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Research Papers

# Envelope Surface Modeling and Tool Path Optimization for Five-Axis Flank Milling Considering Cutter Runout

[+] Author and Article Information
Zhou-Long Li

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: lzl@sjtu.edu.cn

Li-Min Zhu

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: zhulm@sjtu.edu.cn

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received January 20, 2014; final manuscript received April 5, 2014; published online June 5, 2014. Assoc. Editor: Xiaoping Qian.

J. Manuf. Sci. Eng 136(4), 041021 (Jun 05, 2014) (9 pages) Paper No: MANU-14-1025; doi: 10.1115/1.4027415 History: Received January 20, 2014; Revised April 05, 2014

## Abstract

Cutter runout is a common and inevitable phenomenon impacting the geometry accuracy in the milling process. However, most of the works on tool path planning neglect the cutter runout effect. In this paper, a new approach is presented to integrate the cutter runout effect into envelope surface modeling and tool path optimization for five-axis flank milling with a conical cutter. Based on the geometry model of cutter runout which consists of cutter axis and cutter tilt, an analytic expression of cutter edge combined with four runout parameters is derived. Then the envelope surface formed by each cutter edge is constructed using the envelope theory of sphere congruence. Due to the cutter runout effect, the envelope surfaces formed by the cutter edges are different from each other, and the valid envelope surface is the combination of these envelope surfaces which contribute to the final machined surface. To measure the machining errors, the geometry deviations between the valid envelope surface and the design surface are calculated with the distance function. On the basis of the differential property of the distance function, tool path optimization considering cutter runout is modeled as a mixed-integer linear programming (MILP) problem, which can be solved by the branch-and-bound method. Finally, numerical examples are given to confirm the validity and efficiency of the proposed approach. The results show that the geometry errors induced by runout can be reduced significantly using the proposed method.

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## Figures

Fig. 2

Geometry model of a conical cutter

Fig. 1

Geometry model of cutter runout: (a) schematic diagram of cutter runout in milling, (b) coordinate system construction, (c) sectional view of the cutter runout model

Fig. 3

(a) Rotary surfaces generated by each cutter edge and (b) the valid rotary surface

Fig. 4

Geometry of rotary surface generated by the jth cutter edge

Fig. 5

Tool axis trajectory surface defined by two guiding curves

Fig. 6

Cutter swept envelope surfaces formed by the cutter edges

Fig. 7

The valid cutter swept envelope surface

Fig. 8

Distance from a point to the envelope surface formed by each cutter edge

Fig. 9

Surface model of a blade of a centrifugal impeller

Fig. 10

(a) Radius deviation curves contributed by the three cutter edges and (b) the combined radius deviation curve

Fig. 11

Distribution of the deviations of the valid cutter rotary surface from the nominal one

Fig. 12

Distribution of the deviations of the tool envelope surface with runout effect from the design surface

Fig. 13

Distribution of the deviations of the tool envelope surface without runout effect from the design surface

Fig. 14

Straight-line fitting to the discrete points on the valid generatrix of the conical cutter with runout

Fig. 15

Distribution of the deviations of the tool valid envelope surface after LS optimization from the design surface

Fig. 16

Distribution of the deviations of the tool valid envelope surface after min–max–min optimization from the design surface

Fig. 17

Results of the maximum undercut and overcut after each iteration

## Errata

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