歪斜軸齒輪在目前已有許多的應用，尤其以蝸桿蝸輪組與戟齒輪組最被廣泛應用。但是以戟齒輪為例，不同的齒輪製造廠商所製造之齒輪組彼此間卻不能互換使用，不同系統之間缺乏共通性，不利於齒輪研發成果的整合應用，所以歪斜軸齒輪組亟待建立一套通用的數學建構模式。
本研究根據剛體運動理論與共軛嚙合原理，建構歪斜軸齒輪組之參數化數學模式與偏微分拘束方程式。從偏微分拘束方程式求解，提出具有線接觸型態之新型歪斜軸共軛齒形，並對所提出之齒面參數進行定性分析。最後並將所建構之線接觸型態歪斜軸齒輪組，利用運動模擬軟體進行運轉模擬測試，提供作為齒面理論與齒輪性能之佐證。

Presently, there are a lot of applications of gear sets with skew axes, some of them, especially worm gear sets and hypoid gear sets, are widely used. Take hypoid gear as example, gear sets produced by different gear factories can’t fit to each other. Due to the lacking in common properties among different systems, it is disadvantageous to integrated application of development of gear researches. Therefore, a common mathematical constructive model is necessary to be established.
The main content of this thesis is to construct the mathematical parametric model and the partial differential constraint equation according to the rigid-body transformation theory and General Theorem of Conjugate Surfaces. After finding out the solution from the partial differential constraint equation, a new line-contacted type of tooth profile of gear sets with skew axes, quality analyses to the parameters of gear profile rendered are proceeded. Finally, utilize the software of motion simulation to simulate the operating situation of the linear contacted type of gear sets with skew axes constructed, and supply the demonstration of the theory of tooth profile of gear sets and properties of gear sets.

摘要 I
ABSTRACT II
目 錄 III
圖目錄 V
表目錄 VI
符號說明 VII
第一章 緒論 1
1-1 前言 1
1-2 文獻回顧 2
1-3 研究方向與論文結構 4
第二章 歪斜軸齒輪組共軛齒形之數學模式 6
2-1 剛體運動的描述 6
2-2 齒輪組之座標系統定義 8
2-3 瞬間螺旋軸位置參數 9
2-4 N平面的觀念 13
2-5 共軛齒形 19
第三章 歪斜軸齒輪組共軛齒形 23
3-1 歪斜軸齒輪拘束方程式之求解 23
3-2 拘束方程式的簡化 25
3-3 歪斜軸齒輪組共軛齒面的建構 28
3-4 歪斜軸齒輪組共軛齒面產生程式 31
第四章 齒形參數特性分析 33
4-1 理論推導中的限制條件 33
4-2 假設條件的參數特性 36
4-3 參數設定範例 47
4-4 齒面分析 52
第五章 歪斜軸齒輪組齒形之建構實例 54
5-1 齒面建構合成實例說明 54
5-2 運轉測試 61
5-3 其他嚙合條件的齒輪組 64
5-4 非齒輪方面的應用 67
第六章 結論 69
參考文獻 71

