使用分樑法求多重負載懸臂樑震盪模態函

Abstract

本文發明一個新的方法-分樑法(Split Beam Method)，用於更有效率地解決尤拉伯努利樑的特徵質問題。其中利用了楊式模數項的拆解，將拆解後的楊式模數分別對應不同的負載形成子系統，將總系統拆解為多個子系統，並使用線性疊加的方式，疊加出總系統。 接著我們使用兩個例子示範如何使用SBM方法。第一個例子使用兩種不同的分布力，其中，為了解決非齊性特徵值的問題，我們使用了非齊性特徵值問題的公式，求得非齊性特徵值，並帶入非齊性特徵值，解出樑之撓度函數解析解。隨後控制其中一種分布力為定值，並逐步增加另一種分布力的值，當分布力逐漸變大時，我們可以發現增加之分布力所對應的子系統係數會漸漸增大，並主導整個總系統 第二個例子我們引用了關於原子力顯微鏡(AFM)探針的研究，並使用其模型，再稍作修改，將其中一個負載，原子間作用力改為端點力表示，隨後將震盪力設為定值，逐步增加端點力的數值，亦可發現端點力所對應的子系統係數會漸漸增大，並主導整個總系統。 由此兩個例子結果證實樑分離法是可行的，大大簡化了計算的複雜度，避免了一般傅立葉疊加法所導致的過多項數與其無效的計算，另外，也能解決傅立葉疊加法中無法找到主導項的問題。

This study develops a new method, Split Beam Method(SBM). We utilize this method to solve the eigenvalue problem of Euler-Bernoulli Beam Equation more effectively. First of all, we split the Young’s Modulus corresponding to two or more subsystems with different loadings. Then we superimpose these sub-systems using linear superposition method. Following, we show how to apply SBM by two examples. In the first one, we utilize two different distribution loadings. In order to solve the difficulty caused by inhomogeneous eigenvalue problem, we made use of the inhomogeneous eigenvalue problem formula to have the solution. Then, the exact solution of deflection function would be obtained by utilizing inhomogeneous eigenvalue formula. Simultaneously, we control one of a distributed loading as constant and gradually increase the value of another distributed loading. We can find that the coefficient of the related subsystem function will increase when the second loading becomes larger and the second loading will become the dominate item in the total system. In the second one, we quote the paper about atomic force microscope, and use its model. Then, we modify a little to simulate SBM. We make Atomic Force(AFM) as a point force, and fix the oscillating force as constant. Then we find that the coefficient of the related subsystem function will increase when the second point force becomes larger and the point force will become the dominate item in the total system. From the results of these two examples, we ensure SBM is a correct and effective method. It simplify complex calculation and avoid multi- item behavior and useless calculation from Fourier superposition. In addition, SBM also provide one simple way to know which loading is the dominated one which is impossible to know from Fourier superposition.