Hydrodynamic Lubrication Problems Research Articles

Prediction of thrust bearing’s performance in Mixed Lubrication regime

A hydrodynamic thrust bearing could be forced to operate in mixed lubrication regime under various circumstances. At this state, the tribological characteristics of the bearing could be affected significantly and the developed phenomena would have a severe impact on the performance of the mechanism. Until recently, researchers were modeling the hydrodynamic lubrication problem of the thrust bearings either with analytical or with numerical solutions. The analytical solutions are very simple and do not provide enough accuracy in describing the actual problem. To add to that, following only computational methodologies, can lead to time consuming and complex algorithms that need to be repeated every time the operating conditions change, in order to draw safe conclusions. Recent technological advances, especially on the field of computer science, have provided tools that enhance and accelerate the modeling of thrust bearings’ operation. The aim of this study is to examine the application of Artificial Neural Networks as Machine Learning models, that are trained to predict the coefficient of friction for lubricated pad thrust bearings in mixed lubrication regime. The hydrodynamic analysis of the thrust bearing is performed by solving the Average 2-D Reynolds equation numerically. In order to describe the roughness of the profiles, both the flow factors suggested by N. Patir and H.S. Cheng (1978) and the model of J.A. Greenwood and J. H. Tripp (1970) are taken into consideration. Three lubricants, the SAE 0W30, the SAE 10W40 and the SAE 10W60, are tested and compared for a variety of operating velocities and applied coatings. The numerical analysis results are used as training datasets for the machine learning algorithms. Four different ML methods are applied in this investigation: Artificial Neural Networks (ANNs), Multi- Variable Quadratic Polynomial Regression, Quadratic SVM and Regression Trees. The coefficient of determination, R2 is calculated and used to determine the most accurate ML method for the current study. The results showed that ANNs provide very good accuracy in the prediction of friction coefficient compared to the rest of the ML models discussed.

Modified Finite Difference Methods for Reynold Equation With Film Thickness Discontinuity

Abstract In hydrodynamic lubrication problems, the presence of step structures on the surface can cause discontinuities in the film thickness. This article proposes two models for solving the two-dimensional Reynolds equation with film thickness discontinuity using the finite difference method (FDM). In model I, the film thickness variable is defined at the center of the mesh grids, allowing the Reynolds equation to be reformulated in a weak form that eliminates the singularity of film thickness discontinuity and satisfies the flow continuity condition at the film thickness discontinuity region. By considering the step boundary on the surface as the interface, model II is constructed based on the immersed interface method, turning the hydrodynamic lubrication problem into a classical interface problem. The jump conditions across the interface are derived in accordance with the continuous flow requirement. A phase-field function is adopted to describe the interface on the uniform rectangular mesh grids. Numerical experiments are conducted to assess the accuracy and capabilities of the two proposed models for analyzing a step-dimple-textured sealing. The results demonstrate that both modified FDM models can effectively address the thickness discontinuity issue. Model II achieves second-order accuracy for the pressure distribution when dealing with curved interfaces based on Cartesian grids, whereas model I demonstrates first-order accuracy. Both the proposed models exhibit superior accuracy compared to the traditional second-order central FDM when dealing with curved interfaces. Moreover, the performance of model II is further assessed by simulating lubrication problems with complex groove shapes, and the results indicate its flexibility in addressing thickness discontinuity problems with complex curve interface.

Effect of Chamfer Form and Parameters on the Characteristics of Finite Length Journal Bearing under Impact Load

Journal bearings in typical applications are subjected to misalignment due to several causes, such as shaft deformation under load and errors related to the installation and manufacturing processes. Misalignment has well-known severe negative consequences on the performance of the bearings. This paper deals with the bearing chamfer to reduce these consequences of misalignment, and two forms of bearing edge modification are considered in the analysis. These forms are linear and curved chamfering of the bearing edges, where the height of the chamfer in the circumferential direction and the length of the modification in the longitudinal direction are considered as geometrical design parameters. The investigation includes a numerical solution of the hydrodynamic lubrication problem of finite length journal bearing, considering 3D misalignment cases using the finite difference method. This includes the assessment of the chamfer forms and their effects on the bearing performance in terms of the main bearing design parameters. Furthermore, the stability of the chamfered bearings is also investigated under impact load. Results showed that both chamfer forms are beneficial for a certain limit of the design parameters in reducing the maximum pressure and coefficient of friction and in elevating the film thickness levels, extending the range of misalignment in which the journal bearing can operate safely. In addition, the chamfered bearings in both forms showed more stability range in terms of the critical speed and shaft center trajectories under impact load. The bearings with the curved chamfer, where the slope is continuous at the start of modification, showed more uniform film thickness levels, and their shaft center trajectories were closer to the perfectly aligned bearing in the stable operating range of the system.

