不同板理論徑向簡支扇形板之振動理論解

Abstract

本文分別推導極正向性之古典扇行板、Mindlin扇行板和Reddy扇行板之振動理論解，並探討側向剪力變形對扇行板振動之影響。其理論解乃利用Frobenius方法並滿足徑向、環向邊界條件及兩徑向邊之頂點處的正規條件所建構之級數解。 在兩徑向邊之頂點處，藉由力與位移之關係可發現彎矩或剪力有可能趨近無限大，因而造成彎矩奇異性及剪力其異性。本文所建構級數解之精確性可將正向性扇行板簡化為等向性扇行板，並求出無因次化頻率與目前文獻之閉合解比較，可得驗證。 另外本文也分別探討在環向自由端或固定端下，彈性模數及剪力模數對扇行板振動頻率之影響。此外本文也列出改變彈性模數及剪力模數，彎矩奇異和剪力奇異隨著扇行角變化的結果。

This thesis presents the first known analytical solutions for vibrations of a polarly orthotropic sectorial plate with simply-supported radial edges based on the Classical plate theory, Mindlin plate theory and Reddy plate theory. These solutions are series solutions constructed using the Frobenius method and exactly satisfy not only the boundary conditions along the radial and circular edges, but also the regularity conditions at the vertex of radial edges. The moment or shear force singularity at the vertex are exactly considered in these solutions. The correctness of these proposed solutions is confirmed by comparing nondimensional frequencies of isotropic plates obtained from the present solutions corresponding to different plate theories with published data obtained from closed form solutions. This thesis also investigates the effects of elastic and shear moduli on the vibration frequencies of the sectorial plates with free or fixed boundary condition along the circumferential edge. A study is also carried out about the influence of elastic and shear moduli on moment or shear force singularity at the plate origin (r=0) for different vertex angles.