Scattering of flexural wave in a thin plate with multiple circular holes by using the multipole Trefftz method

Abstract

The scattering of flexural wave by multiple circular holes in an infinite thin plate is analytically solved by using the multipole Trefftz method. The dynamic moment concentration factor (DMCF) along the edge of circular holes is determined. Based on the addition theorem, the solution of the field represented by multiple coordinate systems centered at each circle can be transformed into one coordinate system centered at one circle, where the boundary conditions are given. In this way, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the scattering of flexural wave by multiple holes in an infinite plate subject to the incident flexural wave. The formulation is general and is easily applicable to dealing with the problem containing multiple circular holes. Although the number of hole is not limited in our proposed method, the numerical results of an infinite plate with three circular holes are presented in the truncated finite system. The effects of both incident wave number and the central distance among circular holes on the DMCF are investigated. Numerical results show that the DMCF of three holes is larger than that of one, when the space among holes is small and meanwhile the specified direction of incident wave is subjected to the plate. 本文利用多極 Trefftz 法以解析的方式求解含多圓形孔洞之無限域薄板彎曲波散射問題，並計算沿孔洞邊界之動應力集中係數(Dynamic Moment Concentration Factor)。根據加法定理，由多個以各個圓孔中心為原點的座標系統所表示的場解可變換成由單一個座標系統表示，而該座標系統所參考的圓孔邊界條件為給定。依此方式可求得一無限耦合線性代數系統作為一個含多圓形孔洞之無限域薄板受入射彎曲波作用的解析模式。本文理論推導具有一般性可推廣至多圓形孔洞問題。雖然本文所提方法在處理圓形孔洞個數沒有限制，在捨去高次的有限項之代數系統中，本文給出含三個圓形孔洞的數值算例，以探討入射波數與圓形孔洞間距等因素對動應力集中係數的影響。數值結果顯示，當圓形孔洞彼此相當接近且入射彎曲波具有特定入射角時，三圓形孔洞的應力集中係數大於單圓形孔洞的結果。