Identification of the dynamic characteristics of nonlinear structures

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6.4.5 SOME CONSIDERATIONS ON THE COMPUTATIONAL ASPECTS
As shown in the numerical case studies, complete eigensolution of the updated system is
required during each iteration when the measured coordinates are incomplete. Although
computation is becoming cheaper as more powerful computers being produced at lower
cost, this complete eigensolution is often computationally expensive, especially when
systems with big dimensions are considered. It is therefore necessary to discuss some
computational aspects of the eigenvalue problem so that computational effort involved can
be minimised.
If all the eigenvalues and eigenvectors of a matrix are of interest (the complete
eigensolution), the LR and QR algorithms 
which are the most effective of known
methods for the general algebraic eigenvalue problem, can be used. Both methods use a
reduction of the general matrix to triangular form by similarity transforms, but the
reduction is achieved by non-unitary transform in the LR algorithm while it is achieved by
unitary transform in the QR algorithm which is numerically more stable. On the other
hand, if only some of the eigenvalues and their corresponding eigenvectors are of interest,
iterative methods, which are often referred to as 
iteration because only a subset
of the whole eigensolution is of interest, can be employed. In fact, in practical Finite
Element analysis, it is rare for all the modes of the system to be calculated because, in
general, only the lower modes of the system are of interest or, even, valid. The
computational cost of solving the eigenvalue problem is, in general, proportional to the
number of modes which are required. The most effective algorithm used for partial
eigensolution is the Inverse Iteration method 
In the following, it will be shown how
Inverse Iteration method can be used effectively to reduce the computational effort
involved in the eigensolution.
Let the system to be solved be described by matix [A]=[M] 
and suppose
that only the first p modes are of interest. Then, for a given matrix 
with
orthonormal columns (initial estimate of the required eigenvectors), the following Inverse
Iteration generates a sequence of matrices ( 
which will converge
to the first p eigenvectors of interest
For k = 1, 2, . . .
= 
of 
to calculate 
for next iteration)
(6-17)

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Identification of Mathematical Model of Dynamic Structures
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After the eigenvectors 
(which is the converged 
are calculated, the eigenvalues
[‘h.] can be found easily based on the 
Quotient formulation:
= 
(6-18)
The convergence rate of the sequence of 
as shown in 
is proportional to the
ratio 
It should be noted that during the iteration process, only one complex
inverse is required, that being 
and in the case of a free-free system in which
[K] is singular, a shift becomes necessary so that 
in (6-17) becomes 
It is easy to prove 
that systems described by [A] and 
have the same set of
eigenvectors and the 
eigenvalues simply differ by a value of Also, in the
specific case of the Inverse Iteration method where only one eigenvalue and eigenvector
of a system are of interest, should be so chosen such that it is the closest to the
eigenvalue of interest 
In the model updating process, since [AM], [AK] and 
are usually small in the sense
of the Frobenius norm when compared with the original mass [M] and stiffness [K]
matrices, the eigenvalue problem is actually reduced to the problem of dynamic reanalysis
(often referred to as structural modification analysis) and the initial estimation of the
eigenvectors 
can be very accurately chosen as the eigenvectors of the original
system. Due to this accurate choice of initial conditions, it can be expected that the
convergence will in general be very fast.
All this means that if only a partial eigensolution is required, the Inverse Iteration method
is ideal for solving the eigenvalue problem. As shown in 
the whole updated
analytical receptance matrix is required in each iteration of the updating process and,
theoretically, in order to calculate this receptance matrix, all modes should be available.
However, experience shows that the receptance in the lower frequency range can be
accurately approximated by using just a few of the lower modes and by considering the
contribution of higher modes as constant residual terms. For example, for a 
sized practical 
system, if the receptances are of interest only up to the frequency
of the fifth mode, then the estimation of 
based on following equation should be
very accurate when the first 20 modes are included
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