Analysis of misaligned journal bearing with herringbone grooves: consideration of anisotropic slips

Abstract

The effects of anisotropic slip on the lubrication performance of misaligned journal bearings with herringbone grooves are discussed in this study. The modified Reynolds equation, which considered the anisotropic slips, and film thickness equation in misaligned bearings are solved numerically by the finite element method (FEM). The effects of misaligned angle and anisotropic slip on the load capacities and friction force of journal bearings are analyzed. The results show that the existence of slip boundary conditions can dilute the effects of misalignment for journal bearings. Finally, the effects of moment position and dimensionless pressure on the bearing are also discussed.

Appendix: The modified Reynolds equation with anisotropic slips

For the modified Reynolds equation of anisotropic slips and under the usual assumptions of lubrication theory (Hamrock et al. 2004), the Navier–Stokes equation can be reduced by the order of magnitude method, i.e.

$$\begin{aligned} \frac{\partial }{\partial z}\left( {\mu \frac{\partial u}{\partial z}} \right) = \frac{\partial p}{\partial x} \hfill \\ \frac{\partial }{\partial z}\left( {\mu \frac{\partial v}{\partial z}} \right) = \frac{\partial p}{\partial y}. \hfill \\ \end{aligned}$$

(20)

In micro-bearings application, the no-slip boundary condition should be corrected. The anisotropic slip boundary conditions (Jao et al. 2016a, b) are

$$\begin{aligned} u = u_{1x} + b_{1x} \frac{\partial u}{\partial z},\;\;v = v_{1y} + b_{1y} \frac{\partial v}{\partial z},\;{\text{at }}\;z\; = \;0 \hfill \\ u = u_{2x} - b_{2x} \frac{\partial u}{\partial z},\;\;v = v_{2y} - b_{2y} \frac{\partial v}{\partial z},\;{\text{at}}\;z = h, \hfill \\ \end{aligned}$$

(21)

where $\left( {u_{1x}, \, v_{1x} } \right)$ and $\left( {u_{2x}, \, v_{2x} } \right)$ are boundary velocities of surface 1 and 2, respectively. Thus, the velocity profiles between two surfaces can be obtained by solving Eq. (20) with boundary conditions (Eq. 21) as shown in Eqs. (22) and (23)

$$u = u_{1x} + \frac{{z + b_{1x} }}{{h + b_{1x} + b_{2x} }}V_{x} + \frac{1}{\mu }\frac{\partial p}{\partial x}\left[ {\frac{{z^{2} }}{2} - \frac{h}{2}\frac{{\left( {h + 2b_{2x} } \right)\left( {z + b_{1x} } \right)}}{{h + b_{1x} + b_{2x} }}} \right],$$

(22)

$$v = v_{1y} + \frac{{z + b_{1y} }}{{h + b_{1y} + b_{2y} }}V_{y} + \frac{1}{\mu }\frac{\partial p}{\partial y}\left[ {\frac{{z^{2} }}{2} - \frac{h}{2}\frac{{\left( {h + 2b_{2y} } \right)\left( {z + b_{1y} } \right)}}{{h + b_{1y} + b_{2y} }}} \right],$$

(23)

where $V_{x} = u_{2x} - u_{1x}$, and $V_{y} = v_{2y} - v_{1y}$. The modified Reynolds equation can be obtained from the conservation of mass flow rate as shown in Eq. (24).

$$\frac{\partial }{\partial x}\left( {\dot{m}_{x} } \right) + \frac{\partial }{\partial y}\left( {\dot{m}_{y} } \right) = - \frac{\partial }{\partial t}\left( {\rho h} \right),$$

(24)

where the mass flow rates are

$$\dot{m}_{x} = \int_{0}^{h} { \, \rho u \, dz} ,\;\dot{m}_{y} = \int_{0}^{h} { \, \rho v \, dz} .$$

(25)

Thus, the stationary version of the modified Reynolds equation (Chen et al. 2013), which considered the anisotropic slip effect, is

$$\begin{aligned} \frac{\partial }{\partial x}\left( {\frac{{\rho h^{3} }}{12\mu }\frac{\partial p}{\partial x}Q_{px} } \right) + \frac{\partial }{\partial y}\left( {\frac{{\rho h^{3} }}{12\mu }\frac{\partial p}{\partial y}Q_{py} } \right) \hfill \\ = \frac{\partial (\rho h)}{\partial t} + \frac{\partial }{\partial x}\left( {\rho h\frac{{u_{1x} + u_{2x} }}{2}Q_{cx} } \right) + \frac{\partial }{\partial y}\left( {\rho h\frac{{v_{1y} + v_{2y} }}{2}Q_{cy} } \right), \hfill \\ \end{aligned}$$

(26)

where

$$Q_{pi} = \frac{{h^{2} + 4hb_{1i} + 4hb_{2i} + 4b_{1i} b_{2i} }}{{h\left( {h + b_{1i} + b_{2i} } \right)}},\;\quad i = x, \, y,$$

(27)

and

$$Q_{ci} = \frac{{h + 2b_{2i} }}{{h + b_{1i} + b_{2i} }},\;\quad i = x, \, y,$$

(28)

are the Poiseuille flow rate corrector and the Couette flow rate corrector, respectively. The dimensionless form of Eq. (26) is

$$\frac{\partial }{\partial \theta }\left( {\frac{{H^{3} }}{12}\frac{\partial P}{\partial \theta }Q_{px} } \right) + \left( {\frac{d}{b}} \right)^{2} \frac{\partial }{\partial Y}\left( {\frac{{H^{3} }}{12}\frac{\partial P}{\partial Y}Q_{py} } \right) = \frac{\partial }{\partial \theta }\left( {\frac{H}{2}Q_{cx} } \right),$$

(29)

with $\theta = \frac{x}{r}$, $Y = \frac{y}{b/2}$, $H = \frac{h}{c}$, $B_{i} = \frac{{b_{i} }}{c}$, $P = \frac{{pc^{2} }}{{\mu_{0} u_{b} r}}$, $u_{1x} = u_{b}$, $v_{1y} = 0$, $u_{2x} = u_{2y} = 0$, and the Poiseuille and Couette flow rate correctors are shown below, respectively.

$$Q_{px} = \frac{3}{H}\frac{{\left( {H + 2B_{1x} } \right)\left( {H + 2B_{2x} } \right)}}{{H + B_{1x} + B_{2x} }} - 2,$$

(30)

$$Q_{py} = \frac{3}{H}\frac{{\left( {H + 2B_{1y} } \right)\left( {H + 2B_{2y} } \right)}}{{H + B_{1y} + B_{2y} }} - 2,$$

(31)

$$Q_{cx} = \frac{{H + 2B_{2x} }}{{H + B_{1x} + B_{2x} }}.$$

(32)