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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Fig. 1

Gear shaping process

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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. 3

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

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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. 6

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

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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. 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. 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. 14

Experimental case studies

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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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Fig. 17

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




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