具移動質量與裂紋的簡支樑之振動與疲勞分析

Analysis of Vibration and Fatigue of a Simply Supported Beam with a Moving Mass and a Crack

王文龍(Wen-Lung Wang)

指導教授 ： 施延欣

中原大學/工學院/機械工程研究所/碩士(2003年)

https://doi.org/10.6840/cycu200300305

摘要

摘要
在本文中探討一含直裂式裂紋之圓形截面及方形截面的樑，在振動過程中，考慮一個移動質量在樑上做簡諧運動，並依據漢米爾頓(Hamilton)準則求取運動方程式、邊界條件和形狀函數，在開放式裂紋勁度的推導採用破壞力學的理論。
文中，運用Galerkin 的方法，將統御方程式化簡為一個以時為變數的Mathieu方程式和無裂紋時之勁度方程式。而在振動的部份，則使用Runge-Kutta的方法來描述振幅對對負載週期的關係。在疲勞裂紋成長上，採用Modified Forman model方程式來計算出疲勞裂紋成長與次數之間的關係，而得有關振動對疲勞裂紋的影響，裂紋成長對振動的關係也同時被描述及探討。在振動和疲勞裂紋的分析中本文採用呼吸式裂紋模型代替傳統使用開放式裂紋模型。對於頻率的變化、疲勞裂紋成長，使用呼吸式裂紋模型分析可以更客觀的描述振動的過程和裂紋成長的現象。由於文獻及現有分析軟體對耦合分析的缺乏，因此對含裂紋軸的振動與疲勞裂紋成長分析，提供一個完整的分析步驟是本研究主要貢獻。

關鍵字

振動 ； 疲勞裂紋成長 ； 簡諧運動

並列摘要

ABSTRACT
In this study, the small deformations of rectangular cross section beam and circular cross section beam are considered. Including the simple harmonic motion, the moving mass is considered during vibration procedure. The equation of motion and boundary conditions are derived by Hamilton’s principle. The stiffness with opening crack is derived by fracture mechanics. Mathieu equation and the stiffness are without crack derived by Galerkin’s method. For opening and breathing crack, the stiffnesses are used to replace the stiffness without crack. The 4th order Runge – Kutta method to determine the relation of amplitude and time is used. Modified Forman equation is used to calculate the relation of fatigue crack growth and loading cycles. The effect of vibration on fatigue life and the interaction between vibration and fatigue are analyzed by breathing crack theory. That the breathing crack model is applied to describe the phenomenon of frequency response and fatigue crack growth is more realistic. The vibration and fatigue crack growth is lacking in the literature and the commercial software of fatigue, providing a procedure of coupling analysis for the cracked beam is the major accomplishment.
(Keywords: simple harmonic motion, vibration, fatigue crack growth )

並列關鍵字

vibration ； simple harmonic motion ； fatigue crack growth

參考文獻

[12] R. G. Forman, V. E. Kearney and R. M. Engle, Journal of Basic Engineering, Vol.9, pp.459-464, 1968.

[1] A. K. Pandey, M. Biswas and M. Samman, Damage detection from change in curvature model shape, Journal of Sound and Vibration, Vol.145, pp.321-332, 1991.

[2] T. C. Tsai and Y. Z. Wang, Vibration analysis and diagnosis of a cracked shaft, Journal of Sound and Vibration, Vol.192, pp.607-620, 1996.

[3] C. A. Papadopoulos and A. D. Dimarogonas, Coupled longitudinal vibrations of a cracked shaft, Journal of Vibration, Acoustics, Stress and Reliability in Design, Vol.110, pp.1-8, 1988.

[4] T. G. Chondros, A. D. Dimarogonas and J.Yao, A consistent cracked bar vibration theory, Journal of Sound and Vibration, Vol.200, pp.303-313, 1997.