Cutter-Workpiece Engagement Calculation for Milling of Spur Gears Based on Surface Trimming

doi:10.12068/j.issn.1005-3026.2025.20240004

Abstract

Abstract:

A cutter-workpiece engagement （CWE） calculation approach for multi-axis milling of spur gears based on surface trimming was proposed.Firstly， to accurately describe the relative motion relationship between the milling cutter and the workpiece， a gear milling motion model was established.Secondly， based on the established precise mathematical model of the involute gear tooth profile， the constant scallop-height method was applied for tool path planning， with precise calculation of the coordinate positions for each cutter location point.Thirdly， the cutter sweep surface was approximated using a hemispherical surface， and the CWE was defined as a partial sphere constrained by three boundary curves.The hemispherical surface was trimmed using the surface where the boundary curves are located to accurately extract the CWE.Finally， the engagement angles were calculated through an intersection operation between the CWE and a plane perpendicular to the tool axis.A comparative analysis of the simulated and experimentally measured CWE demonstrates the high computational accuracy of the proposed method.Furthermore， compared with solid modeling-based method adopted in the literature， this method has a higher computational efficiency.

Key words: gear milling, cutter-workpiece engagement, tool path planning, surface trimming

CLC Number:

TH 164

Rong-chuang ZHANG, Ming LI. Cutter-Workpiece Engagement Calculation for Milling of Spur Gears Based on Surface Trimming[J]. Journal of Northeastern University(Natural Science), 2025, 46(9): 87-94.

Figures/Tables 16

Fig.1 Coordinate system of spur gear milling

Fig.2 Involute gear tooth profile

Fig.3 Schematic diagram of cutter location point calculation

Fig.4 Machining strategy for gear milling

Fig.5 Flowchart for extracting CWE

Fig.6 Definition of CWE k

Fig.7 Geometric representation of CWE k

Fig.8 Surface trimming process

Fig.9 CWE calculation of different cutter postures

Table 1 Machining parameters

项目	参数	数值
工件参数	模数m_n /mm	6
	齿数z	20
	分度圆压力角 $α n / (°)$	20
	齿顶高系数 $h a *$	1
	顶隙系数 $c *$	0.25
	齿廓残留高度 $h / m m$	0.03
	齿廓半精加工余量T/mm	0.3
刀具参数	刀具半径R/mm	3
	螺旋角 $β / (°)$	15
	刀具刃数	2
切削参数	每齿进给量f/(mm·齿^-1)	0.05
切削参数	刀具偏角 $λ k / (°)$	60

Table 1 Machining parameters

项目	参数	数值
工件参数	模数m_n /mm	6
	齿数z	20
	分度圆压力角 $α n / (°)$	20
	齿顶高系数 $h a *$	1
	顶隙系数 $c *$	0.25
	齿廓残留高度 $h / m m$	0.03
	齿廓半精加工余量T/mm	0.3
刀具参数	刀具半径R/mm	3
	螺旋角 $β / (°)$	15
	刀具刃数	2
切削参数	每齿进给量f/(mm·齿^-1)	0.05
切削参数	刀具偏角 $λ k / (°)$	60

Fig.10 Machining and measurement equipment for test

Fig.11 CWE measurement

Fig.12 Comparison of CWE

Fig.13 Immersion angle

Table 2 Comparison of computational accuracy between surface trimming method and method in reference [11]

刀具路径编号	曲面修剪法面积/mm²	文献[11]方法面积/mm²	前者与后者相对误差/%
2	1.000 821 967	1.000 860 839	-0.003 9
4	0.998 788 367	0.998 803 806	-0.001 5
6	0.996 226 159	0.996 191 599	0.003 5
8	0.992 532 688	0.992 629 893	-0.009 8
10	0.987 684 473	0.987 808 467	-0.012 6
12	0.979 822 484	0.979 945 277	-0.012 5

Fig.14 Comparison of computing time between surface trimming method and method in reference [11]

References 16

[5]	结论
	1 基于铣刀与工件相对运动的数学建模，融合渐开线齿廓理论与等残余高度法，构建了具有严格理论依据的高精度刀具路径规划算法，为切触区域的计算提供了基础.
	2 基于刀齿扫掠面的半球面近似假设，将切触区域严格定义为由 3 条边界曲线约束的局部球面区域.通过引入曲面修剪算法，实现了切触区域的高效精确提取，并计算出切触角度区间.
	3 试验测量结果与仿真数据的高度吻合验证了曲面修剪法在直齿轮铣削切触区域提取中的准确性.计算时间对比显示，该方法在计算效率上具有一定的优越性.
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