Identification of the dynamic characteristics of nonlinear structures

Derivation of Eigenderivatives
2 6 7
 
Using 
 (z) 
can be written as
 
 
 
 
 
 
 
m
N
 
i 
 
j = m+l
i 
(A2.20)
(A2.21)
where 
( 
i
 
a n
we have assumed that 
If the eigenvalues
r
are numbered according to their magnitude in ascending order, then for the class of
problems with a large frequency gap is
 
for j > m
(A2.22)
It is clear that the above approximation is very accurate for 
Thus (A2.22) can be
approximated as
m
 
N
(A2.23)
j = m+l
which can also be written as
m
N
m
 
 
 
 
 
j
 
 _
j
 
j
j # i
(A2.24)
In equation 
only the middle term is unknown and it can be calculated using the
flexibility matrix as shown below.

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Appendix 
q
Derivation of 
2 6 8
Since from the theory of algebraic eigenvalue problems, the flexibility matrix 
can be
calculated from the system’s eigenvalue and eigenvector matrices as
N
= 
=
 
 
(A2.25)
Using this spectral decomposition of 
the middle term of the RHS of (A2.24)
becomes 
(F) and (A2.24) can be written as
m
m
(A2.26)
 
j
j i
We note that the 
term in (A2.26) approximates the effect of higher uncalculated
modal components on eigenvector derivatives of lower modes. With 
being obtained
and 
as expressed in 
the eigenvector derivatives of 
mode can be simply
calculated based on 

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REFERENCES
Ewins, D. J.
 Testing: Theory and Practice”
Research Studies Press, 1984
 of 
 Dynamic characteristics”
Ph.D thesis, 
Eng. Dept., Imperial College, 1983
Titchmarsh, W. C.
“Introduction to the Theory of Fourier Integrals”
Oxford University Press, 1937
Simon, M and Tomlinson, G. R.
“Use 
of the Hilbert 
 in Modal Analysis of Linear
and Nonlinear Structures”
Journal of Sound and Vibration, 1984
He, J. and Ewins, D. J.
“A simple Method of Interpretation for the Modal Analysis
of Nonlinear Systems”
 
Schetzen, M.
‘The 
Volterra 
 Wiener Theories of Nonlinear Systems”
John Wiley and Sons, 1980
Wiener, N.
“Nonlinear Problems in 
 Theory”
Wiley, 1958

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References
2 7 0
Chouychai, T.
 Behaviour of Nonlinear 
Ph.D thesis (in French), 1986, Sain-Ouen, Paris
Ibrahim, S. R. and Mikucik, E. C.
‘The 
 Determination of Vibration Parameters
from Time Response”
Shock and Vibration Bulletin, Vol. 
1976
Jordan, D. W. and Smith, P.
“Nonlinear Ordinary 
 Equations”
Oxford University Press, 1987