Analysis of the thermal elastohydrodynamic lubrication of a finite line contact

A complete numerical solution for the thermal elastohydrodynamic lubrication (EHL) problem of a finite line contact between an infinite plane and an axially profiled cylindrical roller is obtained. The pressure profile and the film shape in the middle part of the roller are very different from those at the ends. However, the finite line contact results in the middle part of the roller are nearly the same as those of the corresponding infinite line contact. It is revealed that, in a finite line contact, the maximum pressure, the highest film temperature and the minimum film thickness all occur at the end regions of the roller. A comparison is made between isothermal and thermal solutions of a finite line contact. In addition, variations in the minimum film thickness, the maximum pressure and the highest temperature with the dimensionless velocity parameter, the slide/roll ratio and the radius of the end profiles of the roller are discussed.     

Edge contact effect on thermal elastohydrodynamic lubrication of finite contact lines

Minimum lubricant film thickness and maximum pressure every so often appear close to roller ends. This study combines the Boussinesq–Cerruti half-space equations with a free boundary correction procedure for precise modeling of edge contact conditions. The thermal EHL model developed associates this representation to a standard finite difference of the energy equation, and to a modified finite difference expansion of the Couette term of the Reynolds equation. To complete the model, the Carreau expression describes the shear-thinning response of the lubricant. The investigation includes different roller profile corrections. The results show that a large radius crowning modification combined with a rounding of the corners constitutes the most effective profile adjustment.     

On the thermal elastohydrodynamic lubrication in opposite sliding circular contacts

Cases of elastohydrodynamic lubrication (EHL) point contacts running under opposite sliding conditions have been studied with consideration of the thermal effect for various loads and entrainment velocities. A thermal EHL solver has been developed and proven to be able to deal even with extreme cases under an infinite slide–roll ratio. Results show that film profiles featuring a dimple can be formed in the contact zone when the slide–roll ratio exceeds a certain level. Moreover, the present study provides theoretical evidence for the lubricating film build up in the case where the two bounding surfaces run with equal but opposite velocities. An effective lubricating film under zero entrainment speeds was experimentally proven by Dyson and Wilson [1] (Proc Instn Mech Engrs, 1968–1969 183(3P) 81) in the 1960's, which, however, cannot be explained by the isothermal EHL theory.     

Thermal elastohydrodynamic lubrication of starved elliptical contacts

For the starved thermal elastohydrodynamic lubrication in elliptical contacts, a numerical solver based on the multi-grid technique was developed and numerical solutions were achieved. The influences of the thickness of oil-supply layer, elliptical ratio, entrainment velocity, slide-roll ratio, and the maximum Hertzian pressure on the lubrication behaviour were investigated. The thermal solutions were compared with isothermal solutions. The numerical results show that, for an oil-starved contact, the central and minimum film thicknesses predicted by both thermal and isothermal solutions change at the same trend as the thickness of the oil-supply layer increases. That is, as the thickness of the oil-supply layer increases, in the beginning both the central and minimum film thicknesses increase rapidly, however, their increasing rates become smaller gradually, at last, when the thickness of the oil-supply layer reaches a certain value, both the central and minimum film thicknesses almost stop increasing. Such situation was called a quasi-fully flooded state in this study. It has been found that the amount of the supplied oil for the quasi-fully flooded state differs with the elliptical ratio, and the maximum amount of the supplied oil is required by a circular contact.     

Inverse approach for calculating pressure and viscosity in elastohydrodynamic lubrication of line contacts

This paper proposes an inverse approach which generates a smoothed pressure distribution based upon a small number of measuring points of the film thickness and overcomes the problems of pressure fluctuations. The Reynolds equation of line contacts is then employed to determine the value of the pressure–viscosity index. To investigate the sensitivity of the results to measurement errors, different errors are deliberately implemented in the numerical calculations. Results show that the traditional direct inverse method requires many measuring points of film thickness to establish the amplitude and location of the pressure spike, but the inverse approach can obtain accurate results using just 29 measuring points. When errors in the film thickness measurements, it is found that the inverse approach still yields a reasonably smooth curve for the pressure profile. When film thickness measurement errors are excluded, the proposed approach is very close to the exact value of the pressure–viscosity index. Even when errors in the film thickness are deliberately introduced, the inverse approach still provides a satisfactory value of the pressure–viscosity index. The resulting apparent viscosity errors are much smaller than those generated when using the direct method. The implemented errors in load and effective elastic modulus have a significant effect on the accuracy of the results, but that the influence of errors in average velocity and in the viscosity at ambient pressure is insignificant. In these implemented errors, the resolution of the film thickness measurement plays the most important role in determining the accuracy of the apparent viscosity.     

