解四階邊界值問題的李群SL(4,R)打靶法之研究

在土木工程領域當中，梁柱等結構元件的振動問題是一個非常重要的研究方向，而其振動可以用一個四階微分方程式來表示。梁與其他支承節點之間的力傳遞關係，便與此微分方程構成了四階邊界值問題。結合保群算法及李群SL(4,R)打靶法，我們可以輕易的求解出邊界值問題的數值解，並且在線性與非線性方程都有很好的表現，除此之外我們也嘗試應用在特徵值問題上，找出彈性梁在自由振動時的自然頻率，以及相對應的振動模態。過去使用李群打靶法求解此類問題時常因為複雜的計算導致求解費時，而本文將改善李群SL(4,R)打靶法的代數結構，進而讓運算變得更加簡單有效率。藉著數值算例顯示，李群SL(4,R)打靶法對於求解梁柱結構元件的振動問題有相當不錯的精度與效率，並且簡單而易於使用。

關鍵字

保群算法；李群打靶法；彈性樑；四階微分方程式；振動問題；特徵值問題

並列摘要

There are a lot of problems in civil and structure engineering science that can be described by ordinary differential equation (ODEs). Especially the vibration problems of beam and column are important issues, which can be written as a Fourth-order differential equation. Structure elements like beam or column with their structure bearings will compose the Boundary Value Problems(BVPs).Group Preserving Schemes (GPS) and Lie-group SL(4,R) shooting methods are numerical methods which can easily solve the linear and nonlinear ODEs. Now we try to use these methods to solve the engineering numerical problem like beam vibration or deflection problems. Furthermore, there are some eigenvalue problems in these ODEs that we will also try to solve. In the past, it took a lot of time to solve these ODEs. We will try a better Lie algebra in Lie-group shooting method and make our numerical analysis more efficiency. Finally, we will observe that these numerical methods have good results in solving these boundary value problems, and we hope it can be applied on the development of the civil engineering or other science in the future.

並列關鍵字

group preserving schemes (GPS) ； Lie-group shooting method ； elastic beam problem ； fourth-order differential equation ； vibration problem ； eigenvalue problem.

參考文獻

in a partial differential equation subject to extra final measurement data.

[5] Ascher Uri M., Mattheij Robert M. M., Russell Robert D., Numerical

[1] Thomson W. T., Theory of vibration with applications. Pretice-Hall

International, Inc., Fourth Edition (1993)

275. 245-57 (2004).

國際替代計量

解四階邊界值問題的李群SL(4,R)打靶法之研究

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