中文摘要
本文探討馬達轉子-軸承-框架系統之結構動態特性，提供動態特性之關係設計過程中，其轉子-軸承系統的動態設計考量方法，利用電腦輔助工程技術，以有限元素方法，先將馬達轉子-軸承-框架系統分離各個部位成為子結構，將子結構個別的數學模型加以離散化，最後以各子結構結合面上節點的動態參數一致性及力系平衡，連結離散化子結構成為整體結構的運動方程式。
在建模型、網格分割及數值求解上，本文使用商用軟體ANSYS，根據分析結果，得到自然頻率與軸承剛度及轉速之關係與模態振形，以及馬達框架之等效剛度等，以此針對馬達轉子-軸承-框架系統之設計提供改善的可行性。
對其馬達轉子進行模態測試實驗，探討分析結果的準確性，並將分析的結果與實驗得到的結果作比對，並根據轉子-軸承系統的物理意義，進行有限元素模型修改，使分析值接近實驗值，以獲得正確的建模方式，提高分析的精準度，以提供給設計者可靠的依據。
關鍵詞：轉子-軸承系統、馬達、有限元素法、模態測試、模態分析、等效剛度。

ABSTRACT
In this thesis, during design processes of finite element model for motor rotor-bearing system, the different design for finite element model of the rotor is described. Researching the rotor-bearing system on the different finite element model, and studying the effects of analysis results of motor rotor-bearing system.
The use of CAE on the analysis of motor rotor-bearing system, During the numeric model processes, the finite element method is used to build the discrete model and analytical theory of the rotor-bearing system. The entire system is divided into several dividable substructures, each of which develops individual systematic mathematic model. Finally, the motion equation of whole structures is combined by the integrity of the dynamic variables and force balances on connections of each substructure.
On numeric solutions, the analysis and calculations are done by the dynamic and static methods from a model built by the commercial finite element software “ANSYS.” In terms of analytical results, such as the relationship and vibration model of natural frequency and the rigidity of the bearing, the feasibility of the design of the above motor rotor-bearing system is recommended.
In order to verify the accuracy of finite element modeling built by ANSYS for the motor rotor-bearing system in this article, during the analyzing, proceeding mode test experiment of motor rotor, and compare the results of both analysis and experiment. The experimental values are the foundation to consider the physical meanings and to suitably modify the modeling in order to decrease the differences between both results, to obtain the correct modeling type, to increase the analysis accuracy, and to provide designers a reliable basis.
Keywords：Rotor-bearing system, Motor, Finite element method, Modal testing, Modal analysis, Equivalent stiffness.

目錄
中文摘要……………………………………………………………..Ⅰ
英文摘要……………………………………………………………..Ⅱ
誌謝…………………………………………………………………..Ⅲ
目錄…………………………………………………………………..Ⅳ
表目錄………………………………………………………………..Ⅵ
圖目錄………………………………………………………………..Ⅶ
第一章 導論…………………………………………………………1
1.1前言…………………………………………………….…..1
1.2文獻回顧……...…………………………………………….3
1.3論文內容簡介….…………………………………………..7
第二章 研究方法……………………………………………………8
2.1以有限元素法建立數學模型………………………………8
2.2有限元素分析…….………………………………………..13
2.2.1靜態分析……………………………………….………14
2.2.2模態分析……………………………………….………15
2.2.3求解……………………….………………….………...15
2.3模態測試實驗………………………………………………16
第三章 馬達轉子軸承系統之模態分析及測試……………………21
3.1自由邊界模態測試實驗……..……………………………..21
3.2自由邊界之轉子模態分析……………………..…………..27
3.2.1三種不同之轉子建模方式…………………………….27
3.2.2分析結果……………………………………………….32
3.3不同軸承剛度及不同轉速下之模態分析…………….…...36
3.3.1 邊界條件………………………………………………36
3.3.2分析結果……….………………………………………37
第四章 馬達框架剛度鑑別…………………………………………46
4.1原理…………………………………………………………46
4.2動剛度………………………………………………………47
4.3等效剛度係數………………………………………………49
4.3.1串聯彈簧之轉子1 DOF系統…………………………49
4.3.2轉子-軸承-框架之2 DOF系統……………………….50
4.3.3等效剛度係數之說明………………………………….52
4.4穩態反應及模態分析………………………………………53
4.4.1建模…………………………………………………….53
4.4.2馬達轉子-軸承-框架系統之動剛度分析……………..58
4.4.3馬達轉子-軸承-框架系統之等效剛度分析.…………..60
4.4.4馬達轉子-軸承-剛性環-框架系統之等效剛度分析….60
第五章結論…………………………………………………………..65
參考文獻……………………………………………………………..67
附錄一………………………………………………………………..72
附錄二………………………………………………………………..73

參考文獻1.Eshleman, R. L., and Eubanks, R. A., “On the Critical Speeds of a Continuous Shaft-Disk System”, ASME Journal of Engineering for Industry, November, pp. 645-652, 1967.2.Eshleman, R. L., and Eubanks, R. A., “On the Critical Speeds of a Continuous Rotors”, ASME Journal of Engineering for Industry, Vol. 91, pp. 1180-1188, 1969.3.Lee, C. W., Katz, R., Ulsoy, A. G., and Scott, R. A., “Modal Analysis of a Distributed Parameter Rotating Shaft”, Journal of Sound and Vibration, 122(1), pp. 119-130, 1988.4.Prohl, M. A., “A General Method for Calculating Critical Speeds of Flexible Rotors”, Journal of Applied Mechanics, Trans ASME, 1945, 12(3):142~148.5.Myklestad, N. O., “A New Method for Calculating Natural Modes of Uncoupled Bending Vibration of Airplane Wings and Other Types of Beams”, Journal of Aero Sci, 1994, 11:153~162.6.Holzer, H., “Die Berechnung der Drehschwingungen”, Julius Springer, 1921, 25.7.Pestel, E. C., Leckie, F. A., “Matrix Methods in Elasto Mechanics”, McGraw-Hill, 1963:51~213.8.Prohl, M. A., “A General Method for Calculating Critical Speeds of Flexible Rotors”, ASME Journal of Applied Mechanics, Vol, 67, pp. A-142-A-148, 1945.9.Lund, J. W., “Stability and Damped Critical Speeds of a Flexible Rotors in Fluid-Film Bearings”, ASME Journal of Engineering for Industry, Vol. 96, pp. 509-516, 1974.10.Bansal, P. N., and Kirk, R. G., “Stability and Damped Critical Speeds of Rotor-Bearing System”, ASME Journal of Engineering for Industry, Vol. 97, pp. 1325-1332, 1975.11.Ruhl, R. L. and Booker, J. F., “A finite element models for distributed parameter turborotor systems”, ASME J. Engineering for Industry, Feb., pp. 126-132, 1972.12.Nelson, H. D. and Mcvaugh, J. M., “The dynamics of rotor-bearing systems using finite elements”, ASME J. Engineering for Industry, 98(2), May, pp. 593-600, 1976.13.Zorzi, E. S. and Nelson, H. D., “Finite element simulation of rotor-bearing systems with internal damping”, ASME J. Engineering for Power, Jan., pp. 71-76, 1977.14.Zorzi, E. S. and Nelson, H. D., “The dynamics of rotor-bearing systems with axial toque – a finite element approach”, ASME J. Mechanical Design, 102, Jan., pp. 158-161, 1908.15.Nelson, H. D., “A finite rotating shaft element using Timoshenko beam theory”, ASME J. Mechanical Design, 102, pp. 793-803, 1980.16.Greenhill, L. M., Bickford, W. B., and Nelson, H. D., “A conical beam finite element for rotor dynamic analysis”, ASME J. Vib. Acoustics, Stress, Reliability in Design, 107, pp. 421-430, Oct., 1985.17. zg ven, H. and zkan, L. Z., “Whirl speeds and unbalance response of multi-bearing rotors using finite elements”, ASME J. Vib. Acoustics, Stress, and Reliability in Design, 106, pp. 72 – 79, 1984.18.Adams, M. L., “Nonlinear Dynamics of Flexible Multi-Bearing Rotors”, Journal of Sound and Vibration ,Vol. 71, pp. 129-144, 1980.19.Jeffcott H H. The Lateral Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed the Effect of Want of Balance. Phil. mag., 1919,37.20.Kang, Y., Shih, Y. P. and Lee, A. C., “Inverstigation on the Steady-state Responses of Asymmetric Rotors”, ASME J. Vib. Acoustics, 114(April). pp. 144-208, 1892.21.Genta, G., “Whirling of Unsymmetrical Rotors：A Finite Element Approach Based on Complex Coordinates”, Journal of Sound and Vibration, 124(1), pp. 27-53, 1988.22.Rieger, N. F., “A Comprehensive guide to computer programs for analysis rotor systems”, Machine design, 22, Jan., pp. 89-95, 1976.23.Firoozian, R. and Stanway, R., “Design and application of a finite element package for modelling turbomachinery vibrations”, J. Sound and Vibration, 134, pp. 115-137, 1989.24.Reddy, V. R. and Sharan, A. M., “The finite element modeled design of lathe spindles: the static and dynamic analyses”, Int. J. Vib. Acoust. Stress Reliabil. Des., 109, pp. 407-415, 1987.25.Wang, W. R. and Chang, C. N., “Dynamic analysis and design of a machine tool spindle-bearing system”, Int. J. Vib. Aeoust., 116, pp. 280-285, 1994.26.Choi, J. K. and Lee, D. G., “Characteristics of a spindle bearing system with a gear located on the bearing span”, Int. J. Mach. Tools Manufact., 37, pp. 171-181, 1997.27.Lin, Y. Cheng, L. and Huang, T. P., “Optimal design of complex flexible rotor-support systems using minimum strain energy under multi-constraint conditions,” Journal of Sound and Vibration, 215(5), pp. 1121-1134, 1998.28.Aini, R., Rahnejat, H. and GOHAR, R., “A five degrees of freedom analysis of vibrations in precision spindles”, Int. J. Mach. Tools Manufact., 30, pp. 1-18, 1990.29.Al-Shareef, K. J. H. and Brandon, J. A., “On the effects of variations in the design parameters on the dynamic performance of machine tool spindle-bearing systems”, Int. J. Mach. Tools Manufact., 30, pp. 431-445, 1990.30.Brandon, J. A. and Al-Shareef, K. J. H., “On the validity of several common assumption in the design of machine tool spindle-bearing systems”, Int. J. Mach. Tools Manufact., 31, pp. 235-248, 1991.31.Conry, T. F., Goglia, P. R., and Cusano, C., “A minimum strain energy approach for obtaining optimal unbalance distribution in flexible rotors,” Journal of Mechanical Design, 104, pp. 875-880, 1982.32.Barrett, L. E., Gunter, E. J., and Allaire, P. E., “Optimum bearing and support damping for unbalance and stability of rotating machinery,” Journal of Engineering for Power, 100(1), pp. 89-94, 1978.33.Brandon, J. A. and Al-Shareef, K. J. H., “Optimization strategies for machine tool spindle-bearing systems: a critical review”, J. Eng. Ind., 114, pp. 244-253, 1992.34.Kang, Y., Chen, H. C., and Liu, J. J., 1995, "Dynamic Analysis of Rotor-Bearing System Using ANSYS," ANSYS Users Conference, Tau-Yuan, Taiwan.35.Kang, Y., Chang, Y. P., Tasi, J.-W., Tang, P.-H, and Chang, Y.-F, 2000, “An Investigation in Stiffness Effects on Dynamics of Rotor-Bearing-Foundation Systems,” Journal of Sound and Vibration, 231(2), pp.343-374.36.Kang, Y., Chang, Y.P.,Tsai,J.W., Chen, S.C.,Yang, L.K., 2001, “Integrated “CAE” Strategies for the Design of Machine Tool Spindle-Bearing-Systems,” Finite Element in Analysis and Design 37, pp.485-511.37.康淵, 2002, “馬達臨界轉速分析及測試方法之建立結案報告”, 2月, 東元電機股份有限公司.