CHAPTER 6 Conclusions and Future Works
6.2 Future works
The tooth surfaces of the convex and concave spherical helical gears generated by a ZN-type hob cutter are indeed new types of gear surfaces. Advanced studies of these kinds of gear for industrial applications are important and necessary. In the future, the following research topics can be extended:
(1) The illustrated approach in Chapter 2 can be further extended to derive the mathematical model for noncircular hobbing locus of the gears, e.g. parabolic or elliptical curved hobbing locii.
(2) The sensitivity analysis can be used to study the surface deviation of the spherical helical gear with respect to hob cutter settings in the manufacturing process.
(3) The curvature analysis of the proposed convex and concave spherical helical gears should be developed for the determination of principal curvatures and directions of the surfaces of convex and concave spherical helical gears.
(4) Real contact ratio, load sharing between the meshing teeth, and transmission error under the given load may be implemented by considering multi-tooth finite
element models. Moreover, the effect of friction force may be investigated by defining the tooth surfaces interaction with friction.
(5) In order to save the preprocess time of the finite element model of other types of gear sets, the developed automatic mesh-generation program of chapter 5 can be further extended to include other types of gear sets, e.g. curvilinear-tooth gear set, bevel gear set, conical gear set, gear set of worm and worm wheel, etc. Moreover, the developed automatic mesh-generation program can be packaged as a plug-in for the software ABAQUS/Standard.
(6) The single flank test and the measurement of noise and vibrations could be performed with the proposed convex and concave spherical helical gear sets.
REFERENCES
[1] Mitome, K., Okuda, T., Ohmachi, T. and Yamazaki, T., “Develop of a New Hobbing of Spherical Gear,” Journal of JSME, Vol. 66 , pp. 1975-1980, 2000.
[2] 山崎隆, “軸断面形状を変え、特色を出した円錐形歯車と球形歯車,” 機械
設計, 第 34 卷, 第 13 号, 第 53-60 頁, 1990.
[3] Litvin, F. L., Theory of Gearing, NASA Reference Publication 1212, Washington D.C., 1989.
[4] Litvin, F. L., Gear Geometry and Applied Theory, PTR Prentice Hall, Englewood Cliffs, New Jersey, 1994.
[5] Litvin, F. L. and Fuentes, A., Gear Geometry and Applied Theory, 2nd Edition, Cambridge University Press, 2004.
[6] Mancuso, J. R., Couplings and Joints, 2nd Edition, Marcel Dekker, 1999.
[7] Yang, S. C., “Mathematical Model of a Ring-Involute-Teeth Spherical Gear with a Double Degree of Freedom,” Journal of Advanced Manufacturing Technology, Vol. 20, pp. 865-870, 2002.
[8] Yang, S. C., “A Rack-Cutter Surface Used to Generate a Spherical Gear with Discrete Ring-Involute-Teeth,” Journal of Advanced Manufacturing Technology, Vol. 27, pp. 14-20, 2005.
[9] Yang, S. C., Chen, C. K. and Li, K. Y., “A Geometric Model of a Spherical Gear with a Double Degree of Freedom,” Journal of Material Processing Technology, Vol. 123, pp. 219-224, 2002.
[10] Yang, S. C., “Study of an Elbow Mechanism Generated by a Conical Cutter,”
Proceedings of the Institution of Mechanical Engineerings Part C: Journal of Mechanical Engineering Science, Vol. 221, pp. 727-738, 2007.
[11] Tsai, Y. C. and Jehng, W. K., “Rapid Prototyping and Manufacturing Technology
Applied to the Forming of Spherical Gear Sets with Skew Axes,” Journal of Materials Processing Technology, Vol. 95, pp, 169-179, 1999.
[12] Chao, L. C. and Tsay, C. B., “Contact Characteristics of Spherical Gears,” 12th World Congress in Mechanism and Machine Science, Vol. 2, pp. 236-241, 2007.
[13] Chao, L. C. and Tsay, C. B., “Contact Characteristics of Spherical Gears,”
Mechanism and Machine Theory, Vol. 43, pp. 1317-1331, 2008.
[14] Chao, L. C. and Tsay, C. B., “Stress Analysis of Spherical Gear Sets,” ASME International Design Engineering Technical Conferences, DETC2009- 86843 (CD-ROM), pp. 1-7, San Diego, California, U.S.A., Aug. 30~Sep. 02, 2009.
[15] Wang, W. S. and Fong, Z. H., “A Dual Face-Hobbing Method for the Cycloidal Crowning of Spur Gears,” Mechanism and Machine Theory, Vol. 43, pp.
1416-1430, 2008.
[16] Mao, K., “Gear Tooth Contact Analysis and its Application in The Reduction of Fatigue Wear,” Wear, Vol. 262, pp. 1281-1288, 2007.
[17] Wang, J. D., and Howard, I. M., “Errors Analysis of Finite Element Modeling of Involute Spur Gears,” ASME Journal of Mechanical Design, Vol. 128, pp. 90-97, 2006.
[18] Chen, W. H. and Tsai, P., “Finite Element Analysis of an Involute Gear Drive Considering Friction Effects,” ASME Journal of Engineering for Industry, Vol.
111, pp. 94-111, 1989.
[19] Ganesan, N. and Vijayarangan, S., “Static Contact Stress Analysis of a Spur Gear Tooth Using the Finite Element Method, Including Frictional Effects,”
Computers and Structures, Vol. 51, pp. 765-770, 1994.
[20] Tsay, C. B., “Helical Gears with Involute Shaped Teeth: Geometry, Computer Simulation, Tooth Contact Analysis, and Stress Analysis,” Journal of
1988.
[21] Litvin, F. L., Fuentes, A., Gonzalez-Perez, I., Carnevali, L. and Sep, T. M., “New Version of Novikov–Wildhaber Helical Gears: Computerized Design, Simulation of Meshing and Stress Analysis,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 5707-5740, 2002.
[22] Chen, Y. C. and Tsay, C. B., “Tooth Contact Analysis and Kinematic Optimization of a Modified Helical Gear Pair with Involute-Teeth Pinion and Modified-Circular-Arc Gear,” Journal of the Chinese Society of Mechanical Engineers, Vol. 21, pp. 537-547, 2000.
[23] Chen, Y. C. and Tsay, C. B., “Stress Analysis of a Helical Gear Set with Localized Bearing,” Finite Element in Analysis and Design, Vol. 38, pp. 707-723, 2002.
[24] Brauer, J., “A General Finite Element Model of Involute Gears,” Finite Element in Analysis and Design, Vol. 40, pp. 1857-1872, 2004.
[25] Colbourne, J. R., “The Contact Stress in Novikov Gear,” Mechanism and Machine Theory, Vol. 24, pp. 223-229, 1989.
[26] Liu, C. C. and Tsay, C. B., “Contact Characteristic of Beveloid Gears,”
Mechanism and Machine Theory, Vol. 37, pp. 333-350, 2002.
[27] Liu, C. C. and Tsay, C. B., “Tooth Undercutting of Beveloid Gears,” ASME Journal of Mechanical Design, Vol. 123, pp. 569-576, 2001.
[28] Tsai, Y. C. and Chin, P. C., “Surface Geometry of Straight and Spiral Bevel Gears,” Journal of Mechanism, Transmissions, and Automation in Design, Vol.
109, pp. 443-449, 1987.
[29] Litvin, F. L., Fuentes, A. and Hayasaka, K., “Design, Manufacture, Stress Analysis, and Experimental Tests of Low-Noise High Endurance Spiral Bevel Gears,” Mechanism and Machine Theory, Vol. 41, pp. 83-118, 2006.
[30] Litvin, F. L., Chen, J. S., and Lu, J., “Application of Finite Element Analysis for Determination of Load Share, Real Contact Ratio, Precision of Motion, and Stress Analysis,” ASME Journal of Mechanical Design, Vol. 118, pp. 561-567, 1996.
[31] Chang, S. L. and Tsay, C. B., “Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured with Shaper Cutters,”
ASME Journal of Mechanical Design, Vol. 120, No.1, pp. 92-99, 1998.
[32] Tseng, R. T. and Tsay, C. B., “Mathematical Model and Undercutting of Cylindrical Gears with Curvilinear Shaped Teeth,” Mechanism and Machine Theory, Vol. 36, pp. 1189-1202, 2001.
[33] Tseng, R. T. and Tsay, C. B., “Contact Characteristics of Cylindrical Gears with Curvilinear Shaped Teeth,” Mechanism and Machine Theory, Vol. 39, pp.
905-919, 2004.
[34] Tseng, J. T. and Tsay, C. B., “Mathematical Model and Surface Deviation of Cylindrical Gears with Curvilinear Shaped Teeth Cut by Hob Cutter,” ASME Journal of Mechanical Design, Vol. 127, pp. 982-987, 2005.
[35] Tseng, J. T. and Tsay, C. B., “Undercutting and Contact Characteristics of Cylindrical Gears with Curvilinear Shaped Teeth Generated by Hobbing,” ASME Journal of Mechanical Design, Vol. 128, pp. 635-643, 2006.
[36] Simon, V., “The Influence of Gear Hobbing on Worm Gear Characteristics,”
ASME Journal of Manufacturing Science and Engineering, Vol. 129, pp.
919-925, 2007.
[37] Janninck, W. K., “Contact Surface Topology of Worm Gear Teeth,” Gear Technology, March/April, pp. 31-47, 1988.
[38] Litvin, F. L., Nava, A., Fan, Q. and Fuentes, A., “New Geometry of Face Worm
Meshing, and Stress Analysis,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 3035-3054, 2002.
[39] Fang, H. S. and Tsay, C. B., “Mathematical Model and Bearing Contacts of the ZN-type Worm Gear Set Cut by Oversize Hob Cutters,” Mechanism and Machine Theory, Vol. 35, pp. 1689-1708, 2000.
[40] Maki, M. and Sakai, T., “A Study on Hourglass Worm Gearing With Developable Tooth Surfaces,” Journal of Mechanical Design, Vol. 100, pp.
451-459, 1978.
[41] Chen, K. Y. and Tsay, C. B., “Mathematical Model and Worm Wheel Tooth Working Surfaces of the ZN-Type Hourglass Worm Gear Set,” Mechanism and Machine Theory, Vol. 44, pp. 1701-1712, 2009.
[42] Simon, V., “Stress Analysis in Double Enveloping Worm Gears by Finite Element Method,” ASME Journal of Mechanical Design, Vol. 115, pp. 179-185, 1993.
[43] Sun, S. C. and Hung, C. H., The Research on Finite Element Simulation of Deformable Bodies Contact, Master thesis, National Chiao Tung University, 2000.
[44] Tsai, Y. C. and Hung, C. H., The Research on FEM Mesh System Applied to 3D Contact Analysis of Elastically Deformable Bodies, Master thesis, National Chiao Tung University, 2002.
[45] Litvin, F. L. and Seol, I. H., “Computerized Determination of Gear Tooth Surface as Envelope to two Parameter Family of Surfaces,” Computer Methods Apply Mechanical Engineering, Vol. 138, pp. 213-225, 1996.
[46] Zienkiewicz, O. C., and Taylor, R. L., The Finite Element Method for Solid and Structural Mechanics, 6th Edition, John Wiley, 2000.
[47] Dudley, D. W., Dudley’s Gear Handbook, 2nd Edition, McGraw-Hill, U.S.A.,
New York, 1992.
[48] ABAQUS/Standard 6.2, User’s Manual, Hibbitt, Karlsson & Sorensen, U.S.A., 2001.
Appendix A
Stress distributions of A Spherical Helical Gear Set with Uniform Element Size along Tooth Thickness Direction
Example A.1 Stress distributions of spherical helical gear sets with convex pinion and convex gear under parallel axes mounting mode and ideal assembly condition
The stress analysis result of the spherical helical gear set with convex pinion and convex gear are illustrated in Fig. A.1. The design parameters of the gear set are the same as those of Table 5.3. Noteworthy, the element sizes along the tooth thickness direction of the convex pinion and convex gear of the gear set are uniform (see Fig.
A.1), moreover, the other settings of element size of the convex pinion and convex gear are the same as those of the spherical helical gear set with convex pinion and convex gear of Example 5.1. The contact stresses of the convex pinion and convex gear at the pinion’s beginning rotation angle are 1225MPa and 1034MPa, respectively, whereas the bending stresses are 76.92MPa and 66MPa, respectively. By comparing the analysis results (Fig. A.1) of Example A.1 with those of Example 5.1 (Fig. 5.9), the contact and bending stresses only have a slight change between the two examples.
However, the analysis time cost of Example A.1 is expensive due to the total elements and nodes are 297228 and 325080, respectively. Therefore, the local refined mesh along the tooth thickness direction of the pinion and gear is also adopted in this study for stress analysis of the spherical helical gear sets under parallel and crossed axes mounting modes.
(a)Convex pinion
(b)Convex gear
Fig. A.1 von-Mises stress distributions on tooth surfaces of the spherical helical gear set with convex pinion and convex gear under the parallel axes mounting
Bending stress: 76.92Mpa Contact stress: 1225Mpa
Bending stress: 66Mpa Contact stress: 1034Mpa
Appendix B
Stress Distributions of A Conventional Helical Gear Set with Axial Misalignments under Parallel Axes Mounting Mode
Example B.1 Stress distributions of conventional helical gear set with axial misalignment v 0.15 under parallel axes mounting mode
Figure B.1 illustrates stress distributions of the conventional helical gear set with 33 teeth helical pinion and 47 teeth helical gear under parallel axes mounting mode.
The normal module and normal pressure angle of the helical pinion and gear are 4 mm/tooth and 20 degrees, respectively. Moreover, the gear set is assembled with axial misalignment v 0.15. It can be found that the conventional helical gear set has edge contacts on tooth flanks of the helical pinion and gear. Therefore, the conventional helical gear set can’t be applied to the assembly condition with a large axial misalignment . v
Example B.2 Stress distributions of conventional helical gear set with axial misalignment h 0.1 under parallel axes mounting mode
The design parameters of the conventional helical gear set of this example are given the same as those of Example B.2. The gear set is assembled with axial misalignment h 0.1 under parallel axes mounting mode. Figure B.2 illustrates stress distributions of the helical pinion and gear set. It can be found that the gear set also has edge contact on the tooth flanks of the helical pinion and gear. Therefore, the conventional helical gear set can’t be applied to the assembly condition with a large axial misalignment . h
(a)Conventional helical pinion
(b)Conventional helical gear
Fig. B.1 von-Mises stress distributions on tooth surfaces of the conventional helical gear set with axial misalignment v 0.15 under the parallel axes
Edge contact Edge contact
(a)Conventional helical pinion
(b)Conventional helical gear
Fig. B.2 von-Mises stress distributions on tooth surfaces of the conventional helical gear set with axial misalignment h 0.1 under the parallel axes mounting mode
Edge contact
Edge contact Edge contact