Elastohydrodynamic lubrication of line contacts

A numerical solution to the problem of thermal elastotydrodynamic lubrication of line contacts was obtained by using a finite-difference formulation. The solution procedure consists of simultaneous solution of the thermal Reynolds equation, the elasticity equation, and the energy equation subject to appropriate boundary conditions. Pressure distribution, film shape, and temperature distribution were obtained for fully flooded conjunctions, a paraffinic lubricant, and various dimensionless speed parameters while the dimensionless load and materials parameters were held constant. Reduction in the minimum film thickness due to thermal effects (as a ratio of thermal to isothermal minimum film thickness) is given by a simple formula as a function of the thermal loading parameter, Q: Plots of pressure distribution, films shape, temperature distribution, and flow are shown for some representative cases.

In this article, the problem of predicting the film thickness in so-called narrow or slender elliptic contacts is revisited. In particular, the variation of the minimum film and central film thickness and their ratio with load and ellipticity are studied. It is shown that the minimum film tends to decrease linearly with increasing ellipticity. The central film thickness, until a certain threshold, decreases very slowly, and then decreases linearly too, with a slope independent of the Hertzian pressure. The ratio central to minimum film thickness is not a constant and varies strongly with ellipticity and load conditions. In this article, it is shown that the ratio of central to minimum film thickness is a linear function of a single nondimensional parameter which has the physical meaning of the ratio of the length of the inlet pressure sweep to the relative width of the contact. Results for narrow contacts published in the engineering literature when represented as a function of this parameter exhibit the same trend. The behaviour is the consequence of the same unifying mechanism that, in the last decade, was discovered to determine aspects of behaviour in elasto-hydrodynamically lubricated (EHL) contacts under time varying conditions, such as surface waviness deformation. For these problems, the single parameter dependence facilitated the development of simple engineering tools (e.g. to predict the roughness deformation). The observation in this article that the ratio central to minimum film thickness is also governed by this parameter opens new perspectives to develop a simple computational engineering tool to predict central and minimum film thickness in EHL contacts.

Recently developed colorimetric interferometry technique was used for the study of both minimum and central film thicknesses for a wide range of operating parameters. Over 300 film thickness maps were obtained for the combination of four values of the materials parameter G, five values of the load parameter W and many values of the speed parameter U. The use of a spacer layer extended the range of film thickness measurement down to 5 nm. An excellent agreement was found between experimental values and data obtained from numerical solution presented by Venner and ten Napel, especially for thin lubrication films. An increase in a speed exponent with increasing material parameter G was observed for both central and minimum film thicknesses. The minimum film thickness and, thereby, the ratio between central and minimum film thickness was confirmed to be of a stronger dependence on material and load dimensionless parameters than Hamrock and Dowson equations predict. Presented at the 54th Annual Meeting Las Vegas, Nevada May 23–27, 1999

The rollers and/or races in cylindrical and tapered roller bearings should be profiled to relieve high edge stress concentrations caused by their finite lengths and misalignment. In this study, a numerical analysis was performed to investigate the elastohydrodynamic lubrication (EHL) of a Lundberg profile-type cylindrical roller. A finite difference method with fully nonuniform grids and the Newton-Raphson method were used to present detailed EHL pressure distributions and film shapes, as well as the variations in the minimum and central film thicknesses with the profile modification coefficient. In the Lundberg profile, the maximum pressure and minimum film thickness always occurred near the edges. Proper modification of the Lundberg profile considerably increased the minimum film thickness.

