應用SCM於奈米碳管之振動分析

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)，應用在求解單壁奈米碳管的自由振動問題，分析其自然振動頻率，亦導入各種不同組合的邊界條件，並增加所取的節點數，觀察其振動頻率與對應之模態，也得以驗證奈米碳管是一個作為超高頻感應器、螺線管或共振器之最佳材料。　　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及其它數值分析方法所得之數值解相互比較，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；單壁奈米碳管；自由振動；自然頻率；數值分析

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 free vibrating problem of single-walled carbon nanotubes to obtain the natural frequency of the system being analyzed. Also, by leading in different combinations of boundary conditions, increasing the numbers of knots, observe the vibrating frequency and corresponding model shape. As for carbon nanotubes, it is clearly illustrated in this thesis that it is a perfect material for the development of ultrahigh frequency sensors, actuators and resonators due to exceptionally high stiffness, high strength and large aspect ratio. At last, results from each example analyzed by SCM will be utilized to compare with theoretical solution derived from Euler-Bernoulli beam model and analytic solution by using other numerical method. 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 ； single-walled nanotubes ； free vibration ； natural frequency ； numerical analysis

參考文獻

[26] 吳兆民，民國95年，應用SCM於梁振動之分析，碩士論文，吳賴雲、鍾立來教授指導，國立台灣大學土木工程學研究所。

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[4] C.W. Bert, A.G. Striz and W. Chen, 1994, “Static Analysis of Structures by the Quadrature Element Method”, Int. J. Solids Structures, Vol. 31, No. 20, 2807-2818.

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應用SCM於奈米碳管之振動分析

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