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 哈尔滨工业大学学报  2019, Vol. 51 Issue (1): 162-169  DOI: 10.11918/j.issn.0367-6234.201802019 0

### 引用本文

JING Chongbo, ZHOU Junjie, LIU Jianhao, LUO Anlin. Self-adapting lubrication characteristics of conical coupling of ball piston hydraulic pump[J]. Journal of Harbin Institute of Technology, 2019, 51(1): 162-169. DOI: 10.11918/j.issn.0367-6234.201802019.

### 文章历史

Self-adapting lubrication characteristics of conical coupling of ball piston hydraulic pump
JING Chongbo, ZHOU Junjie, LIU Jianhao, LUO Anlin
State Key Laboratory of Vehicular Transmission(Beijing Institute of Technology), Beijing 100081, China
Abstract: In this paper, the steady condition of conical spindle distribution and its dynamic response under the changing load is investigated. The lubrication model in spherical coordinate system is developed based on the particular structure, and from force analysis, the dynamic model of conical spindle distribution in axial and radial freedoms is also constructed. The static and dynamic lubricating performance including the shaft eccentricity and distribution gap height is obtained by solving the model with a numerical method. The result shows that the shaft eccentricity and distribution gap height oscillates intensely under the impact loading and gets back to a balance position in a short time. Such a self-adaptive feature of the lubrication performance of conical spindle distribution without requiring external adjustment is significant to ensure the reliability as the ball piston pumps are working.
Keywords: ball piston pump     conical spindle distribution     lubrication characteristics     self - adapting     dynamic characteristics

 1—低压油腔; 2—配流轴; 3—高压油腔; 4—球活塞; 5—作用环; 6—缸体 图 1 球塞泵结构简图 Figure 1 Structure of the ball piston pump

 1—高压平衡油槽; 2—锥形配流轴; 3—低压平衡油槽; 4—缸体; 5—球活塞; 6—支撑轴承; 7—高压窗口; 8—高压油道; 9—低压窗口; 10—低压油道 图 2 球塞泵锥形配流副结构原理图 Figure 2 Schematic of the conical spindle distribution

P.J.Prabhu分别对毛细管和小孔节流的环形腔圆锥轴承的性能进行分析，在一维流状态用解析法导出了Constantinescu球面坐标下封油面上压力分布表达式，并得到了静态承载能力、静态刚度、流量、封油面上的摩擦功耗以及动态刚度、阻尼的解析解，并用差分法计算了环形腔圆锥轴承二维流时的静特性值及动刚度和阻尼系数[13-14]. A.EL.Kagar等对圆锥静压轴承的非牛顿流体工作状态进行了研究，这种静压轴承轴端设有静压腔，在整个圆锥面上油膜厚度连续，分析了锥角、转速、油腔大小端半径比及供油压力对轴承性能的影响，并用实验验证了其理论分析的正确性[15].

1 润滑模型 1.1 雷诺方程推导

 图 3 锥形配流副坐标系 Figure 3 Coordinate system of the conical spindle distribution

 $\frac{{\partial \rho }}{{\partial t}} + \frac{1}{{{r^2}}}\frac{{\partial \left( {\rho {v_r}{r^2}} \right)}}{{\partial r}} + \frac{1}{{r\sin \beta }}\frac{{\partial \left( {\rho {v_\beta }\sin \beta } \right)}}{{\partial \beta }} + \frac{1}{{r\sin \beta }}\frac{{\partial \left( {\rho {v_\theta }} \right)}}{{\partial \theta }} = 0.$

 $\frac{{\partial p}}{{\partial r}} = \frac{1}{{{r^2}}}\frac{\partial }{{\partial \beta }}\left( {\frac{{\mu \partial {v_r}}}{{\partial \beta }}} \right),\;\;\;\;\;\;\frac{{\partial p}}{{\partial \beta }} = 0,$
 $\frac{{\partial p}}{{\partial \theta }} = \frac{{\sin \beta }}{r}\frac{\partial }{{\partial \beta }}\left( {\mu \sin \beta \frac{\partial }{{\partial \beta }}\left( {\frac{{{v_\theta }}}{{\sin \beta }}} \right)} \right) \cong \frac{{\sin \beta }}{r}\frac{\partial }{{\partial \beta }}\left( {\frac{{\mu \partial {v_\theta }}}{{\partial \beta }}} \right).$

 $\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{h^3}\frac{{\partial p}}{{\partial r}}} \right) + \frac{1}{{{r^2}{{\sin }^2}\beta }}\frac{\partial }{{\partial \theta }}\left( {{h^3}\frac{{\partial p}}{{\partial \theta }}} \right) = 6\mu \mathit{\Omega }\frac{{\partial h}}{{\partial \theta }} + 12\mu \frac{{\partial h}}{{\partial t}}.$

