Semi-Analytical Prediction of Starved Line Contacts Considering Oil Transport

Abstract

A very simple and fast solver for the time-dependent iso-viscous-rigid smooth line contact is presented. It accounts for starvation, either geometrical or by lack of lubricant. The upstream oil reserve build-up and transport is also included. The solver boils down to a set of four nonlinear equations. The paper is restricted to the detailed presentation of the solver and some simple examples.

Appendix

This appendix regroups the important equations in their dimensionless form. The code implementing the proposed method uses these dimensionless equations.

$$\begin{aligned} \tilde{P} = U_m I_{1/H^2} - H^* I_{1/H^3} + V I_{X/H^3} \end{aligned}$$

(35)

$$\begin{aligned} \tilde{W} = U_m J_{1/H^2} - H^* J_{1/H^3} + V J_{X/H^3} \end{aligned}$$

(36)

$$\begin{aligned} V = \frac{H_0^{(k+1)} - H_0^{(k)}}{ \Delta } \end{aligned}$$

(37)

The pressure gradient is proportional to:

$$\begin{aligned} U_m H - H^* + V X = 0 \end{aligned}$$

(38)

The flow balance equations are as follows:

$$\begin{aligned} \begin{aligned}&\Delta \left( -2 U_m^{(k+1/2)} H_{oil}^{(k+1/2)} + H^{*\,(k)} + H^{*\,(k+1)} \right) \\& \quad- \left( H_0^{(k+1)} - H_0^{(k)} \right) \left( X_a^{(k)} +X_a^{(k+1)} \right) \\&= 2 \left( I_H(X_a^{(k)}) - I_H(X_a^{(k+1)}) - \left( X_a^{(k)} - X_a^{(k+1)}\right) H_{oil}^{(k+1/2)} \right) \end{aligned} \end{aligned}$$

(39)

$$\begin{aligned} \begin{aligned}&\Delta \left( - 2 U_m^{(k+1/2)} \frac{I_H(X_b^{(k+1)}) - I_H(X_b^{(k)})}{X_b^{(k+1)} - X_b^{(k)}} + H^{*\,(k)} + H^{*\,(k+1)} \right) \\& \quad - \left( H_0^{(k+1)} - H_0^{(k)} \right) \left( X_b^{(k)} + X_b^{(k+1)} \right) \\&= 2 \left( I_H(X_b^{(k+1)}) - I_H(X_b^{(k)}) - \left( X_b^{(k+1)} - X_b^{(k)}\right) H_{0}^{(k)} \right. \\&\left. \quad - U_m \Delta \left( H_b^{(k)} - H_0^{(k)} \right) \right) \end{aligned} \end{aligned}$$

(40)

Finally the two additional interesting quantities:

$$\begin{aligned} H_{out} = \frac{H^* - V X_b}{ \tilde{U} } \end{aligned}$$

(41)

$$\begin{aligned} \tilde{F} = \frac{h_r}{ x_r} \left( \frac{2 U_m}{3} I_{1/H} - \frac{H^*}{2} I_{1/H^2} + \frac{V}{2} I_{X/H^2} \right) \end{aligned}$$

(42)

If the geometry is a simple parabola, the height equation is:

$$\begin{aligned} H = H_0 + \frac{x_r^2}{r h_r} \frac{X^2}{2} \end{aligned}$$

(43)