在本論文中，將討論含開啟裂縫或開合裂縫的齒輪轉子軸承系統的動態特性。首先，發展一個含開啟裂縫的齒輪轉子系統的簡單解析模型來計算含開啟裂縫的齒輪轉子系統的自然頻率、臨界轉速、穩態響應及穩定性。在裂縫模擬方面，將裂縫所造成轉軸局部撓屈的現象，利用能量平衡的觀念以一組線性彈簧取代。因為裂縫的影響，將造成一個週期性的時變系統。利用快速傅立葉轉換(FFT)可以將時域的振動響應轉換成頻域的頻譜圖來研究裂縫的影響。並利用傅羅凱理論(Floquet theory)計算出系統的穩定性。此外，本文發展有限元素模型來分析含開啟或開合裂縫的齒輪轉子軸承系統的自然頻率、臨界轉速、穩態響應及穩定性。

In this thesis, the dynamic analysis of a geared rotor-bearing system with an open crack or a breathing crack is discussed. A simple analytic model of a geared rotor system is developed by taking into account the effect of an open transverse surface crack for determination of natural frequencies, critical speeds, and stability of the system. The existing crack results in a local flexibility of the shaft and it is replaced by a set of springs based on the balance of energy. The effect of the crack results in a periodic time-varying system. From the response time histories, the FFT spectra can be obtained for investigating the effects of crack. The Floquet theory is used to examine the stability of the first variation equations with periodic coefficients. Finite element models are developed for geared rotor-bearing system with an open transverse surface crack and with a breathing transverse surface crack for determination of natural frequencies, critical speeds, dynamic response and stability.
The results show that additional instability regions of the shaft’s spin speed are observed when a crack occurs on the shaft. The influences of the crack on the stability of the system are significant as the depth increases to an evident value.

Anifantis, N. and Dimarogonas, A. D., 1983, “Stability of Columns with a Single Crack Subjected to Follower and Vertical Loads,” International Journal of Solids Structures, Vol. 19, pp. 281-291.
Broek, D., 1986, Elementary Engineering Fracture Mechanics, Martinus Nijhoff Publishers, Boston.
Choi, S. T. and Mau, S. Y., 2001, “Dynamic Analysis of Geared Rotor-Bearing Systems by the Transfer Matrix Method,” ASME Journal of Mechanical Design, Vol. 123, No. 4, pp. 562-568.
Chondros, T. G. and Dimarogonas, A. D., 1989, “ Dynamic Sensitivity of Structures to Cracks,” ASME Journal of Vibration and Acoustics, Stress and Reliability in Design, Vol. 111, pp. 251-256.
Choy, F. K. and Li, W., 1987, “Frequency Component and Modal Synthesis Analysis of Large Rotor-Bearing Systems with Base Motion Induced Excitations,” The Journal of Franklin Institute, Vol. 108, No. 3, pp. 145-156.
Choy, F. K., Tu, Y. K., Zakrjsek, J. J. and Townsend, D. P., 1991, “Effect of Gear Box Vibration and Mass Imbalance on the Dynamics of Multistage Gear Transmission,” ASME Journal of Vibration and Acoustics, Vol. 113, pp. 333-344.
Christides, S. and Barr, A. D. S., 1984, “One-Dimensional Theory of Cracked Bernoulli-Euler Beams,” International Journal of Mechanical Science, Vol. 26, pp. 639-648.
Collins, K. R., Plaut, R. H. and Wauer, J., 1991, “Detection of Cracks in Rotating Timoshenko Shafts Using Axial Impulses,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 113, pp. 74-78.
Collins, K. R., Plaut, R. H. and Wauer, J., 1992, “Free and Forced Longitudinal Vibrations of a Cantilevered Bar with a Crack,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 114, pp. 171-177.
David, J. W. and Mitchell, L. D., 1986, “Linear Dynamic Coupling in Geared Rotor System,” ASME Journal of Vibration, stress and Reliability in Design, Vol. 108, pp.171-176.
