工程科学与技术   2019, Vol. 51 Issue (1): 241-247

Tooth Profile Simplified Modeling and Transmission Angle Analysis of Push-rods Oscillatory Transmission with Needle Gears
FEI Yu, XIE Chao, LI Hua, HUANG Binghong, YAO Jin
School of Manufacturing Sci. and Eng., Sichuan Univ., Chengdu 610065, China
Abstract: The push-rods oscillatory transmission with needle gears has the advantages of simple structure, convenient manufacturing and high machining precision. For this kind of transmission, the profile equation of movable tooth is the basis for tooth analysis and the key issue to guarantee the precision and stability of transmission. Different from the traditional way, where the profile equation is derived by enveloping and the profile curve is approximated by line, a new method for tooth profile equation derivation and simplification was proposed. The profile equation of movable tooth was deduced by the contact condition in the transmission process. Then the equation was simplified by Taylor expansion to facilitate the calculation in the research of tooth profile and meshing characteristics. Sequentially, to guide the rapid design of the transmission structure, an analytical discriminant to guarantee the transmission undistorted was established by the simplified equation. On these bases, the computational formula of transmission angle was derived and the influence of the main design parameters, such as surge wheel’s eccentricity and needle gear’s rotational radius, on the maximum transmission angle were analyzed. Analysis showed that the maximum transmission angle is positively correlated with surge wheel’s eccentricity, yet negatively with needle gear’s rotational radius. Furthermore, the error of simplified equation was analyzed and the result showed that the maximum horizontal and vertical coordinate errors are not exceeding ±0.06 μm. Thus, the accurate tooth profile equation could be replaced by the simplified equation. Finally, the three-dimensional model of the reducer with simplified tooth profile curve was designed and based on it, the fixed speed ratio transmission was verified by simulating. Simulation result showed that the error of speed ratio is about 0.007%. Consequently, the practicability of the simplified equation and the feasibility of oscillatory reducer with simplified movable tooth profile was validated by both theoretical analysis and simulation. The research results could be used as a theoretical basis for structural design and application.
Key words: push-rod oscillatory transmission with needle gears    tooth profile simplified modeling    transmission angle analysis    error analysis

1 推杆针轮活齿传动原理

 图1 推杆针轮活齿传动原理 Fig. 1 Theory of push-rods oscillatory transmission with needle gears

2 活齿齿形的简化建模 2.1 活齿齿形方程推导

 图2 活齿齿形方程建模原理 Fig. 2 Theory of the tooth profile equation modeling

 \left\{ \begin{aligned} & {\varphi _1} = i{\varphi _2},\\ & {\omega _1} = i{\omega _2} \end{aligned} \right. (1)

 ${x_{{O_1}}} = a\cos {\varphi _1} - \sqrt {{b^2} - {a^2}\left( {\sin \;{\varphi _1}} \right){}^2}$ (2)

 ${\left( {{x_0} + l\cos \;{\varphi _2}} \right)^2} + {\left( {{y_0} + l\sin \;{\varphi _2}} \right)^2} = R_{\rm K}^2$ (3)

${{n}} = \left( {{x_0} + l\cos {\varphi _2},{y_0} + l\sin {\varphi _2}} \right)$

 ${{{V}}_{\rm {K}}} \cdot {{n}} = {{{V}}_{\rm {G}}} \cdot {{n}}$ (4)

 \left\{ \begin{aligned} & x = - l\cos \;{\varphi _2}\left( {1 \mp \frac{{{R_{\rm K}}}}{{\sqrt B }}} \right) - {x_{{O_1}}},\\ & y = iA\frac{{{R_{\rm K}}}}{{\sqrt B }} - l\sin \;{\varphi _2}\left( {1 \mp \frac{{{R_{\rm K}}}}{{\sqrt B }}} \right),\\ & A = - a\sin \left( {i{\varphi _2}} \right) + \frac{{{a^2}\sin \left( {2i{\varphi _2}} \right)}}{{2\sqrt {{b^2} - {a^2}{{\left( {\sin \left( {i{\varphi _2}} \right)} \right)}^2}} }},\\ & B = {i^2}{A^2} + {l^2} - 2ilA\sin\; {\varphi _2},\\ & {x_{{O_1}}} = a\cos \left( {i{\varphi _2}} \right) - \sqrt {{b^2} - {a^2}{{\left( {\sin \left( {i{\varphi _2}} \right)} \right)}^2}} \end{aligned} \right. (5)

“–”代表针轮与活齿外啮合，“+”代表针轮与活齿内啮合，因此式（5）取“－”表示本文所求的活齿齿形方程。

2.2 齿形方程的简化

 ${x_{{O_1}}} = a\cos \;{\varphi _1} - b\sqrt {1 - \frac{{{a^2}}}{{{b^2}}}\left( {\sin \;{\varphi _1}} \right){}^2}$ (6)

