Journal of Jilin University(Engineering and Technology Edition) ›› 2019, Vol. 49 ›› Issue (4): 1194-1202.doi: 10.13229/j.cnki.jdxbgxb20180336

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Multi⁃objective optimization design of flexspline profile parameters of double⁃arc harmonic drive

Jia⁃xu WANG1,2(),Qian⁃qian JIANG1,Jun⁃yang LI1,Yan⁃feng HAN1,Lei ZHANG1,Ting TANG1   

  1. 1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China
    2. School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China
  • Received:2018-04-13 Online:2019-07-01 Published:2019-07-16

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Abstract:

The continuous meshing zone and "double conjugate" phenomenon of the circular spline and flexspline under no?load conditions can effectively improve the meshing performance of the double?arc harmonic gear device. For this reason, a multi?objective optimization design method of the tooth profile parameters of the double?circle flexspline is proposed. For the non?public tangential double?arc tooth profile, the conjugate tooth profile was designed by using the relationship between the improved kinematics method and the precise rotation angle. To reduce the difference between the conjugate blank area and the conjugate tooth profile as the optimization goal, a harmonic drive multi?objective optimization model was established. Considering the existence conditions of conjugate tooth profiles, single target analysis and multi?objective optimization are performed. The results show that the two objective functions increase with the increase of the radial deformation coefficient and the double circular arc inclination, and decrease with the increase of the tooth thickness ratio; Radial deformation coefficient and double circular arc inclination have the greatest influence on the two objective functions, and tooth thickness ratio has less influence on the difference of minimum conjugate tooth profile. Convex arc radius coefficient and concave arc radius coefficient have almost no effect on the minimum blank conjugate area; this optimization method can achieve a reasonable selection of parameters of the soft tooth profile, and make the freewheeling state between the circular spline and the flexspline is less than 1.2°, and the difference between the conjugate profiles is about 0.83 μm, which can meet the requirements of engineering applications. Guaranteed no?load continuous meshing interval and "double conjugate" transmission accuracy.

Key words: technology of instrument and meter, harmonic drive, double?arc tooth profile, multi?objective optimization, profile parameters

CLC Number: 

  • TH132.43

Fig. 1

Harmonic gear coordinate system"

Table 1

Harmonic drive parameter meaning."

符 号 意义
ρ O 1点矢径
r m 变形前柔轮的中性层半径
Δφ 矢径ρ与Y 2的夹角
β 轴Y 1相对于轴Y 2的转角
φ 1 矢径ρ与轴Y的夹角
φF 输出端轴YF 相对于轴Y 2的转角
φ 输出端轴YF 与轴Y的夹角
φ w 轴Y相对于Y 2的转角

Fig.2

Coordinate system of double?arc tooth profile"

Table 2

Flexspline profile parameters"

符号 参数 符号 参数
h a * 齿顶高系数

t

y a

x a

y f

x f

m

齿根壁厚
h f * 齿根高系数 凸齿圆心移距量
h 全齿高 凸齿圆心偏移量
ρ a * 凸圆弧半径系数 凹齿圆心移距量
ρ f * 凹圆弧半径系数 凹齿圆心偏移量
γ 双圆弧倾角 模数

Table 3

Flexspline tooth profile parameters"

参 数 数值/mm 参 数 数值/mm
h a * 0.7 t 1 0.7132
h f * 0.8 γ 6°
xa 0.4001 x f 0.2617
ya 0.0671 y f 0.0679
ρ a * 1.5 ρ f * 1.7
k t 1.7 h 0.594

Fig.3

Conjugate area of tooth profile of double?arc flexspline"

Fig.4

Conjugate profile of tooth profile of double?arc flexspline"

Table 4

Relationship between minimum radial deformation coefficient and inclination angle"

序号 w 0 * γ/(?) f 1(X)/(?) f 2(X)×102/mm
1 1.0451 6 1.193892 1.625692
2 1.0356 7 1.366803 2.568918
3 1.0249 8 1.380991 3.557457
4 1.0131 9 1.266604 4.450072
5 1.0004 10 1.265910 5.291463
6 0.9869 11 1.399478 6.388948
7 0.9725 12 1.218316 7.427592
8 0.9575 13 1.205427 8.416463
9 0.9419 14 1.14275 9.359473
10 0.9258 15 1.059737 10.254036

Fig.5

Influence of radial deformation coefficient on single objective function"

Fig.6

Influence of double arc angle on the single objective function"

Fig.7

Influence curve of convex arc radius coefficient on single objective function"

Fig.8

Influence curve of convex arc radius coefficient on single objective function"

Fig.9

Effect of tooth thickness ratio on single objective function"

Table 5

Harmonic gear drive multi-objective optimization results"

序 号 w 0 * γ/(?) ρ a * ρ a * k t f 1(X)/(?) f 2(X)×102/mm
1 1.0451 6 1.5791 1.6668 1.7 1.191279 0.083268
2 1.0356 7 1.6908 1.7289 1.7 1.356309 1.965230
3 1.0249 8 1.5489 1.6754 1.7 1.380993 2.961959
4 1.0131 9 1.5489 1.6753 1.7 1.266431 4.181113
5 1.0004 10 1.5390 1.6535 1.7 1.267915 5.210725
6 0.9869 11 1.5391 1.6534 1.7 1.397736 6.345894
7 0.9725 12 1.5610 1.6296 1.7 1.220946 7.208653
8 0.9575 13 1.5528 1.6194 1.7 1.206309 8.103814
9 0.9419 14 1.5791 1.6050 1.7 1.154419 8.907568
10 0.9258 15 1.5777 1.6001 1.7 1.073299 9.731521
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