﻿ 行星滚柱丝杠副刚度特性分析及试验验证
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (5): 967-973  DOI: 10.11990/jheu.201711108 0

### 引用本文

GUO Jianan, HE peng, HUANG Hongyan, et al. Analysis of planetary roller screw axial stiffness and experiment study[J]. Journal of Harbin Engineering University, 2019, 40(5), 967-973. DOI: 10.11990/jheu.201711108.

### 文章历史

1. 哈尔滨工业大学 能源学院, 黑龙江 哈尔滨 150001;
2. 北京精密机电控制设备研究所, 北京 100076

Analysis of planetary roller screw axial stiffness and experiment study
GUO Jianan 1, HE peng 1, HUANG Hongyan 1, LIU Zhansheng 1, HUANG Yuping 2, DING Weitao 2
1. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;
2. Beijing Research Institute of Precise Mechanical and Electronic Control Equipment, Beijing 100076, China
Abstract: To solve the axial stiffness precision problem of planetary roller screw in servo motor systems, a research for thread contact stiffness, thread stiffness and body stiffness is conducted and a new planetary roller screw stiffness model was established which was based on differential geometry and Hertz contact theory. The effects of load, contact angle and roller number on stiffness are analyzed. The stiffness will increase when the contact angle and roller number increase. Under the same load condition, the growth of stiffness will increase with the increase of contact angle and roller number. A planetary roller screw static stiffness experiment is established, the maximum error between the experiment results and the model presented in this paper is 8.3%. The experiment result verifies the accuracy of the model presented in this paper.
Keywords: planetary roller screw    transmission mechanism    thread contact    differential geometry    contact stiffness    axial stiffness    Hertz contact theory

1 行星滚柱丝杠模型 1.1 行星滚柱丝杠结构模型

 Download: 图 1 行星滚柱丝杠结构模型示意 Fig. 1 Mechanical structure of planetary roller screw
1.2 行星滚柱丝杠运动模型

 $\begin{array}{*{20}{c}} {s = {D_{\rm{S}}} + {D_{\rm{R}}} = {P_{\rm{S}}}\frac{{{\theta _{{\rm{SC}}}}}}{{2{\rm{ \mathsf{ π} }}}} + {P_{\rm{R}}}\frac{{{\theta _{{\rm{RC}}}}}}{{2{\rm{ \mathsf{ π} }}}} = }\\ {\frac{{{\theta _{\rm{S}}}\left( {{r_{\rm{R}}}{P_{\rm{S}}} + {r_{\rm{S}}}{P_{\rm{R}}}} \right)}}{{2{\rm{ \mathsf{ π} }}{r_{\rm{R}}}}}\left( {\frac{{{r_{\rm{S}}} + 2{r_{\rm{R}}}}}{{2{r_{\rm{S}}} + 2{r_{\rm{R}}}}}} \right)} \end{array}$ (1)

1.3 行星滚柱丝杠轴向刚度模型

 Download: 图 2 行星滚柱丝杠三维模型和简化模型示意 Fig. 2 3D and a simplified model for planetary roller screw
1.3.1 行星滚柱丝杠柱体刚度

 $k_{\mathrm{BS}}=\frac{E_{\mathrm{S}} A_{\mathrm{S}}}{p}, \quad k_{\mathrm{BN}}=\frac{E_{\mathrm{N}} A_{\mathrm{N}}}{p}, \quad k_{\mathrm{BR}}=\frac{n_{\mathrm{R}} E_{\mathrm{R}} A_{\mathrm{R}}}{p / 2}$ (2)

1.3.2 行星滚柱丝杠接触刚度

 $x^{2}+z^{2}=r_{\mathrm{p}}^{2}$ (3)

 $\boldsymbol{r}=\left(a \cos \theta, a \sin \theta, \frac{L}{2 \pi} \theta+\sqrt{r_{p}^{2}-a^{2}}-\sqrt{r_{p}^{2}-r_{R}^{2}}+p\right)$ (4)

 ${\mathit{\boldsymbol{N}}_{{\rm{RT}}}} = \left[ {\begin{array}{*{20}{c}} {{S_{{\beta _{\rm{R}}}}}{C_{{\theta _{\rm{R}}}}} + {C_{{\beta _{\rm{R}}}}}{S_{{\alpha _{\rm{R}}}}}{S_{{\theta _{\rm{R}}}}}}\\ {{S_{{\beta _{\rm{R}}}}}{S_{{\theta _{\rm{R}}}}} - {C_{{\beta _{\rm{R}}}}}{S_{{\alpha _{\rm{R}}}}}{C_{{\theta _{\rm{R}}}}}}\\ {{C_{{\beta _{\rm{R}}}}}{C_{{\alpha _{\rm{R}}}}}} \end{array}} \right]$ (5)

