Multi-parameter optimization technology of spiral bevel gear blank
-
摘要:
为适应高功率密度弧齿锥齿轮传动的轻量化需求，提出了一种弧齿锥齿轮轮坯多参数寻优技术。以齿数、模数及齿宽为设计变量，以体积之和最小为目标函数，以工况限制、安装条件及多强度基准为约束，综合遗传算法实现弧齿锥齿轮轮坯参数优化设计。运用功率分别为0.75 MW和5 MW的实例验证所提技术的有效性，结果表明：相较于传统设计方法如适配法和非线性数学规划法，所提技术得益于遗传算法自适应性和可并行性能够实现全局目标优化且成功率为100%；此外，相较于设计需求，0.75 MW实例优化结果的主分直径和齿宽冗余31%、19.23%，5 MW优化结果则是略超出4.69%和2.65%，表明所提技术能够针对不同设计条件为设计者提供有效的设计裕度。
Abstract:In order to meet the lightweight requirements of high-power density spiral bevel gear transmission, a multi-parameter optimization technology for spiral bevel gear blank was proposed. Taking the tooth number, modulus, and tooth width of the gear as design variables, the minimum sum of the volume of the gear pair as the objective function, and the working condition limitations, installation conditions, and multiple strength requirements as constraints, the genetic algorithm was synthesized to realize the optimization design of spiral bevel gear blank parameters. Examples with power of 0.75 MW and 5 MW were used to verify the effectiveness of the proposed technique. The results showed that: compared with traditional design methods such as adaptive method and nonlinear mathematical programming method, the proposed technique could achieve global optimization with a success rate of 100% thanks to the adaptability and parallel ability of genetic algorithm; in addition, compared with the design requirements, the redundancy of pitch diameter and tooth width in the 0.75 MW case was 31% and 19.23%, respectively, while that in the 5 MW case was slightly more than 4.69% and 2.65%, showing that the proposed technique could provide effective design margin for designers under different design conditions.
-
Key words:
- spiral bevel gear /
- gear blank parameters /
- strength benchmark /
- genetic algorithm /
- optimization design
-
图 2 轮坯参数的体积及疲劳强度对比
Figure 2. Comparison of volume and fatigue strength of gear blank parameters
图 3 基于HB和ISO标准的计算安全系数
Figure 3. Calculated safety factors based on HB and ISO standards
图 4 提高应力极限后疲劳强度对比
Figure 4. Comparison of fatigue strength after improving stress limits
图 5 增大轮坯尺寸后疲劳强度对比
Figure 5. Comparison of fatigue strength after enlarging size of blank
图 6 综合优化后体积及疲劳强度对比
Figure 6. Comparison of volume and fatigue strength after comprehensive optimization
