齒輪是許多機械產品中不可或缺的基礎零件，而漸開線正齒輪對(Involute spur gear pair)，具有幾何外廓簡單、製造容易、以及對中心距裝配誤差的低敏感度等優勢，已廣泛地被應用於各種機器中，其亦為平行軸間傳遞動力時最常被使用的機構。但其無法自由地調整背隙(Backlash)，以及在具有軸角度之裝配誤差時，會產生邊緣接觸(Edge contact)與嚙合衝擊(Meshing impact)，導致接觸應力提升、噪音、以及其他傳動上的問題；是以如何使其避免邊緣接觸、嚙合衝擊以及達到背隙可調的目的，一直為齒輪研究之主要方向之一，而其亦為本研究之主要目的。

　　對於邊緣接觸與嚙合衝擊問題，最常見之解決方案為採用具冠狀特性之齒型，但現有之冠狀齒型設計多為以二次加工，或複雜之刀具設定進行生成，於製造上並不簡便。而對於背隙設定問題，最常見之解決方式則為使用直齒錐狀齒輪，但其於裝配誤差下容易產生邊緣接觸之問題。是以本文乃針對此等問題，提出兩型僅由一加工道次即可生成一冠狀齒之新型冠狀正齒輪，以及一可用以避免邊緣接觸問題之冠狀錐形齒輪。其包含有一具漸開線齒廓特性且可避免邊緣接觸發生之新型冠狀正齒輪；一以圓弧齒廓刀具生成，可同時避免嚙合衝擊以及邊緣接觸問題發生之新型雙冠狀正齒輪；以及一具漸開線齒廓特性，且兼具背隙可調與避免邊緣接觸問題之新型冠狀錐形齒輪。

　　本文針對所提三型齒輪，分別介紹其生成機構，並利用Litvin(1989)所提之共軛理論建立其曲面設計、齒印接觸分析以及曲率分析模式。另依據曲率分析之相關推導，本文亦提出該三型齒輪之避免過切條件，推導其各參數避免過切範圍之解析解，以及決定參數避免過切範圍之作法。文中並針對所建立之各模式與作法，分別提出分析實例以觀察所設計齒輪之曲面與嚙合特性，以及進行其參數避免過切範圍的作法與結果之驗證。

　　根據本文所得的結果可知，該三型齒輪皆具有可避免邊緣接觸之特性；其中新型冠狀正齒輪以及新型冠狀錐形齒輪因具有漸開線之特性，能於軸距加大之情況下依然保持定速比之輸出，且新型冠狀錐形齒輪更能於調整背隙後依然保持定速比之傳動；而新型雙冠狀正齒輪則能於各種型態之裝配誤差下避免嚙合衝擊現象之發生，其對於降低實際嚙合時可能產生的噪音與振動問題將具有實質上的幫助。此外，由參數避免過切範圍的驗證結果，可知本文所提決定參數避免過切範圍作法之正確性，其可供作設計者選定參數值時之重要參考。因此，本文所得成果於學術及應用上均有其參考價值。

　Extensive applications of involute spur gears have been found in various machinery; they are also the most popular devices for transmitting power between parallel axes, due to their simple geometry, easy manufacturing, and constant gear-ratio even when the center distance been changed. However, it is found that edge contact and meshing impact of the gear pairs usually occur because of axial misalignments, and the backlash of the gear pair is not adjustable with the specified center distance.

　To avoid edge contact and meshing impact problems, the design concept applied quite often is using the crown-type tooth; and to attain the adjustable backlash, it is applicable to replace the cylindrical involute spur gear pair with a conical one. However, the generation of most of the existing designs having crowned teeth, either requires extra refinements for surface modifications, or needs special controls of cutting tool operations; and the involute conical gear pair has problems of edge contact which need be overcome.

　To deal with the problems mentioned above, three innovative gears with crowned teeth are proposed, which include single-crown spur gears for avoiding edge contact, double-crown spur gears for avoiding both of edge contact and meshing impact, and single-crown conical gears whose backlash can be adjusted and the edge contact can be avoided.

