﻿ 齿根过渡曲面对正交面齿轮弯曲强度影响
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (6): 1066-1072  DOI: 10.11990/jheu.201612080 0

### 引用本文

ZHAO Xinhui, SUN Yongguo, YU Guangbin. The effect of tooth root transition surface on bending strength of orthogonal face gear[J]. Journal of Harbin Engineering University, 2018, 39(6), 1066-1072. DOI: 10.11990/jheu.201612080.

### 文章历史

The effect of tooth root transition surface on bending strength of orthogonal face gear
ZHAO Xinhui, SUN Yongguo, YU Guangbin
School of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin 150080, China
Abstract: To analyze the bending strength of orthogonal face gear effectively, the tooth surface equation and dedendum transition curve surface equation on orthogonal face gear were deduced. The whole tooth profile of the orthogonal face gear was established, and two kinds of 3D entity models of orthogonal face gear with the whole tooth profile were generated by using Pro/Engineer software. Based on the theory of internally tangent parabola, the position of the maximum bending stress of the orthogonal face gear was in the upper half portion of the tooth surface along the tooth height than at the dedendum. Analyses of the bending strength of the three-tooth models of two kinds of face gear cut by slotting cutter from rounded corner and sharp corner at the addendum were carried out using ANSYS software. The tooth root transition curve surface cut by slotting cutter from rounded corner at the addendum improved the bending strength of the orthogonal face gear.
Key words: gear drive    orthogonal face gear    tooth profile equation    addendum arc gear shaper    tooth fillet    dangerous cross-section    bending strength    finite element analysis

1 正交面齿轮几何模型 1.1 加工坐标系与刀具齿面方程的建立

 Download: 图 1 正交面齿轮加工啮合示意图 Fig. 1 Schematic diagram of processing gear of orthogonal face gear
 Download: 图 2 正交面齿轮加工坐标系 Fig. 2 Machining coordinate system of orthogonal face gear

 $\begin{array}{l} \;\;\;\;\;\;{M_{2, S}} = {M_{2, 20}}{M_{20, S0}}{M_{S0, S}} = \\ \left[ {\begin{array}{*{20}{c}} {\cos {\varphi _2}\cos {\varphi _s}}&{ - \sin {\varphi _2}\cos {\varphi _2}}&{ - \sin {\varphi _2}}&0\\ { - \cos {\varphi _s}\sin {\varphi _2}}&{\sin {\varphi _s}\sin {\varphi _2}}&{ - \cos {\varphi _2}}&0\\ {\sin {\varphi _s}}&{\cos {\varphi _s}}&0&0\\ 0&0&0&1 \end{array}} \right] \end{array}$ (1)

 Download: 图 3 刀具齿面参数 Fig. 3 Parameters of the tool tooth surface
 $\begin{array}{l} \;\;\;{\mathit{\boldsymbol{r}}_s}({u_{s, }}{\theta _s}) = {[{x_s}\;\;{y_s}\;\;{z_s}\;\;t]^{\rm{T}}} = \\ \left[ {\begin{array}{*{20}{c}} { \pm {r_{bs}}[\sin ({\theta _{s0}} + {\theta _s}) - {\theta _s}\cos ({\theta _{s0}} + {\theta _s})]}\\ { - {r_{bs}}[\cos ({\theta _{s0}} + {\theta _s}) + {\theta _s}\sin ({\theta _{s0}} + {\theta _s})]}\\ {{u_s}}\\ 1 \end{array}} \right] \end{array}$ (2)