HB接触 Yb 齿根应力纵向分布系数 σh 计算接触应力 mmn 中点法向模数 σh0 计算接触应力基本值 Yθ 弯曲强度温度系数 σhp 许用接触应力 Yx 尺寸系数 Ka 使用系数 Yst 试验齿轮应力修正系数 Kv 动载系数 σflim 弯曲疲劳极限 Khβ 接触强度齿向载荷分布系数 Sfmin 弯曲强度最小安全系数 Zh 节点区域系数 ISO接触 Ze 弹性系数 Khα 接触强度齿间载荷分配系数 Zε 重合度系数 Fn 当量齿轮法向力 Zi 接触强度惯性系数 lbm 中点接触线长度 Zρ 齿廓曲率修正系数 ρrel 垂直于接触线的相对曲率半径 Zc 鼓形系数 Zmb 中点区域系数 dv1 当量齿轮端面分度圆直径 Zls 接触强度载荷分担系数 b 齿面宽 Zk 锥齿轮系数 uv 当量齿数比 Znt 接触强度寿命系数 Zn 寿命系数 Zx 接触强度尺寸系数 Zθ 接触强度温度系数 Zl 润滑系数 σhlim 接触疲劳极限 Zv 速度系数 Shmin 接触强度最小安全系数 Zr 粗糙度系数 HB弯曲 Zw 齿面工作硬化系数 σf 计算弯曲应力 Zhyp 准双曲面系数 σf0 计算弯曲应力基本值 ISO弯曲 σfp 许用弯曲应力 Kfα 弯曲强度齿间载荷分配系数 Kfβ 弯曲强度齿向载荷分布系数 Yε 弯曲强度重合度系数 Fmt 当量齿轮分度圆切向力 Ybs 螺旋角系数 Yfa 齿形系数 Yls 弯曲强度载荷分担系数 Ysa 应力修正系数 Ynt 弯曲强度寿命系数 Yε 加载点系数 Yδrelt 齿根圆角敏感系数 Yγ 载荷分担系数 Yrrelt 齿根表面状况系数 Yi 弯曲强度惯性系数 Ylc 纵向曲率系数 
下载: 导出CSV
表 1 HB和ISO应力计算公式
Table 1. HB and ISO stress calculation formulas
疲劳强度基准 计算公式 HB标准 接触 $ \begin{gathered} {\sigma _{\mathrm {h}}} = {\sigma _{{\mathrm {h}}0}}\sqrt {{K_{\mathrm {a}}}{K_{\mathrm {v}}}{K_{{\text {hβ}}}}} \\ {\sigma _{{\mathrm {h0}}}} = {Z_{\mathrm {h}}}{Z_{\mathrm {e}}}{Z_{\text {ε}}}{Z_{\mathrm {i}}}{Z_{\text{ρ}}}{Z_{\mathrm {c}}}\sqrt {\frac{{{F_{{\mathrm {mt}}}}}}{{{d_{{\mathrm {v}}1}}b}}\cdot\frac{{{u_{\mathrm {v}}} + 1}}{{{u_{\mathrm {v}}}}}} \\ {\sigma _{{\mathrm {hp}}}} = {Z_{\mathrm {n}}}{Z_{\text{θ}}}\frac{{{\sigma _{{\mathrm {hlim}}}}}}{{{S_{{\mathrm {hmin}}}}}} \\ \end{gathered} $ 弯曲 $ \begin{gathered} {\sigma _{\mathrm {f}}} = {\sigma _{{\mathrm {f}}0}}{K_{\mathrm {a}}}{K_{\mathrm {v}}}{K_{{\text {fβ}}}} \\ {\sigma _{{\mathrm {f0}}}} = {Y_{{\mathrm {fa}}}}{Y_{{\mathrm {sa}}}}{Y_{\text {ε}}}{Y_{\text{γ}}}{Y_{\mathrm {i}}}{Y_{{\mathrm {lc}}}}\frac{{{F_{{\mathrm {mt}}}}}}{{b{m_{{\mathrm {mn}}}}}}{Y_{\mathrm {b}}} \\ {\sigma _{{\mathrm {fp}}}} = {Y_{\text{θ}}}{Y_{\mathrm {x}}}{Y_{{\mathrm {st}}}}\frac{{{\sigma _{{\mathrm {flim}}}}}}{{{S_{{\mathrm {fmin}}}}}} \\ \end{gathered} $ ISO标准 接触 $ \begin{gathered} {\sigma _{\mathrm {h}}} = {\sigma _{{\mathrm {h0}}}}\sqrt {{K_{\mathrm {a}}}{K_{\mathrm {v}}}{K_{{\text {hβ}}}}{K_{{\text {hα}}}}} \\ {\sigma _{{\mathrm {h0}}}} = \sqrt {\frac{{{F_{\mathrm {n}}}}}{{{l_{{\mathrm {bm}}}}{\rho _{{\mathrm {rel}}}}}}{Z_{{\mathrm {mb}}}}{Z_{{\mathrm {ls}}}}{Z_{\mathrm {e}}}{Z_{\mathrm {k}}}} \\ {\sigma _{{\mathrm {hp}}}} = {\sigma _{{\mathrm {hlim}}}}{Z_{{\mathrm {nt}}}}{Z_{\mathrm {x}}}{Z_{\mathrm {l}}}{Z_{\mathrm {v}}}{Z_{\mathrm {r}}}{Z_{\mathrm {w}}}{Z_{{\mathrm {hyp}}}} \\ \end{gathered} $ 弯曲 $ \begin{gathered} {\sigma _{\mathrm {f}}} = {\sigma _{{\mathrm {f}}0}}{K_{\mathrm {a}}}{K_{\mathrm {v}}}{K_{{\text {fβ}}}}{K_{{\text {fα}}}} \\ {\sigma _{{\mathrm {f}}0}} = {Y_{{\mathrm {fa}}}}{Y_{{\mathrm {sa}}}}{Y_{\text {ε}}}{Y_{{\mathrm {bs}}}}{Y_{{\mathrm {ls}}}}\frac{{{F_{{\mathrm {mt}}}}}}{{b{m_{{\mathrm {mn}}}}}} \\ {\sigma _{{\mathrm {fp}}}} = {\sigma _{{\mathrm {flim}} }}{Y_{{\mathrm {st}}}}{Y_{{\mathrm {nt}}}}{Y_{{\text{δrelt}}}}{Y_{{\mathrm {rrelt}}}}{Y_{\mathrm {x}}} \\ \end{gathered} $