 $\left\{ \begin{array}{l} {E_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{r}}_{{\theta _{{\rm{bT/bB}}}}}}{\mathit{\boldsymbol{r}}_{{a_{{\rm{bT/bB}}}}}},{F_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{r}}_{{a_{{\rm{bT/bB}}}}}}{\mathit{\boldsymbol{r}}_{{\theta _{{\rm{bT/bB}}}}}}\\ {G_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{r}}_{{\theta _{{\rm{bT/bB}}}}}}{\mathit{\boldsymbol{r}}_{{a_{{\rm{bT/bB}}}}}},{L_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{N}}_{{\rm{bT/bB}}}}{\mathit{\boldsymbol{r}}_{a{a_{{\rm{bT/bB}}}}}}\\ {M_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{N}}_{{\rm{bT/bB}}}}{\mathit{\boldsymbol{r}}_{a{\theta _{{\rm{bT/bB}}}}}},{N_{{\rm{bT/bB}}}} = {\mathit{\boldsymbol{N}}_{{\rm{bT/bB}}}}{\mathit{\boldsymbol{r}}_{\theta {\theta _{{\rm{bT/bB}}}}}} \end{array} \right.$ (6)

 $\begin{array}{*{20}{c}} {{E_{{\rm{ST}}}} = 1 + T_{{\beta _{\rm{S}}}}^2,{F_{{\rm{ST}}}} = - \frac{{{L_{\rm{S}}}{T_{{\beta _{\rm{S}}}}}}}{{2{\rm{ \mathsf{ π} }}}},{G_{{\rm{ST}}}} = {u^2} + {{\left( {\frac{{{L_{\rm{S}}}}}{{2{\rm{ \mathsf{ π} }}}}} \right)}^2},}\\ {{L_{{\rm{ST}}}} = 0,{M_{{\rm{ST}}}} = - {C_{{\beta _{\rm{S}}}}}{S_{{\alpha _{\rm{S}}}}},{N_{{\rm{ST}}}} = - u{S_{{\beta _{\rm{S}}}}}} \end{array}$

κ1κ2为曲面上的2个主曲率，由高斯曲率以及平均曲率定义可以得：

 $K=\kappa_{1} \kappa_{2}=\frac{L N-M^{2}}{E G-F^{2}}$ (7)
 $H=\frac{1}{2}\left(\kappa_{1}+\kappa_{2}\right)=\frac{2 F M-(E N+G L)}{2\left(E G-F^{2}\right)}$ (8)

 $\kappa_{1}=H+\sqrt{H^{2}-K}, \quad \kappa_{2}=H-\sqrt{H^{2}-K}$ (9)

 $R_{1}=\frac{1}{\kappa_{1}}, \quad R_{2}=\frac{1}{\kappa_{2}}$ (10)

1) 赫兹接触变形：

 $\begin{array}{*{20}{c}} {{\mathit{\Theta }_i} + {\chi _i} = \frac{1}{2}\left( {\frac{1}{{{R_{1{R_i}}}}} + \frac{1}{{{R_{2{R_i}}}}} + \frac{1}{{{R_{1i}}}} + \frac{1}{{{R_{2i}}}}} \right)}\\ {{\mathit{\Theta }_i} - {\chi _i} = }\\ {\quad \frac{1}{2}\left[ {{{\left( {\frac{1}{{{R_{1{R_i}}}}} - \frac{1}{{{R_{2{R_i}}}}}} \right)}^2} + {{\left( {\frac{1}{{{R_{1i}}}} - \frac{1}{{{R_{2i}}}}} \right)}^2} + } \right.}\\ {{{\left. {\left( {\frac{1}{{{R_{1{R_i}}}}} - \frac{1}{{{R_{2{R_i}}}}}} \right)\left( {\frac{1}{{{R_{1{R_i}}}}} - \frac{1}{{{R_{2{R_i}}}}}} \right)\cos \left( {2{\gamma _i}} \right)} \right]}^{\frac{1}{2}}}} \end{array}$ (11)

 ${R_{{E_i}}} = \frac{1}{2}{\left( {{\mathit{\Theta }_i}{\chi _i}} \right)^{ - \frac{1}{2}}}$ (12)

 ${\delta _{\rm{H}}} = {F_2}{\left( {\frac{{9{P^2}}}{{16{E^{*2}}{R_{{E_i}}}}}} \right)^{\frac{1}{3}}}\cos \beta \cos \alpha$ (13)
 $\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}$ (14)

