Identification of the dynamic characteristics of nonlinear structures

REVIEW OF FULL MATRIX UPDATING METHODS
The 
term ‘full matrix updating’ is defined here to mean that all elements in the analytical
mass and/or stiffness matrices are considered to be in error during the updating process
and therefore should be corrected using the measured data if possible. Berman’s method
the Error Matrix Method 
and methods presented in references 
belong
to this category. Since the measured data are limited (both measured modes and
coordinates), it will be shown that it is practically very difficult, if not impossible, to
update the analytical model based on these methods. What these methods provide is an
optimised solution which can reproduce the measured modal data to some accuracy, but
which is not the true solution of the problem.

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7
Possibilities and Limitations of Analytical Model Improvement
2 0 8
7.2.1 
BERMAN‘S METHOD
In reference 
Berman developed a systematic method for improving analytical mass
and stiffness matrices based on measured modal data. In general, the method is devised
for the case where the analytical model is in the 
form in which only the
measured degrees of freedom are retained as masters and, as a result, the analytical model
itself becomes fully populated and the measured coordinates are compatible with the
analytical model. In this case, it will be shown that only when all the modes have been
measured, can exact updating become feasible based on Berman’s method. Otherwise the
solution obtained is an optimised one. A brief summary of the method is introduced in
this section for the convenience of discussion and interested readers are referred to the
original paper 
for details.
During the optimisation procedure of Berman’s method, an ‘improved’ analytical mass
matrix is calculated first and then, based on this ‘improved’ mass matrix, an optimised
stiffness matrix is obtained. Although it is difficult to provide a physical interpretation,
the method is developed based on the minimisation of weighted Euclidean norms of the
differences between the analytical and experimental mass matrices and between the
analytical and experimental stiffness matrices respectively as:
 
(7-l)
II [M,] 
Introducing orthogonality and symmetry properties into the experimental mass and
stiffness matrices as physical constraints in the form of Lagrange multipliers, the final
‘improved’ mass and stiffness matrices can be derived from:
[M,] = 
+ 
= 
+ 
+ 
where 
[K,] 
+ 

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7
Possibilities and Limitations of Analytical Model Improvement
209
From equations (7-3) and 
it can be shown that the thus estimated mass matrix 
and stiffness matrix [K,] can never be the exact mass and stiffness matrices of the
structure unless 
the modes have been measured and this point will be further
explained later on.
7.2.2 
THE ERROR MATRIX METHOD (EMM)
The basic theory of EMM 
can be summarised as shown below. Define stiffness
error matrix [AK] to be 
and assume that the [AK], when compared with
[K,], is of second order in the sense of the Euclidean norm. Then, based on matrix
perturbation theory, [AK] can be expressed, to a first order approximation, as:
(7-5)
Again, 
and 
can be approximated using the m 
where m is the number
of measured modes and N is the number of degrees of freedom specified in the analytical
model) corresponding measured and analytical modes and then substituted into (7-5) to
obtain [AK] as: