Identification of the dynamic characteristics of nonlinear structures

N
N
+ 
 + 
for j = 1, N
(6-23)
r = l
r = l
where modal parameters with
are those of the experimental system and modal
parameters without are those of the updated analytical system. Since equation (6-23) is
valid for any then mathematically, we have

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6 Identification of Mathematical Model of Dynamic 
201
a n d
f o r r = l , n
(6-24)
It is not difficult to see that (6-24) is equivalent to following equations
= 
= 
= 
(6-25)
From 
it is clear that the thus-updated analytical model is unique and is the true
solution of the problem because the model reproduces all the eigenvalues and eigenvectors
of the experimental system. Also, it is clear that the number of modes measured is
irrelevant during the development of the above argument and, therefore, the uniqueness of
the updating problem when measured coordinates are complete is independent of the
number of modes measured.
On the other hand, when the measured coordinates are 
incomplete, 
a unique solution can
also be obtained as shown in numerical case studies when the physical connectivity of the
analytical model is employed. This process is intuitively explained as below. From an
information point of view, although the measured coordinates are incomplete, every piece
of measured data contains information about all the modifications (the difference between
the analytical and experimental models) and since the measured data are plentiful in terms
of frequency points and the physical connectivity of the analytical model is imposed as
constraints, the updating problem can, in most cases, still be turned into an
overdetermined one and thus be solved uniquely.
Mathematically, when the measured coordinates are incomplete, the physical connectivity
of the analytical model becomes essential in order to have a unique solution. Suppose one
incomplete column 
of the receptance matrix has been measured (for simplicity,
assume coordinates 
to 
have been measured), then, after the model has been
updated, the
values and their updated analytical counterparts 
should
satisfy 
that is