1.3.3 行星滚柱丝杠螺牙刚度

 $\delta_{T B}=-\frac{\omega a}{C_{8}}\left(\frac{r_{0} C_{9}}{b}-L_{9}\right) \frac{r^{2}}{D} F_{2}+\frac{\omega r_{0}}{b} \frac{r^{3}}{D} F_{3}-\omega \frac{r^{3}}{D} G_{3}$ (15)

1.3.4 行星滚柱丝杠整体轴向刚度

 $\left\{ {\begin{array}{*{20}{l}} {{S_{\rm{N}}} = \Delta {s_{{\rm{NB}}i}} + \Delta {s_{{\rm{NT}}i}} + \Delta {s_{{\rm{NT}}i + 1}} + \Delta {s_{{\rm{NH}}i}}}\\ {{S_{\rm{R}}} = \Delta {s_{{\rm{RB}}i}} + \Delta {s_{{\rm{RT}}i}} + \Delta {s_{{\rm{RT}}i + 1}} + \Delta {s_{{\rm{RH}}i}}} \end{array}} \right.$ (16)

 $\frac{{{Q_{i - 1}} - {Q_i}}}{{\frac{1}{{K_{{\rm{RT}}}^i}} + \frac{1}{{K_{{\rm{NT}}}^i}} + \frac{1}{{K_{\rm{H}}^i}}}} - \frac{{{Q_i} - {Q_{i + 1}}}}{{\frac{1}{{K_{{\rm{RT}}}^{i + 1}}} + \frac{1}{{K_{{\rm{NT}}}^{i + 1}}} + \frac{1}{{K_{\rm{H}}^{i + 1}}}}} = \frac{{{Q_i}}}{{K_{{\rm{NB}}}^{i + 1}}} + \frac{{{Q_i}}}{{K_{{\rm{RB}}}^i}}$ (17)

 ${K_{{\rm{nt}}}} = \frac{{EA}}{L} = \frac{{E{\rm{ \mathsf{ π} }}r_s^2}}{L}$ (18)

 $\frac{1}{K}=\frac{1}{K_{\mathrm{t}}}+\frac{1}{K_{\mathrm{nt}}}$ (19)

 $\frac{1}{K_{\mathrm{RNT}}}=\frac{1}{K_{\mathrm{RT}}^{i}}+\frac{1}{K_{\mathrm{NT}}^{i}}+\frac{1}{K_{\mathrm{H}}^{i}}$ (20)

 $K_{{\rm{RNT}}}^A = K_{{\rm{RNT}}}^1 + K_{{\rm{RNT}}}^2 + \cdots + K_{{\rm{RNT}}}^n$ (21)
2 行星滚柱丝杠刚度特性仿真

 Download: 图 5 不同负载下负载系数随螺纹序号变化曲线 Fig. 5 Effect of roller number on planetary rollar screw axial stiffness

 Download: 图 8 间隙值对行星滚柱丝杠刚度的影响 Fig. 8 Effect of clearance on PRS axial stiffness

 Download: 图 10 滚柱数对行星滚柱丝杠刚度影响 Fig. 10 Effect of roller number on PRS axial stiffness
3 行星滚柱丝杠刚度特性试验研究

 Download: 图 11 行星滚柱丝杠刚度特性试验台示意和实物 Fig. 11 Sketch map and test rig of planctary roller scre waxial stiffness

 Download: 图 12 行星滚柱丝杠轴向刚度试验与仿真结果对比 Fig. 12 Comparation of planetary roller screw axial stiffness experiment results and presented model simulation results

4 结论

1) 相比于传统行星滚柱丝杠刚度模型，本文模型可以分析间隙、接触角以及滚柱数量等参数对行星滚柱丝杠轴向刚度的影响，从仿真结果中可以看出间隙会导致滚柱产生不同的变形，并且随着间隙增大行星滚柱丝杠刚度会减小；螺纹接触角和滚柱数量这2种设计参数越大，行星滚柱丝杠刚度越大；为快速而合理进行行星滚柱丝杠的参数化设计，过大的接触角会影响螺纹的传动，过小的接触角会影响行星滚柱丝杠整体的轴向刚度，因此综合考虑行星滚柱丝杠螺纹接触角45°为最优；滚柱数量会影响丝杠质量、转动惯量并会增大传动过程中的摩擦力，因此滚柱数量也需合理设计。

2) 搭建行星滚柱丝杠轴向刚度特性试验台，刚度特性试验结果与仿真结果误差为8%。通过与刚度试验结果对比，验证了本文所建立的行星滚柱丝杠刚度模型。

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