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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2022, Vol. 56 Issue (9): 1833-1844    DOI: 10.3785/j.issn.1008-973X.2022.09.017
    
Dynamic characteristics of pinion and rack stroke-increment mechanism with grease lubrication
Zhi-qun CHEN1(),Lin-fang QIAN1,2,*(),Yi-cheng ZHU1
1. School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
2. Northwest Institute of Mechanical and Electrical Engineering, Xianyang 712099, China
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Abstract  

In order to accurately obtain the dynamic characteristics of the pinion and rack under grease lubrication condition, a structure-grease film coupling meshing stiffness model was presented, which took both the structural time-varying meshing stiffness and transient thermal elastohydrodynamic lubrication stiffness during the meshing process of the rack and pinion into account, the dynamics equations of the pinion and rack stroke-increment mechanism involving the friction were formulated. The dynamic characteristics of pinion and rack mechanism and grease film were analyzed, the numerical results showed that total meshing stiffness between gear teeth was lower than the time-varying meshing stiffness of structure when the transient thermal elastohydrodynamic effect of grease was considered. With the decreasing of the normal meshing force, the total stiffness also reduced. Generally speaking, with the increase of the equivalent radius of curvature, the center film thickness became more thicker and the center pressure decreased. Both of them showed high frequency fluctuation characteristics. The worst lubrication state occurred on the gear tooth flank near the base circle where the temperature rise and pressure of the grease film were the highest, thickness were the thinnest. The friction coefficient was lower in the middle period with higher rack and pinion drive velocity than that in start and end periods, and it decreased obviously when the meshing point was close to pitch point.

Key words： pinion and rack mechanism      grease lubrication      mesh stiffness      thermal elastohydrodynamic lubrication      oil film characteristics      friction     
Received: 22 September 2021      Published: 28 September 2022
CLC:  TH 132.4  
Fund:  国家自然科学基金资助项目（11472137）
Corresponding Authors: Lin-fang QIAN     E-mail: zqchen_njust@163.com;lfqian@njust.edu.cn
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Zhi-qun CHEN
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Cite this article:

Zhi-qun CHEN,Lin-fang QIAN,Yi-cheng ZHU. Dynamic characteristics of pinion and rack stroke-increment mechanism with grease lubrication. Journal of ZheJiang University (Engineering Science), 2022, 56(9): 1833-1844.

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https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2022.09.017     OR     https://www.zjujournals.com/eng/Y2022/V56/I9/1833


脂润滑齿轮齿条增程机构的动态特性

为了准确地获得脂润滑条件下齿轮齿条的动态特性，考虑齿轮齿条啮合时的结构时变啮合刚度和瞬态热弹流润滑刚度的耦合影响，建立结构?脂膜耦合啮合刚度模型，推导受摩擦影响的齿轮齿条增程机构的动力学方程. 分析齿轮齿条机构及脂膜的动态特性，数值结果表明：在考虑润滑脂的瞬态热弹流效应后，轮齿的啮合总刚度比结构时变啮合刚度低；且法向啮合力越小，总刚度值越低. 中心膜厚、中心压应力均具有高频波动特性，并随着当量曲率半径的增加分别呈上升与下降的趋势. 最恶劣润滑状态出现在齿轮轮齿面上靠近基圆的位置，此处的脂膜温升最高，脂膜压应力最大，脂膜厚度最薄. 摩擦系数在齿轮齿条传动速度较大的中间时段比起始与末端时段的低，在啮合点靠近节点位置时明显下降.