下载: 导出CSV
表 3 ISO最小安全系数
Table 3. ISO minimum safety factor
可靠性 Shmin Sfmin 一般可靠性 1.00～1.10 1.30 较高可靠性 1.12～1.25 1.30 高可靠性 1.50～1.60 1.50
下载: 导出CSV
表 4 最优参数修正系数及应力计算结果
Table 4. Results of correction coefficients and stress of optimal parameters
参数 数值 HB ISO 一般修正
系数使用系数Ka 1 1.00 动载系数Kv 1 2.1815 接触强度齿向载荷分布系数Khβ 1 1.5 接触强度齿间载荷分配系数Khα 1 接触强度
修正系数节点区域系数Zh 2.1307 中点区域系数Zmb 1.0006 弹性系数Ze 192.83 192.83 重合度系数Zε 1.1583 接触强度载荷分担系数Zls 0.8762 接触强度惯性系数Zi 1 齿廓曲率修正系数Zρ 1.0039 鼓形系数Zc 1.225 寿命系数Zn/Znt 1 1 接触强度温度系数Zθ 0.9492 接触强度尺寸系数Zx 1 锥齿轮系数Zk 0.85 润滑系数Zl 1.0199 速度系数Zv 1.0622 粗糙度系数Zr 0.9789 准双曲面系数Zhyp 1.045 中点接触线长度lmb/mm 16.306 垂直于接触线的相对曲率半径ρrel/mm 18.426 弯曲强度
修正系数齿形系数Yfa 2.32 2.247 应力修正系数Ysa 1.7387 1.8432 加载点系数Yε 0.5714 弯曲强度重合度系数Yε 0.625 纵向曲率分布系数Ylc 1.018 试验齿轮应力修正系数Yst 1.775 2 齿根应力纵向分布系数Yb 1.8076 尺寸系数Yx 0.8647 1.0176 载荷分担系数Yγ 0.8139 弯曲强度惯性系数Yi 1 弯曲强度温度系数Yθ 0.9492 螺旋角系数Ybs 1.1789 齿根圆角敏感系数Yδrelt 1.0197 齿根表面状况系数Yrrelt 0.9567 弯曲强度寿命系数Ynt 1 应力结果 计算接触应力基本值σh0/MPa 1266.299 963.771 计算接触应力σh/MPa 1266.299 1743.4 许用接触应力σhp/MPa 1356.027 1773.4 接触强度计算安全系数Sh 1.2 1.02 计算弯曲应力基本值σf0 600.66 407.965 计算弯曲应力σf 600.66 1335 许用弯曲应力σfp 1018.735 1735.546 弯曲强度计算安全系数Sf 2.12 1.3
下载: 导出CSV
表 5 轮坯参数的体积及疲劳强度对比
Table 5. Comparison of volume and fatigue strength of gear blank parameters
组别 Z1 Z2 met/mm b/mm HB ISO V/106 mm3 Sh Sf Sh Sf 1 38 51 2.84 23 1.27 2.50 1.05 1.34 0.8834 2 38 51 2.84 21 1.20 2.12 1.02 1.30 0.8476 3 41 55 2.63 21 1.23 2.31 1.00 1.15 0.8455 4 44 59 2.45 21 1.25 2.35 1.00 1.06 0.8389 5 50 67 2.16 21 1.29 2.42 0.99 0.91 0.8374 6 53 71 2.04 21 1.29 2.43 0.99 0.86 0.8373
下载: 导出CSV
表 6 弧齿锥齿轮轮坯参数实例
Table 6. Parameter examples of spiral bevel gear blank
组别 Z1 Z2 met/mm b/mm βm/（°） $ {\alpha }_{\mathrm{n}}/ $（°） 1 39 53 5.512 60 35 20 2 36 49 5.972 60 35 20 3 31 42 6.935 62 35 20 4 27 37 7.96 62 35 20 5 26 35 8.27 62 35 20 6 25 34 8.6 62 35 20 7 21 29 10 62 35 20
下载: 导出CSV
表 7 提高应力极限后体积及疲劳强度对比
Table 7. Comparison of volume and fatigue strength after improving stress limits
组别 HB ISO V/106 mm3 Sh Sf Sh Sf 1 1.1719 1.7359 0.9641 0.8045 7.8674 2 1.1708 1.7405 0.9666 0.8703 7.9301 3 1.1602 1.7974 0.9768 1.0275 7.9342 4 1.1204 1.7315 1.0288 1.3018 8.2543 5 1.0976 1.6926 1.0811 1.4941 7.8765 6 1.0891 1.6702 1.1318 1.6932 8.1255 7 0.9756 1.5182 1.1949 2.1572 8.0407
下载: 导出CSV
表 8 增大轮坯尺寸后体积对比
Table 8. Comparison of volume after enlarging size of blank
组别 Z1 Z2 met/mm b/mm V/107 mm3 1 39 53 2 36 49 3 31 42 8.31 83.098 1.3827 4 27 37 9.087 90.871 1.2406 5 26 35 8.8 88.002 0.9496 6 25 34 9.113 91.132 0.9617 7 21 29 10.808 108.084 0.9452
