Multi-objective Optimization of Precision Milling Parameters for Variable Cross-section Scrolls

doi:10.3969/j.issn.1004-132X.2025.12.006

Abstract

Abstract:

A multi-objective optimization method for precision milling parameters of variable cross-section scrolls was proposed based on the improved NSGA-Ⅱ algorithm. The experiments were designed using the multi-factor orthogonal experimental method to investigate the interaction effects of milling parameters on surface roughness and main milling forces. The regression models of the objective functions were established and validated through goodness-of-fit tests and residual distribution analyses. A multi-objective mathematical optimization model was formulated based on three objectives： minimizing surface roughness， minimizing main milling forces， and maximizing material removal rates. An adaptive crossover and mutation operator was incorporated into the NSGA-Ⅱ algorithm to obtain the pareto-optimal solution set and determine the best combination of milling parameters. Finally， the effectiveness of the algorithm was validated through milling experiments on scrolls. The experimental results indicate that the improved NSGA-Ⅱ algorithm significantly enhances the machining performance. Compared with the non-optimized parameters， the surface roughness of the scroll is reduced by 8.19%， the milling force is reduced by 12.61%， and the material removal rate is increased by 25.81%.

Key words: variable cross-section scroll, orthogonal test, milling parameter, improved NSGA-Ⅱ algorithm, multi-objective optimization

摘要：

提出了一种基于改进NSGA-Ⅱ算法的变截面涡旋盘精密铣削参数多目标优化方法。利用多因素正交试验法设计实验，分析了铣削参数交互作用对表面粗糙度和主铣削力的影响。建立了目标函数的回归模型，并通过拟合度检验和残差分布分析等方式验证了其有效性。基于表面粗糙度最小、主铣削力最小和材料去除率最大三个优化目标构建了多目标优化数学模型，并在NSGA-Ⅱ算法中引入自适应交叉变异算子，以获取Pareto最优解集，最终确定了最佳铣削加工参数组合。最后通过涡旋盘加工实验验证了算法的有效性。实验结果表明，改进后的NSGA-Ⅱ算法显著优化了加工效果，与优化前相比，涡旋盘表面粗糙度减小8.19%，铣削力减小12.61%，材料去除率增大25.81%。

关键词: 变截面涡旋盘, 正交试验, 铣削参数, 改进NSGA-Ⅱ算法, 多目标优化

CLC Number:

TG455

Xu DANG, Tao LIU, Min YAN, Zhiwei XU. Multi-objective Optimization of Precision Milling Parameters for Variable Cross-section Scrolls[J]. China Mechanical Engineering, 2025, 36(12): 2854-2861.

党旭, 刘涛, 闫敏, 徐智为. 变截面涡旋盘精密铣削参数多目标优化[J]. 中国机械工程, 2025, 36(12): 2854-2861.

Figures/Tables 13

Tab.1 Milling parameters and corresponding levels

水平	a_p/mm	f_z/mm	n/（r·min $- 1$ ）	a_e/mm
$-$ 2	1.0	0.25	4000	0.2
$-$ 1	1.5	0.20	3500	0.4
0	2.0	0.15	3000	0.6
1	2.5	0.10	2500	0.8
2	3.0	0.05	2000	1.0

Tab.1 Milling parameters and corresponding levels

水平	a_p/mm	f_z/mm	n/（r·min $- 1$ ）	a_e/mm
$-$ 2	1.0	0.25	4000	0.2
$-$ 1	1.5	0.20	3500	0.4
0	2.0	0.15	3000	0.6
1	2.5	0.10	2500	0.8
2	3.0	0.05	2000	1.0

