《1 引言》

1 引言

《图1》

Fig.1 Toroidal drive

《2 摩擦系数与膜厚比的关系》

2 摩擦系数与膜厚比的关系

$\begin{array}{l}\overline{p}{}_{\text{a}}\left(x\right)=p{}_{0}\sqrt{1-\left(\frac{x}{b}\right){}^{2}},\\ p{}_{1}\left(x\right)=0\phantom{\rule{0.25em}{0ex}}\left(\lambda \le 0.4\right)‚\\ \overline{p}{}_{\text{a}}\left(x\right)=p{}_{*}\left(3-\lambda \right){}^{m},\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(1\right)\\ p{}_{1}\left(x\right)=\left(p{}_{0}-p{}_{*}\right)\sqrt{1-\left(\frac{x}{b}\right){}^{2}}\left(0.4\le \lambda \le 3\right)‚\\ \overline{p}{}_{\text{a}}\left(x\right)=0,\\ p{}_{1}\left(x\right)=p{}_{0}\sqrt{1-\left(\frac{x}{b}\right){}^{2}}‚\phantom{\rule{0.25em}{0ex}}\left(\lambda \ge 3\right)\end{array}$

$\begin{array}{l}\mu =\mu {}_{\text{a}}\text{ }\left(\lambda \le 0.4\right)\\ \mu =\mu {}_{\text{a}}\frac{p{}_{*}}{p{}_{0}}+\mu {}_{1}\left(1-\frac{p{}^{*}}{p{}_{0}}\right)\phantom{\rule{0.25em}{0ex}}\left(0.4\le \lambda \le 3\right)\text{ }\text{ }\text{ }\left(2\right)\\ \mu =\mu {}_{1}\text{ }\left(\lambda \ge 3\right)\end{array}$

《3 凸峰间摩擦系数的计算》

3 凸峰间摩擦系数的计算

$\text{d}l{}_{\text{t}}=i{}_{01}^{\text{Η}}R\left(\frac{a}{R}+\mathrm{cos}\phi {}_{1}\right)\text{d}\phi {}_{1},\text{ }\text{ }\text{ }\left(3\right)$

$l{}_{\text{t}}=\underset{-\frac{\phi {}_{\text{v}}}{2}}{\overset{\frac{\phi {}_{\text{v}}}{2}}{\int }}i{}_{01}^{\text{Η}}R\left(\frac{a}{R}+\mathrm{cos}\phi {}_{1}\right)\text{d}\phi {}_{1}=i{}_{01}^{\text{Η}}R\phi {}_{\text{v}}\left(\frac{a}{R}-\frac{\mathrm{sin}\frac{\phi {}_{\text{v}}}{2}}{\frac{\phi {}_{\text{v}}}{2}}\right)\text{ }\text{ }\text{ }\left(4\right)$

$l{}_{\text{a}}=\phi {}_{\text{v}}R\sqrt{1+\left(i{}_{01}^{\text{Η}}\right){}^{2}\left(\frac{a}{R}-\frac{\mathrm{sin}\frac{\phi {}_{\text{v}}}{2}}{\frac{\phi {}_{\text{v}}}{2}}\right){}^{2}}\text{ }\text{ }\text{ }\left(5\right)$

$\frac{\text{d}{}^{2}l{}_{\text{t}}}{\text{d}\phi {}_{1}^{2}}=-i{}_{01}^{\text{Η}}R\text{s}\text{i}\text{n}\phi {}_{1}。\text{ }\text{ }\text{ }\left(6\right)$

$l{}_{\text{s}}=2\underset{0}{\overset{\frac{\phi {}_{\text{v}}}{2}}{\int }}\left(-i{}_{01}^{\text{Η}}R\text{s}\text{i}\text{n}\phi {}_{1}\right)\text{d}\phi {}_{1}=2i{}_{01}^{\text{Η}}R\left(1-\mathrm{cos}\frac{\phi {}_{\text{v}}}{2}\right)\text{ }\text{ }\text{ }\left(7\right)$

$\lambda {}_{0}=\frac{l{}_{\text{s}}}{l{}_{\text{a}}}=\frac{2i{}_{01}^{\text{Η}}\left(1-\mathrm{cos}\frac{\phi {}_{\text{v}}}{2}\right)}{\sqrt[\begin{array}{l}\text{ }\\ \phi {}_{\text{v}}\end{array}]{1+\left(i{}_{01}^{\text{Η}}\right){}^{2}\left(\frac{a}{R}-\frac{\mathrm{sin}\frac{\phi {}_{\text{v}}}{2}}{\frac{\text{v}}{2}}\right){}^{2}}}\text{ }\text{ }\text{ }\left(8\right)$

$\lambda {}_{2}=\frac{{l}^{\prime }{}_{\text{s}}}{{l}^{\prime }{}_{\text{a}}}=\frac{2i{}_{21}^{\text{Η}}\left(1-\mathrm{cos}\frac{{\phi }^{\prime }{}_{\text{v}}}{2}\right)}{\sqrt[\begin{array}{l}\text{ }\\ {\phi }^{\prime }{}_{\text{v}}\end{array}]{1+\left(i{}_{21}^{\text{Η}}\right){}^{2}\left(\frac{a}{R}-\frac{\mathrm{sin}\frac{{\phi }^{\prime }{}_{\text{v}}}{2}}{\frac{{\phi }^{\prime }{}_{\text{v}}}{2}}\right){}^{2}}}\text{ }\text{ }\text{ }\left(9\right)$

