含裂紋任意四邊形平板之振動分析

Vibration Analysis of the Arbitrarily Quadrilateral Plates with Crack

林昌弘(Chang- Hong Lin)

指導教授 ： 施延欣

中原大學/工學院/機械工程研究所/碩士(2006年)

https://doi.org/10.6840/cycu200600075

摘要

考慮一個任意四邊形的薄板，而在這薄板含有一單邊裂紋，本文探討裂紋長度對任意四邊形含裂紋薄板振動之影響，此任意四邊形薄板則利用簡支撐的方式將四周固定，施以週期性均勻的平面向負載，來探討含裂紋的任意四邊形平板疲勞裂紋長度對振動的影響，在von Karman大變形理論，推導含裂紋平板的運動方程式，再將其運動方程式，經過總體座標系統(x-y)與自然座標系統(ζ-η)之間的轉換，所得受平面向力作用、不同裂紋長度和縱橫比(aspect ratio)影響的線性自然振動頻率，和已知文獻結果相比較，其結果符合本文所推導之公式，所以本文公式可適用於任意四邊形板，如正方形板、長方形板、梯形板、菱形板、平行四邊形板等等。文中，運用Galerkin的方法，將統御方程式化簡為一個以時間為變數的Mathieu方程式、至於暫態振動的部份，則使用四階Runge-Kutta的方法，將振幅對時間的圖表結果分別繪出，裂紋長度對振動的關係也同時被描述及探討。由於本文中任意四邊形平板可針對含裂紋的正方形板、長方形板、平行四邊形板、菱形版、梯形板作振動分析，提供一個完整的分析步驟是本研究主要的貢獻。

關鍵字

四邊形平板 ； 裂紋長度 ； 線性自然頻率 ； 振動

並列摘要

The analysis of vibration for the arbitrarily quadrilateral plates with an edge crack is subjected to in-plane period forces are presented. The arbitrarily quadrilateral plates are simply supported at all edges .The equations of motions based on von Karman’s plate theory is considered in vibration analysis and transformations between global-coordinate(x - y) and natural-coordinate(ζ-η) are used. The linear natural frequencies , which are affected by in-plane period forces, crack length and aspect ratio, are compared with the results in literature and they are close. The equations of vibration are reduced to an ordinary differential equation by assuming mode shape and Galerkin’s procedure. The natural frequency is determined by this ordinary different equation .The amplitude is determined by means of Runge-Kutta scheme. The results for square,rectangular,rhombic, parallelogram and trapezoidal cracked plates are provided in this study. The effect of vibration on the cracked arbitrary quadrilateral plate is determined and discussed.

並列關鍵字

crack length ； linear natural frequency ； vibration ； cracked plate

參考文獻

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6. R. Solecki, “Bending vibration of a simply supported rectangular plate with a crack parallel to one edge”, Engrg. Frac. Mech., 18(6), pp. 1111-1118, 1983.