Identification of the dynamic characteristics of nonlinear structures

Appendix 
 
 of Eigenderivatives
2 6 5
[G] 
 
(A2.15)
Note that [G] has rank (N-l) (assuming 
is a single eigenvalue) and nullspace 
(nullspace of a matrix [A] is defined as the set of vectors (x) such that [A] (x]=(O)).
This means that for any 
satisfying [G] 
is also a solution, where
is any real number. The approach is first to find some particular solution 
then to
find so that 
A suitable 
can be found by setting one element of
to be zero (since [G] is rank 1 deficient) and solving the remaining elements. This
will work for any (f} in the range of [G] (the range of matrix [A] is defined as the set of
vectors 
y 
such that 
for some {x} and the necessary and sufficient
condition for the existence of exact solution(s) of [A] {x)=(y) is that 
( y 
is in the range
of [A]) as long as the corresponding element of 
 is 
This suggests the
following algorithm:
3)
Find k such that 
II 
 
 
Delete the 
row and column of 
and 
element of {f)+(J)
and let 
and by removing it from (a) 
Solve the reduced linear algebraic problem 
[G] 
= 
Since [G] is rank 1 deficient and {f) is in the range of [G] (since we know 
exists),
the resulting 
automatically satisfies [G] (o)=(f). The reason for choosing such a k
(the eigenvector element with maximum magnitude) is to make sure that the reduced
coefficient matrix 
[G] 
is rank full and numerically well-conditioned so that 
can be
uniquely and accurately determined. This is illustrated as follows. Defining 
and partitioning the matrices, (A2.1) can be written as
L
. . . . . . . . . . . . . . . . .
where 
are the partitioned matrices of 
the k,k element of 
and 
are the partitioned subvectors, and 
the 
component of 
Multiplying out (A2.16) and moving the 
term to the RHS, (A2.16) becomes

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