Identification of the dynamic characteristics of nonlinear structures

Assume 
that 
is normalised such 
that
Multiply both sides of (A2.6) by 
and using (A2.1) and 
then
(A2.7)
(A2.8)
From 
it can be seen that the eigenvalue derivative is determined by the mode
itself. Since as discussed, the complete set of eigenvectors forms a complete linearly-
independent base. Therefore, without any loss of generality, the eigenvector derivative
which is a vector can be expressed as a linear combination of all the eigenvectors of the
system:
(A2.9)
In order to calculate the coefficients 
substitute (A2.9) into (A2.6) and pre-multiply
(A2.6) by 
then
In the case jti, (A2.10) 
 
simplified to solve for 

---

Appendix 
q
Derivation of Eigenderivatives
2 6 4
Pij 
 
When j=i, can be computed from the mass-normalisation condition. Differentiating
(A2.7)
Substitute (A2.9) into 
then becomes
From 
and 
the derivatives of eigenvector are related to all the modes of
the system. Also from (A2.1 it can be seen that the method breaks down when the
eigenvector derivatives of repeated modes are to be calculated because for some j, the
denominator of 
becomes zero.
NELSON’S METHOD
As discussed above, all the eigenvectors of a system are required in order to calculate the
eigenderivatives of one single mode based on the modal method. As mentioned in 
such complete eigensolution can be very expensive when systems of high order are
concerned. In practice, usually only the few lowest modes which are of interest are
available based on a partial eigensolution. In order to cope with such a practical situation,
Nelson developed a method which seeks to derive the eigenderivatives of the mode by
using only those components of the mode itself.
The derivative of the 
is the same as expressed in (A2.8). After the eigenvalue
derivative is calculated, the corresponding eigenvector derivatives can be determined as
shown below. Define 
and [G] as
then equation (A2.6) becomes

---