A mortar-based cavitation formulation using NURBS-based isogeometric analysis

This paper applies NURBS-based isogeometric analysis to hydrodynamic lubrication problems considering cavitation effects. For this purpose, a mass-conserving cavitation formulation analog to the Augmented Lagrangian method is developed. It describes the complementarity constraints between pressure and void fraction and converts the constrained problem to an unconstrained one. This formulation is discretized using a mortar-based approach to avoid oscillations resulting from over-constrained issues. Additionally, both NURBS basis functions and Lagrange polynomials are employed to discretize the fluid domain for comparison. Several numerical examples indicate that the proposed cavitation formulation can provide more accurate and robust solutions when combined with the mortar method and NURBS basis functions. Furthermore, this combination is adopted to investigate systems with rough surfaces and gaps partially filled with lubricants to demonstrate its excellent performance.

FELINE: Finite element solver for hydrodynamic lubrication problems using the inexact Newton method

In this work we present FELINE, a C++ solver of the Reynolds equation for treating hydrodynamic lubrication problems. To correctly describe cavitation regions, FELINE implements the inexact Newton iteration (INE) algorithm within a finite element method (FEM) framework. The solver was tested and validated against known cases in literature and industrially relevant cases of dimpled textures. Furthermore, we provide a benchmark for a complex dimpled texture case to evaluate the performance and robustness of the implementation. FELINE performs very fast when compared with existing implementations and shows a great degree of stability, while providing physically correct solutions thanks to the INE algorithm. Program summaryProgram title: FELINECPC Library link to program files:https://doi.org/10.17632/srkpysb727.1Licensing provisions: Creative Commons by 4.0Programming language: C++Nature of problem: A fast Reynolds equation solver robust enough to correctly solve the linear complementarity problems that arise from proper treatment of the cavitation boundaries.Solution method: An inexact Newton iteration method implemented within a finite element method framework that properly ensures the complementarity conditions are satisfied by introducing a forcing term.

A Hydrodynamic Thrust Bearing Lubricated By A Non-Newtonian Giesekus Fluid

There exists a huge volume of studies of hydrodynamic and elastohydrodynamic lubrication problems for lubricants with Newtonian rheology. Lubricants with Newtonian rheology do not exhibit the usually observed experimentally behavior of having relatively high viscosity for low stresses and relatively low viscosity for high stresses. In this paper we extend the earlier conducted analysis of lubricants with a non-Newtonian Giesekus behavior for the case of thrust bearing modeling. The main goal of the paper is to obtain an analytical solution for a hydrodynamically thrust bearing lubricated by a fluid with the Giesekus rheology. This goal is achieved by careful application of perturbation methods. A three-term approximate analytical solution is obtained and its dependence on the problem input parameters is analyzed.

On the MAOR method for a class of hydrodynamic lubrication problems

In this paper, for solving horizontal linear complementarity problems arising from finite-difference discretizations of hydrodynamic lubrication, the modulus-based accelerated overrelaxation iteration method (MAOR) is analyzed. The convergence range of the relaxation parameters is given, shown to be the generalization of the case of successive overrelaxation. Numerical examples show the efficiency of the MAOR method with the proposed theoretical results.

ЧИСЛЕННЫЙ АНАЛИЗ ГИДРОДИНАМИЧЕСКОЙ СМАЗКИ В ПАРЕ ТРЕНИЯ ИЗ ТВЕРДЫХ МАТЕРИАЛОВ ТОТАЛЬНОГО ЭНДОПРОТЕЗА ТАЗОБЕДРЕННОГО СУСТАВА

A numerical analysis of the hydrodynamic lubrication problem in a friction pair of a hip joint replacement with surfaces made of materials with a high modulus of elasticity, such as metal-metal or ceramic-ceramic, under steady-state walking conditions is presented. The Reynolds equation in spherical coordinates has been solved for the movements of the film with a three-dimensional change in the load and speed experienced in a friction pair of the hip joint during walking. It has been shown that a finite lubricating film thickness can be formed during the walking cycle due to the film squeezing. It has also been found that the design parameters of the spherical sliding pair, such as the radius of the femoral head and the radial clearance between the femoral head and the acetabular element, have a large influence on the predicted film thickness. It was concluded that there is a high likelihood that a continuous hydrodynamic lubricating film may be formed in hip replacement with a ceramic-to-ceramic pair and, possibly, for implants with some well-machined metal-to-metal pairs with a relatively small radial clearance that provides completely fluid friction.

