Identification of the dynamic characteristics of nonlinear structures

MODEL UPDATING USING FRF DATA A NEW
GENERALISED METHOD
6.4.1 DESCRIPTION OF THE METHOD
The development of the model updating method described here is based on the following
mathematical identity:
[[A] + [B]] 
[A] [[A] + [B]] [B] [A] 
(6-l)

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6
Identification of Mathematical Model of Dynamic Structures
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where [A] and 
are two complex matrices satisfying the condition that both [A] and
([A]+[B]) are nonsingular. To check the validity of equation 
premultiply both sides
of (6-l) by ([A]+[B]), since it is nonsingular, so that (6-l) becomes:
Now, if we assume that [A] is the impedance matrix of the analytical system [Z,(o)] and
that ([A]+[B]) is the impedance matrix of the experimental system 
then equation
1) becomes:
= 
(6-3)
Rewriting (6-3) in its more familiar receptance and impedance error form, we have
= 
=
(6-4)
where [AZ(o)] is the impedance error matrix defined as 
In
what follows, it will be shown how equation (6-4) can be used to solve the updating
problem uniquely when one complete column of the receptance matrix is measured and
how it can be extended to cases where the measured coordinates are incomplete.
In the case where one complete column (the 
has been measured, then (6-4) can be
rewritten in terms of measured and analytical receptance terms as
T
1
 
this formula being obtained by simply taking the 
row of both sides of equation (6-4).
In analytical model updating, the physical connectivity of the analytical model should
usually be respected and therefore the updated model should have the same physical
connectivity as that of the analytical model, i.e. the modelling errors can only occur where
the elements of mass or stiffness matrix are 
Further discussions on the necessity
and mathematical validity of preserving the physical connectivity of the analytical model
will be given in Chapter 7. Also, in general, the physical connectivity of the damping
matrix can be assumed to be the same as that of the stiffness. Upon substitution of
[AZ(o)] in (6-S) in terms of [AM], [AK] and 
and transpose, equation (6-5) becomes:
+ 
+ 

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6 Identification of Mathematical Model of Dynamic Structures
180
Consider the physical connectivity of the analytical model and let the design variable
changes in the mass matrix be 
where 
is the total number of independent
design variables in mass matrix which could be individual 
elements, as in the
case of mass spring systems, or coefficients of element mass matrices as in finite element
models), in the stiffness matrix be Ak, (s=l, 
and in the damping matrix to be (s=l,
Then, it can be shown that every element 
where N is the number of
total degrees of freedom specified in the analytical model) of the RHS of 
which is a
vector, can be expressed as a linear combination of the changes of all design variables as: