臺灣博碩士論文加值系統

傳動機械中使用含裂紋軸的安全性問題，在過去二十多年間受到了廣泛的重視。許多工程師及科學家試圖以振動學的方法，來建立含裂紋軸的動態特性。但是對整體的振動之動態特性分析是基於裂紋大小固定的假設。
含裂紋軸受到週期性軸向力與彎矩，裂紋深度隨反覆負載次數會出現裂紋成長的現象。基本上，此一受反覆軸向力與彎矩的軸，是一個強制振動(forced vibration)的問題。
由於裂紋的出現會影響轉軸系統的剛性，而轉軸剛性的變化則進一步改變整個轉軸系統的動態特性。因此自1970年代中期開始，含裂紋轉軸問題的研究就受到廣泛的重視，首先是由Dimarogonas[1]以轉軸柔度量測的方式，計算其局部撓度及轉軸系統勁度(Stiffness)，並據以分析系統之振動反應頻譜; Gasch[2] 及Herry[3]等人進而應用此法於De laval rotor之無質量轉軸系統的反應頻譜分析上，並以此方式建立線上裂縫/振動監測系統（On line crack/Vibration montoring system），希望透過振動訊號的量測與分析，能在轉軸發生異常的初期,即可檢測出破壞的位置及損壞的程度，藉此可避免設備因不可預期的轉軸斷裂，造成無法預估的損失。並藉此加以確保轉軸的可用性及可靠度。
自1960年代末期破壞力學快速發展後，到1967年才由Liebowitz[4] 引用Irwin[5]的應力強度因子理論的觀念，重新以破壞力學觀點定義裂縫對轉軸系統勁度變化。接著如Rice及Levy[6]分別考慮軸力拉伸及彎曲力矩對轉軸勁度的影響。在1980年Diamarogonas及Massouros[7]認為剪應力 (Shear stress) 對轉軸系統勁度的影響頗大。因此又採用 Timoshenko beam theory 轉軸來取代 Euler-Bernoulli beam theory 的轉軸。但綜合以上振動特性之研究及分析了解，當裂縫深度尚淺時，對系統振動特性的改變量微乎其微，幾乎無法精確量測到振動訊號的變化；當裂縫深度達到轉軸半徑的20% 以上時，才能穩定量測系統振動特性的變化。因此如何精進量測技術與設備，成為振動監測系統精確度的重大課題；因此 Iman[8] 採用 Histogram signature analysis technique，由 Cole[9] 所發展之隨機遞減法(Random decrease method)來改善飄移之裂縫高頻訊號的萃取效益。但是以上增進量測技術的方法，成果似乎都不十分顯著。最早期的疲勞壽限評估，就是開始於對不含裂縫之轉軸系統進行完全相反之反覆式負載壽限評估。其所採用的方法不外乎 Wholer的 S-N curve 或是包含評估平均應力效應之 Goodman Diagram，在當時破壞力學尚未萌芽。到 1960 年代末期Liebowitz 以 應力強度因子為參數，建立6*6的局部柔性矩陣(Local compliance matrix) 以模擬轉軸裂縫之局部柔性，由於其所採用裂縫之幾何形狀僅為平板 (Sheet)而非真正轉軸 的破裂模式因而尚有爭議。
施教授[10]曾以破壞力學理論為基礎，建立合理的轉軸裂縫應力強度因子關係式，並據以發展一套含裂縫轉軸疲勞壽限的分析模式，以正確評估轉軸系統在固定及變動之週期性應力作用下的疲勞壽限。
裂紋的疲勞成長速率 ，受結構及裂紋的幾何形狀、材料的性質、負載的型態、大小…等各種因素的影響，是一個相當複雜的問題。1960年，Paris [11] 首先建議以應力強度因子幅，作為評估疲勞裂紋成長的參數，並提出疲勞裂紋成長速率的模式。
含裂紋轉軸，受到週期性軸向力與彎矩，裂紋深度隨反覆負載次數會出現裂紋成長的現象。基本上，此一受反覆軸向力與彎矩轉軸，是一個強制振動(forced vibration)的問題。Papadopoulos and Dimarogonas[12-15]已對此含裂紋軸的橫向與縱向振動作探討。 Huang and Shieh[16]亦曾研究含裂紋的迴轉軸之振動與穩定性，另外，王有任教授[17,18]以Timoshenko連續樑理論為基礎，對含裂紋軸在受到扭力與軸向力作用下的動態行為作探討。以上諸文獻對此一問題之振動行為，已有詳盡的探討，並有良好的數值結果。其中裂紋深度將影響桿件之自然頻率及動態特性。不過上述文獻的分析是基於裂紋長度不會隨著成長的假設。
含裂紋軸的疲勞分析，在NASGRO[19]軟體及其他分析軟體或公式中，裂紋深度受反覆負載次數增加而增加。然而負載頻率與桿件自然頻率相同而產生的共振問題，在實際上，應會影響疲勞裂紋的成長，但此一現象，在現行的疲勞分析中，尚未被考慮。
在本篇文章中，結合振動與疲勞分析，並探討在疲勞裂紋成長的情況下的振動行為，其中含裂紋桿件的自然頻率受裂紋大小而改變，當裂紋成長時，自然頻率是可變的，因此振動的振幅會受到負載頻率的影響。

