工程科学与技术   2018, Vol. 50 Issue (2): 212-219

1. 四川大学 空天科学与工程学院，四川 成都 610065;
2. 重庆大学 机械传动国家重点实验室，重庆 400044;
3. 洪都航空工业集团有限责任公司 综合采购部，江西 南昌 330024

Stiffness Calculation Considering Friction for a Spur Gear Pair with Tooth Wear and Profile Modification
YANG Yong1, WANG Jiaxu1,2, ZHOU Qinghua1, LI Wenguang1, XIONG Lindong3
1. School of Aeronautics and Astronautics,Sichuan Univ.,Chengdu 610065,China;
2. State Key Lab. of Mechanical Transmissions,Chongqing Univ.,Chongqing 400044,China;
3. AVIC Jiangxi Hongdu Aviation Industry Group Corp. Ltd.,Nanchang 330024,China
Key words: mesh stiffness    friction    tooth wear    tooth profile modification    transmission error

1 齿轮啮合刚度建模 1.1 齿廓方程

 \left\{ \begin{aligned}& {x_{1{ r}}} = u,\\& {y_{1{ r}}} = \rho \sin \;\alpha - h - \sqrt {{\rho ^2} - {u^2}} \end{aligned} \right.{{, }}u \in [0,{{ }}\rho \cos \;\alpha ] (1)
 \begin{aligned}& \left\{ \begin{aligned}& {x_{1{ l}}} = u,\\& {y_{1{ l}}} = \cot \;\alpha (u - \rho \cos \;\alpha ) - h;\end{aligned} \right.\\& {{ }}u \in [\rho \cos \;\alpha ,{{ }}(m{h_a^*} + h)\tan \;\alpha + \rho \cos \;\alpha ]\end{aligned} (2)
 图1 齿条形刀具齿形及其关联坐标系 Fig. 1 Tooth shape of a rack-type gear cutter and coordinate systems

 \begin{aligned}& {{{M}}_{21}} = \left[\!\!\!\! {\begin{array}{*{20}{c}}{\cos \;\varphi }&{\sin \;\varphi }&{r(\sin \;\varphi - \varphi \cos \;\varphi )}\\{ - \sin \;\varphi }&{\cos \varphi }&{r(\cos \;\varphi + \varphi \sin \;\varphi )}\\0&0&1\end{array}} \!\!\!\!\right],\\& {{{M}}_{{{f2}}}} = \left[\!\!\!\! {\begin{array}{*{20}{c}}{\cos \;\theta }&{ - \sin \;\theta }&0\\{\sin \;\theta }&{\cos \;\theta }&0\\0&0&1\end{array}} \!\!\!\!\right],\;{{{M}}_{0{ f}}} = \left[\!\!\!\! {\begin{array}{*{20}{c}}0&1&{ - L}\\{ - 1}&0&0\\0&0&1\end{array}} \!\!\!\!\right],\\& \quad\quad\quad\quad\quad\quad {{{M}}_{{{01}}}} = {{{M}}_{0{ f}}}{{{M}}_{{ f}2}}{{{M}}_{21}}\end{aligned} (3)

 \left\{ \begin{aligned}& {x_{AB}} = ( - u + r\varphi )\sin (\varphi - \theta ) + \\&\quad\quad\;\; (r + \rho \sin \;\alpha - h - \sqrt {{\rho ^2} - {u^2}} )\cos (\varphi - \theta ) - L,\\& {y_{AB}} = ( - u + r\varphi )\cos (\varphi - \theta ) - \\&\quad\quad\;\; (r + \rho \sin \;\alpha - h - \sqrt {{\rho ^2} - {u^2}} )\sin (\varphi - \theta ),\\& \varphi (u) = (\rho \sin \;\alpha - h)u/(r\sqrt {{\rho ^2} - {u^2}} ),\\& u \in [0,{{ }}\rho \cos \;\alpha ]\end{aligned} \right.\!\! (4)
 \left\{ \begin{aligned}& {x_{BC}} = ( - u + r\varphi )\sin (\varphi - \theta ) + \\&\quad\quad\;\; [r + \cot \;\alpha (u - \rho \cos \;\alpha ) - h]\cos (\varphi - \theta ) - L,\\& {y_{BC}} = ( - u + r\varphi )\cos (\varphi - \theta ) - \\&\quad\quad\;\; [r + \cot \;\alpha (u - \rho \cos \;\alpha ) - h]\sin (\varphi - \theta ),\\& \varphi (u) = \frac{{u{{\csc }^2}\alpha - (\rho \cot \;\alpha \cos \;\alpha + h)\cot \;\alpha }}{r},\\& u \in [\rho \cos \;\alpha ,{{ }}(m{h_a^*} + h)\tan \;\alpha + \rho \cos \;\alpha ],\end{aligned} \right.\!\! (5)

