Multi-objective Optimization of High-speed Gear Shaft Using Response Surface Method
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摘要: 以提升高速齿轮轴的运行安全性为目的,通过模态分析、参数灵敏度分析和实验设计,建立了高速齿轮轴参数化设计与响应面优化模型,利用多目标优化遗传算法对1阶、2阶临界转速及其对应最大振幅进行寻优,得到了Pareto最优解,实现了多参数耦合下的优化设计,得到了运行安全性较高的转子结构。结果表明:优化后1阶、2阶临界转速与工作转速的间隔降幅分别为22.9%,10.8%,满足转子安全设计要求;通过可靠性校验,1阶、2阶临界转速在优化后的可靠度分别为100%、99.02%,进一步证明了优化方法的正确性。Abstract: For improving the operating safety of high-speed gear shafts, the parametric design and response surface optimization model of the high-speed gear shaft were established by the modal analysis, parameter sensitivity analysis and DOE. The MOGA was used to optimize the first and second critical speed and their corresponding maximum amplitudes, and the Pareto optimal solution set was obtained. The optimization design under the multi-parameter coupling was realized, and the rotor structure with high operation safety was obtained. The optimized results show that the intervals between the first and second order critical speeds and the working speed are reduced by 22.9% and 10.8% respectively, which meet the requirements of rotor safety design. Through reliability verification, the optimized reliability of the first and second order critical speeds are 100% and 99.02% respectively, which further proves the correctness of the optimal method.
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表 1 碳素结构钢材料属性
Table 1. Material properties of carbon structural steel
密度/(kg•m-3) 杨氏模量/MPa 泊松比 屈服强度/MPa 7 850 2×105 0.3 250 表 2 转子的固有频率
Table 2. Natural frequencies of the rotor
转速/(r•min-1) 频率/Hz 第1阶 第2阶 第3阶 BW FW BW FW FW BW 0 136.8 139.2 170.0 173.1 247.1 260.2 5 000 133.3 142.8 164.3 179.1 247.1 260.2 10 000 129.1 147.4 157.6 186.8 247.1 260.2 15 000 124.9 153.2 150.3 194.8 247.1 260.2 20 000 120.8 157.7 144.7 203.2 247.1 260.2 25 000 116.9 162.9 138.8 211.8 247.1 260.2 表 3 设计变量及其取值范围
Table 3. Design variables and their ranges
设计变量 设计初值 设计范围 轴径半径/mm 37.5 [33.75, 41.25] 轴承跨距/mm 235 [211.5, 258.5] 左轮重心位置/mm 110 [99, 121] 右轮重心位置/mm 970 [873, 1 067] 叶轮重量/kg 10 [9, 11] 叶轮转动惯量/(kg•mm2) 60 000 [54 000, 66 000] 轴承刚度/(N•mm-1) 1×106 [9×105, 1.1×106] 轴承阻尼/(N•s•mm-1) 200 [180, 220] 表 4 输入值与输出值的相关系数
Table 4. Design variables and their ranges
设计变量 Ωc1 A1 Ωc2 A2 轴径半径 0.002 0.033 0.016 -0.014 轴承跨距 -0.035 0 0.043 0.154 左轮重心位置 0.363 0.315 0.059 0.116 右轮重心位置 -0.663 -0.685 -0.973 -0.670 叶轮重量 -0.415 -0.420 -0.132 -0.161 叶轮转动惯量 0.072 0.058 -0.035 -0.130 轴承刚度 0.117 0.054 -0.011 -0.050 轴承阻尼 -0.065 -0.028 -0.021 -0.162 表 5 部分样本点
Table 5. Partial sample points
参数名称 1 2 … 25 X1/mm 111.76 114.4 … 112.64 X2/mm 954.48 993.28 … 876.88 X3/kg 9.04 9.84 … 10.16 Ωc1/(r•min-1) 9 217.8 8 975.9 … 8 813.6 Ωc2/(r•min-1) 12 755.7 10 180.7 … 14 564.3 A1/mm 13.902 13.467 … 13.335 A2/mm 16.090 13.367 … 10.503 表 6 输出变量拟合优度
Table 6. Goodness-of-fit results for output variables
误差类型 Ωc1 Ωc2 A1 A2 R2 1 1 1 1 ERMS 5.84×10-6 1.93×10-5 1.4×10-8 6.4×10-8 ERMA 0 0 0 0 表 7 输出变量拟合优度
Table 7. Goodness-of-fit results for output variables
名称 数值 初始种群数 10 000 每次迭代样本数 100 最大Pareto百分比 70% 收敛稳定百分比 2% 最大迭代次数 20 最大候选点数 3 表 8 优化前后结果对比
Table 8. Comparison of results before and after optimization
参数名称 初始点 候选点1 候选点2 候选点3 X1/mm 110 100.27 102.38 101.06 X2/mm 970 1 066.6 1 066.7 1 066.9 X3/kg 10 10.79 10.961 10.992 Ωc1/(r•min-1) 8 776.7 6 491.5 6 488.2 6 569 Ωc2/(r•min-1) 11 337 7 575.2 7 602.7 7 676.6 A1/mm 13.302 8.919 9.152 9.405 A2/mm 14.643 11.798 11.653 11.394 表 9 线性尺寸极限偏差数值表
Table 9. Comparison of results before and after optimization
尺寸范围/mm 精密f 中等m 粗糙c 最粗v [0.5, 3] ±0.05 ±0.1 ±0.2 - (3, 6] ±0.05 ±0.1 ±0.3 ±0.5 (6, 30] ±0.1 ±0.2 ±0.5 ±1 (30, 120] ±0.1 ±0.3 ±0.8 ±1.5 (120, 500] ±0.2 ±0.5 ±1.2 ±2.5 (500, 1 000] ±0.3 ±0.8 ±1.2 ±1.2 (1 000, 2 000] ±0.5 ±1.2 ±3 ±6 表 10 随机变量统计特性表
Table 10. Statistical characteristics of the random variables
参数名称 均值μ 标准差σ 轴径半径/mm 37.5 0.067 轴承跨距/mm 235 0.333 左轮重心位置/mm 100.3 0.2 右轮重心位置/mm 1 066.6 0.533 叶轮重量/kg 10.8 0.2 叶轮转动惯量/(kg•mm2) 6×104 1 200 轴承刚度/(N•mm-1) 1×106 2×104 轴承阻尼/(N•s•mm-1) 200 4 弹性模量/MPa 2×105 6 000 外载荷/N 5 000 100 表 11 Ωc1和Ωc2部分概率列表
Table 11. 11 Partial probability list of Ωc1 and Ωc2
序号 Ωc1/(r·min-1) 可靠度/% σ值 Ωc2/(r·min-1) 可靠度/% σ值 1 6 477.3 0.01 -3.8 7 701.7 0.01 -3.8 2 6 510.5 0.04 -3.3 7 740.4 0.03 -3.5 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 21 7 141.0 97.95 2.0 8 475.9 98.48 2.2 22 7 174.2 98.95 2.3 8 500.0 99.02 2.3 23 7 207.4 99.59 2.6 8 514.6 99.36 2.5 24 7 240.5 99.82 2.9 8 553.4 99.74 2.8 25 7 273.7 99.91 3.1 8 592.1 99.90 3.1 26 7 306.9 99.95 3.3 8 630.8 99.95 3.3 27 7 340.1 99.99 3.8 8 669.5 99.97 3.5 28 8 500.0 100.00 4.7 8 708.2 99.99 3.8 -
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