﻿ RV减速器5自由度纯扭转模型非线性特性分析
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 浙江大学学报(工学版)  2018, Vol. 52 Issue (11): 2098-2109  DOI:10.3785/j.issn.1008-973X.2018.11.008 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2018.11.008
[复制中文]
ZHENG Yu-xin, XI Ying, BU Wang-hui, LI Meng-ru. Nonlinear characteristic analysis of 5-degree-of-freedom pure torsional model of RV reducer[J]. Journal of Zhejiang University(Engineering Science), 2018, 52(11): 2098-2109.
dx.doi.org/10.3785/j.issn.1008-973X.2018.11.008
[复制英文]

### 作者简介

orcid.org/0000-0003-2977-8759.
E-mail: zhengyuxin1989@126.com.

### 通信联系人

orcid.org/0000-0002-2385-7969.
E-mail: yingxi@tongji.edu.cn
.

### 文章历史

RV减速器5自由度纯扭转模型非线性特性分析

Nonlinear characteristic analysis of 5-degree-of-freedom pure torsional model of RV reducer
ZHENG Yu-xin , XI Ying , BU Wang-hui , LI Meng-ru
School of Mechanical Engineering, Tongji University, Shanghai 201804, China
Abstract: In order to study the effect of nonlinear factors on dynamic characteristics of rotate vector (RV) reducer, components of RV reducer were replaced by concentrated masses, the relationship between components were expressed with spring connections. RV reducer system was simplified and a 5-degree-of-freedom pure torsional dynamic model was established by the principle of D’Alembert and micro-displacement method. The dynamic equations were changed through coordinate transformation and dimensionless method. Lunge-Kutta method combined with time-discrete method were used to solve the nonlinear time-varying equations. Dynamic response curves of the system under start-up and stable period and transmission error curve with corresponding spectrum were obtained. The effects of various nonlinear factors on dynamic characteristics of the system during transmission process were studied. A dynamic performance testing platform was built, and experimental transmission error curve and corresponding response spectrum under the condition of 10 r/min input and 30 N·m load were obtained. The experimental results were in agreement with the theoretical results which indicated that the pure torsional model was established and solved correctly. The work provides a theoretical basis for RV reducer dynamics development at some extent.
Key words: RV reducer    pure torsional dynamic model    dynamic response    transmission error    spectrum    experimental platform

1 RV减速器纯扭转动力学模型建立 1.1 模型假设条件

1.2 纯扭转振动模型建立 1.2.1 RV减速器动力学模型图

 1 输入轴；2 太阳轮；3 行星轮；4 曲柄轴；5 行星架；6 针轮；7 摆线轮；8 摆线轮；9 曲柄轴；10行星轮 图 1 两曲柄轴旋转向量减速器模型简图 Fig. 1 Diagram of RV reducer model with two crankshafts

 图 2 两曲柄轴RV减速器动力学模型 Fig. 2 RV reducer dynamic model with two crankshafts
1.2.2 输入轴的动力学方程

 ${J_0}{\ddot \theta _0} = {T_{\rm M}} - {k_{01}}\left( {{\theta _0} - {\theta _1}} \right) - {c_{01}}\left( {{{\dot \theta }_0} - {{\dot \theta }_1}} \right).$ (1)

1.2.3 中心轮的动力学方程

 ${x_{12}}_i = {r_1}{\theta _1} - {r_2}{\theta _{2i}}.$ (2)

 ${\delta _{12i}} = {x_{12i}} - {\zeta _{12i}}.$ (3)

 ${F_{12i}} = {k_{12}}{\delta _{12i}} + {c_{12}}{\dot \delta _{12i}}.$ (4)

 ${J_1}{\ddot \theta _1} = {k_{01}}\left( {{\theta _0} - {\theta _1}} \right) + {c_{01}}\left( {{{\dot \theta }_0} - {{\dot \theta }_1}} \right) - \mathop \sum \limits_{i = 0}^1 {r_1}{F_{12i}}.$ (5)

1.2.4 行星轮和曲柄轴的动力学方程

 ${\sigma _{2i3jY}} = {e_{\rm c}}\left( {{\theta _{2i}} - {\theta _{\rm c}}} \right){\rm{sin }}\;({\theta _{\rm c}} + {\text{π}}j),$ (6)
 ${\sigma _{2i3jX}} = {e_{\rm c}}\left( {{\theta _{2i}} - {\theta _{\rm c}}} \right){\rm{cos }}\;({\theta _{\rm c}} + {\text{π}}j).$ (7)

 $\begin{split}&{\tau _{3j4iY}} = {e_{\rm c}}\left( {{\theta _{3{\rm c}j}} - {\theta _{\rm c}}} \right){\rm{sin}}\;({\theta _{\rm c}} + {\text{π}}j) + \\ &\quad\quad\quad\;{a_0}\left( {{\theta _{3j}} - {\theta _{\rm g}}} \right){\rm{sin}}\;({\theta _{\rm g}} + {\text{π}}i),\end{split}$ (8)
 $\begin{split}&{\tau _{3j4iX}} = {e_{\rm c}}\left( {{\theta _{3{\rm{c}}j}} - {\theta _{\rm{c}}}} \right){\rm{cos }}\;({\theta _{\rm{c}}} + {\text{π}}j) - \\ &\quad\quad\quad\;{a_0}\left( {{\theta _{3j}} - {\theta _{\rm g}}} \right){\rm{cos}}\;({\theta _{\rm g}} + {\text{π}}i).\end{split}$ (9)

 ${\delta _{3j4iY}} = {\sigma _{2i3jY}} - {\tau _{3j4iY}} - {\zeta _{3j4iY}},$ (10)
 ${\delta _{3j4iX}} = {\sigma _{2i3jX}} - {\tau _{3j4iX}} - {\zeta _{3j4iX}}.$ (11)

 ${F_{3j4iY}} = {k_{34}}{\delta _{3j4iY}} + {c_{34}}{\dot \delta _{3j4iY}},$ (12)
 ${F_{3j4iX}} = {k_{34}}{\delta _{3j4iX}} + {c_{34}}{\dot \delta _{3j4iX}}.$ (13)

 $\begin{split}{J_2}{{\ddot \theta }_{2i}} = & {r_2} {F_{12i}} - ({F_{3j4iY(j = 0)}} - {F_{3j4iY(j = 1)}}){e_{\rm c}} {\rm{sin}}\;{\theta _{\rm c}} -\\ & ({F_{3j4iX(j = 0)}} - {F_{3j4iX(j = 1)}}) {e_{\rm c}} {\rm{cos}}\;{\theta _{\rm c}}.\end{split}$ (14)

1.2.5 摆线轮的动力学方程

 ${\tau _{3j6}} = - {r_{\rm c}}'\left( {{\theta _{3j}} - {\theta _{\rm g}}} \right){\rm{sin}}\;{\alpha _{\rm k}} + {e_{\rm c}}({\theta _{3{\rm c}j}} - {\theta _{\rm c}}){\rm{sin}}\;{\alpha _{\rm k}},$ (15)
 ${r_{\rm c}}' = {e_{\rm c}}{z_{\rm c}},$ (16)
 ${\alpha _{\rm k}} = {\text{π}}/2 - {\alpha _{\rm c}} + {\text{π}}j.$ (17)

 ${\delta _{3j6}} = {\tau _{3j6}} - {\zeta _{3j6}}.$ (18)

 ${F_{63j}} = {k_{36}}{\delta _{3j6}} + {c_{36}}{\dot \delta _{3j6}}.$ (19)

 $\begin{split}{J_3}{{\ddot \theta }_{3j}} = & {F_{63j}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - {F_{3j4iX(i = 0)}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\ &{F_{3j4iX(i = 1)}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + {F_{3j4iY(i = 0)}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - \\ &{F_{3j4iY(i = 1)}}{a_0}{\rm{sin }}\;{\theta _{\rm g}}.\end{split}$ (20)

