|
.
.
.
.
(6-8d)
(6-28)
After some mathematical manipulation, equation (6-28) becomes
k=l
k=l
(6-29)
Identification of Mathematical Model of Dynamic Structures
2 0 4
By choosing
such that the maximum magnitude of the elements of the matrix
(k=l,j) is unity, the accuracy of the solution could in some cases be
improved.
6.8 CONCLUSIONS
In this Chapter, a new model updating method has been developed based on the
correlation between the analytical model and measured frequency response function data.
It has been mathematically demonstrated that existing methods based on the correlation
between analytical and measured modal parameters are discrete versions of the new
generalised method presented here and since only the frequency points which are the
natural frequencies of the structure are used in those methods (these points are, in effect,
badly chosen since they contain no information about other modes of the structure), the
classical updating problem in most cases are underdefined. However, as shown in
numerical examples, because of the direct use of measured response function data, the
residual terms involved in the data are taken into account in the new method and the
measured data are always of plentiful in terms of frequency points, and so, the present
method turns the updating problem into an overdetermined one in most cases.
Considering the practical difficulty of measuring FRF data at all the coordinates which are
specified in the analytical model, the present method has been extended to the case where
measured coordinates are incomplete. Mathematically, in these circumstances, the method
is based on a certain form of matrix perturbation analysis and, therefore, an iteration
scheme has to be introduced during the updating process. Some computational aspects
involved in the eigensolution during this iteration have also been discussed so that
computational cost can be reduced.
As for the uniqueness of the updating problem, it is often asked how many modes are
required in order to get a unique (true) solution? Of course, the number of modes required
depends on which modes are chosen and what characteristics the structure possesses if
the updating problem is to be solved based on measured modal data as will be discussed
in next Chapter. However, based on the new method developed in this Chapter, it has
been mathematically proven that in the case where measured coordinates are complete, the
unique (true) solution always exists regardless of the number of modes measured. On the
other hand, when the measured coordinates are incomplete, a unique solution can be
obtained in most cases even when few modes are measured if the physical connectivity of
the analytical model is employed.
6 Identification of Mathematical Model of Dynamic Structures
205
Although experimental modal analysis has been highly developed, there still exist some
problems. In the case where two modes are very close, accurate modal parameters can be
difficult to obtain. Furthermore, measured modes are usually complex as discussed in
Chapter 2 and although some investigation has been undertaken to devise ways of
employing complex modes directly in the correlation
most of the correlation
methods are based on the use of real modes. This means that the measured modes have to
be realised first before they are used. The realisation process not only introduces
numerical errors but at the same time, discards the damping information about the
structure. These problems do not exist in the present generalised FRF method and, as a
by-product of the updating process, the method reveals the damping information about the
structure (damping model).
The method has been extended to the case where the structure under investigation is
nonlinear. It has been shown that in this case, in order to establish the mathematical
model, an accurate linear model as well as the location information of the nonlinearity are,
in general, necessary because the model has to be established based on mode by mode
basis. On the other hand, because of the inconsistency of measured data, it is essential to
use measured FRF data in the correlation, rather than the measured modal data.
Numerical case studies based on an
mass-spring system as well as a 57DOF model
of a frame structure are carried out to assess the practical applicability of the new method
presented in this Chapter. Cases where measured coordinates are both complete and
incomplete are considered and the results have shown marked advantages over the the
existing methods based on the use of measured modal data. Since the method is
developed for the practical case in which both measured modes and coordinates are
incomplete, it is readily applicable to practical correlation analysis.
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