 $h = e\cos \theta - {h_0} = {h_0}\left( {1 - \varepsilon \cos \theta } \right).$

 $h = {h_0}\left( {1 - \varepsilon \cos \theta } \right)\cos \beta .$

1.2 静压边界

 图 4 锥形配流副的流场边界 Figure 4 Flow field boundary of the conical spindle distribution
 图 5 锥形配流副的液阻网络 Figure 5 Hydraulic resistance networks of the conical spindle distribution

Ⅰ流场：

 $\frac{{{p_{\rm{h}}} - {p_1}}}{{{R_3}}} + \frac{{{p_{\rm{2}}} - {p_1}}}{{{R_7}}} = \frac{{{p_1}}}{{{R_5}}} + \frac{{{p_1} - {p_l}}}{{{R_2}}}.$

Ⅱ流场：

 $\frac{{{p_{\rm{h}}} - {p_2}}}{{{R_1}}} = \frac{{{p_{\rm{2}}} - {p_1}}}{{{R_7}}} + \frac{{{p_2}}}{{{R_6}}} + \frac{{{p_2} - {p_l}}}{{{R_4}}}.$

2 动力学模型

 图 6 锥形配流副受力分析 Figure 6 Force analysis of the conical spindle distribution

 $\frac{{{\rm{d}}p}}{{{\rm{d}}t}} = - \frac{1}{{CV}}\frac{{{\rm{d}}V}}{{{\rm{d}}t}}.$ (1)

 $C = \left( {7.25 - \lg \mu } \right) \times {10^{ - 10}}.$ (2)

 ${Q_{{\rm{in}}}} = \frac{{{p_h} - {p_2}}}{{{R_0}}}$ (3)

 ${Q_{{\rm{out1}}}} = \frac{{{p_2}}}{{{R_1}}} + \frac{{{p_2}}}{{{R_2}}},$ (4)
 ${Q_{{\rm{out2}}}} = \frac{{{\rm{d}}V}}{{{\rm{d}}t}},$ (5)

 $\frac{{{\rm{d}}p}}{{{\rm{d}}t}} = \frac{1}{{C{V_s}}}\left( {{Q_{{\rm{in}}}} - {Q_{{\rm{out1}}}} - {Q_{{\rm{out2}}}}} \right).$ (6)

 ${V_s} = \frac{1}{{2\sin \beta }}\left( {{r_4}^2 - {r_7}^2} \right)\left( {\left( {{h_0} - z\tan \beta } \right)\alpha - 2x\sin \frac{\alpha }{2}} \right),$ (7)
 $\frac{{{\rm{d}}{V_s}}}{{{\rm{d}}t}} = \frac{1}{{2\sin \beta }}\left( {{r_4}^2 - {r_7}^2} \right)\left( { - \dot z\tan \beta \alpha - 2\dot x\sin \frac{\alpha }{2}} \right).$ (8)

 $\begin{array}{l} \frac{{{\rm{d}}{p_2}}}{{{\rm{d}}t}} = \frac{\quad {2\sin \beta }\quad }{{C\left( {{r_4}^2 - {r_7}^2} \right)\left( {\left( {{h_0} - z\tan \beta } \right)\alpha - 2x\sin \frac{\alpha }{2}} \right)}} \times \\ \;\;\;\;\;\;\;\;\;\left( {\frac{{{p_h} - {p_2}}}{{{R_1}}} - \frac{{{p_2}}}{{{R_4}}} - \frac{{{p_2}}}{{{R_6}}} - \frac{1}{{2\sin \beta }}\left( {{r_4}^2 - {r_7}^2} \right) \cdot } \right.\\ \;\;\;\;\;\;\;\;\;\left. {\left( { - \dot z\tan \beta \alpha - 2\dot x\sin \frac{\alpha }{2}} \right)} \right). \end{array}$

 ${F_p} = \frac{{\rm{ \mathsf{ π} }}}{2}{d_p}^2{p_h}.$

 ${W_{dz}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_d}\left( {i,j} \right)r{{\sin }^2}\beta {\rm{d}}r{\rm{d}}\theta } } ,$
 ${W_{jz}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_j}\left( {i,j} \right)r{{\sin }^2}\beta {\rm{d}}r{\rm{d}}\theta } } .$