下载: 导出CSV
表 9 增大轮坯尺寸后疲劳强度对比
Table 9. Comparison of fatigue strength after enlarging size of blank
组别 HB ISO Sh Sf Sh Sf 1 2 3 1.331 2.548 1.078 1.300 4 1.259 2.631 1.030 1.310 5 1.120 2.201 1.042 1.378 6 1.120 2.323 1.124 1.661 7 1.120 2.749 1.223 2.308
下载: 导出CSV
表 10 综合优化后体积及疲劳强度对比
Table 10. Comparison of volume and fatigue strength after comprehensive optimization
组别 HB ISO V/107 mm3 Sh Sf Sh Sf 优化前第1组 0.866 1.095 0.852 0.889 0.8254 优化前第2组 0.846 1.063 0.895 1.02 0.7876 优化后第1组 1.1234 1.8734 1.0 1.30 0.9447 优化后第2组 1.12 1.365 1.362 1.925 1.0733
下载: 导出CSV
-
[1] 杨可桢,程光蕴,李仲生,等. 机械设计基础[M]. 6版. 北京: 高等教育出版社,2013. [2] GOLABI S,FESHARAKI J J,YAZDIPOOR M. Gear train optimization based on minimum volume/weight design[J]. Mechanism and Machine Theory,2014,73: 197-217. [3] PATIL M,RAMKUMAR P,MACHINE K S. Multi-objective optimization of the two-stage helical gearbox with tribological constraints[J]. Mechanism and Machine Theory,2019,138: 38-57. doi: 10.1016/j.mechmachtheory.2019.03.037 [4] 曹雪梅,周彦伟,邓效忠,等. 基于遗传算法的高齿弧齿锥齿轮优化设计[J]. 机械传动,2004,28(1): 28-29,47. CAO Xuemei,ZHOU Yanwei,DENG Xiaozhong,et al. The optimization of spiral bevel gear with high tooth based on genetic algorithm[J]. Journal of Mechanical Transmission,2004,28(1): 28-29,47. (in Chinese doi: 10.3969/j.issn.1004-2539.2004.01.009CAO Xuemei, ZHOU Yanwei, DENG Xiaozhong, et al. The optimization of spiral bevel gear with high tooth based on genetic algorithm[J]. Journal of Mechanical Transmission, 2004, 28(1): 28-29, 47. (in Chinese) doi: 10.3969/j.issn.1004-2539.2004.01.009 [5] 刘向阳. 基于机床调整参数的螺旋锥齿轮建模及接触分析[D]. 哈尔滨: 哈尔滨工业大学,2018. LIU Xiangyang. Modeling and contact analysis of spiral bevel gear based on machine tool adjustment parameters[D]. Harbin: Harbin Institute of Technology,2018. (in ChineseLIU Xiangyang. Modeling and contact analysis of spiral bevel gear based on machine tool adjustment parameters[D]. Harbin: Harbin Institute of Technology, 2018. (in Chinese) [6] 许德帮,徐元浩,迟文俊. 基于遗传算法和. NET平台的锥齿轮传动优化设计系统研究[J]. 西南大学学报(自然科学版),2017,39(2): 153-158. XU Debang,XU Yuanhao,CHI Wenjun. Research on optimal design system of bevel gear based on genetic algorithms and. NET platform[J]. Journal of Southwest University (Natural Science Edition),2017,39(2): 153-158. (in ChineseXU Debang, XU Yuanhao, CHI Wenjun. Research on optimal design system of bevel gear based on genetic algorithms and. NET platform[J]. Journal of Southwest University (Natural Science Edition), 2017, 39(2): 153-158. (in Chinese) [7] ZHUO Yaobin,XIANG Xueyan,ZHOU Xiaojun,et al. A method for the global optimization of the tooth contact pattern and transmission error of spiral bevel and hypoid gears[J]. Journal of Zhejiang University: Science A,2017,18(5): 377-392. doi: 10.1631/jzus.A1600240 [8] 姜志宏,王兴卫,池汉佳,等. 