Analysis of rough line contacts operating under mixed elastohydrodynamic lubrication conditions

Heavily loaded machine elements, such as gears, usually operate in the mixed lubrication regime. Surface roughness has a significant effect on the pressure distribution, the subsurface stress field, and the friction coefficient. Based on the superposition of a dry rough and a fully flooded smooth contact, a mixed lubrication model has been developed. The roughness profile is assumed to be known. Surface deformation is calculated by taking into account the pressure distribution that is built up by asperity contacts, asperity interactions, and lubricant flow. Thermal and sliding effects are incorporated into the analysis. Non‐Newtonian lubricant behaviour is considered by using a power‐law rheological model. The pressure distribution, subsurface stress field, and friction coefficient were calculated from the model at several points along the contact path for an FZG type C gear pair. It was shown that a significant part of the load is carried by the contacting asperities. The position of the maximum shear stress is very close to the surface.     

On the shear stress of elastohydrodynamic lubrication in elliptical contacts

Results of mathematical modelling of elastohydrodynamic lubrication of rolling contacts are presented. Effects of dimensionless parameters such as speed, normal load, elliptical parameters and coefficient of limiting shear stress on shear stress distributions have been studied. Moreover, profiles on hydrodynamic pressure and film thickness in EHD contacts have been studied. It has been found that shear stress profiles on two contact surfaces in entraining direction are similar with each other in some way. Shear stresses of fluid film on contact surfaces vary with many factors, which reveals the mechanism of traction in elastohydrodynamically lubricated contacts.     

A deterministic model for line-contact partial elastohydrodynamic lubrication

A deterministic model for partial elastohydrodynamic lubrication (EHL) is presented in this paper. The modelling methodology adopts some of the concepts used in the stochastic modelling of partial EHL and some of the procedures for deterministic calculation of asperity pressures. The model is shown to be capable of simulating the basic process of asperity interaction and solid-to-solid contact within an EHL conjunction of rough surfaces. Deterministic results of transient partial EHL in line contacts are obtained when one pair or multiple pairs of asperities collide. The model may help to gain a fundamental understanding of the transient behaviour of asperity interactions in lubricated concentrated contacts of rough surfaces. Asperity pressures may be calculated more accurately than the conventional analyses under dry and static contact conditions. The work represents a first attempt in deterministic modelling of tribo-contacts operating in the mixed regime of micro-EHL and boundary lubrication. Future work will aim at developing more realistic models incorporating factors such as three-dimensional asperity contacts, asperity plastic deformation, thermal effects and the effect of tribo-chemistry.     

基于弹流知识的球轴承设计

基于点接触弹流知识与赤兹接触理论,对球轴承设计参数沟道半径系数和滚动体直径与节圆直径的比进行了分析,结果表明,沟道半径系数对微型轴承其取值范围较大;对小的沟道半径系数,滚动体直径与节圆直径的比的大小对形成油膜的能力影响较小,并且最大接触压力随沟道半径系数的变化大,而随滚动体直径与节圆直径的比的变化不大。     

Coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication

The coupled effects of surface roughness and flow rheology on elastohydrodynamic lubrication (EHL) circular contact problems are analyzed and discussed. The averaged type Reynolds equation utilizing the average flow model on the interactions between couple stress fluids and surface roughness, the elastic deformation equation, the viscosity–pressure and density–pressure relations equations, and the force balance equation are solved numerically by the multilevel multi-integration (MLMI) algorithm to calculate the pressure distributions and film thickness shapes. The results show that the transverse type roughness and standard deviation of composite roughness enhance the pressure and film thickness in the central contact region. Moreover, the longer the characteristic length of the couple stress fluids is, the smaller the pressure distribution is in the central contact region and the greater the film thickness is in all regions.     

Effect of real longitudinal surface roughness on lubrication film formation within line elastohydrodynamic contact

In many applications (e.g. roller, barrel or needle bearings) surface features exhibit longitudinal alignment to the direction of motion. These features are produced by surface finishing techniques in the circumferential direction and are associated with line or very wide elliptical contact geometries. In such a case, the contact length in the direction of motion is considerably shorter comparing its width and the effect of a longitudinal roughness could significantly influence the lubrication film formation. Recent experimental studies have indicated less severe effect of a longitudinal roughness on lubrication film formation in the comparisons with that observed with transversely orientated roughness caused by the inlet perturbation. Nevertheless, these experimental studies have been focused on the behaviour of artificially produced asperities within a circular contact. The quantitative experimental study of longitudinal real surface roughness within a line contact has not been realized yet. That is why, in this study, the line contact formed between a steel tapered roller and glass disc is observed within an optical test rig and the effects of real surface roughness on lubrication film formation are studied. Experiments carried out under pure rolling conditions have shown that the depth is the key parameter that influences the effect on the film thickness. If the roughness features are shallow, the lubrication film shape within the contact follows the shape of the surface closely. However, the groove having only about 800 nm in depth divided the line contact into two parts that behave as two separate line contacts. Such an effect can increase the risk of the wear of rubbing surfaces as the lubrication film thickness between the real machine components can be significantly lower than expected.     

线接触弹流理论研究的三个方面

本文详细论述了线接触等温全膜弹流、部分膜弹流及热弹流的数值计算方法特点和发展概况；介绍了不同时期提出的各种线接触等温弹流和热弹流膜厚计算公式及其适用范围；列举了各种非牛顿流体模型和粘度—压力—温度关系等流变学关系式并评述了这些关系式的特点及应用场合。关键词     

Transient elastohydrodynamic lubrication film thickness in sliding and rolling line contacts

The contact behavior between cam and follower is greatly influenced by the kinematics and dynamics of the whole valve train system. This is the reason that both shape and thickness of the fluid film in the contact gap are mainly determined by applied loads and relative contact speeds as well as the curvatures of contacting elements. Most of the studies about lubricant film behavior between cam and follower have been performed without a consideration of transient effects in the contact gap. For the computational difficulties of transient effects, most contact conditions such as relative contacting speeds have been regarded as quasi-steady state during the whole operating cycle. In this work, in order to obtain stable convergence, a multigrid multi-level method is used for the computation of load capacity in the lubricant film. Nonlinear valve spring dynamics are also considered in the same way as Hanachi’s. From the computational results, transient EHL film thicknesses under the conditions of different contact geometries are computed for a pushrod type valve train system during an engine cycle. Several results show the squeeze film effect, which is generally not found with conventional EHL computations of the cam and follower contact. The results are also compared with those by the Dowson-Hamrock (D-H) formula, which does not consider the dynamic film effect. Without the dynamic film effect as in D-H’s formula, the minimum film thickness is highly dependent on the entraining lubricant velocity, whereas the minimum film thickness including the squeeze film effect is dependent on the applied load.     

Pure rolling elastohydrodynamic lubrication of short stroke reciprocating motion

The behavior of point contact elastohydrodynamic lubrication (EHL) films under pure rolling short stroke reciprocating motion is investigated using both optical interferometry technique and theoretical analysis. The EHL films are recorded with a high-speed color camera and simulated with multigrid techniques. General variation of oil film under point contact reciprocating motion is explained by a comparison between a fully flooded simulation and an experiment with a frequency of 7.78 Hz. Influence of starvation is considered by another simulation under a higher working frequency of 14.4 Hz and the simulated results are compared with experimental results. Both simulations show good agreements quantitatively with experiments. In addition, general tribo-characteristics of EHL under fully flooded and starved conditions are discussed.     

Monochromatic image analysis of elastohydrodynamic lubrication film thickness by fringe intensity computation

Point contact film thickness in elastohydrodynamic lubrication (EHL) is analyzed by image processing method for the images from an optical interferometer with monochromatic incident light. Interference between the reflected lights both on half mirrorCr coating of glass disk and on super finished ball makes circular fringes depending on the contact conditions such as sliding velocity, applied load, viscosity-pressure characteristics and viscosity of lubricant under ambient pressure. In this situation the film thickness is regarded as the difference of optical paths between those reflected lights, which make dark and bright fringes with monochromatic incident light. The film thickness is computed by numbering the dark and bright fringe orders and the intensity (gray scale image) in each fringe regime is mapped to the corresponding film thickness. In this work, we developed a measuring technique for EHL film thickness by dividing the image patterns into two typical types under the condition of monochromatic incident light. During the image processing, the captured image is converted into digitally formatted data over the contact area without any loss of the image information of interferogram and it is also interpreted with consistency regardless of the observer’ s experimental experience. It is expected that the developed image processing method will provide a valuable basis to develop the image processing technique for color fringes, which is generally used for the measurement of relatively thin films in higher resolution.     