David, J. W. and Park, N. G., 1987, “The Vibration Problem in Gear Coupled Rotor Systems,” American Society of Mechanical Engineers DE-Vol. 2, Rotating Machinery Dynamics, Volume One, Proceeding of the 11th Biennial Conference on Mechanical Vibration and Noise, Boston, U. S. A., pp. 297-303.
Daws, J. W., 1979, An Analytical Investigation of Three Dimensional Vibration in Gear-Coupled Rotor System, Ph. D. Dissertation, Virginia Polytechnic Institute and State University.
Dimarogonas, A. D. and Massouros, G., 1981, “Torsional Vibration of a Shaft with a Circumferential Crack,” Engineering Fracture Mechanics, Vol. 15, pp. 439-444.
Dimarogonas, A. D. and Paipetis, S. A., 1983, Analytical Methods in Rotor Dynamics, Applied Science Publishers, London and New York.
Dimarogonas, A. D. and Papadopoulos, C. A., 1983, “ Vibration of Cracked Shafts in Bending,” Journal of Sound and Vibration, Vol. 91, pp. 583-593.
Gasch, R., 1976, “Dynamic Behavior of a Simple Rotor with a Cross-Sectional Crack,” Vibrations in Rotating Machinery: Institution of Mechanical Engineers, London, pp. 123-148.
Gasch, R., 1993, “A survey of the Dynamic Behaviour of a Simple Rotating Shaft with a Transverse Crack,” Journal of Sound and Vibration, Vol. 160, pp. 313-332.
Grabowski, B., 1980, “The Vibrational Behavior of a Turbine Rotor Containing a Transverse Crack,” ASME Journal of Mechanical Design, Vol. 102, pp. 140-146.
Hagiwara, N., Ida, M. and Kikuchi, K., 1981, “Forced Vibration of a Pinion-Gear System Supported on Journal Bearing,” Proceeding International Symposium of Gearing and Power Transmissions, Tokyo, pp. 85-90.
Hamad, B. M. and Seireg, A., 1979, “Whirl of Geared Systems Supported on Hydrodynamic Bearings,” Proceeding of the International Federation on the Theory of Machine and Mechanisms, Fifth World Congress, Montreal, pp. 1109-1112.
Hamad, B. M. and Seireg, A., 1980, “Simulation of Whirl Interaction in Pinion-Gear Systems Supported on Oil Film Bearings,” ASME Journal of Engineering for Power, pp. 508-510.
Henry, T. A. and Okah-Avae, B. E., 1976, “Vibration in Cracked Shafts,” Vibrations in Rotating Machinery: Institution of Mechanical Engineers, London, pp. 15-19.
Huang, S. C., Huang, Y. M. and Shieh, S. M., 1993, “Vibration and Stability of a Rotating Shaft Containing a Transverse Crack,” Journal of Sound and Vibration, Vol. 162, pp. 387-401.
Iida, H., Tamura, A., Kikuchi, K. and Agata, H., 1980, “Coupled Torsional-Flexural Vibration of Shaft in a Geared System of Rotors,” Bulletin of the Japan Society of Mechanical Engineers, Vol. 23, No. 186, pp. 2111-2117.
Iida, H., Tamura, A. and Yamamoto, H., 1986, “Dynamic Characteristics of a Gear Train System with Softly Supported Shafts,” Bulletin of the Japan Society of Mechanical Engineers, Vol. 29, No. 186, pp. 1811-1816.
Iwatsubo, T., Arii, S. and Kawai, R., 1984a, “Coupled Lateral-Torsional Vibration of Rotor System Trained by Gears,” Bulletin of the Japan Society of Mechanical Engineers, Vol. 27, pp. 271-277.
Iwatsubo, T., Arii, S. and Kawai, R., 1984b, “The Coupled Lateral Torsional Vibration of a Geared Rotor System,” Proceeding of the 3rd International Conference on vibration in Rotating Machinery, Institution of Mechanical Engineers, pp. 59-66.
