應用SCM於非侷域奈米梁之撓曲分析

SCM(Spline Collocation Method)是由Forward Difference 所推導之Spline function，並配合節點佈置(Collocation)的方式，所發展出的一種數值方法；再由各階之Spline function 整理製作出完整的B Spline Value Table，使得在使用SCM 此法分析時，得以用查表的方式大大簡化計算過程，並有系統的編寫電腦程式，得以迅速且方便地求解。本文研究主旨在以SCM (Spline Collocation Method) 延伸發展之MSCM(Modified Spline Collocation Method)，應用在求解非侷域奈米梁變形諧和問題，分析非侷域奈米梁最大撓度值與Timoshenko梁之最大撓度值的差異，並導入各種不同組合的邊界條件、各式載重與Winkler彈性基礎，並增加所取的節點數，可從中看到其求解過程的便利性。且能不侷限於任何形式的載重或邊界條件，僅需結構之控制方程式，即能求得令人滿意之近似解。　　最後，以實例利用SCM進行數值分析，將分析結果與數學上的解析解相互比較，以驗證由SCM所得解之誤差能達到工程上所要求的範圍之內，顯示出SCM 確有其優勢所在，是一種準確、快速、便利且可應用的數值方法，值得作進一步之應用分析研究。

關鍵字

楔形配點法；非侷域奈米梁；變形諧和；最大撓度； Winkler彈性基礎

並列摘要

SCM(Spline Collocation Method) is a numerical analysis method originating from Spline function which is derived from Forward Difference, and also joining the idea of knot-collocating together. By means of developing B Spline Value Table from differential Spline function in various orders, the calculating process can be simplified greatly by simply looking up to the table. Furthermore, it can be used in computer programming in a systematic way. Therefore it becomes much more rapid and convenient when analyzing by using SCM. The purpose of this thesis is to apply MSCM(Modified Spline Collocation Method) which is extended from SCM (Spline Collocation Method) to solve the compatibility of the nonlocal nanobeam, analysis of maximum deflection value and the difference between nonlocal nanobeam and Timoshenko beam . Also, Into a variety of different combinations of boundary conditions, all kinds of loading and Winkler elastic foundation, and take the increase in the number of nodes. We can see the convenience of the solution process. And can not be limited to any form of loading or boundary conditions, only need the structure of the control equation, that is able to obtain satisfactory approximate solution .At last, results from each example analyzed by SCM will be utilized to compare with theoretical solution . Thus it can be verified that the error conducted by SCM is within the standard range of engineering field. It is demonstrated that SCM has its superiority as being one kind of accurate, fast, convenient and applicable numerical method; hence it is indeed worth doing further research.

並列關鍵字

SCM ； nonlocal nanobeam ； maximum deflection ； compatibility ； Winkler elastic foundation

參考文獻

21. 陳一宏〝利用非侷域效應增強禁制帶光孤子之穩定性及移動性〝，國立清華大學物理研究所碩士論文，李瑞光教授指導，民國九十六年六月。

17. Wang, Y. H., Tham, L. G., Cheung, Y. K., “Beams and Plates on Elastic Foundations: A Review”, Progress in Structural Engineering and Materials, Volume 7, Issue 4 , Pages 174 – 182 (2005).

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應用SCM於非侷域奈米梁之撓曲分析

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