工程科学与技术   2019, Vol. 51 Issue (1): 222-228

Load Distribution and Life Prediction of Inverted Planetary Roller Screw
YIN Guofu, SONG Yanxuan, YIN Ming, XIE Luofeng, ZHAO Xiufen
School of Manufacturing Sci. and Eng., Sichuan Univ., Chengdu 610065, China
Key words: inverted planetary roller screw    load distribution    axial deformation    stiffness    contact fatigue life

 图1 IPRS基本结构 Fig. 1 Basic structure of IPRS

1 赫兹变形

 图2 丝杠、滚柱和螺母螺纹的牙型轮廓 Fig. 2 Thread profile of screw, roller and nut

 \begin{aligned}[b] {{{\varPi }}_{\rm{n}}}\left( {\theta ,r} \right)= &\left\{ { r\cos\; \theta ,\;r\sin\; \theta ,} \right.\\ &\left. {\theta {r_{\rm{n}}}\tan\; {\beta _{\rm{n}}} \mp \left( {{{\textit{z}}_{\rm{n}}} - r\tan\; a} \right)} \right\} \end{aligned} (1)
 \begin{aligned}[b] {{{\varPi }}_{\rm{r}}}\left( {\theta ,r} \right) = & \left\{ { r\cos\; \theta ,\;r\sin\; \theta ,\;\; } \right.\\ &\pm \left( {{{\textit{z}}_{\rm{r}}} + \sqrt {r_{\rm{c}}^2 - {{\left( {r - {r_0}} \right)}^2}} } \right) -\theta {r_{\rm{r}}}\tan\; {\beta _{\rm{r}}}\} \end{aligned} (2)
 $\begin{array}{l} {{{\varPi }}_{\rm{s}}}\left( {\theta ,r} \right) = \left\{ { r\cos\; \theta ,\;r\sin\; \theta ,} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \left. {\theta {r_{\rm{s}}}\tan\; {\beta _{\rm{s}}} \pm \left( {{{\textit{z}}_{\rm{s}}} - r\tan\; a} \right)} \right\} \end{array}$ (3)

${p_{{\rm{nr}}}}$ ${p_{{\rm{sr}}}}$ 分别表示一个滚柱节距内螺母与滚柱的啮合点及丝杠与滚柱的啮合点。则该两啮合点在 $r - \theta$ 面上投影如图3所示。基于文献[8]中的计算方法，得：

 图3 接触点相对位置示意 Fig. 3 Relative position of contact point

 ${\phi _{\rm{n}}} = {{\rm arc}\sin ^{ }}\frac{{{r_{\rm{r}}}\left( {\sin \;{\beta _{\rm{r}}} + \sin \;{\beta _{\rm{n}}}} \right)}}{{{r_{\rm{n}}} - {r_{\rm{r}}}}}$ (4)
 ${\phi _{\rm{r}}} = {{\rm arc}\sin ^{ }}\frac{{{r_{\rm{n}}}\left( {\sin\; {\beta _{\rm{r}}} + \sin \;{\beta _{\rm{n}}}} \right)}}{{{r_{\rm{n}}} - {r_{\rm{r}}}}}$ (5)
 ${\phi _{\rm{s}}} = {{\rm arc}\sin ^{ }}\frac{{{r_{\rm{r}}}\left( {\sin\; {\beta _{\rm{r}}} - \sin\; {\beta _{\rm{s}}}} \right)}}{{{r_{\rm{s}}} + {r_{\rm{r}}}}}$ (6)
 ${\phi '_{\rm{r}}} = {{\rm arc}\sin ^{ }}\frac{{{r_{\rm{s}}}\left( {\sin \;{\beta _{\rm{r}}} - \sin \;{\beta _{\rm{s}}}} \right)}}{{{r_{\rm{s}}} + {r_{\rm{r}}}}}$ (7)

 ${\left( {{\rm d}{{\varPi }}} \right)_1} {\left( {{\rm d}{{\varPi }}} \right)_2} = \frac{{\partial {{\varPi }}}}{{\partial \theta }}{\rm d}\theta + \frac{{\partial {{\varPi }}}}{{\partial r}}{\rm d}r$ (8)

 \begin{aligned} & {\rm{d}}\theta /{\rm{d}}r = \\ & \frac{{[ - \left( {EN - GL} \right) \pm \sqrt {{{\left( {EN - GL} \right)}^2} - 4\left( {EM - FL} \right)\left( {FN - GM} \right)} ]}}{{2\left( {EM - FL} \right)}}\end{aligned} (9)