Ju, F. D. and Mirovich, M. E., 1988, “Experimental Diagnosis of Fracture Damage in Structures by the Modal Frequency method,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 110, no. 3, pp. 456-463.
Kahraman, A., Ozguven, H. N., Houser, D. R. and Zakrajsek, J. J., 1992, “Dynamic Analysis of Geared Rotors by Finite Elements,” ASME Journal of Mechanical Design, Vol. 144, pp. 507-514.
Kikids, M. L. and Papadopoulos, C. A., 1992, “Slenderness Ratio Effect on Cracked Beam,” Journal of Sound and Vibration, Vol. 155, pp. 1-11.
Krawczuk, M. and Ostachowicz, W. M., 1993, “Influence of a Crack on the Dynamic Stability of a Column,” Journal of Sound and Vibration, Vol. 167, pp. 541-555.
Lalanne, M. and Ferraris, G., 1990, Rotordynamics Prediction in Engineering, John Wiley and Sons, New York.
Liang, R. Y., Hu, J. and Choy, F., 1992, “Theoretical Study of Crack-Induced Eigenfrequency Changes on Beam Structures,” Journal of Engineering Mechanics, Vol. 118, pp. 384-396.
Liebowitz, H., Vanderveldt, H. and Harris, D. W., 1967, “Carrying Capacity of Notched Column,” International Journal of Solids and Structures, Vol. 3, pp. 489-500.
Lund, J., 1978, “Critical Speed, Stability and Response of a Geared Train of Rotors,” ASME Journal of Mechanical Design, Vol. 100, pp. 535-538.
Mitchell, L. D. and David, J. W., 1985, “Proposed Solution Methodology for the Dynamically Coupled Nonlinear Geared Rotor Mechanics Equation,” ASME Journal of Vibration, stress and Reliability in Design, Vol. 107, pp.112-116.
Mitchell, L. D. and Daws, J. W., 1983, A Basic Approach to Gearbox Noise Prediction, Society of Automotive Engineers Trans 9, pp. 3366-3379.
Mitchell, L. D. and Mellen, D. M., 1975, “Torsional-Lateral Coupling in a Geared High-Speed Rotor System,” ASME paper 75-DET-75
Nelson, H. D. and Nataraj, C., 1986, “The Dynamics of a Rotor System with a Crack shaft,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, pp. 189-196.
Neriya, S. V., Bhat, R. B. and Sanker, T. S., 1984, “Effect of Coupled Torsional- Flexural Vibration of a Geared Shaft System on the Dynamic Tooth Load,” The Shock and Vibration Bulletin, Vol. 54, pp. 67-75.
Neriya, S. V., Bhat, R. B. and Sanker, T. S., 1985, “Coupled Torsional-Flexural Vibration of a Geared Shaft System Using Finite Element Analysis,” The Shock and Vibration Bulletin, Vol. 55, pp. 13-25.
Oda, S. and Miyachika, K., 1985, “Dynamic Behavior of Thin-Rimmed Spur Gears with Various Web Arrangements,” Bulletin of the Japan Society of Mechanical Engineers, pp. 2434-2441.
Ozguven, H. N., 1991, “A Non-Linear Mathematical Model for Dynamic Analysis of Spur Gears Including Shaft and Bearing Dynamics,” Journal of Sound and Vibration, Vol. 145, No. 2, pp. 239-260.
Ozguven, H. N. and Houser, D. R., 1988, “Dynamic Analysis of High Speed Gears by Using Loaded Static Transmission Error,” Journal of Sound and Vibration, Vol. 125, pp. 71-83.
Pandey, A. K. and Biswas, M., 1994, “Damage Detection in Structures Using Changes in Flexibility,” Journal of Sound and Vibration, Vol. 169, pp. 3-17.