1.2.6 行星架的动力学方程

 ${\tau _{54iY}} = {a_0}\left( {{\theta _5} - {\theta _{\rm g}}} \right){\rm{sin}}\;({\theta _{\rm g}} + {\text{π}}i),$ (21)
 ${\tau _{54iX}} = - {a_0}\left( {{\theta _5} - {\theta _{\rm g}}} \right){\rm{cos }}\;({\theta _{\rm g}} + {\text{π}}i).$ (22)

 ${\delta _{54iY}} = - {\tau _{54iY}} - {\zeta _{4i5Y}},$ (23)
 ${\delta _{54iX}} = - {\tau _{54iX}} - {\zeta _{4i5X}}.$ (24)

 ${F_{54iY}} = {k_{54}}{\delta _{54iY}} + {c_{54}}{\dot \delta _{54iY}},$ (25)
 ${F_{54iX}} = {k_{54}}{\delta _{54iX}} + {c_{54}}{\dot \delta _{54iX}}.$ (26)

 $\begin{split}{J_5}{{\ddot \theta }_5} = & \mathop \sum \limits_{i = 0}^1 - {F_{54iX}}{\rm{cos}}\;({\theta _{\rm g}} + {\text{π}}i){a_0} + \\ &\mathop \sum \limits_{i = 0}^1 {F_{54iY}}{\rm{sin}}\;({\theta _{\rm g}} + {\text{π}}i){a_0} - {T_{\rm L}}.\end{split}$ (27)

2 变系数微分方程的求解 2.1 系统状态方程的广义坐标变换

 ${x_{01}} = {\theta _0} - {\theta _1},$ (28)
 ${x_{12i}} = {r_1}{\theta _1} - {r_2}{\theta _{2i}},$ (29)
 ${x_{2i3j}} = {\theta _{2i}} - {\theta _{3j}},$ (30)
 ${x_{3j6}} = {\theta _{3j}},$ (31)
 ${x_{2i5}} = {\theta _{2i}} - {\theta _5}.$ (32)