Numerical simulation of elastohydrodynamic lubrication for an elliptical contact

An elastohydrodynamic problem on elliptical contact is formulated and numerically solved. The mathematical model is described by a system of integro-differential equations and inequalities with boundary conditions. The direction of the rolling velocity vector with respect to the axes of the ellipse has a marked influence on the distribution of pressure and the lubricating film thickness in the contact area. The elastohydrodynamic contact parameters are determined depending on the compression force applied thereto. These results are useful for analyzing processes occurring in lubricated concentrated contacts.     

A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication problems

A fast compound direct iterative algorithm for solving transient line contact elastohydrodynamic lubrication (EHL) problems is presented. First, by introducing a special matrix splitting iteration method into the traditional compound direct iterative method, the full matrices for the linear systems of equations are transformed into sparse banded ones with any half-bandwidth; then, an extended Thomas method which can solve banded linear systems with any half-bandwidth is derived to accelerate the computing speed. Through the above two steps, the computational complexity of each iteration is reduced approximately from O(N ³/3) to O(β ² _N), where _N is the total number of nodes, and β is the half-bandwidth. Two kinds of numerical results of transient EHL line contact problems under sinusoidal excitation or pure normal approach process are obtained. The results demonstrate that the new algorithm increases computing speed several times more than the traditional compound direct iterative method with the same numerical precision. Also the results show that the new algorithm can get the best computing speed and robustness when the ratio, half-bandwidth to total number of nodes, is about 7.5%–10.0% in moderate load cases.     

Numerical evaluation of pressure from experimentally measured film thickness in EHL point contact

This paper is concerned with elastohydrodynamic lubrication, especially the determination of lubricant film thickness and contact pressure within a point contact of friction surfaces of machine parts. A new solution technique for numerical determination of contact pressure is introduced. The direct measurement of contact pressure is very difficult. Hence, input data of lubricant film thickness obtained from the experiment based on colorimetric interferometry are used for the calculation of pressure using the inverse elasticity theory. The algorithm is enhanced by convolution in order to increase calculation speed. The approach described in this contribution gives reliable results on smooth contact and in the future, it will be extended to enable the study of contact of friction surfaces with asperities. Copyright © 2007 John Wiley & Sons, Ltd.     

Effects of surface coating on elastohydrodynamic lubrication of line contacts

A numerical solution to the elastohydrodynamic lubrication (EHL) problem of two coated elastic bodies in line contact is introduced. The non‐Newtonian behavior of the lubricant is incorporated into EHL analysis using Eyring's nonlinear viscous model. The surface elastic deformations are computed from full elasticity analysis of layered elastic half‐space. The iterative Newton–Raphson technique is used in the numerical solution. Subsurface stresses are calculated using two different techniques, the numerical integral scheme based on Fourier transformation and the finite element method. The effects of surface coating (material and thickness) on the pressure profile and subsurface stresses are presented and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Base fluid parameters for elastohydrodynamic lubrication and friction calculations and their influence on lubrication capability

In this study, base fluid parameters for elastohydrodynamic lubrication (EHL) and friction analyses have been determined experimentally. The viscosity at atmospheric pressure, η₀, the pressure‐viscosity index, α, and the EHL friction coefficient, γ, are important parameters in EHL theory and they are crucial in the selection of efficient lubricants for different applications. This investigation focuses on three important lubrication mechanisms: the capability of forming a separating lubricant film, the friction generated in a lubricated contact, and the height of pressure peaks, such as the outlet pressure spike and pressure ripple caused by surface roughness. The influence of different lubricant parameters on these three mechanisms is discussed. The value of α is measured in a Couette high‐pressure viscometer, and the value of γ is obtained from a jumping‐ball device. Other parameters discussed are temperature‐viscosity coefficient, β, bulk modulus, B₀, thermal conductivity at atmospheric pressure, λ₀, and heat capacity unit volume, ρc_p0. A comparison between traditional mineral base oils and environmentally adapted oil based on rapeseed oil and synthetic esters contributes to the further understanding of the performance of these new materials in lubrication applications. It is shown that rapeseed oil and synthetic esters have good lubricating properties and are, in most cases, better than mineral oils.