Papadopoulos, C. A. and Dimarogonas, A. D., 1987a, “Coupled Longitudinal and Bending Vibrations of a Rotating Shaft with an Open Crack,” Journal of Sound and Vibration, Vol. 117, pp. 81-93.
Papadopoulos, C. A. and Dimarogonas, A. D., 1987b, “Coupling of Bending and Torsional Vibration of a Cracked Timoshenko Shaft,” Ingenieur-Archiv, Vol. 57, pp. 257-266.
Papadopoulos, C. A. and Dimarogonas, A. D., 1988, “Stability of Cracked Rotors in the Coupled Vibration Mode,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 110, no. 3, pp. 356-359.
Papadopoulos, C. A. and Dimarogonas, A. D., 1992, “Coupled Vibration of Cracked Shafts,” Journal of Vibration and Acoustics, Vol. 114, pp. 461-467.
Petroski, H. J., 1981, “Simple Static and Dynamic Models for the Cracked Elastic Beam,” International Journal of Fracture, Vol. 17, pp. 71-76.
Petroski, H. J., 1985, “Simple Models for the Stability of a Crack in a Cantilever Beam Subjected to Impact,” Engineering Fracture Mechanics, Vol. 21, pp. 377-381.
Qian, G. L., Gu, S. N. and Jiang, J. S., 1990, “The Dynamic Behaviour and Crack Detection of a Beam with a Crack,” Journal of Sound and Vibration, Vol. 138, pp. 233-243.
Rajab, M. D. and Al-Sabeeh, A., 1991, “ Vibrational Characteristics of Cracked Shafts,” Journal of Sound and Vibration, Vol. 147, pp. 465-473.
Rao, J. S., Chang, J. R. and Shiau, T. N., 1995, “Coupled Bending-Torsional Vibration of Geared Rotors,” American Society of Mechanical Engineers, Design Engineering Technical Conference, Boston, DE- Vol. 84-2, Vol. 3, Part B, pp. 977-991.
Rice, J. R. and Levy, N., 1972, “The Part-Through Surface Crack in an Elastic Plate,” Journal of Applied Mechanics, Vol. 39, pp. 185-194.
Richard, J. A., 1983, Analysis of Periodically Time-Varying Systems, Berlin: Springer-Verlag.
Rizos, P. F., Aspragathos, N. and Dimarogonas, A. D., 1990, “Identification of Crack Location and Magnitude in a Cantilever Beam from the Vibration Modes,” Journal of Sound and Vibration, Vol. 138, pp. 381-388.
Schwibinger, P. and Nordmann, R., 1988, “The Influence of Torsional-Lateral Coupling on the Stability Behavior of Geared Rotor Systems,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 110, pp. 563-571.
Seireg, A. and Houser, D. R., 1970, “ Evaluation of Dynamic Factors for Spur and Helical Gears,” ASME Journal Engineering for Industry, 192, pp. 504-515.
Sekhar, A. S. and Prabhu, B. S., 1992, “Crack Detection and Vibration Characteristics of Cracked Shafts,” Journal of Sound and Vibration, Vol. 157, pp. 375-381.
Sekhar, A. S. and Prabhu, B. S., 1994, “Vibration and Stress Fluctuation in Cracked Shafts,” Journal of Sound and Vibration, Vol. 169, pp. 655-667.
Shen, M. H. and Chu, Y. C., 1992, “Vibrations of Beams with a Fatigue Crack,” Computers and Structures, Vol. 45, pp. 79-93.
Shiau, T. N. and Hwang, J. L., 1993, “Generalized Polynomial Expansion Method for the Dynamic Analysis of Rotor-Bearing Systems,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 115, pp. 209-217.
Simmons, H. R. and Smalley, A. J., 1984, “Lateral Gear Shaft Dynamics Control Torsional Stresses in Turbine-Driven Compressor Train,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 106, pp. 946-951.
Tada, H., Paris, P. C. and Irwin, G. R., 1985, The Stress Analysis of Cracks Handbook, Hellertown, Pensylvania: Del Research Corporation.