Identification of the dynamic characteristics of nonlinear structures

Appendix 
 Derivation of 
266
. . . . . . . . . . . . . . . . . . .
=
k
If 
is 
and assigned a specific value, then subvectors
and 
must be
uniquely determined by solving (A2.17) whose coefficient matrix has rank (N-l) when 
is a single mode. This argument also shows that when is 
the 
column of
the 
matrix in (A2.16) is a linear combination of the remaining (N-l) columns and
the remaining (N-l) columns are therefore linearly independent. Also, from a numerical
point of view, 
should be chosen to be the the element with maximum magnitude so
that the coefficient matrix of (A2.17) and so 
is well-conditioned
inverse.
in terms of its
As can be seen from the above discussion, the process of deriving eigenvector derivatives
based on Nelson’s method preserves the sparseness and banded structure, if any, of [G],
which allows for efficient solution of 
remains to find the scalar 
Substitute 
into 
This completes the solution for
based on Nelson’s method. Note that this
method has the desirable properties of preserving the structure of [M] and [K] (allowing
more efficient solution) and requiring the knowledge of only one eigenvalue-eigenvector
pair. Both properties are important in realistic structural problems where [M] and [K] have
very high order.
IMPROVED MODAL METHOD
The improved modal method seeks to calculate the eigenderivatives approximately using
the lowest m 
pairs of eigenvalues and eigenvectors obtained based on partial
eigensolution. The eigenvalue derivatives are related to the modes themselves and they are
expressed in (A2.8). In order to calculate the eigenvector derivatives, rewrite (A2.9) as

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Appendix 
q