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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 劉進賢(Chein-Shan Liu) | |
| dc.contributor.author | Chih-Yi Lin | en |
| dc.contributor.author | 林芷儀 | zh_TW |
| dc.date.accessioned | 2021-06-16T23:27:57Z | - |
| dc.date.available | 2012-08-01 | |
| dc.date.copyright | 2012-08-01 | |
| dc.date.issued | 2012 | |
| dc.date.submitted | 2012-07-31 | |
| dc.identifier.citation | [1] A. Zettl, Sturm-Liouville theory, mathematical surveys and monographs 121, American Mathematical Society (2005).
[2] B. Chanane, Accurate solutions of fourth order Sturm-Liouville problems, Journal of Computational and Applied Mathematics 234, 3064-3071 (2010). [3] C.S. Liu, A Lie-group shooting method for computing eigenvalues and eigenfunctions of Sturm-Liouville problems, Computer Modeling in Engineering and Sciences 56, 85-112 (2008). [4] C.S. Liu, Computing the eigenvalues of the generalized Sturm-Liouville problems based on the Lie-group SL(2,R), Journal of Computational and Applied Mathematics 236, 4547-4560 (2012). [5] C.S. Liu, Cone of non-linear dynamical system and group preserving schemes, International Journal of Non-Linear Mechanics 36, 1047-1068 (2001). [6] C.S. Liu, Efficient shooting methods for the second order ordinary differential equations, Computer Modeling in Engineering and Sciences 15, 69-86 (2006). [7] C.S. Liu, The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions, Computer Modeling in Engineering and Sciences 13, 149-163 (2006). [8] C.S. Liu, The Lie-group shooting method for singularly perturbed two-point boundary value problems, Computer Modeling in Engineering and Sciences 15, 179-196 (2006). [9] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics 144, American Mathematical Society (2010). [10] H. K. Kim and M. S. Kim, Vibration of beams with generally restrained boundary conditions using Fourier series, Journal of Sound and Vibration 245, 771-784 (2001). [11] H. Y. Lai, J. C. Hsu, and C. K. Chen, An innovative eigenvalue problem solver for free vibration of EulerBernoulli beam by using the Adomian decomposition method, Journal Computers and Mathematics with Applications 56, 3204-3220 (2008). [12] I. Kleiner, The evolution of group theory: a brief survey, Mathematics Magazine 59, NO. 4 (1986) [13] I. N. Herstein, Topics in Algebra, John Wiley and Sons, 2nd edition (1975). [14] J. Gallier, Notes On Group Actions Manifolds, Lie Groups and Lie Algebras, Department of Computer and Information Science, University of Pennsylvania Philadelphia, PA 19104, USA, CIS610 (2005). [15] J. Gallier, Manifolds, Riemannian Geometry Lie Groups and Harmonic Analysis, With Applications, Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 19104, USA, CIS610 (2011). [16] J. C. Hsu, C. K. Chen, H. Y. Lai, Application of the Adomian Modified Decomposition Method to the Free Vibrations of Beams, Department of Mechanical Engineering, National Cheng Kung University Doctoral Dissertation (2009). [17] J. H. Chen, Free vibration analyses of Euler-Bernoulli beams subjected to axial loads and carrying various concentrated elements, Department of System and Naval Mechatronic Engineering, National Cheng Kung University Master Thesis (2007). [18] J. T. S. Wang and C. C. Lin, Dynamic analysis of generally supported beams using Fourier series, Journal of Sound and Vibration 196, 285-293 (1996) [19] L. Greenberg, M. Marletta, The Code SLEUTH for Solving Fourth Order Sturm Liouville Problems, ACM Transactions on Mathematical Software (1997). [20] L. Greenberg, M. Marletta, Oscillation theory and numerical solution of fourth order Sturm Liouville problems, IMA Journal of Numerical Analysis 15, 319-356 (1995). [21] N. M. Auciello, and A. Ercolano, A general solution for dynamic response of axially loaded non-uniform Timoshenko beams, International Journal of Solids and Structures 41, 4861-4874 (2004). [22] R. L. Herman, A Second Course in Ordinary Differential Equations: Dynamical Systems and Boundary Value Problems, Monograph (2008). [23] R. W. Clough, and J. Penzien, Dynamics of Structures, McGraw-Hill Education, International 2 Revised Edition (1993). [24] S. Naguleswaran, Transverse vibration of an uniform Euler-Bernoulli beam under linearly varying axial force, Journal of Sound and Vibration 275, 47-57 (2004). [25] S. Y. Lee, and Y. H. Kuo, Exact solutions for the analysis of general elastically restrained non-uniform beams, Transactions of the ASME: Journal of the Applied Mechanics 59, 205-212 (1992). [26] W. T. Thomson, Theory of vibration with applications, Pretice-Hall International, Inc., Fourth Edition (1993). [27] W. Yeih, J. T. Chen, and C. M. Chang, Applications of dual MRM for determining the natural frequencies and natural modes of an Euler-Bernoulli beam using the singular value decomposition method, Engineering Analysis with Boundary Elements 23, 339-360 (1999). [28] Y. H. Hung, By using the FTIM and Lie-group methods to identify unknown force in Euler-Bernoulli beam, Department of Civil Engineering, National Taiwan University Master Thesis (2011). [29] Y. J. Shin, K. M. Kwon, and J. H. Yun, Vibration analysis of a circular arch with variable cross-section using differential transformation and generalized differential quadrature, Journal of Sound and Vibration 309, 9-19 (2008). | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/65163 | - |
| dc.description.abstract | 梁的振動問題在工程中是一個非常重要的議題，它可以使用四階微分方程表示。在本篇文章中，我們構造一個以李群SL(4,R)為基礎的打靶法以求解其特徵值，而特徵值也就是振動問題中的自然頻率。使用保群算法和李群的封閉性，得到一步保群算法，可由其中一端點的狀態向量轉換到另一端點的狀態向量，再配合廣義中值定理，可得李群SL(4,R)打靶法。此外，我們還可以得到閉合形式的特徵方程。最後，藉著數值算例，顯示這篇文章中提到的方法是可行的、有效的而且易於使用的。 | zh_TW |
| dc.description.abstract | The vibration problem of beam is an important issue in engineering, and it can be expressed as the fourth-order differential equation. In this article, we construct an SL(4,R)-based Lie-group shooting method to find the eigenvalues which are natural frequencies. By using the group preserving scheme (GPS) and the closure property of the Lie group, we develop one-step Lie-group transformation between two ends. Together with the generalized mid-point rule, we have derived an SL(4,R) shooting method. In which, we can develop the analytical characteristic equation to solve the eigenvalues.