An active‐set mixed finite element solver for a transient hydrodynamic lubrication problem in the presence of cavitation

AbstractIn this paper, we study a moving free boundary problem related to the cavitation modeling in lubricated devices. More precisely, a characteristics method combined with a weak formulation in a mixed form is introduced for the Elrod‐Adams model. The formulation is suitable for the use of mixed finite element methods in the numerical approximation. It is proved that the time‐discrete problem and its finite element discretization have unique solutions. Further an efficient primal‐dual active‐set strategy is proposed to solve the resulting algebraic system. The performance of the overall algorithm is illustrated by numerical examples.

Transient Analysis of the Textured Journal Bearing Operating With the Piezoviscous and Shear-Thinning Fluids

In this paper, a mixed lubrication model is presented to analyze the tribological behavior of the textured journal bearings operating from mixed to hydrodynamic lubrication regimes. In particular, the effects of fluid piezoviscosity and the non-Newtonian fluid behavior are also considered. The presented model solves the hydrodynamic lubrication problem by a mass-conserving formation of the Reynolds equation, whereas the metal–metal contact is considered by using the Greenwood and Tripp (GT) contact model which is linked with the hydrodynamic model based on the concept of Johnson's load sharing. As a result, the performance of the textured journal bearing system under different lubrication regimes, including boundary lubrication regime, mixed hydrodynamic lubrication regime, and hydrodynamic lubrication regime, can be evaluated. Using the journal bearing systems operated under the start-up condition as examples, prediction demonstrates the influences of texture distributions on friction and wear. It is found that the friction reducing effect induced by texturing is influenced by the distribution of the texturing zones. In particular, the hydrodynamic friction can be reduced when the eccentricity ratio is changed from high to low. Moreover, it appears that the shear-thinning effect of lubricant cannot be neglected in the transient analysis of journal bearing system.

Stochastic multiscale analysis in hydrodynamic lubrication

SummaryA stochastic multiscale analysis framework is developed for hydrodynamic lubrication problems with random surface roughness. The approach is based on a multi‐resolution computational strategy wherein the deterministic solution of the multiscale problem for each random surface realization is achieved through a coarse‐scale analysis with a local upscaling that is achieved through homogenization theory. The stochastic nature of this solution because of the underlying randomness is then characterized through local and global quantities of interest, accompanied by a detailed discussion regarding suitable choices of the numerical parameters in order to achieve a desired stochastic predictive capability while ensuring numerical efficiency. Finally, models of the stochastic interface response are constructed, and their performance is demonstrated for representative problem settings. Overall, the developed approach offers a computational framework, which can essentially predict the significant influence of interface heterogeneity in the absence of a strict scale separation. Copyright © 2017 John Wiley & Sons, Ltd.

An efficient reduced-order method with PGD for solving journal bearing hydrodynamic lubrication problems

In the present work, a reduced-order method, “Proper Generalized Decomposition (PGD)” is extended and applied to the resolution of the Reynolds equation describing the behavior of the lubricant in hydrodynamic journal bearing. The PGD model is employed to solve the characteristic ‘Reynolds’ partial differential equation using the separation technique through the alternating direction strategy. The resulting separated-dimension system has a low computation cost compared to classical finite-difference resolution. Several numerical benchmark examples are investigated to verify the validity and accuracy of the proposed method. It has been found that numerical results obtained by the PGD method can achieve an improved convergence rate with a very low computation cost.

Development and validation of a gaseous cavitation model for hydrodynamic lubrication

The prediction of cavitation in lubricants is of significance to the design and analysis of hydrodynamic bearings. In this paper, a numerical model for the gaseous cavitation in hydrodynamic lubrication problems is derived based on the cavitation mechanism and origination, which is different from the conventional cavitation conditions. The proposed gaseous cavitation model is then used to calculate the performance of several types of bearings, including a two-axially grooved plain journal bearing, a misaligned journal bearing and parallel thrust bearings with surface textures. The results are validated with experimental data and numerical results calculated by the Half-Sommerfeld, Reynolds, and JFO conditions. The advantage of this new model lies in its accuracy and its independence from the cavitation pressure, which is usually difficult to be accurately defined. Thus this gaseous cavitation model can be useful in the accurate prediction of the cavitation phenomenon and performance of hydrodynamic bearings under steady state.

Application of the fluid–structure interaction technique for the analysis of hydrodynamic lubrication problems

Fluid–structure interaction technique seems to be one of the most promising possibilities for theoretical analysis of lubrication problems. It allows coupling of different physical fields in one computational task, taking into account the interaction between them. In this article, two sets of fluid–structure interaction analyses focusing on the bearing performance evaluation are presented. One analysis was applied to a water-lubricated journal bearing and the other to a hydrodynamic thrust bearing lubricated with oil. Steady-state operation was considered in both cases. In the presented cases of fluid–structure interaction analyses, all important phenomena accompanying bearing operation are considered, e.g. lubricant flow, structure movements and their deformations as well as heat transfer in case of thrust bearing. The problems encountered during modelling are discussed in this article, as well as the results of calculations: hydrodynamic pressures, gap geometries or temperature profiles.