关键词： 齿轮齿条机构,  脂润滑,  啮合刚度,  热弹流润滑,  油膜特性,  摩擦 
Fig.1 Configuration of pinion and rack stroke-increment mechanism
Fig.2 Mesh stiffness coupling model of structure-grease
参数 数值 参数 数值 参数 数值
$ L_1^{{\rm{p}}} $ 3.922 6 $ Q_3^{{\rm{p}}} $ 0.409 5 $ T_2^{{\rm{r}}} $ ?0.285 7
$M_1^{{\rm{p}}}$ 1.625 4 $ R_3^{{\rm{p}}} $ 0.330 8 $ U_2^{{\rm{r}}} $ 0.104 0
$ P_1^{{\rm{p}}} $ 2.697 9 $ S_3^{{\rm{p}}} $ 0.236 8 $ V_2^{{\rm{r}}} $ 0.577 7
$ Q_1^{{\rm{p}}} $ 0.434 7 $ T_3^{{\rm{p}}} $ 0.085 1 $ QH_2^{{\rm{r}}} $ 0.104 0
$ L_2^{{\rm{p}}} $ ?0.511 4 $ U_3^{{\rm{p}}} $ ?0.080 4 $ MH_2^{{\rm{r}}} $ 0.577 7
$ M_2^{{\rm{p}}} $ 0.339 7 $ V_3^{{\rm{p}}} $ 1.685 9 $ L_3^{{\rm{r}}} $ 1.440 4
$ P_2^{{\rm{p}}} $ 0.330 8 $ L_1^{{\rm{r}}} $ ?1.248 0 $ M_3^{{\rm{r}}} $ 0.095 2
$ Q_2^{{\rm{p}}} $ 0.330 8 $ M_1^{{\rm{r}}} $ ?0.071 4 $ Q_3^{{\rm{r}}} $ ?0.096 7
$ R_2^{{\rm{p}}} $ 0.330 8 $ P_1^{{\rm{r}}} $ 0.322 6 $ S_3^{{\rm{r}}} $ ?0.118 2
$ S_2^{{\rm{p}}} $ 0.236 8 $ Q_1^{{\rm{r}}} $ 2.007 9 $ T_3^{{\rm{r}}} $ 0.285 7
$ T_2^{{\rm{p}}} $ ?0.085 1 $ LH_1^{\rm{r}} $ 0.357 1 $ U_3^{{\rm{r}}} $ ?0.104 0
$ U_2^{{\rm{p}}} $ 0.080 4 $ MH_1^{{\rm{r}}} $ 2.007 9 $ V_3^{{\rm{r}}} $ 0.577 7
$ V_2^{{\rm{p}}} $ 1.685 9 $ L_2^{{\rm{r}}} $ 1.440 4 $ QH_3^{{\rm{r}}} $ ?0.104 0
$ L_3^{{\rm{p}}} $ ?0.511 4 $ M_2^{{\rm{r}}} $ ?0.095 2 $ MH_3^{{\rm{r}}} $ 0.577 7
$ M_3^{{\rm{p}}} $ ?0.399 7 $ Q_2^{{\rm{r}}} $ 0.096 7 — —
$ P_3^{{\rm{p}}} $ 0.330 8 $ S_2^{{\rm{r}}} $ ?0.118 2 — —
Tab.1 Constant coefficient terms of structural coupling stiffness
Fig.3 Deformation compatibility condition considering coupling phenomenon
Fig.4 Schematic diagram of grease lubrication thermal elastohydrodynamic film at meshing point
参数 数值 参数 数值
齿轮质量 ${m_{{\rm{p}}} }/{{\rm{kg}}}$ 4.91 齿宽 $b/{{\rm{mm}}}$ 20
齿轮转动惯量 ${I}_{\mathrm{p} }/(\rm{kg} \cdot {{\rm{m}}}^{2})$ 4.2×10?4 弹性模量 $ E/{\text{GPa}} $ 207
上侧齿条质量 ${m_{{\rm{r}}} }/{ {\rm{kg} } }$ 12.91 泊松比 $\; \mu $ 0.3
模数 ${\rm{ m}}/{{\rm{mm}}}$ 3 齿侧间隙 $2\bar D/{{\rm{mm}}}$ 0.10
齿轮齿数 $ z $ 20 — —
Tab.2 Structural parameters of pinion and rack
名称 c/
(J·kg·K?1) 		k/
(W·m?1·K?1) 		ρ/
(kg·m?3) 		ρT/
(m·k?1) 		φ0/
(Pa·sn) 		n
齿轮 470 46 7850 — — —
齿条 470 46 7850 — — —
润滑脂 1646 0.14 880 0.00065 8.634 0.754
Tab.3 Rheological parameters of grease with temperature of 303 K and thermodynamic parameters of rack and pinion drive systems
Fig.5 Flowchart of solving dynamic model of pinion and rack with grease lubrication
Fig.6 Displacement curve and velocity curve of upper rack
Fig.7 Meshing state characteristics during second tooth meshing cycle of gear and upper rack
Fig.8 Meshing state characteristics during eighth tooth meshing cycle of gear and upper rack
Fig.9 Total friction coefficient of upper rack
Fig.10 Variation of friction coefficient during second tooth meshing process
Fig.11 Finite element model of engagement of tooth and pinion engagement
Fig.12 Comparison of meshing stiffness of second tooth
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