 ${\mathit{\boldsymbol{n}}_s} = \left[ {\begin{array}{*{20}{c}} {{n_{sx}}}\\ {{n_{sy}}}\\ {{n_{sz}}} \end{array}} \right] = \frac{{\partial {\mathit{\boldsymbol{r}}_s}/\partial {\theta _s} \times \partial {\mathit{\boldsymbol{r}}_s}/\partial {u_s}}}{{\left| {\partial {\mathit{\boldsymbol{r}}_s}/\partial {\theta _s} \times \partial {\mathit{\boldsymbol{r}}_s}/\partial {u_s}} \right|}} =\\ \left[ {\begin{array}{*{20}{c}} { \mp \cos ({\theta _{s0}} + {\theta _s})}\\ { - \sin ({\theta _{s0}} + {\theta _s})}\\ 0 \end{array}} \right]$ (3)
1.2 正交面齿轮齿面方程的建立

 ${\mathit{\boldsymbol{r}}_s} = {[{x_s}\;\;{y_s}\;\;{z_s}]^{\rm{T}}} = {x_s}{\mathit{\boldsymbol{i}}_s} + {y_s}{\mathit{\boldsymbol{j}}_s} + {z_s}{\mathit{\boldsymbol{k}}_s}$ (4)

 ${\mathit{\boldsymbol{v}}^{(S, 2)}} = \left[ \begin{array}{l} {v_x}^{(S, 2)}\\ {v_y}^{(S, 2)}\\ {v_z}^{(S, 2)} \end{array} \right] = {\omega _s}\left[ {\begin{array}{*{20}{c}} { - {y_s} - {z_s}{q_{2s}}\cos {\varphi _s}}\\ {{x_s} + {z_s}{q_{2s}}\sin {\varphi _s}}\\ {{q_{2s}}({x_s}\cos {\varphi _s} - {y_s}\sin {\varphi _s})} \end{array}} \right]$ (5)

 ${\mathit{\boldsymbol{n}}_s}\cdot{\mathit{\boldsymbol{v}}^{(S, 2)}} = 0$ (6)

 $f({\theta _s}, {\varphi _s}, {u_s}) = {r_{bs}} - {u_s}{q_{2s}}\cos {\varphi _\theta } = 0$ (7)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{x}}_2} = {\mathit{\boldsymbol{r}}_{bs}}\left[ {\cos {\varphi _2}\left( {(\sin {\varphi _\theta } \mp {\theta _s}\cos {\varphi _\theta }) - \frac{{\sin {\varphi _2}}}{{{q_{2s}}\cos {\varphi _\theta }}}} \right)} \right]\\ {y_2} = - {\mathit{\boldsymbol{r}}_{bs}}\left[ {\sin {\varphi _2}\left( {(\sin {\varphi _\theta } \mp {\theta _s}\cos {\varphi _\theta }) + \frac{{\cos {\varphi _2}}}{{{q_{2s}}\cos {\varphi _\theta }}}} \right)} \right]\\ {z_2} = - {r_{bs}}(\cos {\varphi _\theta } \pm {\theta _s}\sin {\varphi _\theta }) \end{array} \right.$ (8)

1.3 正交面齿轮齿根过渡曲面及其齿廓的生成

 Download: 图 4 两种插齿刀结构示意图 Fig. 4 Schematic diagram of the structure of gear slotting cutter

 $\left\{ \begin{array}{l} {r_{2t}}({\varphi _s}, {\gamma _A}, {u_s}) = [{M_{2t, t}}]{r_t}({\gamma _A}, {u_s})\\ \;\;\;\;\;\;\;\;f({u_s}, {\varphi _s}, {\gamma _A}) = 0 \end{array} \right.$ (9)

 $\left\{ \begin{array}{l} {X_{2t}} = {r_c}\cos {\varphi _2}\left( {\cos B - \cos A} \right) + {r_A}\cos {\varphi _2}\sin C - \\ \;\;\; - \frac{{{r_A}\sin {\varphi _2}\cos ({\alpha _A} + {\gamma _A}) - {r_A}\sin {\varphi _2}\cos {\alpha _A}}}{{{q_{2s}}\left( {\cos A - \cos B} \right)}}\\ {Y_{2t}} = {r_c}\sin {\varphi _2}\left( {\cos B - \cos A} \right) - {r_A}\sin {\varphi _2}\sin C - \\ \;\;\;\;\frac{{{r_A}\cos {\varphi _2}\cos ({\alpha _A} + {\gamma _A}) - {r_A}\cos {\varphi _2}\cos {\alpha _A}}}{{{q_{2s}}\left( {\cos A - \cos B} \right)}}\\ {Z_{2t}} = {r_c}\sin A - {r_c}\sin B - {r_A}\cos C \end{array} \right.$ (10)