$\phantom{\rule{0.25em}{0ex}}\text{d}W\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}F{}_{\text{n}\text{i}}\mu {}_{\text{g}}\text{d}\theta \phantom{\rule{0.25em}{0ex}}+\phantom{\rule{0.25em}{0ex}}F{}_{\text{n}\text{i}}\mu {}_{\text{s}}\text{d}l\phantom{\rule{0.25em}{0ex}}。\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(10\right)$

$\begin{array}{l}W=\mu {}_{\text{a}}F{}_{\text{n}\text{i}}l{}_{\text{a}}=\underset{0}{\overset{l{}_{\text{s}}}{\int }}F{}_{\text{n}\text{i}}\mu {}_{\text{s}}\text{d}l+\underset{0}{\overset{\theta }{\int }}\mu {}_{\text{g}}F{}_{\text{n}\text{i}}\text{d}\theta =\\ \mu {}_{\text{s}}F{}_{\text{n}\text{i}}l{}_{\text{s}}+\mu {}_{\text{g}}F{}_{\text{n}\text{i}}\frac{l{}_{\text{g}}}{r}‚\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(11\right)\end{array}$

$\mu {}_{\text{a}01}=\mu {}_{\text{s}}\lambda {}_{0}+\mu {}_{\text{g}}\frac{1}{r}\left(1-\lambda {}_{0}\right),\text{ }\text{ }\text{ }\left(12\right)$

$\mu {}_{\text{a}21}=\mu {}_{\text{s}}\lambda {}_{2}+\mu {}_{\text{g}}\frac{1}{r}\left(1-\lambda {}_{2}\right)。\text{ }\text{ }\text{ }\left(13\right)$

《4 液体润滑摩擦系数的计算》

4 液体润滑摩擦系数的计算

$F\approx F{}_{\text{s}}={\int }_{-b}^{b}\tau \left(y=0,\text{或}y=h\right)\text{d}x,\text{ }\text{ }\text{ }\left(14\right)$

$\tau =\eta \left(u{}_{2}-u{}_{1}\right)/Η{}_{\mathrm{min}}‚\eta =\eta {}_{0}e{}^{\alpha {}_{0}p}‚p=p{}_{0}\sqrt{1-\left(x/b\right){}^{2}}\approx p{}_{0}\left(1-x{}^{2}/b{}^{2}\right)$, 代入式 (14) 得:

$F=\frac{b\left(u{}_{2}-u{}_{1}\right)}{Η{}_{\mathrm{min}}}\eta {}_{0}e{}^{\alpha p{}_{0}}\sqrt{\frac{\pi }{\alpha p{}_{0}}}。\text{ }\text{ }\text{ }\left(15\right)$

$\mu {}_{1}=\frac{F}{Ρ}=\frac{4}{\sqrt{\pi }}\overline{u}\eta {}_{0}\frac{e{}^{\alpha p{}_{0}}}{p{}_{0}Η{}_{\mathrm{min}}\sqrt{\alpha p{}_{0}}}\frac{\text{Δ}u}{\overline{u}},\text{ }\text{ }\text{ }\left(16\right)$

μ1=FP=4

πη0eαp0p0Hminαp0λ0, 2 (ττc)

μ1=FP=4

πη0eαp0p0Hminαp0λc (ττc) (17)

《5 结果分析》

5 结果分析

《5.1超环面行星蜗杆传动啮合副的滑动率分析》

5.1超环面行星蜗杆传动啮合副的滑动率分析

《图2》

《图3》

《5.2摩擦系数随润滑状态及传动参数的变化规律》

5.2摩擦系数随润滑状态及传动参数的变化规律

《图4》

Fig.2 Dependence of friction coefficient μl on parameters of the drive 曲线1—行星轮与定子; 曲线2—行星轮与蜗杆

1) 随参数a/R增大, 行星轮与定子之间摩擦系数大幅度减小, 而行星轮与蜗杆之间的摩擦系数略有增加;行星轮与定子之间的摩擦系数大于行星轮与蜗杆之间的摩擦系数, 随参数a/R增大, 两者数值逐渐接近。

2) 随参数iH01增大, 行星轮与定子之间摩擦系数及行星轮与蜗杆之间摩擦系数均大幅度增大。

3) 随参数iH21增大, 行星轮与定子之间的摩擦系数略有减小, 而行星轮与蜗杆之间的摩擦系数略有增加。

4) 随参数R/r增大, 行星轮与定子之间摩擦系数及行星轮与蜗杆之间摩擦系数均大幅度增大。

1) 随参数a/R增大, 行星轮与定子之间的摩擦系数略有减小。

2) 随参数iH01增大, 行星轮与定子之间的摩擦系数大幅度增大。

《图5》

Fig.3 Dependence of friction coefficient μa on parameters of the drive

《图6》

Fig.4 Dependence of friction coefficient μ on λ 曲线1—凸峰接触; 曲线2—液体润滑; 曲线3—混