 ${\rho _1} 、{\rho _2} = H \pm \sqrt {{H^2} - K}$ (10)

 $K = \frac{{LN - {M^2}}}{{EG - {F^2}}}$ (11)
 $H = \frac{{LG - 2MF + NE}}{{2\left( {EG - {F^2}} \right)}}$ (12)

 $f\left( \rho \right) \!=\! \frac{{\sqrt {{{\left( {{\rho _1} \!-\! {\rho _2}} \right)}^2} \!+\! {{\left( {\rho '\!\!{_1} \!-\! \rho '\!\!{_2}} \right)}^2} \!+\! 2\left( {{\rho _1} \!-\! {\rho _2}} \right)\left( {\rho '\!\!{_1} \!-\! \rho '\!\!{_2}} \right)\cos \; \omega } }}{{\displaystyle\sum \rho \!=\! \left[ {\left( {2 - {e^2}} \right)L\left( e \right) - 2\left( {1 - {e^2}} \right)K\left( e \right)} \right]/\left[ {{e^2}L\left( e \right)} \right]}}$ (13)

 $\delta \left( {{N_{\rm n}}} \right) \!=\! \left( {1 \!-\! {\nu ^2}} \right)K\left( e \right){\left( {\sum \rho } \right)^{\left( {1/3} \right)}}{{N_{\rm n}^{\left( {2/3} \right)}} \Big/ {\left[ {{\text{π}} {m_{\rm a}}EE_0^{\left( {1/3} \right)}} \right]}} \!\!\!\!\!\!\!$ (14)

2 载荷分布及刚度模型

 图4 IPRS螺纹啮合段变形状态 Fig. 4 IPRS deformation conditions of thread meshing section

 ${l_{{\rm {n}}i}} = 4pn\cos\; a\cos\; {\beta _{\rm{r}}} \div \left\{\left[ {{\text{π}} \left( {D_0^2 - d_{{\rm{ne}}}^2} \right){E_{\rm{n}}}} \right]\sum\limits_{j = i + 1}^m {{N\!\!_j}}\right\}$ (15)
 ${l_{{\rm{rn}}i}} = 2p\cos \;a\cos \;{\beta _{\rm{r}}} \div \left\{ {\left( {{\text{π}} d_{{\rm{re}}}^2{E_{\rm{r}}}} \right)\left[ {2\sum\limits_{j = 1}^i {\left( {N{\!\!_j} - N'{\!\!\!_j}} \right)} + {N_i}^\prime } \right]} \right\}$ (16)
 ${l_{{\rm{rs}}i}} = 2p\cos \;a\cos \;{\beta _{\rm{r}}} \div \left\{ {\left( {\text{π} d_{{\rm{re}}}^2{E_{\rm{r}}}} \right)\left[ {2\sum\limits_{j = 1}^i {\left( {N{\!\!_j} - N'{\!\!\!_j}} \right)} + {N_{{i + 1} }}} \right]} \right\}$ (17)
 ${l_{{\rm{s}}i}}\left| {_{{\text{异侧承载}}}} \right. = 4pn\cos\; a\cos\; {\beta _{\rm{r}}} \div \left\{ {\left( {\text{π} d_{{\rm{se}}}^2{E_{\rm{s}}}} \right)\sum\limits_{j = 1}^i {N'{\!\!\!_j}} } \right\}$ (18)
 ${l_{{\rm{s}}i}}\left| {_{{\text{同侧承载}}}} \right. = 4pn\cos\; a\cos\; {\beta _{\rm{r}}} \div \left\{ {\left( {\text{π} d_{{\rm{se}}}^2{E_{\rm{s}}}} \right)\sum\limits_{j = i + 1}^m {N'{\!\!\!_j}} } \right\}$ (19)