　Based on gearing theory, models for surface design, tooth contact analysis (TCA), and curvature analysis of the three types of gear pairs are developed. Moreover, the nonundercutting criteria, and two ways to determine design-parameter-ranges for avoiding undercutting are also proposed based on the curvature analysis model. Examples are included to verify the correctness of the models developed and to demonstrate gear characteristics.

　According to the results of this study; all the three innovative gear pairs can avoid edge contact with various types of assembly errors; and both the single-crown spur gears and conical gears provide constant gear-ratio when there is no assembly error, or there are errors that cause change of center distance only, the conical one can even retain the constant gear ratio when the backlash been adjusted; and the double-crown spur gear can avoid meshing impact with various types of assembly errors. Besides, it is verified that, the proposed methodology for determining design-parameter-ranges for avoiding undercutting can give accuracy ranges of the design parameters of the three innovative gear pairs. It is believed that the results are valuable in both of the academic field and industrial applications.

Asakawa, T., Mitome, K.-I., and Ohmachi, T., 2004, "Hobbing of Concave Conical Gear," (In Japanese) Transactions of the Japan Society of Mechanical Engineers, Part C, Vol. 70, No. 1, pp. 315-320.
Barrett, J., 1995, "Conical Helical Gears Control Backlash," Eureka: Engineering Materials and Design, Vol. 15, No. 6, pp. 22-23.
Baxter, M. L., 1948, "Curvature-Acceleration Relations for Plane Cams," ASME Transactions, Vol. 70, pp. 483-489.
Baxter, M. L., 1962, Gear Handbook (Edited by D. W. Dudley), McGraw-Hill, New York, U.S.A., pp. 1.1-1.21.
Beam, A. S., 1954, "Beveloid Gearing," Machine Design, Vol. 26, No. 12, pp. 220-238.
Brauer, J., 2002, "Analytical Geometry of Straight Conical Involute Gears," Mechanism and Machine Theory, Vol. 37, No. 1, pp. 127-141.
Brauer, J., 2005, "Transmission Error in Anti-Backlash Conical Involute Gear Transmissions: a Global-Local FE Approach," Finite Elements in Analysis and Design, Vol. 41, No. 5, pp. 431-457.
Chakraborty, J., and Dhande, S. G., 1977, Kinematics and Geometry of Planar and Spatial Cam Mechanisms, Wiley Eastern Limited, New Delhi, India.
Chang, S.-L., and Tsay, C.-B., 1998, "Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured with Shaper Cutters," Transactions of the ASME, Journal of Mechanical Design, Vol. 120, pp. 92-99.
Chang, S.-L., Tsay, C.-B., and Tseng, C.-H., 1997, "Kinematic Optimization of a Modified Helical Gear Train," Transactions of the ASME, Journal of Mechanical Design, Vol. 119, No. 2, pp. 307-314.
Chang, S.-L., and Tseng, H.-C., 2005, "Design of a Novel Cutter for Manufacturing Helical Cutting Tools," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, Vol. 219, No. 4, pp. 395-408.
Chen, C.-F., Lin, H.-T., and Tsay, C.-B., 2002, "Mathematical Model and Undercutting Analysis of Modified Helical Gears with Smaller Number of Teeth," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 23, No. 4, pp. 301-311.
Chen, C.-F., and Tsay, C.-B., 2004, "Computerized Tooth Profile Generation and Analysis of Characteristics of Elliptical Gears with Circular-arc Teeth," Journal of Materials Processing Technology, Vol. 148, No. 2, pp. 226-234.
Chen, C.-H., 1979, "Boundary Curve, Singular Solution, Complementary Conjugate Surfaces and Conjugation Analysis in Theory of Conjugate Surface," Proceedings of the 5th World Congress on the Theory of Machines and Mechanisms, Montreal, Canada, pp. 1478-1481.
