A conjecture of Renshaw and Mote concerning gyroscopic systems with parameters predicts the eigenvalue locus in the neighborhood of a double-zero eigenvalue. In the present paper, this conjecture is reformulated in the language of generalized eigenvectors, angular splitting, and analytic behavior of eigenvalues. Two counter-examples for systems of dimension two show that the conjecture is not generally true. Finally, splitting or analytic behavior of eigenvalues is characterized in terms of expansion of the eigenvalues in fractional powers of the parameter.

Issue Section:
Brief Notes
Topics:
Dimensions, Eigenvalues, Stability
1.
Baumga¨rtel, H., 1985, Analytic Perturbation Theory for Matrices and Operators, Section 9.3, Birkha¨user Verlag, Basel (OT 15), Switzerland.
2.
Lancaster, P., and Tismenetsky, M., 1985, The Theory of Matrices, Academic Press, Orlando, FL.
3.
Renshaw
A. A.
, and
Mote
C. D.
,
1996
, “
Local Stability of Gyroscopic Systems Near Vanishing Eigenvalues
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
63
, pp.
116
–
120
.
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Copyright © 1999
 by The American Society of Mechanical Engineers
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