分樑法與非齊性特徵值解於溫克勒地基模型與多重負載的歐拉-伯努力樑之應用

Abstract

本文延續實驗室發展的新方法__分樑法(Split Beam Method)，用於更加快速地解決在承受多種負載情況下的歐拉-伯努力樑方程(Euler-Bernoulli Beam Equation)問題。分樑法原理是對楊氏模量(Young's modulus)進行分解，分解後的楊氏模量分別對應不同的負載形成基函數方程式，將這些方程式求解得出基函數，再將基函數做線性疊加代回原歐拉-伯努力樑方程並利用瑞利-里茲方法(Rayleigh-Ritz Method)與非齊性特徵值公式求出特徵值及線性疊加的係數，最後將係數代回即得樑方程的模態振型函數(Mode shape function)。 然後用兩個模型去示範分樑法的應用及特性。第一個模型選用溫克勒地基模型(Winkler Foundation Model)，我們把此模型的彈簧當成負載來分解，分解成兩個或多個只作用於一點的彈簧，這些彈簧的彈性係數依照分解出的基函數方程數量去做平分，然後把分解的數量慢慢增加來觀察模態振型函數及特徵值的收斂情況，能發現分解成四個以上之後，便能有與溫克勒地基模型解析解不到百分之一的誤差，也代表著我們用四個彈簧就能模擬出無限個彈簧對樑之影響。而第二個模型放了三個不同大小的單點外力，然後把這三個外力大小加總並平均分成四個或八個單點力放置於樑的各處，隨後調整組合外力之位置，以觀察係數之變化，隨著組合外力慢慢右移，可以看出放在較左邊的基函數係數也隨之變小而右邊的基函數係數則反之，從係數的變化中能清楚地看出樑在各位置受力的變化。從兩個例子中能證實比起傅立葉展開法能用更少的基函數就能達到很好的精準度，並且在分解負載時能依照不同情況去作改變，並從係數結果得到不同的物理意義。

This article continues the development of a new method in the laboratory called the Split Beam Method, which is used to solve the Euler-Bernoulli Beam Equation more rapidly under multiple load conditions. The principle of the Split Beam Method involves decomposing the Young's modulus and forming separate basic function equations corresponding to different loads. These equations are solved to obtain the basic functions, which are then linearly combined and substituted back into the original Euler-Bernoulli Beam Equation. The Rayleigh-Ritz Method and the non-homogeneous eigenvalue formula are used to calculate the eigenvalues and the coefficients of the linear combination. Finally, the coefficients are substituted back to obtain the mode shape function of the beam equation. Two models are used to demonstrate the application and characteristics of the Split Beam Method. In the first model, the Winkler Foundation Model is chosen, and the springs in this model are decomposed to represent the loads. The springs are decomposed into two or more springs that act on a single point. The elastic coefficients of these springs are evenly distributed based on the number of decomposed basic function equations. By gradually increasing the number of decompositions, the convergence of the mode shape function and eigenvalues can be observed. It is found that when decomposed into four or more springs, the error is less than a certain percentage compared to the analytical solution of the Winkler Foundation Model. This means that using only four springs can simulate the influence of an infinite number of springs on the beam. In the second model, three different point forces of varying magnitudes are applied to the beam. The sum of these three forces is divided equally into four or eight point forces placed at different positions on the beam. The positions of the three forces are adjusted to observe the changes in the coefficients. As the combined forces gradually move to the right, it can be observed that the coefficients of the basic functions on the left decrease, while those on the right increase. The changes in the coefficients clearly reflect the variations in the forces acting on the beam at different positions. These two examples demonstrate that the Split Beam Method can achieve high accuracy with fewer basic functions compared to the Fourier Expansion Method. Furthermore, the Split Beam Method allows for adjustments in the load decomposition according to different situations, and the coefficient results provide different physical interpretations.