Fig.1 Milling principle and experiment of scroll plate

Fig.2 Measuring equipment

Tab.2 Orthogonal test results

序号	a_p/mm	f_z/mm	n/（r·min $- 1$ ）	a_e/mm	Ra/μm	F_X /N	F_Y /N	F_Z /N
1	1.0	0.25	3500	0.6	0.722	149.72	85.61	41.64
2	1.0	0.20	3000	0.8	0.672	137.50	93.39	37.74
3	1.0	0.15	2500	1.0	0.632	115.05	102.14	34.11
4	1.0	0.10	2000	0.2	0.526	80.74	43.23	22.63
5	1.0	0.05	4000	0.4	0.401	58.32	42.61	14.95
6	1.5	0.25	3000	1.0	0.822	186.27	137.87	58.00
7	1.5	0.20	2500	0.2	0.729	141.75	68.83	45.08
8	1.5	0.10	4000	0.6	0.570	109.21	75.50	29.87
9	1.5	0.05	3500	0.8	0.461	85.54	79.48	22.68
10	1.5	0.15	2000	0.4	0.693	130.52	88.06	39.23
11	2.0	0.25	2500	0.4	0.880	202.56	117.26	65.32
12	2.0	0.20	2000	0.6	0.845	193.51	136.02	59.14
13	2.0	0.15	4000	0.8	0.730	157.62	126.93	46.20
14	2.0	0.10	3500	1.0	0.645	153.78	138.66	45.66
15	2.0	0.05	3000	0.2	0.482	78.94	49.37	23.34
16	2.5	0.05	2500	0.6	0.550	116.14	88.33	34.70
17	2.5	0.10	3000	0.4	0.678	157.88	88.79	49.43
18	2.5	0.15	3500	0.2	0.793	157.71	72.15	51.60
19	2.5	0.20	4000	1.0	0.869	217.12	167.33	64.78
20	2.5	0.25	2000	0.8	0.995	272.57	170.05	80.12
21	3.0	0.10	2500	0.8	0.745	176.38	139.78	55.61
22	3.0	0.15	3000	0.6	0.833	206.90	132.11	61.16
23	3.0	0.05	2000	1.0	0.593	132.29	124.19	39.78
24	3.0	0.20	3500	0.4	0.901	231.46	117.61	70.46
25	3.0	0.25	4000	0.2	0.945	237.26	91.69	73.69

Tab.2 Orthogonal test results

序号	a_p/mm	f_z/mm	n/（r·min $- 1$ ）	a_e/mm	Ra/μm	F_X /N	F_Y /N	F_Z /N
1	1.0	0.25	3500	0.6	0.722	149.72	85.61	41.64
2	1.0	0.20	3000	0.8	0.672	137.50	93.39	37.74
3	1.0	0.15	2500	1.0	0.632	115.05	102.14	34.11
4	1.0	0.10	2000	0.2	0.526	80.74	43.23	22.63
5	1.0	0.05	4000	0.4	0.401	58.32	42.61	14.95
6	1.5	0.25	3000	1.0	0.822	186.27	137.87	58.00
7	1.5	0.20	2500	0.2	0.729	141.75	68.83	45.08
8	1.5	0.10	4000	0.6	0.570	109.21	75.50	29.87
9	1.5	0.05	3500	0.8	0.461	85.54	79.48	22.68
10	1.5	0.15	2000	0.4	0.693	130.52	88.06	39.23
11	2.0	0.25	2500	0.4	0.880	202.56	117.26	65.32
12	2.0	0.20	2000	0.6	0.845	193.51	136.02	59.14
13	2.0	0.15	4000	0.8	0.730	157.62	126.93	46.20
14	2.0	0.10	3500	1.0	0.645	153.78	138.66	45.66
15	2.0	0.05	3000	0.2	0.482	78.94	49.37	23.34
16	2.5	0.05	2500	0.6	0.550	116.14	88.33	34.70
17	2.5	0.10	3000	0.4	0.678	157.88	88.79	49.43
18	2.5	0.15	3500	0.2	0.793	157.71	72.15	51.60
19	2.5	0.20	4000	1.0	0.869	217.12	167.33	64.78
20	2.5	0.25	2000	0.8	0.995	272.57	170.05	80.12
21	3.0	0.10	2500	0.8	0.745	176.38	139.78	55.61
22	3.0	0.15	3000	0.6	0.833	206.90	132.11	61.16
23	3.0	0.05	2000	1.0	0.593	132.29	124.19	39.78
24	3.0	0.20	3500	0.4	0.901	231.46	117.61	70.46
25	3.0	0.25	4000	0.2	0.945	237.26	91.69	73.69

