Extremum properties of the generalized Rayleigh quotient associated with flutter instability

Huseyin, K.; Plaut, R. H.

doi:10.1090/qam/430427

Authors: K. Huseyin and R. H. Plaut
Journal: Quart. Appl. Math. 32 (1974), 189-201
MSC: Primary 34C99; Secondary 49G99
DOI: https://doi.org/10.1090/qam/430427
MathSciNet review: 430427
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The extremum properties of the generalized Rayleigh quotient related to flutter instability are investigated. It is shown that, in addition to the well-known stationary property, under certain circumstances the quotient exhibits maximum-minimum properties which are in contrast to those of the classical Rayleigh quotient. One consequence is that an approximate method of stability analysis using these results leads to a lower bound as opposed to an upper bound in the classical case. The results are applied to multiple-parameter systems and a physical interpretation is given for the generalized Rayleigh quotient, leading to the proof of a convexity theorem.

John William Strutt Rayleigh Baron, The Theory of Sound, Dover Publications, New York, N. Y., 1945. 2d ed. MR 0016009
F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
S. H. Gould, Variational methods for eigenvalue problems. An introduction to the methods of Rayleigh, Ritz, Weinstein, and Aronszajn, Mathematical Expositions, No. 10, University of Toronto Press, Toronto, 1957. MR 0087019
A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, Arch. Rational Mech. Anal. 1 (1957), no. 1, 233–241. MR 1553461, DOI https://doi.org/10.1007/BF00298007
Peter Lancaster, Lambda-matrices and vibrating systems, International Series of Monographs in Pure and Applied Mathematics, Vol. 94, Pergamon Press, Oxford-New York-Paris, 1966. MR 0210345

K. Huseyin and R. H. Plaut, The elastic stability of two-parameter nonconservative systems, J. Appl. Mech. 40, 175–180 (1973)

R. H. Plaut, Determining the nature of instability in nonconservative problems, AIAA J. 10, 967–968 (1972)

P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math. 6 (1964), 377–387. MR 171375, DOI https://doi.org/10.1007/BF01386087
R. H. Plaut and K. Huseyin, Derivatives of eigenvalues and eigenvectors in non-self-adjoint systems, AIAA J. 11 (1973), 250–251. MR 320024, DOI https://doi.org/10.2514/3.6740

P. F. Papkovich, Works on the structural mechanics of ships, vol. 4, Moscow, 1963 (in Russian)

J. D. Renton, Buckling of frames composed of thin-wall members, in A. H. Chilver, editor, Thin-walled structures, Wiley, New York, 1967, pp. 1–59

K. Huseyin and J. Roorda, The loading-frequency relationship in multiple eigenvalue problems, Trans. ASME Ser. E. J. Appl. Mech. 38 (1971), 1007–1011. MR 292352
K. Huseyin and H. H. E. Leipholz, Divergence instability of multiple-parameter circulatory systems, Quart. Appl. Math. 31 (1973/74), 185–197. MR 436698, DOI https://doi.org/10.1090/S0033-569X-1973-0436698-0

F. Buckens, Über Eigenwertscharen, Öst. Ing. Arch. 12, 82–93 (1958)