Chen, C.-K., Chiou, S.-T., Fong, Z.-H., Lee, C.-K., and Chen, C.-H., 2001, "Mathematical Model of Curvature Analysis for Conjugate Surfaces with Generalized Motion in Three Dimensions," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, Vol. 215, No. 4, pp. 487-502.
Chen, C.-K., and Wang, C.-Y., 2001, "Compensating Analysis of a Double Circular-Arc Helical Gear by Computerized Simulation of Meshing," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, Vol. 215, No. 7, pp. 759-771.
Chen, Y.-C., and Tsay, C.-B., 2000, "Tooth Contact Analysis and Kinematic Optimization of a Modified Helical Gear Pair with Involute-Teeth Pinion and Modified-Circular Arc Teeth Gear," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 21, No. 6, pp. 537-547.
Chen, Y.-C. and Tsay, C.-B., 2001a, "Mathematical Model and Undercutting Analysis of Modified Circular-Arc Helical Gears," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 22, No. 1, pp. 41-51.
Chen, Y.-C., and Tsay, C.-B., 2001b, "Bearing Contact of a Helical Gear Pair with Involute Teeth Pinion and Modified Circular-arc Teeth Gear," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, Vol. 215, No. 10, pp. 1175-1187.
Colbourne, J. R., 1987, The Geometry of Involute Gears, Springer-Verlag, New York, U.S.A., pp. 229-238.
Colbourne, J. R., 1989, "The Curvature of Helicoids," Mechanism and Machine Theory, Vol. 24, No. 3, pp. 213-221.
De Donno, M., and Litvin, F. L., 1999, "Computerized Design, Generation and Simulation of Meshing of a Spiroid Worm-Gear Drive with a Ground Double-Crowned Worm," Transactions of the ASME, Journal of Mechanical Design, Vol. 121, No. 2, pp. 264-273.
Dhande, S. G., Bhadoria, B. S., and Chakraborty, J., 1975, "A Unified Approach to the Analytical Design of Three Dimensional Cam Mechanisms," Transactions of the ASME, Journal of Engineering for Industry, Vol. 97B, No. 1, pp. 327-333.
Dhande, S. G., and Chakraborty, J., 1976a, "Curvature Analysis of Surface in Higher Pair Contact; Part 1: An Analytical Investigation," Transactions of the ASME, Journal of Engineering for Industry, Vol. 98, No. 2, pp. 397-402.
Dhande, S. G., and Chakraborty, J., 1976b, "Curvature Analysis of Surface in Higher Pair Contact; Part 2: Application to Spatial Cam Mechanisms," Transactions of the ASME, Journal of Engineering for Industry, Vol. 98, No. 2, pp. 403-409.
Denavit, J., and Hartenberg, R. S., 1955, "A Kinematics Notation for Lower Pair Mechanisms Based on Matrices," Transactions of the ASME, Journal of Applied Mechanics, Vol. 22, No. 2, pp. 215-221.
Dooner, D. B., and Seireg, A. A., 1995, The Kinematic Geometry of Gearing, John Wiley & Sons, New York, U.S.A..
Dudley, D. W., 1968, Gear Handbook: The Design, Manufacture, and Application of Gears, McGraw-Hill, New York, U.S.A..
Feng, P.-H., Litvin, F. L., Townsend, D. P., and Handschuh, R. F., 1999, "Determination of Principal Curvatures and Contact Ellipse for Profile Crowned Helical Gears," Transactions of the ASME, Journal of Mechanical Design, Vol. 121, No. 1, pp. 107-111.
Fong, Z.-H., and Tsay, C.-B., 1990, "Tooth Contact Analysis of Spiral Bevel Gears," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 11, No. 6, pp. 538-544.
Fong, Z.-H., and Tsay, C.-B., 1992, "The Undercutting of Circular-Cut Spiral Bevel Gears," Transactions of the ASME, Journal of Mechanical Design, Vol. 114, pp. 317-325.
Fong, Z.-H., and Tsay, C.-W., 2000, "Study on the Undercutting of Internal Cycloidal Gear with Small Tooth Difference," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 21, No. 4, pp. 359-367.
Griffis, M., 2003, "A Study of Curvature for Single Point Contact," Mechanism and Machine Theory, Vol. 38, No. 12, pp. 1391-1411.
Gunbara, H., Shimachi, S., Kobayashi, T., and Kawada, H., 1992, "A Study on Hourglass-Worm Gearings Designed to Concentrate Surface Normals: Design for Worm Gearings with a Large Lead Angle," ASME, Proc. 1992 International Power Transmission and Gearing Conf., Vol. 1, pp. 153-159.
Hori, K., Hayashi, I., and Iwatsuki N., 1998, "The Ideal Tooth Profiles of Conical-External and -Internal Gears Meshing with Cylindrical -Involute Gears over the Entire Tooth Width," JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, Vol. 41, No. 4, pp. 901-911.
Innocenti, C., 1997, "Analysis of Meshing of Beveloid Gears," Mechanism and Machine Theory, Vol. 33, No. 3, pp. 363-373.
Jang, J.-R., Chiou, S.-T., and Li, J.-L., 2004, "On the Curvature Analysis of Variable Pitch Lead Screw Mechanisms with Meshing Elements in Both Reciprocation and Oscillation for Avoiding Undercutting," Transactions of the Canadian Society for Mechanical Engineering, Vol. 28, No. 3-4, pp. 491-509.
Kang, Y.-H., Liu, J.-Y., and Yan, H.-S., 1994, "Curvature Analysis of Variable Pitch Cylindrical Cams with Conical Meshing Elements," Mathematical and Computer Modelling, Vol. 19, No. 5, pp. 51-64.
Kang, Y.-H., and Yan, H.-S., 1996, "Curvature Analysis of Variable Pitch Lead Screws with Cylindrical Meshing Elements," Transactions of the Canadian Society for Mechanical Engineering, Vol. 20, No. 2, pp. 139-157.
Kawasaki, K., and Tamura, H., 1997, "Method for Cutting Hypoid Gears (Duplex Spread-Blade Method)," JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, Vol. 40, No. 4, pp. 768-775.
Kin, V., 1990, "Limitation of Worm and Worm Gear Surfaces in Order to Avoid Undercutting," Gear Technology, pp. 30-35.
Komatsubara, H., Mitome, K.-I., and Ohmachi, T., 2002, "Development of Concave Conical Gear Used for Marine Transmissions (1st Report, Principle of Generating Helical Concave Conical Gear)," JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, Vol. 45, No. 1, pp. 371-377.
Lin, C.-Y., Tsay, C.-B., and Fong, Z.-H., 1996, "Tooth Contact Analysis of Hypoid Gears," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 17, No. 3, pp. 241-249.
Litvin, F. L., 1968, Theory of Gearing, (in Russian) 2nd Ed., Nauka, Moscow, Russia.
Litvin, F. L., 1989, Theory of Gearing, NASA, Washington, D.C., U.S.A..
Litvin, F. L., 1994, Gear Geometry and Applied Theory, PTR Prentice-Hall, Englewood Cliffs, New Jersey, U. S. A..
Litvin, F. L., Argentieri, G., De Donno, M., and Hawkins, M., 2000, "Computerized Design, Generation and Simulation of Meshing and Contact of Face Worm-Gear Drives," Computer Methods in Applied Mechanics and Engineering, Vol. 189, No. 3, pp. 785-801.
Litvin, F. L., Fuentes, A., Fan, Q. and Handschuh, R. F., 2002a, "Computer Design, Simulation of Meshing, and Contact and Stress Analysis of Face-Milled Formate Generated Spiral Bevel Gears," Mechanism and Machine Theory, Vol. 37, No. 5, pp. 441-459.
