Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T01:27:30.099Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear eigenvalue problems for the whirling of heavy elastic strings

Published online by Cambridge University Press:  14 November 2011

Stuart S. Antman
Stuart S. Antman
Affiliation:
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A. and Department of Mathematics, University of Maryland, College Park, Maryland, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 59 - 85
Copyright
Copyright © Royal Society of Edinburgh 1980

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

1Coddington, E. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
2Courant, R. and Hilbert, D.. Methods of Mathematical Phyics, vol. I (New York: Interscience, 1953).Google Scholar
3Crandall, M. G. and Rabinowitz, P. H.. Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19 (1970), 1083–1102.Google Scholar
4Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Functional Analysis 8 (1971), 321–340.CrossRefGoogle Scholar
5Dancer, E. N.. A note on bifurcation from infinity. Quart. J. Math. Oxford Ser. 25 (1974), 81–84.CrossRefGoogle Scholar
6Kolodner, I. I.. Heavy rotating string—a nonlinear eigenvalue problem. Comm. Pure Appl. Math. 8 (1955), 395–408.CrossRefGoogle Scholar
7Krasnosel'skii, M. A.. Topological methods in the Theory of Nonlinear Integral Equations (New York: Macmillan, 1964).Google Scholar
8Olmstead, W. E.. Extent of the left branching solution to certain bifurcation problems. SIAM J. Math. Anal. 8 (1977), 392–401.CrossRefGoogle Scholar
9Pimbley, G. H.. A sublinear Sturm-Liouville Problem. J. Math. Mech. 11 (1962), 121–138.Google Scholar
10Pimbley, G. H.. A super-linear Sturm-Liouville problem. Trans. Amer. Math. Soc. 103 (1962), 229–248.CrossRefGoogle Scholar
11Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487–513.CrossRefGoogle Scholar
12Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973), 161–202.CrossRefGoogle Scholar
13Rabinowitz, P. H.. On bifurcation from infinity. J. Differential Equations 14 (1973), 462–475.CrossRefGoogle Scholar
14Stuart, C. A.. Solutions of large norm for nonlinear Sturm-Liouville problems. Quart. J. Math. Oxford Ser. 24 (1973), 129–139.CrossRefGoogle Scholar
15Stuart, C. A.. Spectral theory of rotating chains. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 199–214.CrossRefGoogle Scholar
16Stuart, C. A.. Steadily rotating chains. In Applications of methods of functional analysis to problems in mechanics (Germain, P. and Nayroles, B. eds) Lecture Notes in Mathematics 503, 490–499 (Berlin: Springer, 1976).Google Scholar
17Toland, J. F.. Asymptotic linearity and nonlinear eigenvalue problems. Quart. J. Math. Oxford Ser. 24 (1973), 241–250.CrossRefGoogle Scholar
18Toland, J. F.. A duality principle for non-convex optimization and the calculus of variations, Arch. Rational Mech. Anal. 71 (1979), 41–61.CrossRefGoogle Scholar
19Truesdell, C.. The Rational Mechanics of Flexible or Elastic Bodies 1638–1788, L. Euleri Opera Omnia, Ser. II, Vol. 11 2, Füssli, Zürich, 1960.CrossRefGoogle Scholar
20Turner, R. E. L.. Superlinear Sturm-Liouville problems. J. Differential Equations 13 (1973), 157–171.CrossRefGoogle Scholar
21Wolkowisky, J. H.. Nonlinear Sturm-Liouville problems. Arch. Rational Mech. Anal. 35 (1969), 299–320.CrossRefGoogle Scholar