Identification of the dynamic characteristics of nonlinear structures

incomplete 
but the modelling errors are 
in the measured coordinates, a direct
solution of the updating problem is still possible based on the Eigendynamic Constraint

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7
Possibilities and Limitations of Analytical Model Improvement
2 2 5
Method. In general, however, when the measured coordinates are incomplete, a direct
solution is not possible and some kind of perturbation or sensitivity analysis based on an
iterative scheme has to be sought. In this case, an Inverse Eigensensitivity Analysis
Method is employed to establish the criterion of how many modes and coordinates should
be measured in order to solve the updating problem.
7.5.1 
THE EIGENDYNAMIC CONSTRAINT METHOD
The problem of reconstructing system matrices from identified eigenvalues and
eigenvectors has been considered by some authors 
Assuming a certain form of
the modification matrix such as diagonal matrix, and using all the identified eigenvalues
of the modified system, the modification matrix can be calculated 
Applications
have been found for such studies in the solution of inverse Stum-Liouville problems and
nuclear spectroscopy 
In reference 
introduced and extended the
theory of 
to the identification of vibrational systems by using both measured
eigenvalues and eigenvectors to reconstruct the mass and stiffness matrices. However,
his analysis is restricted to his specific fixed-free mass-spring chain system because
during the development of his arguments, he assumed-that the system’s eigenvalue
problem is in the form of Jacobian matrices 
Based on this simple system, he
established the necessary and sufficient conditions for a given vector to be one of the
eigenvectors of the system and pointed out that it is possible to reconstruct the mass and
stiffness matrices of the system by using two identified modes which satisfy certain
conditions although the thus reconstructed system is not unique in the sense that it can be
scaled by an arbitrary factor. Ibrahim 
later extended Gladwell’s theory into
analytical model updating of dynamic structures. The Eigendynamic Constraint Method
described below is similar to Ibrahim’s method. However, since it makes use of the
mass-normalisation properties of measured modes, the problem of uniqueness of the
identified system is resolved.
The method is formulated based on the eigendynamic equation and the mass
normalisation properties of measured modes as mentioned in 
[AM] 
+ [AK] 
As discussed before, the physical connectivity of the analytical model should be
preserved during the updating process, and so the updated model should have the same

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7
Possibilities and Limitations of Analytical Model Improvement
2 2 6
connectivity as that of the analytical model. When the connectivity information is
employed, (7-7) and (7-8) can be combined and turned into standard linear algebraic
equations in terms of an unknown vector 
consisting of all the design variable
changes in the mass and stiffness matrices in a similar way to that discussed in Chapter 6,
as
(7-15)
where N is the number of degrees of freedom specified in the analytical model, L is the
number of total independent design variables in the mass and stiffness matrices,
and 
are the coefficient matrix and vector formed using the
analytical model and 
measured mode properties.
To 
how the coefficient matrix 
and 
can be obtained, again consider the 2DOF
mass-spring system as mentioned in Chapter 6. When the first mode is used, equation 
7) becomes
+ 
+ 
After further mathematical manipulation, (7-7a) becomes
0
Similarly, (7-8) becomes
(7-7a)
(7-7b)

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7 Possibilities and Limitations of Analytical Model Improvement
2 2 7
Further, (7-8a) can be written as:
(7-8b)
Combining (7-7b) and 
coefficient matrix 
and 
can be obtained.
Equation (7-15) is obtained based on the 
measured mode; when m measured modes
are available, it is not difficult to see that the dimension of the coefficient matrix [A]
becomes 
and 
becomes m(N+l)xl, that is:
 = 
(7-16)
In general, when 
L, equation (7-16) becomes overdetermined and the SVD
technique can be used to solve the unknown vector {P}. After (P} has been calculated,
the updated mass and stiffness matrices can be reconstructed by using the physical
of the analytical model.
7.5.2 INVERSE EIGENSENSITIVITY ANALYSIS
As mentioned previously, when the measured coordinates are incomplete, direct solution
of the updating problem is generally not possible and some kind of perturbation or
sensitivity analysis has to be employed based on an iterative scheme. In this case, in
order to establish a criterion concerning how many modes and coordinates should be
measured in order to solve uniquely the updating problem, the Inverse Eigensensitivity
Analysis method 
can be employed to calculate the design variable changes given
differences between analytical and measured natural frequencies and modeshapes. The
method was first introduced to analytical model improvement by 
et al 
Later,
modified the procedure described in 
by introducing matrix
perturbation to avoid the eigensolution required in every iteration. Lallement 
recently
extended the method to pinpoint where the significant modelling errors are first and then

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7
Possibilities and Limitations of Analytical Model Improvement
2 2 7
Further, 
(7-8a) can be written as:
(7-8b)
Combining (7-7b) and 
coefficient matrix 
and 
can be obtained.
Equation 
(7-15) is obtained based on the 
measured mode; when m measured modes
are available, it is not difficult to see that the dimension of the coefficient matrix [A]
becomes 
and 
becomes m(N+l)xl, that is:
 =
(7-16)
In general, when 
L, 
equation (7-16) becomes overdetermined and the SVD
technique can be used to solve the unknown vector {P}. After (P} has been calculated,
the updated mass and stiffness matrices can be reconstructed by using the physical
of the analytical model.
7.5.2 INVERSE EIGENSENSITIVITY ANALYSIS
As mentioned previously, when the measured coordinates are incomplete, direct solution
of the updating problem is generally not possible and some kind of perturbation or
sensitivity analysis has to be employed based on an iterative scheme. In this case, in
order to establish a criterion concerning how many modes and coordinates should be
measured in order to solve uniquely the updating problem, the Inverse Eigensensitivity
Analysis method 
can be employed to calculate the design variable changes given
differences between analytical and measured natural frequencies and modeshapes. The
method was first introduced to analytical model improvement by 
et al 
Later,
modified the procedure described in 
by introducing matrix
perturbation to avoid the eigensolution required in every iteration. Lallement 
recently
extended the method to pinpoint where the significant modelling errors are first and then

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Possibilities and Limitations of 
Model 
2 2 8
to reduce the number of unknowns to improve the solution condition. Derivation of the
eigenvalue and eigenvector derivatives which are required in the formulation of updating
problem is explained in the Appendix II of this thesis.
From the theory of the algebraic eigenvalue problem, a system’s eigenvalues and
eigenvectors are implicit functions of its design variables. Hence, based on the Taylor
series representation, the relationship between the change of modal parameter (6 can be
the change of any eigenvalue or of any eigenvector element) and the vector of design
variable change (P) can be expressed as