遗传算法下弧齿锥齿轮齿面接触特性优化[J]. 机械科学与技术,2022,41(3): 409-413. JIANG Zhihong,WANG Xingwei,CHI Hanjia,et al. Optimization of tooth surface contact characteristics of spiral bevel gears by genetic algorithm[J]. Mechanical Science and Technology for Aerospace Engineering,2022,41(3): 409-413. (in ChineseJIANG Zhihong, WANG Xingwei, CHI Hanjia, et al. Optimization of tooth surface contact characteristics of spiral bevel gears by genetic algorithm[J]. Mechanical Science and Technology for Aerospace Engineering, 2022, 41(3): 409-413. (in Chinese) [9] International Organization for Standardization. Calculation of load capacity of bevel gears: Part 1 introduction and general influence factors: ISO 10300-1-2014[S]. Geneva,Switzerland: International Organization for Standardization,2014. [10] International Organization for Standardization. Calculation of load capacity of bevel gears: Part 2 calculation of surface durability (pitting): ISO 10300-2[S]. Geneva,Switzerland: International Organization for Standardization,2014. [11] International Organization for Standardization. Calculation of load capacity of bevel gears: Part 3 calculation of tooth root strength: ISO 10300-3[S]. Geneva,Switzerland: International Organization for Standardization,2014. [12] American Gear Manufacturers Association. Rating the pitting resistance and bending strength of generated straight bevel,zerol bevel and spiral bevel gear teeth: ANSI/AGMA 2003-B97[S]. Alexandria,US: American Gear Manufacturers Association,1997. [13] 中华人民共和国航空工业部. 航空锥齿轮 承载能力一般系数计算: HB/Z 89.1-1985[S]. 北京: 中华人民共和国航空工业部,1985. [14] 中华人民共和国航空工业部. 航空锥齿轮 齿面接触疲劳强度计算: HB/Z 89.2-1985[S]. 北京: 中华人民共和国航空工业部,1985. [15] 中华人民共和国航空工业部. 航空锥齿轮 轮齿弯曲疲劳强度计算: HB/Z 89.3-1985[S]. 北京: 中华人民共和国航空工业部,1985. [16] 中华人民共和国航空工业部. 航空锥齿轮 胶合承载能力计算: HB/Z 89.4-1985[S]. 北京: 中华人民共和国航空工业部,1985. [17] 吴昌林,吕云霏. ISO与AGMA渐开线圆柱齿轮强度计算标准的比较[J]. 中国机械工程,2011,22(12): 1418-1423,1465. WU Changlin,LÜ Yunfei. Comparison between ISO and AGMA gear strength rating methods for involute cylindrical gears[J]. China Mechanical Engineering,2011,22(12): 1418-1423,1465. (in ChineseWU Changlin, LÜ Yunfei. Comparison between ISO and AGMA gear strength rating methods for involute cylindrical gears[J]. China Mechanical Engineering, 2011, 22(12): 1418-1423, 1465. (in Chinese) [18] 黎向宇,刘少军. HB与AGMA锥齿轮强度计算标准比较[J]. 