Numerical analysis is performed to study the transient behavior of EHL finite line contact of a cylindrical roller and flat plane under load impulse. In the present work, effect of pressure on the density and viscosity of lubricant is considered. Finite difference method is used to discretize the governing equations. Multilevel Multi-integration method is used to calculate the elastic deformation. Moreover, Multigrid method is implemented to accelerate the convergence process. Uniqueness of this finite line contact analysis is that it provides an ability to determine the transient behavior of lubricated contact even at the edges of roller. Results show that the load impulse causes squeezing and separation movement within the contact that develops film dimple and pressure ripples at the inlet region, which propagate towards the exit region due to the entrainment motion. It is noticed that the time taken by oil film [Formula: see text] to travel the Hertzian contact width and the time period [Formula: see text] of load impulse decides the behavior of lubricated contacts. Firstly, under a relatively heavy load when the contact width is large enough so that [Formula: see text], then a significant rise in central film thickness (CFT), central minimum film thickness (CMFT) and minimum film thickness (MFT) occurs after the execution of load impulse. Further, under the light load generating a relatively small contact width such that [Formula: see text], then comparatively a small rise in CFT occurs right during the load impulse. Lastly, for a given load if the time period of impulse [Formula: see text] is large enough satisfying the condition [Formula: see text], then a considerable reduction in CFT, CMFT and MFT takes place during the application of load impulse. Moreover, as compare to other cases, for [Formula: see text] the steady state condition is reestablished after a relatively more number of time cycles. It is observed that the maximum pressure and MFT occurs at the contact edges of roller which can be controlled by a proper choice of the radius of end profile [Formula: see text].

A numerical solution of an isothermal elastohydrodynamic contact, between two infinitely long waved surface cylinders, is presented. For given material properties of both cylinders, lubricants with pressure dependent viscosity, for known speeds and elastohydrodynamic load, for waviness surface parameters (wavelength and amplitude), solutions for the pressure distribution and film, thickness are obtained by simultaneous solution of elasticity equation and Reynolds equation. No side leakage is postulated. The effect of surface waviness on pressure distribution and central and minimum film thickness is studied.

In this research, a rigid ball impacting on an elastic half-space with lubricant in between is modeled by using the transient modified Reynolds equation, which considers the coupled effects of Navier-slip and flow rheology, the ball motion equation, and the deformation equations. In the impact-plasto-elastohydrodynamic lubrication (PEHL) point contact problems, the deformation mechanism consists of the elastic deformation based on linear elasticity, and the plastic deformation based on the flow rule and hardening model, as the von Mises stress is greater than the yield strength. The elastic-perfectly plastic (EPP) model is used first for basic studies, and the linear hardening model is used for further discussion. The impact-elastohydrodynamic lubrication (impact-EHL) model overestimates the central pressure, impact load, and minimum film thickness, while underestimating central film thickness and the position of minimum film thickness occurrence as compared to the present results from the impact-plasto-elastohydrodynamic lubrication (PEHL) model. As the slip length increases or the flow index decreases, the central film thickness decreases. For shear-thickening fluid (n > 1), the effects of slip weaken the shear-thickening effects, and make the central film thickness closer to that of a Newtonian fluid without slip. The effects of slip on the pressure distributions seem insignificant due to the force balance, and thus the central pressure and the elastic/plastic deformation. Moreover, the strain-hardening effect shows the inhomogeneous yield strength after one impact.

Experimental study of central and minimum film thickness in elastohydrodynamic elliptic contacts

The Newton-Raphson method was applied to solve the thermal EHD lubrication model of line contacts. By accounting for thermal effects in the Newton-Raphson scheme, a very stable numerical approach was obtained. Two models with viscosity constant and variable across the oil film were developed. The results under extremely heavy conditions of dimensionless load W = 52 * 10−5 (pH = 2 GPa) and dimensionless rolling velocity U = 20 * 10−11 are presented. They show that even for pure rolling, but under heavy load and high rolling velocity conditions, the thermal effects significantly reduce the minimum film thickness. The distributions of pressure, film thickness, and temperature for two rolling velocities and various loads are presented. They indicate that under high rolling velocity conditions the thermal effects have a strong influence on a pressure spike.

