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Research Papers

Analytical Expression of the Swept Surface of a Rotary Cutter Using the Envelope Theory of Sphere Congruence

[+] Author and Article Information
LiMin Zhu1

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P.R.Chinazhulm@sjtu.edu.cn

XiaoMing Zhang, Gang Zheng, Han Ding

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P.R.China

1

Corresponding author.

J. Manuf. Sci. Eng 131(4), 041017 (Jul 15, 2009) (7 pages) doi:10.1115/1.3168443 History: Received January 19, 2009; Revised June 01, 2009; Published July 15, 2009

Based on the observation that many surfaces of revolution can be treated as a canal surface, i.e., the envelope surface of a one-parameter family of spheres, the analytical expressions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spatial motions are derived by using the envelope theory of sphere congruence. For the toroidal cutter, two methods for determining the effective patch of the envelope surface are proposed. With the present model, it is shown that the swept surfaces of a torus and a cylinder can be easily constructed without complicated calculations, and that the minimum distance (between the swept surface and a simple surface) and the signed distance (between the swept surface and a point in space) can be easily computed without constructing the swept surface itself. An example of global tool path optimization for flank milling of ruled surface with a conical tool, which requires to approximate the tool envelope surface to the point cloud on the design surface, is given to confirm the validity of the proposed approach.

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Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 6

Grazing point on the toroidal cutter

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Figure 7

Spatial B-spline curve and associated moving Frenet frame

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Figure 8

Surface swept by the conical cutter

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Figure 9

Surface swept by the drum-shaped cutter

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Figure 10

Surface swept by the toroidal cutter

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Figure 11

Surface model of a blade of an impeller

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Figure 12

Interference between the tool envelope surface and the design surface before optimization

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Figure 13

Interference between the tool envelope surface and the design surface after optimization

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Figure 14

Tool motion defined by the two curves

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Figure 1

Envelope of the two-parameter family of spheres

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Figure 2

Geometry of the conical cutter

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Figure 3

Geometry of the drum-shaped cutter

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Figure 4

Geometry of the toroidal cutter

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Figure 5

Characteristic curve on the toroidal cutter

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