Line Contact Problem Research Articles

Numerical analysis of the Thermal EHL line contact problem under intermittent motion

Abstract A mathematical model of time-varying thermal elastohydrodynamic lubrication (EHL) is developed using a sleeve chain as the object of study. The effects of thermal effect, load, speed, rest time and equivalent radius of curvature on its EHL are investigated using theoretical simulations. The results show that during intermittent motion, a portion of oil is entrapped in the contact zone at the end of the deceleration phase, after which this entrapped oil gradually moves out of the contact zone with the onset of speed, and a lower film of oil at the lubrication inlet passes through the center of the contact zone. In simple sliding intermittent motion, the temperature rise plays a crucial role in the variation of the oil film, particularly during the motion phases, and is an unavoidable factor. An increase in stop time reduces the film thickness and increases the following friction coefficient in the acceleration stage. An increase in the equivalent radius of curvature increases the film thickness and thermal rise in the contact zone as well as decreases the friction coefficient. Since the sleeve chain is one of the most commonly used mechanism in industry, high priority should be given to the lubrication design of this structure.

Newton-GMRES Method for Thermal Elasto-hydrodynamic Lubrication of Line Contact Problems

Newton-GMRES Method for Thermal Elasto-hydrodynamic Lubrication of Line Contact Problems

Newton-GMRES-Method for the Scritinization of electric double layer and surface roughness on EHL line contact problem

The electric double layer phenomenon exists on the solid interface under the water-liquid condition. The water molecules are ionized and adhered in the interface forming the sturn layer is a diffused layer in which molecules can move with the movement of bulk of molecules. Because of these two characteristics, the boundary layer of water molecules is called the electric double layer. The aim of present study is to explore the impact of two fraction surfaces of electric double layer (EDL) on a thin water lubricating film on an elastohydrodynamic lubrication (EHL) line contact problem with a sinusoidal surface roughness. The governing modified Reynolds and film thickness equations are based on mathematical model of electro-viscosity of asymmetrical electrical double layer is analyzed numerically. The viscosity-pressure relation of water and theoretical evaluation pertaining to the effect of electric double layer on film-thickness and pressure distribution of EHL with water film of line contact problem is discussed in detail. The effect of zeta potential on film thickness and pressure is determined using Newton’s-GMRES method with Daubechies D6 wavelet as a pre-conditioner. The results predict that, EDL has less impact on pressure distribution and significant impact on film thickness. The obtained results are compared with results of Dowson and Higginson which are comparable.

A Fast Thermoelastic Model Based on the Half-Space Theory Applied to Elastohydrodynamic Lubrication of Line Contacts Involving Free Boundaries

Abstract This study allows contact models based on semi-analytical methods including the impacts of thermoelastic deformations in contacts of finite dimension bodies. The proposed method controls heat flows crossing free boundaries. A comparison with finite element analysis (FEA) reveals that the proposed method can reduce the calculation times by more than 98%. The paper introduces the thermoelasticity effects into thermal-elastohydrodynamic lubrication (TEHL) modeling of line contact problems. The analysis reveals that including thermoelastic deformations changes the pressure profile and tends to localize the pressure close to the distribution center. Compared to TEHL simulations, the examined configurations caused an overall increase in the maximum pressure by about 9%, an overall film thickness reduction of about 7%, and an overall temperature increase of about 2 K.

FFT-ASSISTED ALGORITHMS FOR 3D LINE-CONTACT PROBLEMS

The line-contact is a particular type of contact with a contact length much greater than its width. Such contact scenarios can be treated in the frame of a two-dimensional plane-strain problem if the contacting surfaces can be considered nominally smooth. However, surface irregularities inherent to any manufacturing technique lead to a discontinuous contact area that differs from the one derived on the basis of the smooth profile assumption. It is therefore tantalizing to pursue the solution of a line-contact problem using an intrinsically three-dimensional (3D) model, which can only be numerical due to lack of general analytical solutions in contact mechanics. Considering the geometry of the line-contact, a major challenge in its numerical modelling is that the expected contact area is orders of magnitude larger in one direction compared to the other. This may lead to an unreasonably large number of grids in the contact length direction, which translates to a prohibitive computational burden. An alternative approach, employed in this paper, is to treat the line-contact as non-periodic in the contact width direction, but periodic in the contact length direction, with a period equal to the window required to capture and replicate the surface specific texture. This periodicity encourages the contact problem solution by spectral methods based on the fast Fourier transform (FFT) algorithm. Based on this idea, two methods are derived in this paper from the existing Discrete Convolution Fast Fourier Transform (DCFFT) technique, which was previously developed for purely non-periodic contact problems. A first algorithm variant employs a special padding technique for pressure, whereas a second one mimics the contribution of multiple pressure periods by summation of the influence coefficients over a domain a few times larger than the target domain. Both techniques are validated against the existing analytical Hertz solution for the line-contact and a good agreement is found. The advanced methods seem well adapted to the simulation of contact problems that can be approximated as periodic in one direction and non-periodic in the other.

A Galerkin/hyper-reduction technique to reduce steady-state elastohydrodynamic line contact problems

A novel model order reduction technique for the elastohydrodynamic problem consisting of the Reynolds equation and the elasticity equation is proposed that drastically reduces the dimensionality of the problem and the computing time. The method is developed for stationary Newtonian isothermal line contacts. It employs the Galerkin projection method for the structural part combined with a hyper-reduction technique for the Reynolds equation. The model order reduction strategy consists of two phases. A first offline phase where snapshots of the full order model solution are computed using a selective refinement technique; and an online phase where the precomputed bases are used to compute the reduced order model. The method leads to a large reduction in the dimensionality of the full order model which results in a considerable reduction of the computing time, in a o(103) speedup factor, with an excellent correlation to the results obtained with the full order model.

Reflections on early studies of elasto-hydrodynamic lubrication

It is almost 50 years since theoretical work on elastohydrodynamic lubrication commenced in the Department of Mechanical Engineering of The University of Leeds. Details of the development of numerical solutions to the line contact problem during the 6-year period (1956–1962) that the authors worked together on the problem will be outlined. The computing aids available during the 18-month period involved in generating the first solution consisted of two hand-operated mechanical calculating machines, with the first digital computer at Leeds being installed in 1959. The general research environment during the period will be recalled and a number of significant events recorded. It is appropriate to record at this Symposium aspect of these initial developments in a subject that has dominated research in tribology throughout the latter part of the 20th century and into the early years of the 21st century. The excitement of being involved in taking some of the first steps in a field described by the late Professor F.T. Barwell (1970) as ‘The major event in the development of lubrication science since Reynolds's own paper’, will be recalled.

Elastohydrodynamic Lubrication Line Contact of a Functionally Graded Material Coated Half-Plane

Abstract The elastohydrodynamic lubrication line contact problem between a functionally graded material (FGM) coated half-plane and a rigid cylindrical punch is investigated. The inhomogeneous elastic properties of the FGM coating are expressed by the exponential model. The lubricant between two solids is supposed to be the Newtonian fluid. The fluid viscosity and density are considered to be dependent on the fluid pressure. To determine the unknown film thickness and fluid pressure at the lubricant contact region, a numerical iterative method is employed to simultaneously solve the flow rheology equation, Reynolds equation, load balance equation, and film thickness equation. Influences of the stiffness ratio of the FGM coating, the resultant normal load, the punch radius, and the entraining velocity on the lubricant film thickness and fluid pressure are analyzed.

A Computational Fluid Dynamics Study on Shearing Mechanisms in Thermal Elastohydrodynamic Line Contacts

A computational fluid dynamics (CFD) model of the thermal elastohydrodynamically lubricated (EHL) line contact problem has been developed for the purpose of exploring the physical processes that occur inside a thin EHL film subjected to shearing motion. The Navier–Stokes equations are solved by using the finite volume method (FVM) in a commercial CFD software, ANSYS Fluent. A set of user-defined functions (UDF) are used for computing viscosity, density, heat source, temperature of moving surfaces and elastic deformation of the top roller according to well-established equations commonly used in the EHL theory. The cavitation problem is solved by taking into account multiphase mixture flow. The model combinations of Houpert and Ree–Eyring and of Tait and Carreau were used for modeling the non-Newtonian behavior of Squalane and the results were compared. Both rheological models suggest the existence of shear-band and plug-flow at high fluid pressure. Due to the differences in viscosity at GPa-level pressure, the chosen model has substantial influence on the computed shear stress and temperature distributions in the high-pressure region. This shows the importance of using correct rheology information in the whole range of pressure, temperature, and shear strain rate.

On the Proportionality Between Area and Load in Line Contacts

The relative contact area of rough surface contacts is known to increase linearly with reduced pressure, with proportionality factor kappa. In its common definition, the reduced pressure contains the root-mean-square gradient (RMSG) of the surface. Although easy to measure, the RMSG of the entire surface does not coincide, at small loads, with the RMSG over the actual contact area {bar{g}}_{text {r}}, which gives a better description of the contact between rough surfaces. It was recently shown that, for Hertzian contacts, linearity between area and load is indeed obtained only if the RMSG is determined over the actual contact area. Similar to surface contacts, in line contacts, numerical data are often studied using theories that predict linearity by design. In this work, we revisit line contact problems and examine whether or not the assumption of linearity for line contacts holds true. We demonstrate, using Green’s function molecular dynamics simulations, that kappa for line contacts is not a constant: It depends on both the reduced pressure and the Hurst exponent. However, linearity holds when the RMSG is measured over the actual contact area. In that case, we could compare kappa for line and surface contacts and found that their ratio is approximately 0.9. Finally, by analytically deriving the proportionality factor using {bar{g}}_{text {r}} in the original model of Greenwood and Williamson, a value is obtained that is surprisingly in good agreement with our numerical results for rough surface contacts.

A novel numerical scheme for the analysis of effects of surface roughness on EHL line contact with couple stress fluid as lubricant

In this article, a Jacobian-free Newton Multigrid (JFNMG) method is used for obtaining the solution of isothermal, steady and compressible elastohydrodynamic lubrication (EHL) line contact problem with surface roughness. The lubricant is a couple stress fluid. A finite difference scheme is used for the solution of EHL equations. The proposed JFNMG method, for the solution of resulting nonlinear system of algebraic equations, comprises nonlinear Newton iterations on the outer loop and linear multigrid iterations on the inner loop. It overcomes the limitations of conventional schemes for the investigation of the problems covering wide range of physical parameters of interest. For increasing values of couple stress parameter, there is an increase in minimum film thickness and considerable decrease (in height as well as spread) of pressure spike. Also, the sensitivity of height and spread of pressure spike as a function of load, couple stress parameter and other parameters are investigated.

A multigrid method for EHL line contact problem with Grease as lubricant

The paper presents an isothermal Elastohydrodynamic lubrication (EHL) of line contact problem with grease as a lubricant, described by Herschel-Bulkley model. The simulation of the governing problem is analyzed using Multigrid method with full approximation scheme (FAS) for varying load, speed and rheological index n Herschel-Bulkley model. The EHL problem considered comprises of Reynolds equation and film thickness understudy for different parameters of interest. The minimum film thickness and pressure spike decrease with a decrease in rheological index n. The present paper traces out the comparison of minimum film thickness with the standard results of Dowson and Higginson. The results obtained are comparable and are presented in terms of graphs and tables.

Jacobian free Newton-GMRES method for the solution of elastohydrodynamic grease lubrication in line contact using wavelet based pre-conditioners

Jacobian Free Newton-GMRES (JFN-GMRES) method is used in the present study for the analysis of isothermal, steady and incompressible elastohydrodynamic lubrication line contact problem. The lubricant is grease, described by Herschel–Bulkley constitutive equation. The problem is investigated for smooth surface as well as surface with asperity. Governing equations are discretized on uniform grid. The resulting system of algebraic equations is solved using JFN-GMRES method for wide range of parameters of interest. For the convergence of restarted GMRES wavelet based pre-conditioners are used. Pressure profiles, film shapes and details of Petrusevich pressure spikes are given for moderate to high load and speed. The proposed scheme is stable and fast converging. Results presented in tables and graphs confirm these findings. The minimum film thickness increases for decrease in n, a parameter characterizing grease lubricant, with other parameters held constant. There is a noticeable decrease (in height as well as spread) of pressure spike for small values of n, which arrests bearings fatigue life.

Wavelet preconditioned Newton-Krylov method for elastohydrodynamic lubrication of line contact problems

In this paper, wavelet preconditioned Newton-Krylov method is used for the solution of elastohydrodynamic lubrication line contact problem. A finite difference method is used for the solution of EHL equations. The method comprise the non-linear Newton iteration on the outer loop and a linear iteration on the inside loop where wavelet based preconditioned GMRES (Generalized Minimum Residual Method of Krylov subspaces) is used. In the scheme, the Jacobian matrix-vector product is approximated accurately with much ease (without forming Jacobian explicitly and requiring no extra storage). It overcomes the limitations of conventional schemes for the investigation of the EHL problems, for computing pressure and film thickness distributions, covering wide range of physical parameters of practical interest. Also, the sensitivity of height and spread of pressure spike on load and other parameters are presented. To confirm and validate the solutions obtained, by the present method, are compared with those obtained by full approximation scheme of multigrid method.