机械强度,2015,37(5): 917-923. LI Xiangyu,LIU Shaojun. Comparison between HB and AGMA bevel gear strength calculation standard[J]. Journal of Mechanical Strength,2015,37(5): 917-923. (in ChineseLI Xiangyu, LIU Shaojun. Comparison between HB and AGMA bevel gear strength calculation standard[J]. Journal of Mechanical Strength, 2015, 37(5): 917-923. (in Chinese) [19] LISLE T J,SHAW B A,FRAZER R C. External spur gear root bending stress: a comparison of ISO 6336: 2006,AGMA 2101-D04,ANSYS finite element analysis and strain gauge techniques[J]. Mechanism and Machine Theory,2017,111: 1-9. doi: 10.1016/j.mechmachtheory.2017.01.006 [20] BUCUR M,CANANAU S,MIRICA R F,et al. Comparative analysis concerning the state of stress for involute gears using ISO,AGMA standards and finite element method[J]. IOP Conference Series: Materials Science and Engineering,2022,1262(1): 012002. doi: 10.1088/1757-899X/1262/1/012002 [21] 滕文爽,郭梅,宋振海. HB与ISO锥齿轮齿面接触疲劳强度计算标准比较[J]. 机械传动,2021,45(4): 156-159,176. TENG Wenshuang,GUO Mei,SONG Zhenhai. Comparison between HB and ISO standard for calculating bevel gear tooth surface contact fatigue strength[J]. Journal of Mechanical Transmission,2021,45(4): 156-159,176. (in ChineseTENG Wenshuang, GUO Mei, SONG Zhenhai. Comparison between HB and ISO standard for calculating bevel gear tooth surface contact fatigue strength[J]. Journal of Mechanical Transmission, 2021, 45(4): 156-159, 176. (in Chinese) [22] 滕文爽,郭梅,宋振海,等. HB与ISO标准中锥齿轮轮齿弯曲疲劳强度计算标准比较[J]. 航空发动机,2022,48(3): 65-69. TENG Wenshuang,GUO Mei,SONG Zhenhai,et al. Comparison of calculation standards for bending fatigue strength of bevel gear teeth in HB and ISO standards[J]. Aeroengine,2022,48(3): 65-69. (in ChineseTENG Wenshuang, GUO Mei, SONG Zhenhai, et al. Comparison of calculation standards for bending fatigue strength of bevel gear teeth in HB and ISO standards[J]. Aeroengine, 2022, 48(3): 65-69. (in Chinese) [23] 罗潘,梁尚明,蒋立茂,等. 基于多目标遗传算法的弧齿锥齿轮多学科优化设计[J]. 机械设计与制造,2012(8): 6-8. LUO Pan,LIANG Shangming,JIANG Limao,et al. Multidisciplinary design optimization of spiral bevel gear based on multi-objective genetic algorithm[J]. Machinery Design & Manufacture,2012(8): 6-8. (in Chinese doi: 10.3969/j.issn.1001-3997.2012.08.003LUO Pan, LIANG Shangming, JIANG Limao, et al. Multidisciplinary design optimization of spiral bevel gear based on multi-objective genetic algorithm[J]. Machinery Design & Manufacture, 2012(8): 6-8. (in Chinese) doi: 10.3969/j.issn.1001-3997.2012.08.003 -
点击查看大图
计量
- 文章访问数: 465
- HTML浏览量: 367
- PDF量: 28
- 被引次数: 0