 $\begin{split}{J_0}{{\ddot x}_{01}} = & {T_{\rm M}} - {k_{01}}{x_{01}} - {c_{01}}{{\dot x}_{01}} - \\ &({J_0}/{J_1})[{k_{01}}{x_{01}} + {c_{01}}{{\dot x}_{01}} - {r_1}{f_{12}} - {r_1}{f_{12'}}],\end{split}$ (33)
 $\begin{split}({J_1}/{r_1}) {{\ddot x}_{12}} = &{k_{01}}{x_{01}} + {c_{01}}{{\dot x}_{01}} - {r_1}{f_{12}} - {r_1}{f_{12'}} - \\ & ({J_1}{r_2}/{J_2}{r_1})[{r_2}{f_{12}} - ({F_{34Y}} - {F_{3'4Y}}){e_{\rm c}}{\rm{sin}}\;{\theta _{\rm c}} - \\ & ({F_{34X}} - {F_{3'4X}}){e_{\rm c}}{\rm cos} \;{\theta _{\rm c}}],\end{split}$ (34)
 $\begin{split}({J_1}/{r_1}){{\ddot x}_{12'}} = & {k_{01}}{x_{01}} + {c_{01}}{{\dot x}_{01}} - {r_1}{f_{12}} - {r_1}{f_{12'}} - \\ & ({J_1}{r_2}/{J_2}{r_1})[{r_2}{f_{12'}} - ({F_{34'Y}} - {F_{3'4'Y}}){e_{\rm c}}{\rm{sin}}\;{\theta _{\rm c}} - \\ & ({F_{34'X}} - {F_{3'4'X}}){e_{\rm c}}\cos\; {\theta _{\rm c}}],\end{split}$ (35)
 $\begin{split}{J_2}{{\ddot x}_{23}} = & {r_2}{f_{12}} - \left( {{F_{34Y}} - {F_{3'4Y}}} \right){e_{\rm c}}{\rm{sin }}\;{\theta _{\rm c}} - \\ &\left( {{F_{34X}} - {F_{3'4X}}} \right){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}} - ({J_2}/{J_3})[{F_{63}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - \\ &{F_{34X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + {F_{34'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\ &{F_{34Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - {F_{34'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}}],\quad\quad\quad\quad\quad\quad\quad(36)\end{split}$
 $\begin{split}{J_2}{{\ddot x}_{23'}} =& {r_2}{f_{12}} - \left( {{F_{34Y}} - {F_{3'4Y}}} \right){e_{\rm c}}{\rm{sin }}\;{\theta _{\rm c}} - \\&\left( {{F_{34X}} - {F_{3'4X}}} \right){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}} - ({J_2}/{J_3})[{F_{63{\rm{'}}}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - \\&{F_{3'4X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + {F_{3'4'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\&{F_{3'4Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - {F_{3'4'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}}],\quad\quad\quad\quad\quad\quad(37)\end{split}$ (37)
 $\begin{split}{J_2}{{\ddot x}_{2'3}} = &{r_2}{f_{12'}} - \left( {{F_{34'Y}} - {F_{3'4'Y}}} \right){e_{\rm c}}{\rm{sin }}\;{\theta _{\rm c}} - \\&\left( {{F_{34'X}} - {F_{3'4'X}}} \right){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}} - ({J_2}/{J_3})[{F_{63}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - \\&{F_{34X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + {F_{34'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\&{F_{34Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - {F_{34'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}}],\quad\quad\quad\quad\quad\quad(38)\end{split}$ (38)
 $\begin{split}{J_2}{{\ddot x}_{2'3'}} = &{r_2}{f_{12'}} - \left( {{F_{34'Y}} - {F_{3'4'Y}}} \right){e_{\rm c}}{\rm{sin }}\;{\theta _{\rm c}} - \\&\left( {{F_{34'X}} - {F_{3'4'X}}} \right){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}} - ({J_2}/{J_3})[{F_{63{\rm{'}}}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - \\&{F_{3'4X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + {F_{3'4'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\&{F_{3'4Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - {F_{3'4'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}}],\quad\quad\quad\quad\quad(39)\end{split}$ (39)
 $\begin{split}{J_3}{{\ddot x}_{36}} =& {F_{63}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - {F_{34X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} +{F_{34'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\& {F_{34Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} - {F_{34'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}},\quad\quad\quad\quad\quad\quad\;(40)\end{split}$ (40)
 $\begin{split}{J_3}{{\ddot x}_{3'6}} =& {F_{63{\rm{'}}}}{r_{\rm c}}'{\rm{cos }}\;{\alpha _{\rm c}} - {F_{3'4X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} +{F_{3'4'X}}{a_0}{\rm{cos }}\;{\theta _{\rm g}} + \\ & {F_{3'4Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}} -{F_{3'4'Y}}{a_0}{\rm{sin }}\;{\theta _{\rm g}},\end{split}$ (41)
 $\begin{split}{J_5}{{\ddot x}_{25}} =& - \left( {{F_{54'X}} - {F_{54X}}} \right){a_0}{\rm{cos }}\;{\theta _{\rm g}} \!-\!\left( {{F_{54Y}} - {F_{54'Y}}} \right){a_0}{\rm{sin }}\;{\theta _{\rm g}} + \\& {T_{\rm L}} +({J_5}/{J_2})\Big[{r_2}{f_{12}} - ({F_{34Y}} - {F_{3'4Y}}){e_{\rm c}}{\rm{sin }}\;{\theta _{\rm c}} - \\& ({F_{34X}} - {F_{3'4X}}){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}}\Big],\quad\quad\quad\quad\quad\quad\quad\quad\quad(42)\end{split}$ (42)
 $\begin{split}{J_5}{{\ddot x}_{2'5}} =& - \left( {{F_{54'X}} - {F_{54X}}} \right){a_0}{\rm{cos }}\;{\theta _{\rm g}} - \left( {{F_{54Y}} - {F_{54'Y}}} \right){a_0} \cdot \\& {\rm{sin }}\;{\theta _{\rm g}} +{T_{\rm L}} + ({J_5}/{J_2})\Big[{r_2}{f_{12'}} - ({F_{34'Y}} - {F_{3'4'Y}}){e_{\rm c}}\cdot \\& {\rm{sin }}\;{\theta _{\rm c}} -({F_{34'X}} - {F_{3'4'X}}){e_{\rm c}}{\rm{cos }}\;{\theta _{\rm c}}\Big].\quad\quad\quad\quad\quad\quad(43)\end{split}$ (43)