Elrod–Adams cavitation model for a new nonlinear Reynolds equation in piezoviscous hydrodynamic lubrication

In previous studies, different cavitation models have been incorporated into the classical Reynolds equation in piezoviscous regimes. The advantages of the Elrod–Adams cavitation model compared with the Reynolds model have been demonstrated in this classical framework. Recently, a new nonlinear Reynolds equation was rigorously justified [15] for lubricated line contact problems by introducing the piezoviscous Barus law into the departure Navier–Stokes equations before passing to the thin film limit. In addition, the corresponding nonlinear first order ordinary differential equation (ODE) has been proposed.In the present study, we incorporate the Elrod–Adams model for cavitation and we pose the free boundary problem associated with the nonlinear first order ODE, which involves a multivalued Heaviside operator for the relationship between the lubricant pressure and saturation. After analyzing the qualitative properties of the solution, we propose suitable numerical techniques for solving the problem as well as obtaining the lubricant pressure, saturation, and viscosity. Finally, we give some numerical results to illustrate the performance of the proposed numerical methods as well as comparisons with alternative models.

Elastohydrodynamic Lubrication Analysis of Finite Line Contact Problem With Consideration of Two Free End Surfaces

Elastohydrodynamic lubrication (EHL) analysis in finite line contacts is usually modeled by a finite-length roller contacting with a half-space, which ignores effect of the two free boundaries existing in many applications such as gears or roller bearings. This paper presents a semi-analytical method, involving the overlapping method and matrix formation, for EHL analysis in the finite line contact problem to consider the effect of two free end surfaces. Three half-spaces with mirrored loads to be solved are overlapped to cancel out the stresses at expected surfaces, and three matrices can be obtained and reused for the same finite-length space. The isothermal Reynolds equation is solved to obtain the pressure distribution and the fast Fourier transform (FFT) is used to speed up the elastic deformation and stress related calculation. Different line contact situations, including straight rollers, tapered rollers, and Lundberg profile rollers, are discussed to explore the effect of free end surfaces.

Multigrid method for the solution of EHL line contact with bio-based oils as lubricants

Abstract The paper presents Elastohydrodynamic lubrication line contact problem with bio-based oil as lubricants for an isothermal case. The fast convergence method for the solution of Elastohydrodynamic lubrication line contact problem with seed oil as lubricant is analyzed using Multigrid, Multilevel Multi-Integration with the influence of different load and speed. The result shows that the use of these oils has the potential to substitute the function of common lubricant so as to reduce dependence on conventional oil lubricants. The results obtained are comparable and the pressure spikes are smooth as compared to the earlier findings which are shown in terms of graphs and tables.

Influence of lubrication starvation and surface waviness on the oil film stiffness of elastohydrodynamic lubrication line contact

Stiffness properties of interfacial engineering surfaces are of great importance to the dynamic performance of relevant mechanical systems. Normal contact stiffness and oil film stiffness of line contact problems are studied in this work analytically and numerically. The Hertzian contact theory and the Yang–Sun method are applied to predict the contact stiffness, while the empirical elastohydrodynamic lubrication (EHL) film thickness method and the complete numerical EHL model are used to predict the oil film stiffness. The numerical model mainly consists of the Reynolds equation; the film thickness equation, in which the regular surface roughness is taken into consideration; the force balance equation; and the viscosity-pressure equation. The effects of the normal load, rolling speed, regular surface waviness, and starved lubrication level on the oil film stiffness are investigated.

Wavelet Based Numerical Solution of Elasto-hydrodynamic Lubrication Problems via Lifting Scheme

Wavelet theory is a relatively new and an emerging area in mathematical research, particularly wavelets are successfully used in fast algorithms for easy implementation. Elasto-hydrodynamic lubrication model involves coupled system as well as highly nonlinearity, is very difficult to solve numerically by standard methods to obtain better accuracy with low CPU time. The wavelet based schemes are appropriate to get a sufficient accuracy in low CPU time. In this paper, we present the wavelet based numerical solution of elastohydrodynamic lubrication line and point contact problems using lifting scheme. The proposed scheme gives higher accuracy in terms of super convergence in less CPU time, which has been illustrated through the test problems.

Modeling of Finite-Length Line Contact Problem With Consideration of Two Free-End Surfaces

Finite-length line contact conditions, existing in applications such as gears or roller bearings, lead to subsurface stress distribution influenced by the free boundaries. This paper presents a semi-analytical method (SAM) for the finite-length line contact problem, based on the overlapping concept and matrix formation, to consider the effect of two free-end surfaces. In order to obtain two free surfaces, three half-spaces with mirrored loads to be solved are overlapped to cancel out the stresses at expected surfaces. The error introduced by this method is analyzed and proven to be negligible. The conjugate gradient method (CGM) is used to solve the pressure distribution, and the fast Fourier transform (FFT) is used to speed up the elastic deformation and stress-related calculation. The model is verified by finite element method (FEM) and shows a high conformity and efficiency. Besides, the line contact situations are discussed to explore the effect of free surfaces.