 $u = u(\varphi ) = \varphi r{\sin ^2}\alpha + (\rho {\cos ^2}\alpha + h\sin \;\alpha )\cos \;\alpha$ (6)
 图2 齿轮加工坐标变换 Fig. 2 Coordinate transformation by rotation of gear machining

1.2 能量法求啮合刚度

 图3 轮齿啮合受力的梁模型 Fig. 3 Beam model of a spur gear tooth

 \begin{aligned}{U_{ a}} = & \int_0^{{d_{ i}}} {\frac{{{F_x^2}}}{{2E{A_x}}}} { d}x = \int_0^{\rho \cos \;\alpha } {\frac{{{F_x^2}}}{{4EB{y_{AB}}(u)}} \cdot \frac{{{ d}{x_{AB}}(u)}}{{{ d}u}}} { d}u + \\& \int_{\rho \cos \;\alpha }^{u(\theta - \gamma )} {\frac{{{F_x^2}}}{{4EB{y_{BC}}(u)}} \cdot \frac{{{ d}{x_{BC}}(u)}}{{{ d}u}}} { d}u\end{aligned} (7)
 \begin{aligned}{U_{ b}}= & \int_0^{{d_{ i}}} {\frac{{{{\left[ {{F_y}({d_{ i}} - x) - {F_x}{h_{ i}}} \right]}^2}}}{{2E{I_x}}}} { d}x=\\& \int_0^{\rho \cos \alpha } {\frac{{{{\left[ {{F_y}({d_{ i}} - {x_{AB}}(u)) - {F_x}{h_{ i}}} \right]}^2}}}{{\displaystyle\frac{4}{3}EBy_{AB}^3(u)}} \cdot \frac{{{ d}{x_{AB}}(u)}}{{{ d}u}}} { d}u + \\& \int_{\rho \cos \alpha }^{u(\theta - \gamma )} {\frac{{{{\left[ {{F_y}({d_{ i}} - {x_{BC}}(u)) - {F_x}{h_{ i}}} \right]}^2}}}{{\displaystyle\frac{4}{3}EBy_{BC}^3(u)}} \cdot \frac{{{ d}{x_{BC}}(u)}}{{{ d}u}}} { d}u\end{aligned} (8)
 \begin{aligned}{U_{ s}} = & \int_0^{{d_{ i}}} {\frac{{1.2{F_y^2}}}{{2G{A_x}}}} { d}x{{ = }}\int_0^{\rho \cos \alpha } {\frac{{1.2{F_y^2}}}{{4GB{y_{AB}}(u)}} \cdot \frac{{{ d}{x_{AB}}(u)}}{{{ d}u}}} { d}u + \\ & \int_{\rho \cos \alpha }^{u(\theta - \gamma )} {\frac{{1.2{F_y^2}}}{{4GB{y_{BC}}(u)}} \cdot \frac{{{ d}{x_{BC}}(u)}}{{{ d}u}}} { d}u\end{aligned} (9)

 ${F_x} = F\sin \;\beta + f\cos \;\beta,{{ }}{F_y} = F\cos \;\beta - f\sin \;\beta$ (10)

 ${F_x} = F\sin \;\beta - f\cos \;\beta {{; }}{F_y} = F\cos \;\beta + f\sin \;\beta$ (11)

Yang等[14]发现：在齿轮啮合过程中，赫兹接触刚度为常数，并可由式（12）求得：

 ${k_{ h}} = \frac{{{\text{π}} b}}{2}{\left( {\frac{{1 - {v_1}^2}}{{{E_1}}}{{ + }}\frac{{1 - {v_2}^2}}{{{E_2}}}} \right)^{ - 1}}$ (12)

 ${k_{ f}} = \frac{{Eb}}{{{{\cos }^2}\beta \{ L \times{{(\displaystyle\frac{{{u_{ f}}}}{{{S_{ f}}}})}^2} + M\times(\displaystyle\frac{{{u_{ f}}}}{{{S_{ f}}}}) + P\times(1 + Q \times{{\tan }^2}\beta )\} }}$ (13)

