Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-25T13:55:39.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear eigenvalue problems for the whirling of heavy elastic strings, II: new methods of global bifurcation theory

Published online by Cambridge University Press:  14 November 2011

J. C. Alexander ,
Stuart S. Antman  and
Shi-tao Deng
J. C. Alexander
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, U.S.A.
Stuart S. Antman
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, U.S.A.
Shi-tao Deng
Affiliation:
Chinese University of Science and Technology, Hofei, Anwhei, People's Republic of China
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper treats the global qualitative behaviour of all bifurcating configurations of whirling nonlinearly elastic strings with ends fixed on the axis of rotation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 3-4 , 1983 , pp. 197 - 227
Copyright
Copyright © Royal Society of Edinburgh 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alexander, J. C.. A primer on connectivity. Proc. Conf. on Fixed Point Theory, 1980, ed. Fadell, E. & Fournier, G.. Lecture Notes in Mathematics 886, 455–483 (Berlin: Springer, 1981).Google Scholar
2Alexander, J. C. and Antman, S. S.. Global behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems. Arch. Rational Mech. Anal. 76 (1981), 339–354.CrossRefGoogle Scholar
3Alexander, J. C. and Antman, S. S.. Global behavior of solutions of nonlinear equations depending on infinite-dimensional parameters. Indiana Univ. Math. J., in press.Google Scholar
4Alexander, J. C. and Yorke, J. A.. The implicit function theorem and global methods of cohomology. J. Functional Analysis 21 (1976), 330–339.CrossRefGoogle Scholar
5Antman, S. S.. Multiple equilibrium states of nonlinearly elastic strings. SIAM J. Appl. Math. 37 (1979), 588–604.Google Scholar
6Antman, S. S.. The equations for large vibrations of strings. Amer. Math. Monthly 87 (1980), 359–370.CrossRefGoogle Scholar
7Antman, S. S.. Nonlinear eigenvalue problems for the whirling of heavy elastic strings. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 59–85.Google Scholar
8Antman, S. S. and Brezis, H.. The existence of orientation-preserving deformations in nonlinear elasticity. In Nonlinear Analysis and Mechanics, ed. Knops, R., Pitman Res. Notes in Math. 27, 1–29 (London, 1978).Google Scholar
9Ball, J. M., Knops, R. and Marsden, J. E.. Two examples in nonlinear elasticity. In Journées d'Analyse Non Linéaire, ed. Bénilan, Ph. and Roberts, J.. Lecture Notes in Mathematics 665, 41–48 (Berlin: Springer, 1978).Google Scholar
10Böhme, R.. Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme. Math. Z. 127 (1972), 105–126.CrossRefGoogle Scholar
11Browne, R. C.. Dynamic stability of one-dimensional visco-elastic bodies. Arch. Rational Mech. Anal. 68 (1979), 231–262.Google Scholar
12Dancer, E. N.. Global structure of the solutions of non-linear real analytic eigenvalue problems. Proc. London Math. Soc. 27 (1973), 747–765.Google Scholar
13Dold, A.. Lectures on Algebraic Topology (Berlin: Springer, 1972Google Scholar
14Dunford, N. and Schwartz, J. T.. Linear Operators, Part I (New York: Wiley-Interscience, 1958).Google Scholar
15Kato, T.. Perturbation Theory for Linear Operators (Berlin: Springer, 1966).Google Scholar
16Klainerman, S. and Majda, A.. Formation of singularities for wave equations including the nonlinear vibrating string. Comm. Pure Appl. Math. 33 (1980), 241–263.Google Scholar
17Knops, R. and Wilkes, E. W.. Theory of Elastic Stability. Handbuch der Physik, VIa/3, 125–302 (Berlin: Springer, 1973).Google Scholar
18Kolodner, I. I.. Heavy rotating string—a nonlinear eigenvalue problem. Comm.Pure Appl. Math. 8 (1955), 395–408.Google Scholar
19Krasnosel'skii, M. A. and Rutitskii, Ya. B.. Convex Functions and Orlicz Spaces, (in Russian), Fizmatgiz, 1958; English transl. by Boron, L. F. (Groningen: Noordhoff, 1961)Google Scholar
20Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J.Math. 3 (1973), 161–202.Google Scholar
21Reeken, M.. The equation of motion of a chain. Math. Z. 155 (1977), 219–237.Google Scholar
22Reeken, M.. Classical solutions of the chain equation I, II. Math. Z. 165 (1979), 143–169; 166 (1979), 67–82.Google Scholar
23Reeken, M.. Rotating chain fixed at two points vertically above each other. Rocky Mountain J. Math. 10 (1980), 409–427.Google Scholar
24Sather, D.. Branching of solutions of nonlinear equations in Hilbert space. Rocky Mountain Math. J. 3 (1973), 203–250.Google Scholar
25Stuart, C. A.. Spectral theory of rotating chains. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 199–214.Google Scholar
26Toland, J. F.. A duality principle for non-convex optimization and the calculus of variations. Arch. Rational Mech. Anal. 71 (1979), 41–61.Google Scholar
27Toland, J. F.. On the stability of rotating heavy chains. J. Differential Equations 32 (1979), 15–31.Google Scholar
28Whyburn, G. T.. Topological Analysis, Rev. Edn (Princeton Univ. Press, 1964).Google Scholar
29Wu, C.-H.. Whirling of a string at large angular speeds—a nonlinear eigenvalue problem with moving boundary layers. SIAM J. Appl. Math. 22 (1972), 1–13.Google Scholar
30Young, L. C.. Lectures on the Calculus of Variations and Optimal Control Theory (Philadelphia: Saunders, 1969).Google Scholar