Abstract
A circular ring rotates about a diameter in space. Deformations of the ring depend on a non-dimensional parameterJ which represents the relative importance of centrifugal forces to flexural rigidity. The solutions are found by three methods: series expansion for smallJ, matched asymptotic expansions for largeJ and exact numerical solution using both quasi-Newton and homotopy methods.
Zusammenfassung
Ein Kreisring rotiert um einen Durchmesser im Raum. Verformungen des Ringes hängen vom dimensionslosen ParameterJ ab, der die relative Bedeutung der zentrifugalen Kräfte im Verhältnis zur Biegesteifigkeit angibt. Die Lösungen wurden durch drei Methoden gefunden: Reihenentwicklung für kleinesJ, asymptotische Expansion für großesJ, und exakte numerische Integration sowohl mit einer quasi-Newtonschen als auch mit einer homotopischen Methode.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. E. Dennis and J. J. More,Quasi—Newton methods-motivation and theory. SIAM Rev.21, 46–79 (1977).
L. T. Watson and C. Y. Wang,A homotopy method applied to elastica problems. Int. J. Solids Struct.17, 29–37 (1981).
L. F. Shampine and M. K. Gordon, Computer Solution of Ordinary Differential Equations: The Initial Value Problem. Freeman, San Francisco 1975.
W. G. Bickley,The heavy elastica. Phil. Mag. Ser. 7,17, 603–622 (1934).
F. Odeh and I. Tadjbakhsh,A nonlinear eigenvalue problem for rotating rods. Arch. Rat. Mech. Anal.20, 91–94 (1965).
N. Bazley and B. Zwahlen,Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem. Arch. Rat. Mech. Anal.28, 51–58 (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wang, C.Y., Watson, L.T. Free rotation of a circular ring about a diameter. Z. angew. Math. Phys. 34, 13–24 (1983). https://doi.org/10.1007/BF00962612
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00962612