Identification of the dynamic characteristics of nonlinear structures

 
+ 
 
Pi Pj + 
i = l
i = l
(7-17)
where and 
are the first- and second-order sensitivity coefficients. Suppose n out of
N coordinates have been measured for the 
mode then, based on 
to a first order
approximation, we have:
 
(7-18)
 
where 
is the sensitivity matrix for n measured coordinates and one measured
eigenvalue of the 
mode, and can be calculated in a way as illustrated in Appendix II of
this thesis, and 
is the difference vector between the measured and analytical
eigenvector and eigenvalue of the 
mode. If m measured modes are available, (7-18a)
can be rewritten as
m(n+l)xL 
(7-19)
When m(n+l) L, (7-19) becomes a set of 
algebraic equations and
the SVD technique can be used to solve 
Since (7-19) is formulated based on first-
order approximation, the exact solution of 
cannot be obtained directly and an iterative
procedure has to be introduced as illustrated in Fig.7.16. Again, after (P) has been
calculated, the updated model can be reconstructed using the the analytical model.

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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