Identification of the dynamic characteristics of nonlinear structures

mxn
 
 
Applying SVD to [A], we obtain
or
or
(A1.9)
(A1.lO)
(Al.ll)
 
(A1.12)
mxl
(A1.13)
From 
can be easily solved. Having calculated 
the vector 
can be recovered from (A 1.12) as
(A1.14)

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APPENDIX 
DERIVATION OF EIGENDERIVATIVES
Sensitivity analysis, the study of changes in system dynamic characteristics with respect
to parameter variation, is being used in a variety of engineering disciplines ranging from
automatic control theory to the analysis of large-scale physiological systems [ 
Some
of the specific areas where sensitivity analysis has been applied include (i) system
identification 
(ii) development of insensitive control systems 
(iii) use in
gradient-based mathematical programming methods 
(iv) assessment of design
changes on system performance 
and (v) approximation of system response to a
change in a system parameter 1 
In this appendix, the derivation of eigenderivatives
for eigenvalues and eigenvectors with respect to the system’s design parameter changes is
discussed. Eigenderivatives with respect to design variables have been found to be
particularly useful in certain analysis and design applications, e.g., approximating a new
vibration modeshape due to a perturbation in design variable, determining the effect of
design changes on the dynamic behaviour of a structure, and tailoring modeshapes to
minimise displacements at certain points on a structure. Here, the eigenderivatives are
derived for the purpose of analytical model improvement as discussed in Chapter 7.
Methods for calculating eigenderivatives include the modal method 
 
Nelson’s
method 
and an improved modal method 
The modal method is
straightforward in theory but requires all the modes of system which is sometimes
computationally expensive especially when systems with large dimensions considered.
Nelson’s method seeks to calculate the eigenderivatives of the mode by just using the
modal parameters of that mode. However, one matrix inverse of system dimension (in
fact, of dimension (N-l) where N is the dimension of the system) is required for each
mode in order to solve the linear algebraic equations involved. The improved modal

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