Identification of the dynamic characteristics of nonlinear structures

Appendix 
q
Derivation of Eigenderivatives
2 6 2
method aims to derive the eigenderivatives approximately (the eigenderivatives themselves
are only the first-order approximation) by using the calculated lower modes and the
flexibility matrix. In the following, all three methods are discussed and their advantages
and disadvantages in terms of computational cost and numerical accuracy are examined.
However, before discussing the methods in detail, it is necessary to mention that although
discussions have been made on the derivation of eigenderivatives of repeated modes
[ 
only the eigenderivatives of distinct modes are presented in this appendix. On the
other hand, as it will be needed in later discussions, we state here that for any static
structure whose mass and stiffness matrix are symmetric and whose mass matrix 
definite (which is the case of interest here), the complete set of eigenvectors of the system
forms a complete linearly-independent base and so any vector of the same dimension can
be expressed as a linear combination of all these eigenvectors. This argument is briefly
proved as below. The matrix representation of the vibration eigenvalue problem is
Since
(A2.1)
[M] is symmetric and positive-definite, [M] can be decomposed as 
= 
(A2.2)
where [L] is a non-singular lower triangular matrix. Upon substitution (A2.2) into
(A2. 
(A2.3)
Since [L] is non-singular, pre-multiply both sides of (A2.3) by 
so that (A2.3)
becomes
(A2.4)
Let 
and substitute into 
then
(A2.5)
Since 
[K] 
is real and symmetric, the complete set of eigenvectors of 
forms
a complete orthogonal base, regardless of the existence of repeated modes 
On the

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Appendix 
q
Derivation of 
263
other hand, since 
is non-singular, 
= 
[z] forms a complete linearly-
independent base.
THE MODAL METHOD
Differentiating 
with respect to the
design variable