes are almost the same (and, indeed, for all 52 modes, the


 Possibilities and Limitations of 
Model 
2 2 3
eigenvalues are the same up to 14 digits when double precision computation is
considered). In addition, the eigenvectors of the true model and the model with 25
elements are the same (up to 14 digits) for all the corresponding modes and coordinates.
The MAC values of the corresponding eigenvectors of the ‘true’ model and the 
reduced model and of the true model and the model with 25 elements are shown in
Fig.7.16. This suggests that it is possible for a constant coefficient model to represent the
structure accurately while at the same time preserving its physical connectivity, a feature
which is essential from the point of view of analytical model updating.
 
element of exact model
A
element of reduced model
S = 0.004 
I = 0.001 
E = 5.0 x 
= 2800 
Fig.7.15 Free-free Beam Structure
mode no.
1
2
3
4
39.0937
107.763
211.259
349.224
exact natural frequency of the structure (Hz)
natural frequency of the 
model (Hz)
natural frequency of the model with 25 elements (Hz)
5
521.683
521.759
521.759
1901.57
1905.02
1905.02
Table 7.1 Comparison of Natural Frequencies of Three Different Models


7
Possibilities and Limitations of Analytical Model Improvement
2 2 4
10
20
30
40
50
Mode Number
‘true’ and 
element models, . . 
‘true’ and 
 
MAC values indicating the correlation of corresponding eigenvectors
7.5
MINIMUM DATA REQUIRED TO UPDATE AN
ANALYTICAL MODEL
So far, it has been established that in order to update an analytical model using limited
measured data, it is essential that the physical connectivity of the analytical model should
be preserved. The mathematical validity of preserving such physical connectivity has
been demonstrated. With this connectivity information available, it will be shown that
although the measured data may be quite limited in practice, the updating problem can
become overdetermined in most cases. In what follows, the criteria for the minimal
measured data required to solve an updating problem uniquely are discussed.
Although many different methods have been developed to update an analytical model
using measured data, there are no clear rules concerning just how much data should be
measured (how many modes and coordinates) in order to solve the updating problem
uniquely. Such a criterion is important not only because modal testing is costly, but more
importantly because it is required for the analyst to judge whether an available set of
modal data is enough to obtain a unique solution. In the following, a model updating
method based on eigendynamic properties (Eigendynamic Constraint Method ECM) is
developed and, based on this method, a criterion of how many modes should be
measured in order to have a unique solution of the updating problem is established for the
case when the measured coordinates are complete. When the measured coordinates are

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