 $\lambda = \frac{a}{b},p = \lambda \sin\; {\varphi _1}$ (7)

 ${x_{{O_1}}} = a\cos \;{\varphi _1} - b\sqrt {1 - p{}^2}$ (8)

 $\sqrt {1 - {p^2}} = 1 - \frac{1}{2}{p^2} - \frac{1}{8}{p^4} - \cdots$ (9)

 $\;\;\;\;\;\;\;\;\;\;{x_{{O_1}}} = b[ - 1 + \frac{1}{4}{\lambda ^2} + \lambda \cos \;{\varphi _1} - \frac{1}{4}{\lambda ^2}\cos \left( {2{\varphi _1}} \right)]$ (10)

 $\frac{{{\rm d}{x_{O1}}}}{{{\rm d}t}} = b{\omega _1}\left( {\frac{1}{2}{\lambda ^2}\sin \left( {2{\varphi _1}} \right) - \lambda \sin \;{\varphi _1}} \right)$ (11)

 \left\{ \begin{aligned} & x = - l\cos \;{\varphi _2}\left( {1 - \frac{{{R_{\rm K}}}}{{\sqrt B }}} \right) - {x_{O1}},\\ & y = iA\frac{{{R_{\rm K}}}}{{\sqrt B }} - l\sin \;{\varphi _2}\left( {1 - \frac{{{R_{\rm K}}}}{{\sqrt B }}} \right),\\ & A = b\left( {\frac{1}{2}{\lambda ^2}\sin (2i{\varphi _2}) - \lambda \sin (i{\varphi _2})} \right),\\ & B = i_{}^2{A^2} + {l^2} - 2ilA\sin \;{\varphi _2},\\ & {x_{{O_1}}} = b\left( { - 1 + \frac{1}{4}{\lambda ^2} + \lambda \cos (i{\varphi _2}) - \frac{1}{4}{\lambda ^2}\cos \left( {2i{\varphi _2}} \right)} \right),\\ & \lambda = a/b \end{aligned} \right. (12)

2.3 传动不失真条件

 $\rho = \frac{{{{\left( {{{x'}^2} + {{y'}^2}} \right)}^{3/2}}}}{{x'y{'\!'} - x{'\!'}y'}}$ (13)

 $- {i^2}a(1 - \lambda )\frac{{{R_{\rm K}}}}{l} + (l - {R_{\rm K}})\ge 0$ (14)

3 传动角分析

 图3 活齿对中心轮传动角解析 Fig. 3 Transmission angle of movable tooth to the center wheel

 $\alpha = \frac{{\text{π}} }{2} - \beta$ (15)

 图4 不同偏心距 ${{a}}$ 对传动角 ${{\alpha}}$ 的影响 Fig. 4 Influence of different eccentricities on the transmission angle

 图5 不同回转半径 ${{l}}$ 对传动角 ${{\alpha}}$ 的影响 Fig. 5 Influence of different rotational radius on the transmission angle

4 齿形误差分析与定传动比验证

4.1 齿形简化方程误差分析

 图6 活齿齿形曲线对比 Fig. 6 Comparison of two tooth profile curves

 图7 齿形简化方程误差 Fig. 7 Errors of simplified tooth profile equation

4.2 定传动比仿真分析

 图8 推杆针轮活齿减速器实体建模 Fig. 8 3D model of the push-rods with needle gears oscillatory speed reducer

 图9 中心轮输出角速度 Fig. 9 Output angular velocity of the center wheel

5 结　论

1）利用活齿传动过程中的接触条件，解出接触点坐标，从而对推杆针轮活齿传动中活齿的齿形方程进行推导。与传统使用包络原理的方法相比推导过程更简便，推导出的方程形式更直观，这种方法可以为活齿齿形曲线的推导提供新思路。

2）在活齿齿形方程推导过程中，利用泰勒展式得到活齿齿形的简化方程，该方程可以简化在研究活齿齿形与啮合特性如曲率和传动角等问题时的数学计算。利用该简化方程推导出活齿传动不失真的参数判别式，该规律有助于快速对活齿结构进行设计。

3）传动角分析表明，推杆针轮活齿传动的最大传动角与激波器偏心距成正相关，与针轮回转半径成负相关，并且激波器偏心距对最大传动角的影响更为显著，这一规律可为结构参数的优化选择提供依据。

4）对活齿齿形简化方程进行误差分析，其结果表明横、纵坐标误差最大不超过±0.06 μm，简化方程可以替代准确方程进行研究，验证了活齿简化方程的实用性。通过仿真验证其定传动比传动的功能，传动比平均误差约为0.007%，从而验证了采用活齿简化方程构建活齿减速器的可行性。

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