The shaft serves as the power transmission in the dynamic machine, and is the major component of rotational mechanism. The shaft applies extensively in industry, nuclear energy, steampower or hydropower electric factory. For example, turbine shaft, pump shaft, rotor in electric factory are the important part in whole facilities. The shaft may occur crack after long term, continuous and high speed operating, so it requires overhaul. While the shaft damage suddenly in operating, it may further damage other circumferential facilities and results in time-consuming overhaul or even affects human being life and property. Therefore, in order to ensure integrity of the whole facilities, it is the important work to estimate availability of the cracked shaft.
The appearance of crack influences the rigidity of the shaft system and variation of shaft rigidity further changes dynamic property of the whole shaft system. The research of cracked shaft has received wide attention since the medium-term in 1970. Dimarogonas [1] used shaft compliance measurement to compute local flexibility and stiffness of the shaft system and the vibration of the system was studied also. Gasch [2] and Herry [3] applied this method in de laval rotor analysis of no mass shaft system and used this way to found on line crack/vibration montoring system. They hoped by way of measuring and analyzing vibration information could overhaul the location of fracture and the degree of damage when the shaft occurred unusual in the initial stage. By this way, it could avoid causing unable computing loss of the facilities due to against expectation of the shaft fracture and in addition to ensure availability and reliability of the shaft.
Fracture mechanics has developed quickly since the last phase in 1960. In 1967, Liebowitz [4] cited Irwin’s [5] theory of stress intensity factor and based on fracture mechanics defining the crack versus the variation of stiffness in the shaft system. Rice and Levy [6] considered axial force and bending moment effected on the shaft stiffness. In 1980, Dimarogonas and Massouros [7] recognized shear stress had a great influence on the shaft stiffness of the system. So, they used Timoshenko’s beam theory shaft replaced Euler- Bernoulli’s beam theory shaft. Synthesizing above research and analysis of vibration property, the variation of vibration property in system is very small and it almost unable measures accurately the variation of vibration information when the crack depth is shallow. One can measure stably the variation of vibration property in system when the crack depth over 20% of the shaft radius. Therefore, improving measuring technology and facilities becomes significant topic of Vibration Montoring System. Iman [8] used histogram signature analysis technique. Random Decrease Method developed by Cole [9] that improved high frequency information collecting benefit of the floating crack. Achievement is similar to not conspicuous above the method of improving measuring technology. Fatigue life estimation in the most early stage is beginning to proceed life estimation of complete opposite repeated load in the shaft system. The method is not beyond the scope of S-N curve of Wholer or Goodman Diagram of estimating average stress effect when fracture mechanics is not sprout. Up to the last phase in 1960, Liebowitz used stress intensity factor as parameter to found 6*6 local compliance matrix and imitated local compliance in the cracked shaft. It had argument because he used geometric shape of the crack was only sheet and was not real fracture form of the shaft.
Shih [10] has founded reasonable relative formula of stress intensity factor of the cracked shaft based on fracture mechanics. According to it developing a suit analytic form of fatigue life of the cracked shaft and estimating fatigue life correctly for the cracked shaft system with firm and variant periodical stress.
The effects on the fatigue crack growth rate of several parameters such as structure, geometric shape of the crack, material character, type and size of load is a very complicated problem. In 1960, Paris [11] suggested to use Stress Intensity Factor range as the parameter of estimating fatigue crack growth and presented the form of fatigue crack growth rate.
The cracked shaft with alternating axial force and bending moment is analyzed in the present work. The crack depth grows with repeated load. In the main, the shaft with repeated axial force and bending moment is the forced vibration problem. Papadopoulos and Dimarogonas [12-15] have already to probe into the cracked shaft with the longitudinal and transverse vibration. Huang, Huang and Shieh [16] also have studied the vibration and stability of the cracked shaft. Besides, Wang [17, 18] has researched the dynamic behavior of the cracked shaft with axial force and bending moment based on Timoshenko continuous beam theory. The above-mentioned literature is directed against the vibration behavior of this problem have detailed research and good numerical results. The crack depth has influence on the natural frequency and dynamic character of the shaft.
In NASGRO [19] software and other analytic software or formula, the crack depth increases with the increment number of cyclic load. However, resonance occurs when loading frequency is the same as natural frequency, it should have influence on fatigue crack growth in practice. This resonance phenomenon has not been considered in fatigue analysis at present.
The present study combines the vibration and fatigue analysis to discuss the vibration behavior of the cracked shaft with the crack growth. The natural frequency of the shaft depends on the crack size. The natural frequency is changed when the crack is growing. Therefore the amplitude of vibration is affected by the loading frequency.

Chapter1 Introduction
Chapter2 Equation of motion
Chapter3 Local flexibility matrix
Chapter4 Analytical procedure
Chapter5 Fatigue crack growth
Chapter6 Result and discussion
Chapter7 Conclusion

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