Utilizing the theory developed by the authors in an earlier publication, the influence of the ellipticity parameter, the dimensionless speed, load, and material parameters on minimum film thickness was investigated. The ellipticity parameter was varied from one (a ball on a plate configuration) to eight (a configuration approaching a line contact). The dimensionless speed parameter was varied over a range of nearly two orders of magnitude. The dimensionless load parameter was varied over a range of one order of magnitude. Conditions corresponding to the use of solid materials of bronze, steel, and silicon nitride and lubricants of paraffinic and naphthenic mineral oils were considered in obtaining the exponent in the dimensionless material parameter. Thirty-four different cases were used in obtaining the minimum film thickness formula given below as H¯min=3.63U0.68G0.49W−0.073(1−e−0.68k) A simplified expression for the ellipticity parameter was found where k=1.03RyRx0.64 Contour plots were also shown which indicate in detail the pressure spike and two side lobes in which the minimum film thickness occurs. These theoretical solutions of film thickness have all the essential features of the previously reported experimental observations based upon optical interferometry.

Minimum, central and average film thicknesses have been calculated for the isothermal E.H.L. point contact case, for a variety of load, rolling speed, and material parameters. The equations governing this problem were solved using a Multigrid method. This technique offers the possibility to work with a very fine grid, obtaining detailed and accurate solutions, at the cost of moderate cpu times and storage requirements, on medium size computers. Computations for low loads, requiring a large inlet zone, have been carried out using local grid refinements. The fluid in these calculations is assumed to be compressible and its viscosity-pressure behavior is described by either the Roelands equation, or the Barus equation. The ratio between the calculated minimum film thickness and the central value varied with the parameters governing the problem, but for low loads, a value of 3/4 was obtained. The film thickness behavior at these low loads can be accurately described in terms of the minimum film thickness. For higher loads, however, a description based on a film thickness, averaged over the Hertzian contact, is more appropriate to be compared with the asymptotic solution (Ertel, Grubin).

Transient thermal elastohydrodynamic lubrication (EHL) line contact simulations are carried out to study the traction behavior during accelerated motion considering realistic shear-thinning behavior. Using three lubricants with different inlet viscosity and shear-thinning parameters, the application of present analysis for lubricant selection is demonstrated. Owing to squeeze film action, the film evolution is delayed, and EHL traction during acceleration is found to increase much above the designed value. This effect decreases with increasing starting speed. The most shear-thinning test oil considered here yields the lowest traction coefficient with minimum variation in its value desirable for smooth and vibration-free operation.

The lubrication performance of rib/roller end contacts of cylindrical roller bearings was studied both theoretically and experimentally for end-crowned rollers and inclined ribs. A partial EHL program was used to calculate the film thickness and the friction in the rib/roller end contact. Calculated minimum film thickness shows a strong load dependence although the central film thickness is still a weak function of the load. The influence of the contact position on the film thickness was also investigated for roller skewness and design tolerance. It was found that the contact location affects the minimum film thickness strongly in spite of a weak influence on the central film thickness. Friction and scuffing experiments were conducted on a special rig, which can achieve arbitrary slide/roll ratio to simulate the rib/roller end contacts. Good agreements were found between measured and calculated friction based on the assumption that the lubricant was Newtonian. In scuffing experiments, scuffing propagation was observed on the rib contact surface. The critical load for scuffing is strongly correlated with the sliding velocity. The critical load at high speeds is lower than that at low speeds although the friction at high speeds is lower. These data imply the importance of the contact location and the wear process for film breakdown.

Recent numerical results have indicated that the relationship between the film thickness and the speed may not always follow the simple power law, especially under severe conditions. This paper is aimed at obtaining experimental results at high contact stresses and low speeds to study the thin film behavior. Ultrathin lubricant films were observed at maximum Hertz pressures of 0.52, 1.01, and 1.54 GPa by using an optical test rig. Central and minimum film thickness values were obtained with thin film colorimetric interferometry from chromatic interferograms. The nonlinear behavior of both central and minimum film thicknesses in log-log coordinates was observed as rolling speed and thereby film thickness decreased. This tendency became more obvious at higher contact pressures.

Analysis of the thermal elastohydrodynamic lubrication of a finite line contact