含內部點支撐矩形板的自由振動分析

Abstract

本文探討了含內部任意點支撐矩形板的自由振動。使用單一區間的分析方 式，將點支撐的集中反力用雙傅立葉級數展開，再利用Levy的平板振動解 ，推導出具有任意點支撐矩形板自由振動的級數解。此一單區間的分析方 式，有別於一般多區間的分析方法，可以省去將矩形板分成多個子區間來 分析的麻煩。本文分析的平板，其兩個對邊為簡支，另外兩邊為簡支、自 由或固定三種中的一種，而內部點支撐則為任意數目、任意位置。本文所 推導出來的解，解決了控制方程式中所包含的由點支撐所造成的不連續反 力，為一相當簡便的作法。在數值結果方面，則包括了各種參數及各式邊 界條件下的自然頻率、振動模態和收斂性的討論，與多區間解的自然頻率 亦有所比較。

A single domain application of the impulse function approach is used to study the free, flexural vibration problems of Levy- type plates with arbitrary interior point supports. This analysis is based on the representation of the concentrated reactions at the point support locations by a double Fourier sine series expansion of the impulse function. This study provides an extension of the classical Levy's solution in the vibration analysis of plate with interior point supports. The extended solution allows one to solve the problem in a single domain in contrast to the multiple domain solutions available in the literature. This new approach can greatly reduce the complexities of the problems involving arbitrary multiple point supports. The versatility and ability of this approach is demenstrated by considering a rectangular plate with two opposite edges simply supported and classical boundary conditions (free, clamped or simply supported) on the other two edges. The simulation of interior point supports in plates provides a method for presenting the discrete discontinuities in the force field, in the interior of plate, into the governing differential equation and obtaining a concise representation of the solution in a single domain of plate. Numerical results are presented for a number of specific problems, including the convergence and accuracy of the approach, which include natural frequencies and mode shapes of some typical plates with interior point supports. The reults obtained by using the single domain approach are compared with those obtained by using the multiple domain approaches in the literature.