2.2 系统动力学参数 2.2.1 输入轴参数计算

 $\mathop \sum \nolimits \phi_u = \mathop \sum \limits_{u = 1}^3 \frac{{T{l_u}}}{{G{I_u}}} = \mathop \sum \limits_{u = 1}^3 \frac{{32T{l_u}}}{{G{\text{π}}{{\left( {2{r_u}} \right)}^4}}}\,.$ (44)

 ${k_{01}} = \frac{T}{\mathop \sum \nolimits {\phi_u} . }$ (45)

 ${c_{01}} = 2\xi \bigg[{{{{k_{01}}}}/\Big({{{\rm{1/}}{J_{\rm{0}}} + {\rm{1/}}{J_{\rm{1}}}}}}\Big)\bigg]^{1/2} .$ (46)

2.2.2 太阳轮和行星轮啮合参数

 $c' = 1/s.$ (47)

 $\begin{array}{l}s = 0.047 \; 23 + {{0.155 \; 51} / {{z_1}}} + {{0.257 \; 91} / {{z_2}}} - 0.006 \; 35{\gamma _1}\;\\\quad\;\; - 0.116 \; 54{{{\gamma _1}} / {{z_1}}} - 0.001 \; 93{\gamma _2} - 0.241 \; 88{{{\gamma _2}} / {{z_2}}} + \\\quad\;\; 0.005 \; 29\gamma _1^2 + 0.001 \; 82\gamma _2^2.\end{array}$ (48)

 ${k_{\rm{r}}} = (0.75{\varepsilon _{\rm a}} + 0.25)c'.$ (49)

 ${\varepsilon _{\rm a}} = [{z_1}(\tan\; {\alpha _{{\alpha _1}}} - \tan\; \alpha ) + {z_2}(\tan\; {\alpha _{{\alpha _2}}} - \tan\; \alpha ){\rm J}/2{\text{π}}.$ (50)

 ${k_{12}} = {k_{\rm{r}}} b \times {10^6}.$ (51)

 ${c_{12}} = 2{\xi _{{\rm{12}}}} \bigg[{{k_{12}}{{r_1^2r_2^2{J_1}{J_2}}}/\Big( {{r_1^2{J_1} + r_2^2{J_2}}}}\Big)\bigg] ^{1/2}.$ (52)

2.2.3 摆线轮和针轮啮合参数

 ${k_\nu } = \frac{{{\text{π}}bE{\rho _{\nu {\rm r}}}{\rho _{\rm c}}}}{{4(1 - {\mu ^2})({\rho _{\nu {\rm r}}} + {\rho _{\rm c}}){\rho _\nu }}}.$ (53)

 ${\rho _{\nu {\rm{r}}}} = \frac{{\left( {{r_{\rm{z}}} + {\varOmega _1}} \right){{\left( {1 + {k^2} - 2k{\rm{cos }}\;{\varphi _\nu }} \right)}^{3/2}}}}{{k\left( {{z_{\rm{p}}} + 1} \right){\rm{cos }}\;{\varphi _\nu } - \left( {1 + {z_{\rm{p}}}{k^2}} \right)}} + \left( {{r_{\rm rp}} + {\varOmega _2}} \right),$ (54)
 ${\rho _c} = {r_{\rm rp}},$ (55)
 ${\rho _\nu } = \frac{{{\rho _{\nu {\rm{r}}}}{\rho _{\rm{c}}}}}{{{\rho _{\nu {\rm{r}}}} + {\rho _{\rm{c}}}}}.$ (56)