1.Chang, H. L., 1992, A Study of Parametric Tooth Profiles for Spur Gears and Bevel Gears, Ph.D. Thesis, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.
2.Chang, H. L., and Tsai, Y. C., 1991, “An Investigation on the Design Space for Parametric Tooth Profiles,” Proceedings of the 8th World Congress on the Theory of Machines and Mechanisms, Prague, pp. 539 – 542.
3.Chang, H. L., and Tsai, Y. C., 1992, “A Mathematical Model of Parametric Tooth Profiles for Spur Gears,” ASME Journal of Mechanical Design, Vol. 114, No. 1, pp. 8 – 16.
4.Colbourne, J. R., 1982, “The Kinematics of Crossed Helical Gears,” ASME Journal of Mechanical Design, Vol. 104, No. 1, pp. 83 – 89.
5.Dareing, D. W., and Chen, C. M., 1973, “Kinematics and Evaluation of Flat-Toothed Hypoid Gears,” ASME Journal of Engineering for Industry, Vol. 95, No. 4, pp. 1171 – 1177.
6.Denavit, J., and Hartenberg, R.S., 1955, “A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices,” Trans. ASME, J. Appl. Mech. Ser. E., Vol. 22, No. 2, pp. 215 ~ 221.
7.Dooner, D. B., and Seirey, A. A., 1995, The Kinematic Geometry of Gearing, John Wiley & Sons, Inc., New York.
8.Hunt, K. H., 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford.
9.Ishida, K., Ueda, H., Ohashi, S., and Fukui, Y., 1978, “Theoretical and Experimental Investigation of a New Plane Toothed Wheel and Its Enveloping Hourglass Worm,” ASME Journal of Mechanical Design, Vol. 100, No. 3, pp. 460 – 469
10.Jehng, W. K., 1999, A Study of the Tooth Profiles of Gear Sets with Skew Axes, Ph.D. Thesis, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.
11.Jehng, W. K., and Tsai, Y. C., 1998, “Curvature Analysis and Synthesis of Skew Gear Sets with Kinematic Parametric Models,” Transactions of the Canadian Society for Mechanical Engineering, Vol. 22, No. 4A, pp. 341 – 368.
12.Jehng, W. K., and Tsai, Y. C., 1999, “Kinematics Parametric Method to Generate New Tooth Profiles of Gear Sets with Skew Axes,” Mechanism and Machine Theory, Vol. 34, No. 6, pp. 857 – 876.
13.Lauwerier, H. A., and Koiter, W. T., 1978, Applied Mathematics and Mechanics, North-Holland Publishing Company, Oxford.
14.Litvin, F. L., 1989, Theory of Gearing, NASA Reference Publication, 1212, AVSCOM Technical Report, 88-C-035.
15.Litvin, F. L., 1994, Gear Geometry and Applied Theory, PTR Prentice Hall, Inc., Englewood Cliffs, New Jersey.
16.Litvin, F. L., Chaing, W. S., Kuan, C., Lundy, M., and Tsung, W. J., 1991b, “Generation and Geometry of Hypoid Gear-Member with Face –Hobbed Teeth of Uniform Depth,” International Journal of Machine Tools and Manufacture, Vol. 31, No. 2, pp. 167 – 181.
17.Litvin, F. L., and Gutman, Y., 1981a, “Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform” –part 1. Calculations for Machine Settings for Member Gear Manufacture of the Formate and Helixform Hypoid Gears,” ASME Journal of Mechanical Design, Vol. 103, No. 1, pp. 83 – 88.
18.Litvin, F. L., and Gutman, Y., 1981b, “Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform” –part 2. Machine Setting Calculations for the Pinions of Formate and Helixform Gears,” ASME Journal of Mechanical Design, Vol. 103, No. 1, pp. 89 – 101.
19.Litvin, F. L., and Gutman, Y., 1981c, “Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform” –part 3. Analysis and Optimal Synthesis Methods for Mismatch Gearing and its Application for Hypoid Gears of “Formate” and “Helixform”,” ASME Journal of Mechanical Design, Vol. 103, No. 1, pp. 102 – 113.
20.Litvin, F. L., and Kin, V., 1992, “Computerized Simulation of Meshing and Bearing Contact for Single-Enveloping Worm-Gear Drives”, ASME Journal of Mechanical Design, Vol. 114, No. 2, pp. 313 – 316.
21.Litvin, F. L., Petrov, K. M., and Ganshin, V. A., 1974, “The Effect of Geometrical Parameters of Hypoid and Spiroid Gears on Its Quality Characteristics,” ASME Journal of Engineering for Industry, Vol. 96, No. 1, pp. 330 – 334.
22.Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, “Determination of Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel Gears,” ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, No. 4, pp. 495 – 500.
23.Manfredo, P., 1976, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
24.Millman, R. S., and Parker, G. D., 1977, Elements of Differentical Geometry, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
25.Nakada, T., and Yoshimoto, I., 1973, “Design Formula for Involute Profile Shifted Gears Applied to a Skew Shaft Gear-Box,” Mechanism and Machine Theory, Vol. 8, No. 3, pp. 325 – 338.
26.Sakai, T., and Maki, M., 1974, “An Investigation on secondary Action on Skew Gears,” ASME Journal of Engineering for Industry, Vol. 96, No. 1, pp. 25 – 32.
27.Sung, L. M., 1995, A Study on the Parametric Conjugate Tooth Profiles of Gear Sets with Skew Axes, Ph.D. Thesis, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.
28.Tsai, Y. C., and Sung, L. M., 1993, “A Kinematic Study for Gear Sets with Skew Axes,” Journal of Applied Mechanisms and Robotics, Vol. 1, No. 1, pp. 36 – 44.
29.鄧靜華,林輝龍, 1984, 微分幾何學, 國立編譯館, 台北。
30.孫家樂等編, 2000, 偏微分方程, 中央圖書出版社, 台北。
31.趙良五等編, 1985, 微分方程, 台灣商務印書館, 台北。