Identification of the dynamic characteristics of nonlinear structures

7.5.3 NUMBER OF MODES AND COORDINATES REQUIRED
TO 
UPDATE AN ANALYTICAL MODEL
When the measured coordinates are complete, it can be seen from equation (7-16) that in
order to have a unique solution of the updating problem, the following condition has to
be satisfied:
L
(7-20)
where, as mentioned, m is the number of measured modes, N is the number of degrees
of freedom specified in the analytical model and L is the number of independent design
variables. Therefore the number of modes required (m) in order to solve the updating
problem is 
or the minimum number of modes is 
where E(x)
is a mathematical function such that when it is applied to a real number x, it returns an
integer value which is x E(x)
When the measured coordinates are 
incomplete, 
the Eigendynamic Constraint Method
cannot generally be applied directly. However, in the special case in which all the
modelling errors are localised in the measured coordinates, the unmeasured coordinates
can be interpolated exactly based on the analytical model itself using Kidder’s expansion
method, as discussed before. The thus-expanded modes can then be used in the
Eigendynamic Constraint Method to update the analytical model. For this case, the
criterion for the minimum number of measured modes is therefore the same as is given in
(7-20).
In general, when the measured coordinates are incomplete, there exists a criterion
concerning the number of modes and coordinates which should be measured in order to
update the analytical model uniquely. From 
in order to have a unique solution of
the updating problem, the following criterion has to be satisfied:
m(n+l) L
(7-21)
where n is the number of measured coordinates. Expression (7-21) is the necessary
condition, and as will be shown in the numerical case study, it is also sufficient when
only mass or stiffness modelling errors are considered. When the modelling errors are
present in both the mass and stiffness matrices, the situation becomes more complicated
and it is suggested that in this case an error location procedure be employed in order to

---

7
Possibilities and Limitations of Analytical Model Improvement
231
reduce the number of unknowns in {P) before the calculation can be carried out based on
the Inverse Eigensensitivity Analysis.
7.5.4 NUMERICAL CASE STUDIES
In order to verify numerically the criteria presented above, a mass-spring system shown
in Fig.7.17 is considered. The system has 10 degrees-of-freedom and consists of 21
design variables 
10 mass elements and 11 stiffness elements).
x i
x i
x i
Fig.7.17 A 10 DOF Mass-spring System
When the measured coordinates are complete, according to equation 
This means that for this specific mass-spring system, two measured
modes with complete coordinates are in principle sufficient to identify all these 10 mass
elements and 11 stiffness elements. Numerical results for the identification based on the
Eigendynamic Constraint Method using the first and second ‘measured modes are
shown in Table 7.2 (kg for mass and N/m for stiffness).
Identified
12.00000 11.00000 10.00001 100000.1 100000.1 300000.2 100000.1
100000.0 300000.0 100000.0 100000.0
1OOOOo.c 3oOOOO.c
Table 7.2 Identification Results
Although, in general, the ECM is not directly applicable to the case when the measured
coordinates are incomplete, it has been noted that in the special case when the 

---

7
Possibilities and Limitations of Analytical Model Improvement
2 3 2
errors are 
in the measured coordinates, a direct 
of the updating problem
is still possible because in this case, the unmeasured coordinates can be exactly
interpolated based on the analytical model itself. In this numerical example, the exact
stiffness modelling errors are shown in 
with a 50% stiffness modification in
and 
(figure 7.17). Coordinates 
x x x and 
are supposed to be measured,
thereby including the stiffness errors introduced between coordinates 
and 
Again, the first two modes are used and the identified stiffness error matrix is exact, as
shown in 
(a) exact stiffness error matrix
(b) identified stiffness error matrix
Exact and Identified Stiffness Error Matrix (coordinates complete)
When mass modelling errors are considered, 10 design variables are taken into account in
the Inverse Sensitivity Analysis 
for 10 mass elements). Half the coordinates are
supposed to be measured (all the odd-numbered coordinates). According to 
the
number of modes required in order to update the mass matrix can be calculated as
In the calculation, the first and second ‘measured’ modes are used. The
mass modelling errors are introduced by modifying 
and 
to 50% of their
original values. The exact mass error matrix and the iteration results are illustrated in
and 

---

 and Limitations of Analytical Model Improvement
233
exact error mass matrix
Exact Mass Error Matrix
first iteration estimation
third iteration estimation

---

7 Possibilities and Limitations of Analytical Model Improvement
234
5th iteration estimation
Iteration Results (mass error case)
Similarly, in the case when stiffness modelling errors are considered, 11 design variables
(11 stiffness elements) are taken into account in the Inverse Sensitivity Analysis. Again,
all the odd numbered coordinates are assumed to be measured. According to 
the
number of modes required in order to update the stiffness matrix can be calculated as
m=E( 1 
The first two ‘measured’ modes are used in the calculation. The stiffness
modelling errors are introduced by increasing 
and by 100% of the original
values. The exact stiffness error matrix and the iteration results are shown in 
and 
exact error stiffness matrix
Exact Mass Error Matrix

---

7
Possibilities and Limitations of Analytical Model Improvement
235
first iteration estimation
second iteration estimation
third iteration estimation
Iteration Results (stiffness error case)
GENERALISATION OF THE CRITERION
The criteria concerning how much data should be measured in order to solve the updating
problem uniquely have been developed based on the ECM and the Inverse
Eigensensitivity Analysis method. Since, as discussed in detail in Chapter 6, the new

---

7
Possibilities and Limitations of Analytical 
Model Improvement
236
method developed based on the measured FRF data is a 
version of the model
updating methods based on measured modal data, it is therefore expected that such
criteria can be generalised when measured FRF data are used in the updating process. In
fact, as has been demonstrated in Chapter 6, a unique solution of the updating problem
can always be obtained when measured coordinates are complete regardless of the
number of measured modes, based on the method developed in Chapter 6. In the case
when measured coordinates are incomplete, the method presented in Chapter 6 is based
on a form of perturbation analysis and, as mentioned, such a perturbation analysis based
the use of FRF data can be regarded as a generalisation of the Inverse Eiegensensitivity
Analysis method presented in this Chapter. Therefore, the number of measured modes
required to solve the updating problem using the method presented in Chapter 6 can be
expected to be less than that required by the ECM in the case where measured coordinates
are complete and, the Inverse Eigensensitivity Analysis in the case where measured
coordinates are incomplete.
Also, it is perhaps worth mentioning that in the case when both mass and stiffness
modelling errors exist, the criterion given in (7-21) based on the Inverse Eigensensitivity
Analysis is not sufficient. It has been found that although (7-19) becomes largely
overdetermined (e.g. the number of equations is twice as many as the number of
unknowns), the condition of [S] in terms of its inverse is generally very poor when both
mass and stiffness modelling errors are considered. This is probably because given
modal parameter changes in certain modes might be achieved either by mass modification
or stiffness modification and, as a result, the Inverse Eigensensitivity Analysis becomes
mathematically uncertain. As shown in the numerical case studies of 
this problem
does not exist when FRF data are used in the method developed in Chapter 6.
7.5.6 
APPLICATION OF THE METHODS TO THE
GARTEUR STRUCTURES
As 
mentioned in Chapter 6, when a Finite Element model is considered, the mass and
stiffness error matrices can in general be expressed as: