Research Papers

Virtual Model of Gear Shaping—Part I: Kinematics, Cutter–Workpiece Engagement, and Cutting Forces

[+] Author and Article Information
Andrew Katz

Precision Controls Laboratory,
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada

Kaan Erkorkmaz

Precision Controls Laboratory,
Department of Mechanical and Mechatronics
University of Waterloo,
Waterloo, ON N2L 3G1, Canada
e-mail: kaane@uwaterloo.ca

Fathy Ismail

Precision Controls Laboratory,
Department of Mechanical and Mechatronics
University of Waterloo,
Waterloo, ON N2L 3G1, Canada

1Corresponding author.

Manuscript received July 28, 2017; final manuscript received March 5, 2018; published online April 16, 2018. Assoc. Editor: Laine Mears.

J. Manuf. Sci. Eng 140(7), 071007 (Apr 16, 2018) (15 pages) Paper No: MANU-17-1483; doi: 10.1115/1.4039646 History: Received July 28, 2017; Revised March 05, 2018

Gear shaping is, currently, the most prominent method for machining internal gears, which are a major component in planetary gearboxes. However, there are few reported studies on the mechanics of the process. This paper presents a comprehensive model of gear shaping that includes the kinematics, cutter–workpiece engagement (CWE), and cutting forces. To predict the cutting forces, the CWE is calculated at discrete time steps using a tridexel discrete solid modeler. From the CWE in tridexel form, the two-dimensional (2D) chip geometry is reconstructed using Delaunay triangulation (DT) and alpha shape reconstruction. This in turn is used to determine the undeformed chip geometry along the cutting edge. The cutting edge is discretized into nodes with varying cutting force directions (tangential, feed, and radial), inclination angles, and rake angles. If engaged in the cut during a particular time-step, each node contributes an incremental force vector calculated with the oblique cutting force model. Using a three-axis dynamometer on a Liebherr LSE500 gear shaping machine tool, the cutting force prediction algorithm was experimentally verified on a variety of processes and gears, which included an internal spur gear, external spur gear, and external helical gear. The simulated and measured force profiles correlate closely with about 3–10% RMS error.

Copyright © 2018 by ASME
Topics: Gears , Cutting , Kinematics
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Klocke, F. , and Kobialka, C. , 2000, “ Reducing Production Costs in Cylindrical Gear Hobbing and Shaping,” Gear Technol., Mar./Apr., pp. 26–31.
Klocke, F. , and Köllner, T. , 1999, “ Hard Gear Finishing With a Geometrically Defined Cutting Edge,” Gear Technol., Nov./Dec., pp. 24–29.
Armarego, E. , and Uthaichaya, M. , 1977, “ A Mechanics of Cutting Approach for Force Prediction in Turning Operations,” J. Eng. Prod., 1(1), pp. 1–18.
Reddy, R. G. , Kapoor, S. G. , and DeVor, R. E. , 2000, “ A Mechanistic Force Model for Contour Turning,” ASME J. Manuf. Sci. Eng., 122(3), pp. 398–405. [CrossRef]
Meyer, R. , Köhler, J. , and Denkena, B. , 2012, “ Influence of the Tool Corner Radius on the Tool Wear and Process Forces During Hard Turning,” Int. J. Adv. Manuf. Technol., 58(9–12), pp. 933–940. [CrossRef]
Altintas, Y. , and Lee, P. , 1996, “ A General Mechanics and Dynamics Model for Helical End Mills,” CIRP Ann.—Manuf. Technol., 45(1), pp. 59–64. [CrossRef]
Budak, E. , Altintas, Y. , and Armarego, A. , 1996, “ Prediction of Milling Force Coefficients From Orthogonal Cutting Data,” ASME J. Manuf. Sci. Eng., 118(2), pp. 216–224. [CrossRef]
Omar, O. , El-Wardany, T. , and Elbestawi, M. , 2007, “ An Improved Cutting Force and Surface Topography Prediction Model in End Milling,” Int. J. Mach. Tools Manuf., 47(7–8), pp. 1263–1275. [CrossRef]
Khoshdarregi, M. R. , and Altintas, Y. , 2015, “ Generalized Modeling of Chip Geometry and Cutting Forces in Multi-Point Thread Turning,” Int. J. Mach. Tools Manuf., 98, pp. 21–32. [CrossRef]
Chandrasekharan, V. , Kapoor, S. , and DeVor, R. , 1998, “ A Mechanistic Model to Predict the Cutting Force System for Arbitrary Drill Point Geometry,” ASME J. Manuf. Sci. Eng., 120(3), pp. 563–570. [CrossRef]
de Lacalle, L. L. , Rivero, A. , and Lamikiz, A. , 2009, “ Mechanistic Model for Drills With Double Point-Angle Edges,” Int. J. Adv. Manuf. Technol., 40(5–6), pp. 447–457. [CrossRef]
Sutherland, J. , Salisbury, E. , and Hoge, F. , 1997, “ A Model for the Cutting Force System in the Gear Broaching Process,” Int. J. Mach. Tools Manuf., 37(10), pp. 1409–1421. [CrossRef]
Ozturk, O. , and Budak, E. , 2003, “ Modeling of Broaching Process for Improved Tool Design,” ASME Paper No. IMECE2003-42304.
Imani, B. , Sadeghi, M. , and Elbestawi, M. , 1998, “ An Improved Process Simulation System for Ball-End Milling of Sculptured Surfaces,” Int. J. Mach. Tools Manuf., 38(9), pp. 1089–1107. [CrossRef]
Merdol, S. D. , and Altintas, Y. , 2008, “ Virtual Cutting and Optimization of Three-Axis Milling Processes,” Int. J. Mach. Tools Manuf., 48(10), pp. 1063–1071. [CrossRef]
Spence, A. D. , Abrari, F. , and Elbestawi, M. , 2000, “ Integrated Solid Modeller Based Solutions for Machining,” Comput.-Aided Des., 32(8–9), pp. 553–568. [CrossRef]
Budak, E. , Ozturk, E. , and Tunc, L. , 2009, “ Modeling and Simulation of 5-Axis Milling Processes,” CIRP Ann.—Manuf. Technol., 58(1), pp. 347–350. [CrossRef]
Hosseini, A. , and Kishawy, H. , 2013, “ Parametric Simulation of Tool and Workpiece Interaction in Broaching Operation,” Int. J. Manuf. Res., 8(4), pp. 422–442. [CrossRef]
Klocke, F. , Gorgels, C. , Schalaster, R. , and Stuckenberg, A. , 2012, “ An Innovative Way of Designing Gear Hobbing Processes,” Gear Technol., May, pp. 48–53.
Tapoglou, N. , and Antoniadis, A. , 2012, “ CAD-Based Calculation of Cutting Force Components in Gear Hobbing,” ASME J. Manuf. Sci. Eng., 134(3), p. 031009.
Fetvaci, C. , 2010, “ Generation Simulation of Involute Spur Gears Machined by Pinion-Type Shaper Cutters,” Strojniski Vestnik—J. Mech. Eng., 56(10), pp. 644–652.
Tsay, C.-B. , Liu, W.-Y. , and Chen, Y.-C. , 2000, “ Spur Gear Generation by Shaper Cutters,” J. Mater. Process. Technol., 104(3), pp. 271–279. [CrossRef]
Shunmugam, M. S. , 1982, “ Profile Deviations in Internal Gear Shaping,” Int. J. Mach. Tool Des. Res., 22(1), pp. 31–39. [CrossRef]
Chang, S.-L. , and Tsay, C.-B. , 1998, “ Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured With Shaper Cutters,” ASME J. Mech. Des., 120(1), pp. 92–99. [CrossRef]
König, W. , and Bouzakis, K. , 1977, “ Chip Formation in Gear-Shaping,” Ann. CIRP, 26(1), pp. 17–20.
Erkorkmaz, K. , Katz, A. , Hosseinkhani, Y. , Plakhotnik, D. , Stautner, M. , and Ismail, F. , 2016, “ Chip Geometry and Cutting Forces in Gear Shaping,” CIRP Ann.—Manuf. Technol., 65(1), pp. 133–136. [CrossRef]
Katz, A. , Erkorkmaz, K. , and Ismail, F. , 2018, “ Virtual Model of Gear Shaping Part II: Elastic Deformations and Virtual Gear Metrology,” ASME J. Manuf. Sci. Eng., in press.
Katz, A. , 2017, “ Cutting Mechanics of the Gear Shaping Process,” Ph.D. thesis, UWSpace, Waterloo, ON, Canada.
Erkorkmaz, K. , and Altintas, Y. , 2001, “ High Speed CNC System Design—Part I: Jerk Limited Trajectory Generation and Quintic Spline Interpolation,” Int. J. Mach. Tools Manuf., 41(9), pp. 1323–1345. [CrossRef]
Altintas, Y. , 2012, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, Cambridge, UK.
Stabler, G. V. , 1964, “ The Chip Flow Law and Its Consequences,” Fifth Machine Tool Design and Research Conference, Birmingham, UK, Sept., pp. 243–251.
Benouamer, M. O. , and Michelucci, D. , 1997, “ Bridging the Gap Between CSG and Brep Via a Triple Ray Representation,” SMA'97 Fourth ACM Symposium on Solid Modeling and Applications, Atlanta, GA, May 14–16, pp. 68–79.
Edelsbrunner, H. , Kirkpatrick, D. , and Seidel, R. , 1983, “ On the Shape of a Set of Points in the Plane,” IEEE Trans. Inf. Theory, 29(4), pp. 551–559. [CrossRef]
Berg, M. D. , Cheong, O. , Kreveld, M. V. , and Overmars, M. , 2008, Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag TELOS, Santa Clara, CA.
Watson, D. , 1980, “ Computing the n-Dimensional Delaunay Tessellation With Application to Voronoi Polytopes,” Comput. J., 24(2), pp. 167–172. [CrossRef]


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Fig. 3

Reciprocating motion kinematics: (a) slider-crank mechanism and (b) stroke length and tool overrun

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Fig. 2

Kinematic components and coordinate systems in gear shaping process: (a) reciprocating feed, (b) rotary and radial feed, and (c) coordinate systems

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Fig. 1

Gear shaping process

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Fig. 6

Rake face model of (a) spur and (b) helical gear shapers

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Fig. 4

Experimental validation of feed drive axis kinematic model: (a) position, velocity, and acceleration of r(t) and (b) position of r(t), ϕc(t), ϕg(t), and z(t)

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Fig. 5

Oblique cutting force model

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Fig. 7

Projection of transverse nodes onto rake face and definition of tooth angle: (a) projection of nodes for spur shaper, (b) projection of nodes for helical shaper, and (c) definition of tooth angle (γ)

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Fig. 14

Experimental case studies

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Fig. 8

Cutting direction calculation

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Fig. 9

Distribution of inclination and rake angles on single gear tooth with cutter rake angle of 5deg and helical angle of 25deg in helical gear shaper case

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Fig. 10

Cutter–workpiece engagement using dexel representation: (a) cutter–workpiece engagement and (b) chip in dexel representation

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Fig. 11

Typical chip geometry in helical gear shaping case

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Fig. 17

Gouges and scraping as seen on the external helical gear (after roughing pass)

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Fig. 12

Reconstruction of two-dimensional chip cross section: (a) Delaunay triangulation, (b) alpha shape reconstruction, (c) 2D chip geometry, and (d) triangle-node association

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Fig. 13

Projection of triangles onto plane normal to tangential direction

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Fig. 15

Error contour plot for AISI 1141 steel at τ=805.6 N/mm2

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Fig. 16

Simulated and experimental cutting forces



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