 ${k_{36}} = \sum\limits_{\nu = n}^m {{k_\nu }l_{{\rm o}v}^2},$ (57)
 ${l_{{\rm o}\nu }} = {r_{\rm c}}'{{({\rm{sin}}\;{\varphi _\nu })}}/({{{1 + {k^2} - 2k{\rm{cos}}\;{\varphi _\nu }}}})^{1/2}.$ (58)

 ${c_\nu } = 2{\xi _{{\rm{36}}}}\left({{{{{k_\nu }r_{\rm c}}{'^2}{r_{{\rm{rp}}}}^2{J_3}{J_6}}}/\big({{{r_{\rm c}}{'^2}{J_3} + {r_{{\rm{rp}}}}^2{J_6}}}}\big)\right)^{1/2}.$ (59)

 ${c_{36}} = \sum\limits_{\nu = n}^m {{c_\nu }} .$ (60)
2.2.4 转动惯量

 ${m_{\bar \omega} } = {\rho _{{\bar \omega}}}{\text{π }}{r_{\bar \omega}}^2{l_{\bar \omega} },$ (61)
 ${J_{\bar \omega}} = {\rho _{{\bar \omega}}}{\text{π}}{r_{\bar \omega}}^4{l_{\bar \omega}}/2.$ (62)

2.2.5 动力参数计算结果

2.3 纯扭转模型计算结果 2.3.1 加速状态中的系统动态响应

RV减速器驱动电机的启动机械特性曲线如图3所示. 连续动作区域是指电机连续不间断的工作方式，并且不受时间和发热的限制. 瞬间动作区域是指短时间断工作方式，如点动启停. 考察RV减速器运动特性是在连续动作区域中进行，即在图3中ABC曲线的下方区域中进行.

 图 3 电机驱动机械特性曲线 Fig. 3 Mechanical characteristic curve of motor driver

 ${T_{\rm M}} = {\partial _1} + {\partial _2}\omega .$ (63)

 ${\omega _{\rm H}} = {n_{\rm H}}{\text{π}}/30.$ (64)

 ${\omega _0} = {n_{\rm 0}}{\text{π}}/30.$ (65)

 ${T_{\rm M}} = 23.9 - 0.045\;86\omega .$ (66)

AB段力矩恒为14.3 N·m，超过BC段的驱动力矩为0 N·m.

2.3.2 加速到稳定状态中的系统动态响应

 图 4 RV减速器各部件的转动角、转动角速度关系 Fig. 4 Rotational angle and rotational angular velocity relationship between various components of RV reducer

 图 5 RV减速器第1级传动部分传动误差 Fig. 5 Transmission error of first transmission part of RV reducer

 图 6 RV减速器第2级传动部分传动误差 Fig. 6 Transmission error of second transmission part of RV reducer
3 非线性因素对系统动态特性影响 3.1 第1级直齿轮非线性因素的影响

 ${F_{12t}}{\rm{ = }}{c_{{\rm{12}}}}\dot x(t) + {k_{12}}tx(t).$ (67)

$x(t)$ 的表达式为

 $x(t) = \left\{ {\begin{array}{*{20}{c}}{{x_\beta } - \eta ,\;\;\;\;\;\;\;\;\;\;{x_\beta } > \eta };\\{\;\;\;\;0,\;\;\;\;\;\; - \eta \leqslant {x_\beta } \leqslant \eta \;};\\{{x_\beta }{\rm{ + }}\eta ,\;\;\;\;\;\;\;\;\;{x_\beta } < - \eta ,\;}\end{array}} \right.$ (68)
 ${x_\beta } = {x_{12}} - \varsigma (t).$ (69)