Last but not least, in this article, several numerical examples show that the mentioned method is feasible, efficient and easy to use. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T23:27:57Z (GMT). No. of bitstreams: 1 ntu-101-R99521204-1.pdf: 7265009 bytes, checksum: 04e69f921fa16620f1f6978952b60c61 (MD5) Previous issue date: 2012 | en |
| dc.description.tableofcontents | CONTENTS
口試委員會審定書.............................................# 誌謝.......................................................i 摘要......................................................ii ABSTRACT.................................................iii CONTENTS..................................................iv LIST OF FIGURES..........................................vii LIST OF TABLES............................................ix Chapter1 Introduction.....................................1 1.1 Motivation and Preface................................1 1.2 Background............................................1 1.3 Organization..........................................3 Chapter2 SL(4,R)-based Lie-group shooting method..........5 2.1 Group.................................................5 2.2 Lie Group.............................................7 2.3 Group Preserving Scheme (GPS).........................8 2.3.1 Cayley Transformation...........................10 2.3.2 Exponential Mapping.............................10 2.4 One-Step Group Preserving Scheme.....................11 2.5 A Generalized Mid-point Rule.........................12 2.6 An SL(4,R) Shooting Method...........................13 Chapter3 By Using the Lie-group SL(4,R) Method to Solve the Free Vibration Problems of Elastic Beam.........14 3.1 Beam Vibration Equation..............................14 3.2 Fourth-order Generalized Sturm-Liouville Problem.....17 3.3 Lie-group Symmetry, and Self-adjoint System..........19 3.4 Special Linear Group G...............................20 3.4.1 No Axial Force ( s(x) = 0 ).....................21 3.4.2 With Axial Force ( s(x) ≠ 0 )...................25 3.5 An SL(4,R) Shooting Method...........................70 3.5.1 Simple Beam.....................................70 3.5.2 Cantilever Beam.................................73 3.5.3 Free-free Beam..................................75 3.5.4 Clamped-clamped Beam............................76 3.5.5 Clamped-hinged Beam.............................78 3.5.6 Hinged-free Beam................................80 3.5.7 Sliding-hinged Beam.............................82 3.5.8 Sliding-sliding Beam............................85 3.5.9 Clamped-sliding Beam............................87 3.5.10 Sliding-free Beam...............................90 Chapter4 Numerical Examples..............................93 4.1 Fourth-order Sturm-Liouville Problems................93 4.1.1 Example1-Fourth-order Sturm-Liouville Problem1..93 4.1.2 Example2-Fourth-order Sturm-Liouville Problem2..97 4.2 Uniform beams.......................................100 4.2.1 Example3–Uniform Simple Beam...................100 4.3 Non-uniform beams...................................103 4.3.1 Example4–Non-uniform Beam......................103 4.3.2 Example5–Non-uniform Clamped-hinged Beam.......104 4.3.3 Example6–Non-uniform Clamped-hinged Beam with Axial Force........................107 Chapter5 Conclusions and Future Works...................108 REFERENCE................................................109 | |
| dc.language.iso | en | |
| dc.subject | 彈性梁的自由振動問題 | zh_TW |
| dc.subject | 廣義四階Sturm-Liouville問題 | zh_TW |
| dc.subject | R)為基礎的打靶法 | zh_TW |
| dc.subject | 李群SL(4 | zh_TW |
| dc.subject | 特徵方程 | zh_TW |
| dc.subject | 特徵值 | zh_TW |
| dc.subject | R)-based Lie-group Shooting Method | en |
| dc.subject | Free Vibration Problems of the Elastic Beam | en |
| dc.subject | Fourth-order Generalized Sturm-Liouville Problem | en |
| dc.subject | Eigenproblem | en |
| dc.subject | Eigenvalues | en |
| dc.subject | Characteristic Equation | en |
| dc.subject | SL(4 | en |
| dc.title | "以李群SL(4,R)的方法求解彈性梁的自由振動問題" | zh_TW |
| dc.title | By Using the Lie-group SL(4,R) Method to Solve the Free Vibration Problems of the Elastic Beam | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 100-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 范佳銘,張致文 | |
| dc.subject.keyword | 彈性梁的自由振動問題,廣義四階Sturm-Liouville問題,特徵值,特徵方程,李群SL(4,R)為基礎的打靶法, | zh_TW |
| dc.subject.keyword | Free Vibration Problems of the Elastic Beam,Fourth-order Generalized Sturm-Liouville Problem,Eigenproblem,Eigenvalues,Characteristic Equation,SL(4,R)-based Lie-group Shooting Method, | en |
| dc.relation.page | 112 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2012-07-31 | |
| dc.contributor.author-college | 工學院 | zh_TW |
| dc.contributor.author-dept | 土木工程學研究所 | zh_TW |
| 顯示於系所單位： | 土木工程學系 | |
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