Fig.3 Ra varies with the interaction of milling parameters

Fig.4 FX varies with the interaction of milling parameters

Fig.5 Surface roughness target model evaluation

Fig.6 Main milling force target model evaluation

Fig.7 Optimization result Pareto frontier solution set

Fig.8 Surface integrity distribution in multi-objective simultaneous optimization

Tab.1 Pareto partial optimal solution set and objective function value

序号	方法	Ra/μm	降低百分比/%	F_X /N	降低百分比/%	φ_MRR/（mm³·min $- 1$ ）	增大百分比/%
A	NSGA-Ⅱ	0.490	0.204	91.509	2.519	899.145	4.786
A	改进NSGA-Ⅱ	0.489	0.204	89.203	2.519	942.174	4.786
B	NSGA-Ⅱ	0.758	1.187	189.085	3.195	4598.992	7.958
B	改进NSGA-Ⅱ	0.749	1.187	183.041	3.195	4964.975	7.958
C	NSGA-Ⅱ	0.955	1.361	270.562	1.668	9323.064	5.866
C	改进NSGA-Ⅱ	0.942	1.361	266.048	1.668	9869.984	5.866
D	NSGA-Ⅱ	0.624	1.122	123.858	3.427	2388.414	2.017
D	改进NSGA-Ⅱ	0.617	1.122	119.613	3.427	2436.581	2.017
E	NSGA-Ⅱ	0.889	1.349	243.257	3.188	7720.609	3.444
E	改进NSGA-Ⅱ	0.877	1.349	235.742	3.188	7986.509	3.444
平均			1.045		2.803		4.814

Tab.1 Pareto partial optimal solution set and objective function value

序号	方法	Ra/μm	降低百分比/%	F_X /N	降低百分比/%	φ_MRR/（mm³·min $- 1$ ）	增大百分比/%
A	NSGA-Ⅱ	0.490	0.204	91.509	2.519	899.145	4.786
A	改进NSGA-Ⅱ	0.489	0.204	89.203	2.519	942.174	4.786
B	NSGA-Ⅱ	0.758	1.187	189.085	3.195	4598.992	7.958
B	改进NSGA-Ⅱ	0.749	1.187	183.041	3.195	4964.975	7.958
C	NSGA-Ⅱ	0.955	1.361	270.562	1.668	9323.064	5.866
C	改进NSGA-Ⅱ	0.942	1.361	266.048	1.668	9869.984	5.866
D	NSGA-Ⅱ	0.624	1.122	123.858	3.427	2388.414	2.017
D	改进NSGA-Ⅱ	0.617	1.122	119.613	3.427	2436.581	2.017
E	NSGA-Ⅱ	0.889	1.349	243.257	3.188	7720.609	3.444
E	改进NSGA-Ⅱ	0.877	1.349	235.742	3.188	7986.509	3.444
平均			1.045		2.803		4.814

Fig.9 The actual processing of the scroll plate

Tab.4 Milling parameter optimization results

参数	优化前	优化后	对比/%
Ra/μm	0.708	0.650	$-$ 8.19
F_X /N	155.470	135.863	$-$ 12.61
φ_MRR/（mm³·min $- 1$ ）	2100	2642	25.81

Tab.4 Milling parameter optimization results

参数	优化前	优化后	对比/%
Ra/μm	0.708	0.650	$-$ 8.19
F_X /N	155.470	135.863	$-$ 12.61
φ_MRR/（mm³·min $- 1$ ）	2100	2642	25.81

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