 ${k_{ t}} = \displaystyle\frac{1}{{\displaystyle\frac{1}{{{k_{ b1}}}} + \displaystyle\frac{1}{{{k_{ s1}}}} + \displaystyle\frac{1}{{{k_{ a1}}}} + \displaystyle\frac{1}{{{k_{ f1}}}} + \displaystyle\frac{1}{{{k_{ b2}}}} + \displaystyle\frac{1}{{{k_{ s2}}}} + \displaystyle\frac{1}{{{k_{ a2}}}} + \displaystyle\frac{1}{{{k_{ f2}}}} + \displaystyle\frac{1}{{{k_{ h}}}}}}$ (14)
1.3 考虑轮齿误差的啮合刚度

 ${\delta _1} + {E_{{ p}1}} + {E_{{ g}1}} = {\delta _2} + {E_{{ p}2}} + {E_{{ g}2}} + E_{ g}^{ s} - E_{ p}^{ s}$ (15)

 ${E_{12}} = {\delta _1} - {\delta _2} = {E_{{ p}2}} + {E_{{ g}2}} + E_{ g}^{ s} - E_{ p}^{ s} - {E_{{ p}1}} - {E_{{ g}1}}$ (16)

 ${E_{ t}} = \delta + {E_{ p}} + {E_{ g}}$ (17)
 图4 双齿啮合时含齿形误差齿对沿啮合线的啮合位置图 Fig. 4 Schematic diagram of double-tooth engagement along line of action with tooth profile errors

 ${F_n} = {F_1} + {F_2},{{ }}{F_1} = {k_1}{\delta _1},{{ }}{F_2} = {k_2}{\delta _2},{{ }}{\delta _1} \ge 0,{{ }}{\delta _2} \ge 0$ (18)

 $k = {F_n}/\max ({\delta _1},{{ }}{\delta _2})$ (19)

 \left\{\begin{aligned}& k = \frac{{{k_1} + {k_2}}}{{1 + {k_2}{E_{12}}/{F_n}}},{{ }}{\delta _1} - {\delta _2} = {E_{12}} > 0;\\& k = \frac{{{k_1} + {k_2}}}{{1 - {k_1}{E_{12}}/{F_n}}},{{ }}{\delta _1} - {\delta _2} = {E_{12}} < 0\end{aligned}\right. (20)

 \left\{\begin{aligned}& {R_1} = \frac{{{F_1}}}{{{F_n}}} = \frac{{{k_1}}}{{{k_1} + {k_2}}}(1 + \frac{{{k_2}{E_{12}}}}{{{F_n}}}),\\& {R_2} = \frac{{{F_2}}}{{{F_n}}} = \frac{{{k_2}}}{{{k_1} + {k_2}}}(1 - \frac{{{k_1}{E_{12}}}}{{{F_n}}})\end{aligned}\right. (21)
2 结果与讨论

2.1 摩擦力对齿轮啮合刚度的影响

 图5 不同摩擦系数下的齿轮啮合刚度与载荷分配系数 Fig. 5 Evolutions of gear mesh stiffness and load sharing ratio for different coefficients of friction

2.2 非均匀磨损齿轮

 \left\{ \begin{aligned} & {{x'}\!\!\!_{CD}} = {x_{CD}} - W\Delta S\sin \;{\beta _{ u}},\\ & {{y'}\!\!\!_{CD}} = {y_{CD}} - W\Delta S\cos \;{\beta _{ u}}\end{aligned} \right. (22)

 图6 磨损轮齿示意图 Fig. 6 Schematic diagram of a wear gear tooth

 图7 不同磨损量对相对啮合刚度 Fig. 7 Relative mesh stiffness of single tooth pair with different wear values

 图8 不同磨损量的齿轮 Fig. 8 Gear pair with different wear values

2.3 修形齿轮

 图9 修形轮齿示意图 Fig. 9 Schematic diagram of a gear tooth with modification

 图10 不同修形长度的齿轮 Fig. 10 Gear pair with different lengths of profile modification

3 结　论

1）摩擦力使啮入阶段齿对啮合刚度增大，啮出阶段齿对啮合刚度减小，摩擦系数越大，刚度变化量越大。

2）对于含有齿形偏差的齿轮副双齿啮合时，啮合刚度随载荷增大而增大，齿形误综合差较大的齿对载荷分担系数随载荷增大而增大。

3）齿面非均匀磨损量会显著降低双齿啮合区刚度并减小重合度，在轻载条件下尤其明显。

4）载荷大于修形设计载荷值时，修形效果不明显；载荷小于修形设计载荷值时，可能出现刚度不足、重合度减小和加载传动误差显著增大等问题。

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