Litvin, F. L., Fuentes, A., Gonzalez-Perez, I., Carvenali, L., Kawasaki, K., and Handschuh, R. F., 2003, "Modified Involute Helical Gears: Computerized Design, Simulation of Meshing and Stress Analysis," Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 33-34, pp. 3619-3655.
Litvin, F. L., Fuentes, A., Gonzalez-Perez, I., Carvenali, L., and Sep, T. M., 2002b, "New Version of Novikov-Wildhaber Helical Gears: Computerized Design, Simulation of Meshing and Stress Analysis," Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 49-50, pp. 5707-5740.
Litvin, F. L., Fuentes, A., Zanzi, C., and Pontiggia, M., 2002c, "Design, Generation, and Stress Analysis of Two Versions of Geometry of Face-Gear Drives," Mechanism and Machine Theory, Vol. 37, No. 10, pp. 1179-1211.
Litvin, F. L., and Gutman, Y., 1981a, "Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform”," Transactions of the ASME, Journal of Mechanical Design, Vol. 103, No. 4, pp. 83-88.
Litvin, F. L. and Gutman, Y., 1981b, "A Method of Local Synthesis of Gears Grounded on the Connections Between the Principal and Geodetic Curvatures of Surfaces," Transactions of the ASME, Journal of Mechanical Design, Vol. 103, No. 4, pp. 114-125.
Litvin, F. L. and Kim, D. H., 1997, "Computerized Design, Generation and Simulation of Meshing of Modified Involute Spur Gears with Localized Bearing Contact and Reduced Level of Transmission Errors," Transactions of the ASME, Journal of Mechanical Design, Vol. 119, No. 1, pp. 96-100.
Litvin, F. L., Krylov, N. N., and Erikhov, M. L., 1975, "Generation of Tooth Surface by Two-Parameter Enveloping," Mechanism and Machine Theory, Vol. 10, No. 5, pp. 365-373.
Litvin, F. L., Kuan, C., Kieffer, J., Bossler, R., and Handschuh, R. F., 1991, "Straddle Design of Spiral Bevel and Hypiod Pinions and Gears," ASME Transactions, Journal of Mechanical Design, Vol. 113, No. 4, pp. 422-426.
Litvin, F. L., Peng, A., and Wang, A., 1999, "Limitation of Gear Tooth Surfaces by Envelopes to Contact Lines and Edge of Regression," Mechanism and Machine Theory, Vol. 34, No. 6, pp. 889-902.
Litvin, F. L., Seol, I. H., Kim, D., Lu, J., Wang, A. G., Egelja, A., Zhao, X., and Handschuh, R. F., 1996, "Kinematic and Geometric Models of Gear Drives," Transactions of the ASME, Journal of Mechanical Design, Vol. 118, No. 4, pp. 544-550.
Litvin, F. L. and Tsay, C.-B., 1985, "Helical Gears with Circular Arc Teeth: Simulation of Conditions of Meshing and Bearing Contact," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 107, No. 4, pp. 556-564.
Litvin, F. L., Vecchiato, D., Demenego, A., Karedes, E., Hansen, B., and Handschuh, R., 2002d, "Design of One Stage Planetary Gear Train with Improved Conditions of Load Distribution and Reduced Transmission Errors," Transactions of the ASME, Journal of Mechanical Design, Vol. 124, No. 4, pp. 745-752.
Litvin, F. L., Vecchiato, D., Fuentes, A., and Gonzalez-Perez, I., 2004, "Automatic Determination of Guess Values for Simulation of Meshing of Gear Drives," Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 33-35, pp. 3745-3758.
Litvin, F. L., Zhang, J., and Handschuh, R. F., 1988a, "Crowned Spur Gears: Methods for Generation and Tooth Contact Analysis- Part 1: Basic Concepts, Generation of the Pinion Tooth Surface by a Plane," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, No. 3, pp. 337-342.
