Research Papers

Free-Form Conjugation Modeling and Gear Tooth Profile Design

[+] Author and Article Information
Bowen Yu

Research Assistant
e-mail: byu42@students.tntech.edu

Kwun-lon Ting

Center for Manufacturing Research, Tennessee
Technological University,
Cookeville, TN 38501
e-mail: kting@tntech.edu

1Correspondence author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received December 14, 2011; final manuscript received August 10, 2012; published online October 1, 2012. Assoc. Editor: David Dooner.

J. Mechanisms Robotics 5(1), 011001 (Oct 01, 2012) (9 pages) Paper No: JMR-11-1144; doi: 10.1115/1.4007490 History: Received December 14, 2011; Revised August 10, 2012

The paper presents the first theory and practice of free-form conjugation modeling. It introduces the concept of master-slave that broadens the traditional computer aided single geometry modeling to dual geometry modeling. The concept is then applied to gear geometry to establish the first free-form conjugation modeling technique, in which a fictitious free-form rack-cutter or free-form contact path is proposed as the master geometry. Simple geometric relationship suitable for conjugation modeling and undercutting analysis is found between the master and the conjugate profiles. With a free-form master geometry, free-form conjugation modeling has desirable properties of guaranteed continuity, flexibility, and controllability. It offers unlimited representation capability crucial for conjugation modification and optimization. Undercutting is examined rigorously and extensively under the free-form technique via differential geometry. For general planar conjugation, including noncircular gearing, necessary and sufficient undercutting conditions in terms of the master geometry are obtained. Simple geometric explanation of undercutting is demonstrated in terms of the distance between a contact path and two centrode centers. The technique is demonstrated with B-spline as the master geometry.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Litvin, F., and Fuentes, A., 2004, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge.
Tsay, C., and Fong, Z., 1989, “Tooth Contact Analysis for Helical Gears With Pinion Circular Arc Teeth and Gear Involute Shaped Teeth,” ASME J. Mech., Transm., Autom. Des., 111(2), pp. 278–284. [CrossRef]
Tsay, C., and Fong, Z., 1991, “Computer Simulation and Stress Analysis of Helical Gears With Pinion Circular Arc Teeth and Gear Involute Teeth,” Mech. Mach. Theory, 26(2), pp. 145–154. [CrossRef]
Zhang, G., Xu, H., and Long, H., 1995, “Double Involute Gear With Ladder Shape of Tooth,” Chin. J. Mech. Eng., 31, pp. 47–52. Available at http://en.cnki.com.cn/Article_en/CJFDTOTAL-JXXB506.008.htm
Ariga, Y., and Nagata, S., 1985, “Load Capacity of a New WN Gear With Basic Rack of Combined Circular and Involute Profile,” ASME J. Mech., Transm., Autom. Des., 107(4), pp. 565–572. [CrossRef]
Komori, T., Ariga, Y., and Nagata, S., 1990, “A New Gears Profile Having Zero Relative Curvature at Many Contact Points (LogiX Tooth Profile),” J. Mech. Des., 112, pp. 430–436. [CrossRef]
Luo, S., Wu, Y., and Wang, J., 2008, “The Generation Principle and Mathematical Models of a Novel Cosine Gear Drive,” Mech. Mach. Theory, 43(12), pp. 1543–1556. [CrossRef]
Wang, J., Luo, S., and Wu, Y., 2010, “A Method for the Preliminary Geometric Design of Gear Tooth Profiles With Small Sliding Coefficients,” J. Mech. Des., 132, p. 054501. [CrossRef]
Jing, L., 2009, “A Pressure Angle Function Method for Describing Tooth Profiles of Planar Gears,” J. Mech. Des., 131(5), p. 51005. [CrossRef]
Dooner, D., 1997, “Use of Noncircular Gears to Reduce Torque and Speed Fluctuations in Rotating Shafts,” J. Mech. Des., 119, pp. 299–306. [CrossRef]
Figliolini, G., and Angeles, J., 2003, “The Synthesis of Elliptical Gears Generated by Shaper-Cutters,” J. Mech. Des., 125, pp. 793–801. [CrossRef]
Figliolini, G., and Angeles, J., 2005, “Synthesis of the Base Curves for n-Lobed Elliptical Gears,” J. Mech. Des., 127, pp. 997–1005. [CrossRef]
Shih, Y., and Chen, S., 2011, “A Flank Correction Methodology for a Five-Axis CNC Gear Profile Grinding Machine,” Mech. Mach. Theory, 47, pp. 31–45. [CrossRef]
Shih, Y.-P., and Chen, S.-D., 2012, “Free-Form Flank Correction in Helical Gear Grinding Using a Five-Axis Computer Numerical Control Gear Profile Grinding Machine,” J. Mech. Des., 134, p. 041006. [CrossRef]
Buckingham, E., 1988, Analytical Mechanics of Gears, Dover Pubns, Mineola, NY.
Dudley, D., 1969, The Evolution of the Gear Art, American Gear Manufacturers Association, Alexandria, VA.
Dong, H., Ting, K., Liu, J., and Wang, D., 2010, “A Unified Conjugation Theory for Planar Transmission,” ASME Conference, Paper No. DETC2010-29000.
Dong, H., Ting, K., Yu, B., Liu, J., and Wang, D., 2012, “Differential Contact Path and Conjugate Properties of Planar Gearing Transmission,” ASME J. Mech. Des., 134, p. 061010. [CrossRef]
Atkinson, K., 2009, An Introduction to Numerical Analysis, Wiley-India.
Piegl, L., and Tiller, W., 1997, The NURBS Book, Springer-Verlag, New York.
Dudley, D., 1994, Handbook of Practical Gear Design, CRC, Boca Raton.


Grahic Jump Location
Fig. 1

Fixed system Ff, moving system Fm, and instant system (Frenet system) FI. The straight-line cm is the moving centrode of the rack-cutter, while the curve cf is the fixed centrode of a gear.

Grahic Jump Location
Fig. 2

Frenet frames of cf and rf

Grahic Jump Location
Fig. 3

Gear 1 rotates counterclockwise while gear 2 rotates clockwise. The rack-cutter translates to the left.

Grahic Jump Location
Fig. 5

Gear profile design with B-splines. There are a pair of gears, a rack-cutter, and a contact path. The straight segments of contact path and cutter profile indicate that gears are involute.

Grahic Jump Location
Fig. 4

Contact paths of circular gears. Gear centrodes and radii are shown in the figures. The contact paths go closer to one gear center and further away from the other gear center. In the left figure, the red lines become shorter and shorter from the top to the bottom of the contact path, while the blue lines become longer and longer.

Grahic Jump Location
Fig. 6

The first two examples are valid modified gear tooth profiles. The third one has two contact points. There are cusps in the last one.

Grahic Jump Location
Fig. 7

The first two examples are valid modified gear tooth profiles. The third one has two contact points. There are cusps in the last one.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In