A Study on the Vibration of an Elastic Sphere

彈性球體振動之研究

黃禮棟(Li-Duhng Huarng) ； 陳必超(Pictiaw Chen)

《中華農業研究》 43卷4期 (1994/12) Pp. 439-452

https://doi.org/10.29951/JARC.199412.0010

摘要

在球體座標系中，利用常量及向量方程式代入之分離變數法於理論上分析彈性球體之振動可以得到三角幾何方程式、雷詹德方程式及白賽爾方程式等三種方程式。其中之第一類雷詹德聯合方程式主宰振動模式，而第一類球體座標白賽爾方程式則可導出該振動模式之無次元頻率。一個均質彈性球體最簡單之振動模式為純壓縮模式及純剪切模式。本文中闡釋純壓縮模式及純剪切模式中在某些情況與條件下之無次元頻率之解析方法。雖然各種混合振動模式之無次元頻率之解答因為過於複雜以致於無法使用理論分析而必須借重電子計算機以數值方法計算以求取近似值，但文中仍然列入了由藍姆(Lamb)於1882年所發表之解析方法以及庫克(Cooke)與藍德(Rand)二人利用電子計算機以數值分析法計算所得到溥松比(Poisson's ratio)之值分別為0.1，0.2，0.3及0.4等條件下之無次元頻率的數值供值參考。

關鍵字

振動模式 ； 無次元頻率 ； 彈性球體

並列摘要

A theoretical analysis of the vibration of an elastic sphere in spherical coordinates using the scalar and vector potential equations by the separation of variables method leads to the solutions of the trigonmetric equation, the Legendre equation and the Bessel equation. The associated Legendre polynomials of the first kind govern the mode shapes of vibration while the spherical Bessel functions of the first kind lead to the non-dimensional frequencies of the vibration modes The simplest vibration modes of a homogeneous elastic sphere are the pure compressional modes and the pure shear modes. The procedures to solve the non-dimensional frequencies for some cases of both the pure compressional modes and the pure shear modes are illustrated in the contents. Although, the solutions for the cases of the mixed modes of vibration are too complicated to be solved analytically, a different approach method published by Lamb in 1882 are included. A table for the non-dimensional frequencies of the mixed class vibration obtained numerically with a computer program for Poisson's ratio; v; equals to 0.1, 0.2, 0.3 and 0.4, given by Cooke and Rand are also included for quick reference.

並列關鍵字

Vibration mode ； Non-dimensional frequency ； Elastic Sphere