利用Ritz法分析具有V型缺口之矩形薄板振動

Abstract

應力奇異點之問題常發生於工程力學的分析計算中。本論文以薄板理論為基礎，利用Ritz法分析具有V型缺口之矩形板振動，在分析過程中使用兩組允許函數序列，分別為：(1)多項式函數，其本身可構成一組完備之序列；(2)角函數，滿足V型缺口兩自由邊緣之邊界條件，並可精確地描述缺口尖端之應力奇異特性。本論文之研究案例包含完全自由與懸臂矩形板，先以完整的收斂性分析驗證角函數能夠有效地加速自然振動頻率之收斂速度，並探討不同幾何及位置之V型缺口對矩形板振動行為之影響。本論文為首次研究具有V型缺口之矩形板振動，此研究結果可提供後人研究參考與比較。

This thesis presents a novel method for accurately determining the natural frequencies of rectangular plates with an edge V-notch. Based on the well-known Ritz method, two sets of admissible functions are used simultaneously: (1) algebraic polynomials, which form a complete set of functions; (2) corner functions, which are the general solutions of bi-harmonic equation, duplicate the boundary conditions along the edges of the notch, and describe the stress singularities at the sharp vertex of the V-notch exactly. The rectangular plates under consideration are either completely free or cantilevered. The effects of corner functions on the convergence of solutions are demonstrated through comprehensive convergence studies. Accurate numerical results and nodal patterns are tabulated for V-notched plates having various notch angle, depths and locations. These are the first known frequency and nodal pattern results of V-notched rectangular plates in the published literature.