Buckling of a critically tapered rod: global bifurcation

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Stuart C.A and Vuillaume G 2003Buckling of a critically tapered rod: global bifurcationProc. R. Soc. Lond. A.4591863–1889http://doi.org/10.1098/rspa.2002.1092SectionRestricted accessBuckling of a critically tapered rod: global bifurcation C.A Stuart C.A Stuart Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author and G Vuillaume G Vuillaume Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author C.A Stuart C.A Stuart Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author and G Vuillaume G Vuillaume Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (; ) Google Scholar Find this author on PubMed Search for more papers by this author Published:08 August 2003https://doi.org/10.1098/rspa.2002.1092AbstractThis paper, which can be considered as a continuation of the papers Buckling of a heavy tapered rod (2001, J. Math. Pures Appl. 80, 281–337) and On the spectral theory of a heavy tapered rod (2002, Proc. R. Soc. Edinb. A132, 729–764), is concerned with the study of buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem{A(s)u′(s)}′ + μsinu(s) = 0 for all s ∈ (0,1), <BR></BR>u(1) = lims→0A(s)u′(s) = 0and ∫01A(s)u′(s)2 ds < ∞, where A(s) ∈ C([0,1]) is such that A(s) > 0 for all s > 0 and lims→A(s)/sp = L for some constants p ⩾ 0 and L ∈ (0,∞). We deal with the critical case p = 2 and study the set of all solutions of the problem. In particular, we find the points μ ∈ R+ such that bifurcation occurs at (μ,0).As was shown in the former reference, (μ,u) is a solution of the problem if and only if u ∈ HA and u = μGA(u), where HA denotes the Hilbert space of all admissible configurations and GA : HA → HA is a completely continuous operator. If 0 ⩽ p < 2, GA is Fréchet differentiable and the 2001 paper contains results about global bifurcation derived from the general theory. In the case p = 2, GA is not Fréchet differentiable. We look for the bifurcation points in this critical case, where the abstract theory cannot be invoked. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited byStuart C (2021) Global bifurcation at isolated singular points of the Hadamard derivative, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379:2191, Online publication date: 22-Feb-2021. Stuart C (2020) Qualitative properties and global bifurcation of solutions for a singular boundary value problem, Electronic Journal of Qualitative Theory of Differential Equations, 10.14232/ejqtde.2020.1.90:90, (1-36) Castro H (2017) The essential spectrum of a singular Sturm-Liouville operator, Mathematische Nachrichten, 10.1002/mana.201600176, 291:4, (593-609), Online publication date: 1-Mar-2018. Stuart C (2016) Bifurcation points of a singular boundary-value problem on (0,1), Journal of Differential Equations, 10.1016/j.jde.2015.12.040, 260:7, (6267-6321), Online publication date: 1-Apr-2016. Stuart C (2015) Bifurcation without Fréchet differentiability at the trivial solution, Mathematical Methods in the Applied Sciences, 10.1002/mma.3409, 38:16, (3444-3463), Online publication date: 15-Nov-2015. Castro H Oscillations in a semi-linear singular Sturm–Liouville equation, Asymptotic Analysis, 10.3233/ASY-151318, 94:3-4, (363-373) Castro H (2014) Bifurcation analysis of a singular nonlinear Sturm–Liouville equation, Communications in Contemporary Mathematics, 10.1142/S0219199714500126, 16:05, (1450012), Online publication date: 1-Oct-2014. Ram Y (2014) Nonlinear eigenvalue problems of the elastica, Mechanical Systems and Signal Processing, 10.1016/j.ymssp.2013.12.006, 45:2, (408-423), Online publication date: 1-Apr-2014. Evéquoz G and Stuart C On differentiability and bifurcation Advances in Mathematical Economics, 10.1007/4-431-30899-7_6, (155-184) Stuart C and Vuillaume G (2004) Buckling of a critically tapered rod: properties of some global branches of solutions, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460:2051, (3261-3282), Online publication date: 8-Nov-2004. This Issue08 August 2003Volume 459Issue 2036 Article InformationDOI:https://doi.org/10.1098/rspa.2002.1092Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/08/2003Published in print08/08/2003 License: Citations and impact Keywordsessential spectrumEuler elasticabifurcation