彈性板受動態載重之研究

Abstract

在工程應用之觀點，有關板之問題是一個很重要的課題。板承受動態外力發生振動需要解決之諸問題，需先求得板之自由振動頻率和振態然始可進一步分析研究。而處理板之有關文獻中大致上可分為古典力學理論與有限元素法兩種，其中應用古典力學理論在板之形狀規則與邊界條件簡單時，尚不難處理，如遇板之形狀不規則或邊界條件較複雜時就惟有應用有限元素法來求解，而本文之目的即是應用有限元素法來求得動態載重下板之變形情形，並由古典力學之解來印證其準確性。 首先求得板元素之勁度矩陣及質量矩陣，再合成為板之勁度矩陣及質量矩陣，並代入邊界條件，其次由解(eigenvalue and eigenvector)求得板之自由振動頻率和振態。並假設各種不同型態之載重，假設板上受一時間有關之外力，其位移方程式為此等振態之線性組合，用振態重疊原理求得位移與時間之關係，並分析當彈性基礎考慮時，對板之變形之影響。 最後將有限元素法所得之結果，與薄板理論所推導之解相比較，以確定其可行性，並且將板依不同之切割法，分別求得其自由振動頻率，互相比較，以觀察其不同分割法之精度。

In engineering applications, solving the problems of the vibration of plate caused by dynamic loading exerted on plate, in general, is required to obtain the natural vibration frequency and mode shape of the plate first, and then the further analysis. Generally, the literature on plate can be distinguished into two main catalogs, which are classic theory and finite element method. Classic theory can be employed to solve problems with simple geometry which associate with simple shape and boundary of plate. On the other hand, when shape and boundary of plate become complicated, then the classic theory becomes inadequate and needs to finite element method must be adopted for solution. Thus, the main purpose of this study is to use finite element method to obtain the displacement in plate when dynamic load is exerted on it, applying the classic theory to see how the accuracy of the finite element method model, for some simple cases. To use finite element method, one should first generate stiffness matrices and mass matrices of the elements, then combine these of stiffness matrices and mass matrices into global matrices by inducing boundary conditions. And then, natural vibration frequencies and mode shapes of plate through solving the eigenvalue and eigenvector of free vibration of the plate, for the dynamic analysis. Finally, the results obtained through finite element method will be compared with that using classic theory in order to evaluate the accuracy of finite element approach.