 ${W_{dj1}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_d}\left( {i,j} \right)r\cos {\theta _{i,j}}\sin \beta \cos \beta {\rm{d}}r{\rm{d}}\theta } } ,$
 ${W_{dj2}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_d}\left( {i,j} \right)r\sin {\theta _{i,j}}\sin \beta \cos \beta {\rm{d}}r{\rm{d}}\theta } } ,$
 ${W_{dj}} = \sqrt {{W_{dj1}}^2 + {W_{dj2}}^2} ,$
 ${W_{jj1}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_j}\left( {i,j} \right)r\cos {\theta _{i,j}}\sin \beta \cos \beta {\rm{d}}r{\rm{d}}\theta } } ,$
 ${W_{jj2}} = \sum\limits_{i = 1}^{i = m} {\sum\limits_{j = 1}^{j = n} {{p_j}\left( {i,j} \right)r\sin {\theta _{i,j}}\sin \beta \cos \beta {\rm{d}}r{\rm{d}}\theta } } ,$
 ${W_{jj}} = \sqrt {{W_{jj1}}^2 + {W_{jj2}}^2} .$

 $\left\{ \begin{array}{l} {W_{dz}} + {W_{jz}} - {F_p} = m\ddot z,\\ {W_{dj}} + {W_{jj}} = m\ddot x. \end{array} \right.$ (9)
3 数值计算结果分析

 图 7 Δp =2 MPa高压平衡槽建压过程 Figure 7 Pressure building process of balanced high-pressure grooves(Δp=2 MPa)

 图 8 Δp =2 MPa时的轴心轨迹 Figure 8 Track of shaft axis (Δp =2 MPa)

 $\varepsilon = \frac{{x\cos \beta }}{{{h_0} + z\sin \beta }}.$

 $h = {h_0} + z\sin \beta .$

 图 9 偏心率动态变化过程 Figure 9 Dynamic change of eccentricity
 图 10 偏心率动态变化过程局部放大 Figure 10 Partial amplification of dynamic change of eccentricity

 图 11 配流间隙的动态变化过程 Figure 11 Dynamic change of distribution gap
 图 12 配流间隙的动态变化局部放大图 Figure 12 Partial amplification of dynamic change of distribution gap

 ${h_{\min }} = h\left( {1 - \varepsilon } \right).$

 图 13 最小间隙动态变化过程 Figure 13 Dynamic change of minimum gap

 图 14 配流间隙动态变化过程 Figure 14 Dynamic change of distribution gap
 图 15 偏心率动态变化过程 Figure 15 Dynamic change of eccentricity

 图 16 最小间隙动态变化过程 Figure 16 Dynamic change of minimum gap
4 试验验证

 1—驱动电机; 2—转矩转速传感器; 3—锥形配流副组件; 4—被试件; 5—回油管; 6—信号采集系统; 7—进油管; 8—上位机; 9—流量压力传感器; 10—泵站驱动电机; 11—出口溢流阀; 12—高压泵站 图 17 锥形配流副试验系统原理图 Figure 17 Experimental schematic diagram of the conical spindle distribution
 1—管接头; 2—后端盖; 3—位移传感器; 4—壳体; 5—缸体; 6—锥形配流轴; 7—输入轴 图 18 锥形配流副模型试验示意图 Figure 18 Schematic diagram of test model of the conical spindle distribution

 图 19 配流间隙动态变化理论与试验对比 Figure 19 Theoretical and experimental comparison of dynamic change of the distribution gap

1) 被试件中锥形配流轴在径向上的位移受到管接头和后端盖支口的限位，实际偏心率与理论值相比较小，动态变化过程中相应的动压承载力小于理论值，导致轴向上受到的油液推力减小，行程较短.

2) 配流轴与管接头、后端盖间隙配合处靠O型圈密封，O型圈受挤压，在轴向上对配流轴有较大的摩擦力，在配流轴轴向受力较小的情况下摩擦力不仅限制了配流间隙变化的极值，也产生阻尼效应缩短动态变化时间.

5 结论

1) 在受到冲击载荷时，高压平衡槽压力经历迅速上升、超调、稳定调节三个过程达到稳定，相比于工作压力的上升具有滞后性，从而导致了配流轴复杂的受力变化，在轴向与径向发生运动.

2) 锥形配流副在受到冲击载荷时，在径向方向上不断向下运动，并伴随着剧烈波动，轴向上向外运动，将配流轴推离缸体；在达到一定位置后，逐渐恢复到平衡位置，具有自适应的调节特性.

3) 冲击载荷越大，配流轴的动态变化越剧烈，配流间隙和偏心率峰值越大，最小间隙值越小，工作可靠性越低；工作转速越高，配流间隙峰值越大，偏心率峰值越小，最小间隙值越大，稳定性越高.因此，为保证可靠性，球塞泵在工作中应尽量避免大冲击、低转速的工况.

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