Identification of the dynamic characteristics of nonlinear structures

N
N
+ 
 + 
for j = 1, n
(6-26)
 
Equation (6-26) is equivalent to the following equations
a n d
f o r r = l , n
(6-27)

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6 Identification of Mathematical Model of Dynamic Structures
2 0 2
From 
it is clear that the updated model has the same natural frequencies and
damping loss factors as those of the experimental model, but the modeshape matrix is not
completely the same due to incompleteness of the measured coordinates as shown in
Fig.6.15 (only the submatrix of the whole eigenvector matrix corresponding to the
measured coordinates has been determined). Hence, the updated model cannot, in
general, be unique. However, if we impose the physical connectivity of the analytical
model so that the updated model preserves the physical connectivity also, then the
solution, will in most cases, become unique.
NXN
determined eigenvector elements
arbitrary eigenvector elements
Fig.6.15 Illustration of the Determination of Eigenvector Elements
6.7 
BALANCE OF COEFFICIENT MATRICES
As mentioned in 
when FRF data at j different frequency points are used, equations
in the case of complete coordinates and (6-l 1) in the case of incomplete coordinates
become
(6-8c)

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6
Identification of Mathematical Model of Dynamic Structures
2 0 3
 
 
.
 
 
q
(6-lla)
Due to the different magnitudes of 
data at different frequency points (e.g., data
around resonance and anti-resonance frequencies), the RHS coefficient matrices of
equations 
and (6-lla) could in some cases be poorly-conditioned in terms of its
generalised inverse and this ill-conditioning could cause numerical difficulty in the
updating process, especially when FRF data used are contaminated by measurement
errors. Although such ill-conditioning problem can be overcome by properly choosing
frequency points, an alternative numerical technique is presented here.
Consider equation (6%) and rewrite it as
The least-square solution of 
is