"以李群SL(4,R)的方法求解彈性梁的自由振動問題"

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。

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.