矩形板自由振動之傅立葉級數求解

Abstract

本研究利用傅立葉餘弦級數求解矩形板之振動，以多項式作為擴充函數修正傅立葉級數逐項微分所造成的問題。本研究所提之級數解，利用兩種方式滿足矩形板各邊界條件及控制方程，求解自然振動頻率：(1)積分轉換為傅立葉級數域、(2)配點法。透過收斂性分析，與文獻結果比較，驗證分析方法及電腦程式之正確性；並比較兩方法優劣，依據結果顯示，轉換於傅立葉級數域之方法收斂性較快，進一步探討不同邊界條件、長寬比對板振動之影響。

This study establishes analytical solutions of vibrations of rectangular plates using Fourier cosine series supplemented with polynomial functions. The transverse displacement of a thin plate is expressed by a double Fourier cosine series and two single Fourier cosine series multiplied with four polynomial functions, respectively, so that the derivatives of the transverse displacement can be obtained from term-by-term differentiations of these series. The unknown coefficients in the series solution are determined by approximately satisfying the boundary conditions and governing equation in the sense of Fourier series expansion or point matching. It is found that the former approach gives faster convergence of natural frequencies than the latter approach after the validity of these two approaches and correctness of developed computer programs are confirmed through convergence studies and comparisons with published results. The approaches are further applied to investigate the vibrations of rectangular plates with different boundary conditions and aspect ratios.