Litvin, F. L., Zhang, J., and Handschuh, R. F., 1988b, "Crowned Spur Gears: Methods for Generation and Tooth Contact Analysis- Part 2: Generation of the Pinion Tooth Surface by a Surface of Revolution," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, No. 3, pp. 343-347.
Liu, C.-C., and Tsay, C.-B., 2001, "Tooth Undercutting of Beveloid Gears," Transactions of the ASME, Journal of Mechanical Design, Vol. 123, pp. 569-576.
Liu, C.-C., and Tsay, C.-B., 2002a, "Mathematical Model and Contact Simulations of Concave Beveloid Gears," Transactions of the ASME, Journal of Mechanical Design, Vol. 124, No. 4, pp. 753-760.
Liu, C.-C., and Tsay, C.-B., 2002b, "Contact Characteristic of Beveloid Gears," Mechanism and Machine Theory, Vol. 37, No. 4, pp. 333 -350.
Merritt, H. E., 1954, Gears, 3rd ed., Isaac Pitman & Sons, London, U.K..
Mitome, K.-I., 1983a, "Conical Involute Gear: Part 1, Design and Production System," Bulletin of JSME, Vol. 26, No. 212, pp. 299-305.
Mitome, K.-I., 1983b, "Conical Involute Gear: Part 2, Design and Production System of Involute Pinion-Type Cutter," Bulletin of JSME, Vol. 26, No. 212, pp. 306-312.
Mitome, K.-I., 1985, "Conical Involute Gear: Part 3, Tooth Action of A Pair fo Gears," Bulletin of the JSME, Vol. 28, No. 245, pp. 2757 -2764.
Mitome, K.-I., 1991, "Conical Involute Gear (Design of Nonintersecting -Nonparallel-Axis Conical Involute Gear)," JSME International Journal, Series III: Vibration, Control Engineering, Engineering for Industry, Vol. 34, No. 2, pp. 265-270.
Mitome, K.-I., 1993, "Infeed Grinding of Straight Conical Involute Gear," JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, Vol. 36, No. 4, pp. 537-542.
Mitome, K.-I. , 1995, "Design of Miter Conical Involute Gears Based on Tooth Bearing," JSME International Journal, Series C: Mechanical Systems, Machine Elements and Manufacturing, Vol. 38, No. 2, pp. 307-311.
Mitome, K.-I., Ohmachi, T., and Komatsubara, H., 2003, "Conical Involute Gear –Development, Applications, and View of Tomorrow," Proceedings of the ASME Design Engineering Technical Conference, Vol. 4B, pp. 771-778.
Mitome, K.-I., Yamazaki, T., 1996, "Design of Conical Involute Gear Engaged with Profile Shifted Spur Gear on Intersecting Shafts," (In Japanese) Transactions of the Japan Society of Mechanical Engineers, Part C, Vol. 62, No. 598, pp. 2436-2441.
Norton, R. L., 2004, Design of Machinery, 3rd ed., McGraw-Hill, New York, U.S.A., Chap. 8-Chap. 9.
Nutbourne, A. W., and Martin, R. R., 1988, Differential Geometry Applied to Curve and Surface Design, Vol. 1, Ellis Horwood Limited, Chichester, England.
Pedrero, J. I., Artés, M., and García-Masiá, C., 2004, "Determination of the Effective Path of Contact of Undercut Involute Gear Teeth," Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, Vol. 218, No. 7, pp. 413-434.
Raven, F. H., 1959, "Analytical Design of Disk Cams and Three- Dimensional Cams by Independent Position Equations," Transactions of the ASME, Journal of Applied Mechanics, Vol. 81, pp. 18-24.
Shtipelman, B. A., 1978, Design and Manufacture of Hypoid Gears, John Wiley and Sons, New York, U. S. A., pp. 148-176.
Simon, V., 1989, "Optimal Tooth Modifications for Spur and Helical Gears," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 111, No. 4, pp. 611-615.
Struik, D. J., 1988, Lectures on Classical Differential Geometry, Dover Publications, New York, U.S.A..