IPRS螺牙变形指各组件螺牙受到法向载荷时由于弯矩、剪切、根部倾斜、根部剪切、丝杠及滚柱径向收缩、螺母径向扩大等因素而产生的变形。该变形与所受法向载荷呈线性关系，计算方式见文献[11]。

 \begin{aligned}[b] & {l_{{\rm{n}}i}} - {\delta _{{\rm{n}}i}} - {\varepsilon _{{\rm{n}}i}} + {\delta _{{\rm{n}}\left( {i + 1} \right)}} + {\varepsilon _{{\rm{n}}\left( {i + 1} \right)}}=\\ & \quad {l_{{\rm{r}}i}} + {\delta _{{\rm{r}}i}} + {\varepsilon _{{\rm{r}}i}} - {\delta _{{\rm{r}}\left( {i + 1} \right)}} - {\varepsilon _{{\rm{r}}\left( {i + 1} \right)}} \end{aligned} (20)
 \begin{aligned}[b] &{{l'}\!\!_{{\rm{r}}i}} - {{\delta '}\!\!_{{\rm{r}}i}} - {{\varepsilon '}\!\!_{{\rm{r}}i}} + {{\delta '}\!\!_{{\rm{r}}\left( {i + 1} \right)}} + {{\varepsilon '}\!\!_{{\rm{r}}\left( {i + 1} \right)}}=\\ & \quad{l_{{\rm{s}}i}} + {\delta _{{\rm{s}}i}} + {\varepsilon _{{\rm{s}}i}} - {\delta _{{\rm{s}}\left( {i + 1} \right)}} + {\varepsilon _{{\rm{s}}\left( {i + 1} \right)}} \end{aligned} (21)

 $F = n\sum\limits_{i = 1}^m {{N_i}\cos\; a\cos\; {\beta _{\rm{s}}}}= n\sum\limits_{i = 1}^m {{{N'}\!\!_i}\cos\; a\cos\; {\beta _{\rm{s}}}}$ (22)

IPRS的载荷分布可由式（20）～（22）联立求得。以 $\varDelta$ $\varDelta$ ′为异侧承载和同侧承载时IPRS承载端的轴向变形，得承载端轴向变形表达式如下：

 \begin{aligned}[b] \varDelta = & \sum\limits_{i = 1}^{m - 1} {{l_{{\rm{r}}i}}} + {\delta _{{\rm{nm}}}} + {\delta _{{\rm{sm}}}} + {\delta _{{\rm{rm}}}} + {{\delta '}\!\!_{{\rm{rm}}}} +\\ & {\varepsilon _{{\rm{nm}}}} + {\varepsilon _{{\rm{sm}}}} + {\varepsilon _{{\rm{rm}}}} + {{\varepsilon '}\!\!_{{\rm{rm}}}} \end{aligned} (23)
 \begin{aligned}[b] \varDelta ' = & \; {\delta _{{\rm{nm}}}} + {\delta _{{\rm{sm}}}} + {\delta _{{\rm{rm}}}} + {{\delta '}\!\!_{{\rm{rm}}}} + \quad \\ & {\varepsilon _{{\rm{nm}}}} + {\varepsilon _{{\rm{sm}}}} + {\varepsilon _{{\rm{rm}}}} + {{\varepsilon '}\!\!_{{\rm{rm}}}} \end{aligned} (24)
3 寿命模型

IPRS的失效形式分为3种：偶然失效，磨损失效及疲劳失效[2]。在润滑良好的情况下，接触表面疲劳失效是机构失效的主要原因[12]。从断裂力学的角度来看，表面疲劳破坏通常经历裂纹萌生和裂纹拓展两个阶段。通常裂纹的萌生寿命是构成接触面全寿命的主要部分[13]。为了建立IPRS接触疲劳寿命的计算模型，有以下假设：1）IPRS机构在稳定运转时，螺纹牙间载荷分布与静压力下的分布相同；2）IPRS机构裂纹扩展寿命在全寿命中很小，在寿命预测时可忽略不计；3）接触椭圆长半轴位于滚道法平面上。

 $M = A\sqrt[f]{{ - {{\textit{z}}_0^h}\left( {\frac{{{\rm ln}\left( S \right)}}{{V{\tau _0^c}}}} \right)}}\quad$ (25)

 $V = 4a{{\textit{z}}_0}\left( {{\text{π}} r\sec\; \beta } \right) \quad$ (26)
 ${{\textit{z}}_0} = \frac{b}{{\left( {t + 1} \right)\sqrt {2t - 1} }} \quad$ (27)
 ${\tau _0} = \frac{{3{N_{\rm{n}}}\sqrt {2t - 1} }}{{4{\text{π}} abt\left( {t + 1} \right)}}\quad$ (28)

