Bifurcation Analysis of Euler Buckling Problem from the viewpoint of Singularity Theory

• Material Type: 博士論文
• Title: Bifurcation Analysis of Euler Buckling Problem from the viewpoint of Singularity Theory
• Author/Editor: ATIA, AFROZ
• Author Heading: ATIA, AFROZ
• Publication, Distribution, etc.: 埼玉大学大学院理工学研究科
• Publication Date: 2019
• Publication Date (W3CDTF): 2019
• Alternative Title: 特異点論の観点からのオイラー座屈問題の分岐解析
• Periodical title: 博士論文（埼玉大学大学院理工学研究科（博士後期課程））
• Degree grantor/type: 埼玉大学
• Date Granted: 2019-09-20
• Date Granted (W3CDTF): 2019-09-20
• Dissertation Number: 甲第1143号
• Degree Type: 博士(学術)
• Conferring No. (Dissertation): 甲第1143号
• Text Language Code: eng
• Target Audience: 一般
• Note (General): type:text; We consider buckling of rod which is subjected to compressive force λ. In 1757, L. Euler’s found the critical load of the system, and this problem is often called Euler buckling problem. This is actually a celebrated example of pitchfork bifurcation. M. Golubitsky and D. Schaeffer considered a modified version of Euler buckling problem in the variational formulation using strain energy and potential energy, and they show that this modified problem is a versal unfolding of the original problem. We consider a rather more general problem in the context of variational set-up and discuss smoothness of the problem, which is not discussed by M. Golubitsky and D. Schaeffer. This is important to apply Lyapunov-Schmidt reduction. We also describe 3-jets of the equations which define the bifurcation set B and the hysteresis set H, which enable us to draw figures of B and H approximately under suitable set-up.; Declaration of Authorship iiiAbstract vAcknowledgements vii1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries 52.1 Sobolev Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Versal Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 P-K equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Basic Bifurcation theorem . . . . . . . . . . . . . . . . . . . . . 92.5 Bifurcation Set and Hysteresis Set . . . . . . . . . . . . . . . . . 103 Euler Buckling Problem 133.1 Buckling of the rod with pinned ends . . . . . . . . . . . . . . 133.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Energy formulation of Euler buckling problem . . . . . . . . . 153.4 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . 164 Smoothness of Φ 194.1 Differentiability of Φ . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Φ is C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Φ is C∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Taylor coefficients of Φ 295.1 Taylor expansion of Φ . . . . . . . . . . . . . . . . . . . . . . . 295.2 First order derivative of (L)u . . . . . . . . . . . . . . . . . . . 305.3 Derivatives of the coefficient of α1 . . . . . . . . . . . . . . . . 315.4 Second order derivative of (L)u . . . . . . . . . . . . . . . . . . 325.5 Third order derivative of (L)u . . . . . . . . . . . . . . . . . . . 326 Lyapunov-Schmidt reduction 356.1 The first order derivatives of W . . . . . . . . . . . . . . . . . . 366.2 The second order derivatives of W . . . . . . . . . . . . . . . . 377 Bifurcation equation F = 0 and its Taylor coefficients 417.1 The first order derivatives of F . . . . . . . . . . . . . . . . . . 417.2 The second order derivatives of F . . . . . . . . . . . . . . . . . 427.3 The third order derivatives of F . . . . . . . . . . . . . . . . . . 428 Versality 458.1 P-K-versality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Bifurcation set and hysteresis set 479.1 Equation of B and H . . . . . . . . . . . . . . . . . . . . . . . . 479.2 Figures of B and H . . . . . . . . . . . . . . . . . . . . . . . . . 52A Appendix 63A.1 Inverse Function Theorem and Implicit Function Theorem . . 63A.2 Second variation of Energy equation of Euler buckling problem 63A.3 Reduction Method of Lyapunov-Schmidt . . . . . . . . . . . . 68Bibliography 71; 主指導教員 : 福井敏純
• DOI: 10.24561/00019024; https://doi.org/10.24561/00019024
• Persistent ID (NDL): info:ndljp/pid/11551870; https://dl.ndl.go.jp/pid/11551870
• Collection: 障害者向け資料
• Collection (Materials For Handicapped People:1): テキストデータ
• Collection (particular): 国立国会図書館デジタルコレクション > デジタル化資料 > 博士論文; https://dl.ndl.go.jp/collections/A00014
• Acquisition Basis: 博士論文（自動収集）
• Date Accepted (W3CDTF): 2020-10-06T21:18:06+09:00
• Date Created (W3CDTF): 2020-07-21
• Format (IMT): application/pdf
• Access Restrictions: 国立国会図書館内限定公開
• Service for the Digitized Contents Transmission Service: 図書館・個人送信対象外
• Availability of remote photoduplication service: 可
• Periodical Title (URI): https://doi.org/10.24561/00019024; http://id.nii.ac.jp/1586/00019024/
• Data Provider (Database): 国立国会図書館 : 国立国会図書館デジタルコレクション; https://dl.ndl.go.jp

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2023-09-04 再収集