This paper considers the calculation of eigenvalue and eigenvector derivatives when the eigenvalues are repeated. An extension to Nelson’s method is used to calculate the first order derivatives of eigenvectors when the derivatives of the associated eigenvalues are also equal. The continuity of the eigenvalues and eigenvectors is discussed, and the discontinuities in the eigenvectors, when they are regarded as functions of two or more design parameters, is demonstrated. The validity of Taylor series for the eigenvalues and eigenvectors is examined and the use of these series critically assessed.

1.
J. E.
, and
Friswell
M. I.
,
1993
, “
Model Updating in Structural Dynamics: A Survey
,”
Journal of Sound and Vibration
, Vol.
167
, No.
2
, pp.
347
375
.
2.
H. M.
, and
Haftka
R. T.
,
1986
, “
Sensitivity Analysis of Discrete Structural Systems
,”
AIAA Journal
, Vol.
24
, No.
5
, pp.
823
832
.
3.
Fox
R. L.
, and
Kapoor
M. P.
,
1968
, “
Rates of Change of Eigenvalues and Eigenvectors
,”
AIAA Journal
, Vol.
6
, No.
12
, pp.
2426
2429
.
4.
Lallement, G., and Kosanek, J., 1993, “Parametric Correction of Self Adjoint Finite Element Models in the Presence of Multiple Eigenvalues,” Proceedings of Modern Practice in Stress and Vibration Analysis, Sheffield, England, April, 593–603.
5.
Juang
J.-N.
,
Ghaemmaghami
P.
, and
Lim
K. B.
,
1989
, “
Eigenvalue and Eigenvector Derivatives of a Nondefective Matrix
,”
Journal of Guidance, Control, and Dynamics
, Vol.
12
, No.
4
, pp.
480
486
.
6.
Bernard
M. L.
, and
Bronowicki
A. J.
,
1994
, “
Modal Expansion Method for Eigensensitivity with Repeated Roots
,”
AIAA Journal
, Vol.
32
, No.
7
, pp.
1500
1506
.
7.
Akgu¨in
M. A.
,
1994
, “
New Family of Modal Methods for Calculating Eigenvector Derivatives
,”
AIAA Journal
, Vol.
32
, No.
2
, pp.
379
386
.
8.
Lim
K. B.
,
Juang
J.-N.
, and
Ghaemmaghami
P.
,
1989
, “
Eigenvector Derivatives of Repeated Eigenvalues using Singular Value Decomposition
,”
Journal of Guidance, Control, and Dynamics
, Vol.
12
, No.
2
, pp.
282
283
.
9.
Nelson
R. B.
,
1976
, “
Simplified Calculation of Eigenvector Derivatives
,”
AIAA Journal
, Vol.
14
, pp.
1201
1205
.
10.
Ojalvo
I. U.
,
1987
, “
Efficient Computation of Mode-Shape Derivatives for Large Dynamic Systems
,”
AIAA Journal
, Vol.
25
, No.
10
, pp.
1386
1390
.
11.
Mills-Curran
W. C.
,
1988
, “
Calculation of Eigenvector Derivatives for Structures with Repeated Eigenvalues
,”
AIAA Journal
, Vol.
26
, No.
7
, pp.
867
871
.
12.
Mills-Curran
W. C.
,
1990
, “
Comment on ‘Eigenvector Derivatives with Repeated Eigenvalues’
,”
AIAA Journal
, Vol.
28
, No.
10
, pp.
1846
1846
.
13.
Dailey
R. L.
,
1989
, “
Eigenvector Derivatives with Repeated Eigenvalues
,”
AIAA Journal
, Vol.
27
, No.
4
, pp.
486
491
.
14.
Wilkinson, J., 1965, The Algebraic Eigenvalue Problem, Oxford University Press.
15.
Haftka, R. T., Gu¨rdal, Z., and Kamat, M. P., 1990, Elements of Structural Optimisation, Second Edition, Kluwer Academic Publishers.
16.
Hou
G. J. W.
, and
Kenny
S. P.
,
1992
, “
Eigenvalue and Eigenvector Approximate Analysis for Repeated Eigenvalue Problems
,”
AIAA Journal
, Vol.
30
, No.
9
, pp.
2317
2324
.
17.
Pesˇek, L., 1994, “The Updating of Mechanical Models with Repeated Eigenvalues by the Inverse Sensitivity Method,” Proceedings of the 19th International Seminar on Modal Analysis, Leuven, Belgium, September, pp. 305–314.
18.
Chen
T.-U.
,
1993
, “
Design Sensitivity Analysis for Repeated Eigenvaues in Structural Design
,”
AIAA Journal
, Vol.
31
, No.
12
, pp.
2347
2350
.
19.
Friswell, M. I., Garvey, S. D., and Penny, J. E. T., 1993, “Finite Element Model Updating of Rotor-Bearing Systems,” Conference on Modern Practice in Stress and Vibration Analysis, Sheffield, UK, April, pp. 557–568, Sheffield Academic Press.
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