Takahashi, K., and Ito, N., 1986, "Third-Order Surface Application to Determine the Tooth Contact Pattern of Hypoid Gears," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 108, No. 2, pp. 263-269.
The MathWorks, 1996, MATLAB, The Language of Technical Computing; Using MATLAB, Version 5, The MathWorks, Inc., Maryland, U.S.A..
Tsai, Y.-C., and Jehng, W.-K., 1999, "A 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.
Tsay C.-B., 1988, "Helical Gears with Involute Shaped Teeth: Geometry, Computer Simulation, Tooth Contact Analysis, and Stress Analysis," Transactions of the ASME, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, No. 4, pp. 482-491.
Tsay, C.-B., Sheu., W.-L., and Wu, C.-H., 1990, "Spur Gears with Crowned Teeth," Transactions of the Chinese Institute of Engineers, Journal of the Chinese Society of Mechanical Engineers, Vol. 11, No. 2, pp. 121-133.
Tseng, R.-T. and Tsay, C.-B., 2001, "Mathematical Model and Undercutting of Cylindrical Gears with Curvilinear Shaped Teeth," Mechanism and Machine Theory, Vol. 36, No. 11-12, pp. 1189-1202.
Tseng, R.-T. and Tsay, C.-B., 2004, "Contact Characteristics of Cylindrical Gears with Curvilinear Shaped Teeth," Mechanism and Machine Theory, Vol. 39, No. 9, pp. 905-919.
Wang, W. H., Tseng, C. H., and Tsay, C. B., 1997, "Surface Contact Analysis for a Spatial Cam Mechanism," Transactions of the ASME, Journal of Mechanical Design, Vol. 119, No. 2, pp. 169-177.
Wildhaber, E., 1956, "Surface Curvature," Product Engineering, Vol. 27, No. 5, pp. 184-191.
Wildhaber, E., 1962, "Method and Machine for Producing Crowned Teeth," USA Patent, 3,046,844.
Yan, H.-S. and Cheng, W.-T., 1999, "Curvature Analysis of Spatial Cam-Follower Mechanisms," Mechanism and Machine Theory, Vol. 34, No. 2, pp. 319-339.
Zhang, Y., and Fang, Z., 1999, "Analysis of Tooth Contact and Load Distribution of Helical Gears with Crossed Axes," Mechanism and Machine Theory, Vol. 34, No. 1, pp. 41-57.
Zhang, Y., and Wu, Z., 1997, "Offset Face Gear Drives: Tooth Geometry and Contact Analysis," Transactions of the ASME, Journal of Mechanical Design, Vol. 119, No. 1, pp. 114-119.
Zhang, Y., and Xu, H., 2003, "Pitch Cone Design and Avoidance of Contact Envelope and Tooth Undercutting for Conical Worm Gear Drives," Transactions of the ASME, Journal of Mechanical Design, Vol. 125, No. 1, pp. 169-177.
李肇祿，2001，"兩型一次包絡變導程螺桿機構之設計參數的避免過切範圍研究"，碩士論文，國立成功大學機械工程學系，台南，台灣。
邱顯堂，陳志弘，1998，”一次包絡曲面曲率分析及其於變導程螺桿機構之應用”，第一屆全國機構與機器設計學術研討會論文集，台南市，台灣，205-212頁。
陳志新，1985，共軛曲面基本原理，科學出版社，北京，中國。
陳志弘，1997，"多重包絡及其應用於變導程螺桿機構的曲面設計之研究"，碩士論文，國立成功大學機械工程學系，台南，台灣。
陳朝光，唐余勇，吳宏業，1998，微分幾何及其在機械工程中的應用，哈爾濱工業大學出版社，北京，中國。
蔡錫錚，吳思漢，2002，"平行軸錐形齒輪對齒面寬之最佳設計"，第五屆全國機構與機器設計學術研討會論文集，高雄市，台灣，244-250頁。