 Download: 图 5 正交面齿轮插齿理论齿廓 Fig. 5 Theoretical gear profile of orthogonal face gear by pinion cutter

 Download: 图 6 两种插齿刀切制的正交面齿轮齿根过渡曲面 Fig. 6 Two kinds of tooth root transition curved surface cut by angle cutter & corner cutter

 Download: 图 7 正交面齿轮全齿实体模型 Fig. 7 3D model of orthogonal face gear
2 正交面齿轮弯曲强度分析 2.1 正交面齿轮危险截面的计算

 Download: 图 8 正交面齿轮受力图 Fig. 8 Force diagrams of orthogonal face gear

 $\left\{ \begin{array}{l} C = - {r_{bs}}\left[ {\sin {\varphi _2}(\sin {\varphi _\theta } \mp {\theta _s}\cos {\varphi _\theta }) + \frac{{\cos {\varphi _2}}}{{{q_{2s}}\cos {\varphi _\theta }}}} \right]\\ x = {r_{bs}}\left[ {\cos {\varphi _2}(\sin {\varphi _\theta } \mp {\theta _s}\cos {\varphi _\theta }) - \frac{{\sin {\varphi _2}}}{{{q_{2s}}\cos {\varphi _\theta }}}} \right]\\ y = {r_{bs}}(\cos {\varphi _\theta } \pm {\theta _s}\sin {\varphi _\theta }) - {r_{bs}} - \frac{d}{2}\tan \gamma \end{array} \right.$ (11)

 Download: 图 9 正交面齿轮齿面承受最大弯曲应力位置 Fig. 9 Maximum bending stress position on tooth surface
2.2 正交面齿轮有限元分析模型

 Download: 图 10 尖角插齿刀切制的面齿轮弯曲应力云图 Fig. 10 Bending stress nephogram of face gear by angle cutter
 Download: 图 11 圆角插齿刀切制的面齿轮弯曲应力云图 Fig. 11 Bending stress nephogram of face gear by corner cutter

 Download: 图 12 两种正交面齿轮的弯曲应力曲线 Fig. 12 Bending stress curves of two kinds of orthogonal face gears
3 结论

1) 根据面齿轮的加工啮合原理和插齿理论齿廓包络面方程，研究了正交面齿轮齿根过渡曲面的理论推导过程，并利用Matlab软件和Pro/E软件建立了正交面齿轮的工作齿面和齿根过渡曲面，准确给出了面齿轮的三维实体模型。进而比较了两种插齿刀切制的正交面齿轮齿根过渡曲面的结构，从结构分析了带有齿顶圆角的插齿刀所切制的正交面齿轮的过渡曲面是光滑连续的，不易产生应力集中现象，理论上使得其承载能力得到加强。

2) 利用内切抛物线法从理论的角度分析了正交面齿轮轮齿的最大弯曲应力位置，正交面齿轮轮齿在受到作用于齿顶的集中载荷时，其最大弯曲应力位置在沿齿高方向的齿面的上半部分。

3) 基于ANSYS软件验证了理论推导的正确性。通过有限元分析结果可以得出采用齿顶圆角插齿刀切制的正交面齿轮的最大弯曲应力小于采用齿顶尖角插齿刀所切制的面齿轮，即齿顶圆角插齿刀切制的齿根过渡曲面提高了正交面齿轮的弯曲强度。

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