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Research Papers

# Mathematical Modeling and Simulation of the External and Internal Double Circular-Arc Spiral Bevel Gears for the Nutation Drive

[+] Author and Article Information
Ligang Yao1

School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, 350108, P. R. Chinaylgyao@fzu.edu.cn

Bing Gu, Shujuan Haung

School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, 350108, P. R. China

Guowu Wei, Jian S. Dai

School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, 350108, P. R. China; Department of Mechanical Engineering, Kings College, London, WC2R 2LS, UK

1

Corresponding author.

J. Mech. Des 132(2), 021008 (Feb 09, 2010) (10 pages) doi:10.1115/1.4001003 History: Received April 06, 2009; Revised December 03, 2009; Published February 09, 2010; Online February 09, 2010

## Abstract

The purpose of this paper is to propose a pair of external and internal spiral bevel gears with double circular-arc in the nutation drive. Based on the movement of nutation, this paper develops equations of the tooth profiles for the gear set, leading to the mathematical modeling of the spiral bevel gear with a constant helical angle gear alignment curve, enabling the tooth surface to be generated, and permitting the theoretical contacting lines to be produced in light of the meshing function. Simulation and verification are carried out to prove the mathematical equations. Numerical control (NC) simulation of machining the external and internal double circular-arc spiral bevel gears is developed, and the spiral gears were manufactured on a NC milling machine. The prototype of the nutation drive is illustrated in the case study at the end of this paper.

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## Figures

Figure 1

Sketch of the two-stage nutation drive

Figure 2

Kinematic diagrams of the two-stage nutation drive

Figure 3

Sketch of tooth generation of the bevel gear

Figure 4

Meshing coordinate systems between the crown and external spiral bevel gears

Figure 5

Meshing coordinate systems between the crown and internal spiral bevel gears

Figure 6

Basic tooth profile of the double circular-arc profile

Figure 7

Coordinate system Sn on the actual center-gear alignment curve

Figure 8

Sketch of the gear alignment curve of the spiral bevel gear

Figure 9

Gear alignment curves prior to and after rotating

Figure 10

Actual boundaries of the tooth surface

Figure 11

Actual center-gear alignment curves

Figure 12

Exploded virtual assembly of the nutation drive reducer

Figure 13

Computer modeling of the external and internal spiral bevel gears

Figure 14

Kinematical simulation results

Figure 15

Sketch of NC milling machining for spiral bevel gears

Figure 16

Simulation machining of the spiral bevel gears

Figure 17

The NC machining experiment of the external and internal spiral bevel gears

Figure 18

Double circular-arc spiral bevel gear

Figure 19

Flow chart of the machining process of the double circular-arc spiral bevel gear

Figure 20

Assembling of the two external gears and input shaft

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