圓曲樑在彈性基礎上之動力特性

Abstract

這篇論文是在討論矩形斷面曲樑之自由振動，此曲樑放於彈性基礎上，沿著樑的中心線上每一個點上考慮彎矩、扭轉、剪力。可以把彎矩和扭轉表示θ(角度)和t(時間)之函數，再藉由三個方向之動力平衡方程式可另外三個方程式，經過整合之後可以得到兩條控制方程式。 文中假設位移函數中的時間變化部份為 ，則方程式可簡化成為非時間函數之方程式。考慮符合邊界條件，代入只考慮角度之位移方程式，把六條方程式寫成矩陣的型式，組成一個6×6的系統矩陣[B]，因為方程式必須有解故其矩陣行列式值為零，利用此特性可以求得其系統頻率，帶回其方程式可得其振態。 之後還有分析完整的一個環，分析步驟和前面相同，只是在邊界條件變連續條件，同樣可以組成6×6的系統矩陣，利用行列式值為零特性可以得到其頻律和振態。最後使用有限元素軟體ANSYS和其理論結果做比較。

In this thesis, we presented the method to obtain the natural frequencies and respective modal shapes of circular curved beams on elastic foundations. The bending moment and the twisting moment can be expended as functions in form of θ(angle) and t(time). According to the dynamic equilibrium of the curved beam, we have three equations. After combining the equations, we can attain two governing equations. Considering the boundary conditions, we can obtain a 6×6 matrix. Because non-trivial solutions should exist, it is necessary that the determinant of the matrix equals zero. Therefore, the natural frequencies of the curved beam can be obtained and the respective modal shapes are also found. To do this, the boundary conditions are replaced with continuity conditions. We also analyzed a complete ring. We also employed the finite element software ANSYS to find the frequencies and respective modal shapes of curved beam in order to compared with the solution of the theory derived above and compare to my theory solutions.