 $a = {m_a}{\left[ {{{3{N_{\rm{n}}}{E_0}} / {\left( {2\sum \rho } \right)}}} \right]^{{1 / 3}}}$ (29)
 $b = {m_b}{\left[ {{{3{N_{\rm{n}}}{E_0}} / {\left( {2\sum \rho } \right)}}} \right]^{{1 / 3}}}$ (30)
 ${m_b} = {\left\{ {{{2\sqrt {1 - {e^2}} L\left( e \right)} / {\text{π}} }} \right\}^{1/3}}$ (31)

 ${\gamma _{\rm{n}}} = {{{k^2}} / {\left( {2k + 2} \right)}} \quad$ (32)
 ${\gamma _{\rm{r}}} = {{\left( {k - 1} \right)\left( {k + 2} \right)} / {\left( {2k + 2} \right)}}$ (33)
 ${\gamma _{\rm{s}}} = {{k\left( {k + 2} \right)} / {\left( {2k + 2} \right)}} \quad$ (34)

 ${S\!\!_{\rm{n}}} = \exp \left[ { - k{{\left( {{{{\gamma _{\rm{n}}}M} / A}} \right)}^f}\sum {{V_{{\rm{n}}i}}} \times }{{{{\tau _{{\rm{n}}i}}\!\!\!\!^c} / {{{\textit{z}}_{{\rm{n}}i}}\!\!\!\!^h}}} \right]$ (35)
 \begin{aligned}[b] {S\!_{\rm{r}}} = & \exp \left[ { - {{\left( {{\gamma _{\rm{r}}}{M / A}} \right)}^f}\left( {\sum {{V_{{\rm{r}}i}}} \times } \right.} \right.\\ & \left. \displaystyle{\left. {{{{\tau _{{\rm{r}}i}}\!\!\!\!^c} / {{{\textit{z}}_{{\rm{r}}i}}\!\!\!\!^h}} + \sum {{{V'}\!\!\!_{{\rm{r}}i}}} {{{{\tau '}\!\!_{{\rm{r}}i}}\!\!^c} / {{{{\textit{z}}'}\!\!_{{\rm{r}}i}}\!\!^h}}} \right)} \right] \end{aligned} (36)
 ${S\!_{\rm{s}}} = \exp \left[ { - k{{\left( {{{{\gamma _{\rm{s}}}M} / A}} \right)}^f}\sum {{V_{{\rm{s}}i}}} \times }{{{{\tau _{{\rm{s}}i}}\!\!\!^c} / {{{\textit{z}}_{{\rm{s}}i}}\!\!\!^h}}} \right]$ (37)

 \begin{aligned}[b] M' = & A\left[ {n\gamma _{\rm{n}}^f\sum {\left( {{{{V_{{\rm{n}}i}}\tau _{{\rm{n}}i}^c} / {{\textit{z}}_{{\rm{n}}i}^h}}} \right)} } \right. + n\gamma _{\rm{s}}^f \times \\ & \sum {\left( {{{{V_{{\rm{s}}i}}\tau _{{\rm{s}}i}^c} / {{\textit{z}}_{{\rm{s}}i}^h}}} \right)} + \gamma _{\rm{r}}^f\sum {\left( {{{{V_{{\rm{r}}i}}\tau _{{\rm{r}}i}^c} / {{\textit{z}}_{{\rm{r}}i}^h}}} \right)} + \\ & {\left. {{\gamma '}\!\! _{\rm{r}}\!\!^{f}\sum {\left( {{{{{V'}\!\!\!_{{\rm{r}}i}}{\tau '} \!\!_{{\rm{r}}i}\!\!^{c}} / {{{\textit{z}}'}\!\!_{{\rm{r}}i}\!\!^{h}}}} \right)} } \right]^{{{ - 1} / f}}}{\left[ { - \ln \left( S \right)} \right]^{{1 / f}}} \end{aligned} (38)
4 模型验证及讨论 4.1 模型验证

 图5 承载端轴向变形 Fig. 5 Axial deformation between working sides

4.2 结构参数对偏载率的影响

 图6 各参数对偏载率的影响 Fig. 6 Influence of various parameters on load distribution rate

4.3 承载形式对载荷分布的影响

4.4 结构参数对轴向变形的影响

 图8 各参数对轴向变形的影响 Fig. 8 Influence of various parameters on axial deformation

4.5 结构参数对接触疲劳寿命的影响

 图9 各参数对接触疲劳寿命的影响 Fig. 9 Influence of various parameters on contact fatigue life

 图10 丝杠、螺母及所有滚柱的可靠性 Fig. 10 Reliability of the screw, nut and all rollers

5 结　论

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