 $\varsigma (t) =\sum\limits_{{q_1} = 1}^{{{{x }}_1}}{{\varsigma _{q_1}}\sin\; ({q_1}{\omega _{\rm m}}t + {\psi _{{\varsigma }{q_1}}})} .$ (70)

 ${k_{12}}(t) = {k_{12}} + \Delta {k_{12}}(t),$ (71)
 $\Delta {k_{12}}(t) =\sum\limits_{{q_2} = 1}^{{{{x }}_2}}{{k_{12}}_{q_2}\cos\; (q{\omega _{\rm m}}t + {\psi _{{k}{q_2}}})} .$ (72)

 图 7 第1级传动部分非线性因素对RV减速器系统动力特性的影响 Fig. 7 Effects of nonlinear factors of first transmission part on system dynamic characteristics of RV reducer
3.2 第2级摆线轮非线性因素的影响

 ${k'_{36}}(t) = {k'_{36}} + \Delta {k'_{36}}(t).$ (73)

4 RV减速器综合检测实验平台 4.1 搭建实验台

 图 9 RV减速器动力性能综合检测实验平台示意图 Fig. 9 Sketch map of experimental platform for comprehensive testing of dynamic performance of RV reducer

 图 10 RV减速器动力性能综合检测实验平台 Fig. 10 Comprehensive experimental platform for dynamic performance of RV reducer
4.2 实验台运行结果与理论对比

 图 11 RV减速器负载扭矩 Fig. 11 Load torque of RV reducer

 图 12 RV减速器运行过程中的噪声 Fig. 12 RV reducer’s noise during operation

RV减速器的传动误差是指在个周期内传动误差的最大数值和最小数值之间的差值. 被测减速器的传动角度误差曲线如图13(a)所示. 图13(a)为输出轴旋转2圈得到的结果，0~2π rad为第1周期，2π~4π rad为第2周期，分别分析每个周期. 在0~2π rad中，分析最大和最小传动误差的差值，在2π~4π rad中，再次分析最大和最小传动误差的差值. 起点不同的2组实验得到的传动误差数值不同，但是2次数据的整体变化趋势相同，并且第1周期和第2周期变化趋势相同. 被测RV减速器在第1组数据的第1个周期内的最大传动误差为53″，第1组数据的第2个周期内的最大传动误差为54″，第2组数据的第1个周期内的最大传动误差为52″，第2组数据的第2个周期内的最大传动误差为50″，说明实验在2个周期内运行平稳，没有出现周期性干扰或者误差. 实验数据与图6(c)中不考虑时变刚度、啮合间隙和啮合误差时的最大传动误差18″和图7(f)8(d)中考虑非线性因素时的最大传动误差26″较接近，说明理论模型的计算精度较高. 但是实验和理论计算之间仍有一定的差异，主要是因为理论模型是建立在一定假设条件下的理想模型，考虑的影响因素较少，实际中的传动系统中的影响因素众多，传动误差大于理论计算结果是可以接受的. 如图13(b)所示为整机传动误差的频谱图. 系统在0.000 86、0.002 772、0.067 58、0.105 7、0.136 9、0.178 5、0.206 2、0.270 3、0.554 5、0.605 6、1.185、1.456、1.801 Hz等处都出现了明显的峰值，其中在0.000 86、0.067 58、0.105 7 Hz处峰值最大，0.107 5 Hz与理论计算中的0.103 5 Hz非常接近，进一步说明理论计算和实验结果较吻合. 实验中的多峰值频率主要是由系统中的不稳定输入和输出引起其他频谱.

 图 13 RV减速器运行过程中的整机传动误差 Fig. 13 RV reducer’s transmission error during operation
 图 8 第2级非线性因素对RV减速器系统动力特性影响 Fig. 8 Effects of nonlinear factors of second transmission part on system dynamic characteristics of RV reducer
5 结　语

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