A new more consistent Reynolds model for piezoviscous hydrodynamic lubrication problems in line contact devices

Hydrodynamic lubrication problems in piezoviscous regime are usually modeled by the classical Reynolds equation combined with a suitable law for the pressure dependence of viscosity. For the case of pressure–viscosity dependence in the Stokes equation, a new Reynolds equation in the thin film limit has been proposed by Rajagopal and Szeri. However, these authors consider some additional simplifications. In the present work, avoiding these simplifications and starting from a Stokes equation with pressure dependence of viscosity through Barus law, a new Reynolds model for line contact lubrication problems is deduced, in which the cavitation phenomenon is also taken into account. Thus, the new complete model consists of a nonlinear free boundary problem associated to the proposed new Reynolds equation.Moreover, the classical model, the one proposed by Rajagopal and Szeri and the here proposed one are simulated through the development of some numerical algorithms involving finite elements method, projected relaxation techniques, duality type numerical strategies and fixed point iteration techniques. Finally, several numerical tests are performed to carry out a comparative analysis among the different models.

Analytical Solution to the Hydrodynamic Lubrication of Fan-Shaped Thrust Step Bearings

Step bearings are frequently used in industries for better load capacities. Analytical solutions to the Rayleigh step bearing and a rectangular slider with a finite width are available in literature, but none for a fan-shaped thrust step bearing. This study starts with a known solution to the Laplace equation in a cylindrical coordinate system, which is in the form of an infinite summation. A set of analytical solutions to pressure, load capacity, flow rate, and torque loss is derived in this paper for hydrodynamic lubrication problems encountered in the fan-shaped step bearing. These analytical solutions are compared with those for the rectangular slider and the Rayleigh step bearing to reveal relationships among them. When the inner radius becomes smaller, the load capacity increases, almost linearly in a certain region. The effects of inner radius, step height, and step location on pressure distribution and load capacity are studied in general and under a specific set of bearing geometry as an example. The presented solutions can be useful for designers to maximize bearing performance as well as for researchers to benchmark numerical lubrication models.

Thermo-hydrodynamic study of the journal bearing under static load

A number of studies have been carried out on the hydrodynamic lubrication of journal bearings. This article seeks to summarize this work, presenting it in the form of a flow chart. The treatment of rupture and reformation of the film fluid in the contact region, for a steady loaded circular journal bearing, are accomplished using the Reynolds and the Jakobsson—Floberg and Olsson (JFO) boundary conditions. A flow chart taking into account the resolution of some of the hydrodynamic lubrication problems is elaborated. A computer code is carried out for solving the Reynolds equation and for calculating the most important parameters relating to a hydrodynamic contact. The obtained results for a finite-length bearing are compared with those obtained from the short-bearing hypothesis. In addition, a global thermal study is carried out in order to evaluate the effect of temperature.

The homogenization process of the Reynolds equation describing compressible liquid flow

This paper summarizes the homogenization process of rough, hydrodynamic lubrication problems governed by the Reynolds equation used to describe compressible liquid flow. Here, the homogenized equation describes the limiting result when the wavelength of a modeled surface roughness goes to zero. The lubricant film thickness is modeled by one part describing the geometry/shape of the bearing and a periodic part describing the surface topography/roughness. By varying the periodic part as well as its wavelength, we can try to systematically investigate the applicability of homogenization on this type of problem. The load carrying capacity is the target parameter; deterministic solutions are compared to homogenized by this measure. We show that the load carrying capacity rapidly converges to the homogenized results as the wavelength decreases, proving that the homogenized solution gives a very accurate representation of the problem when real surface topographies are considered.

A Study on Squeeze Films Between Porous Rubber Surface and Rigid Surface: Analysis Based on the Viscoelastic Continuum Model

The squeeze-film effect between a porous rubber surface and a rigid surface has been investigated as a problem of viscoelastic hydrodynamic lubrication (VEHL). An axisymmetric continuum model is introduced to estimate the surface deformation of a cylindrical porous rubber block based on a three-element viscoelastic model. The coefficients in the viscoelastic constitutive equation are experimentally determined and then the deformation of the porous rubber block is calculated by the FEM technique. The approximate solutions for the fluid pressures in the squeeze film are obtained from the equation of motion, which includes both local and convective inertia terms, and the continuity equation, using the perturbation method. Experimental data on the squeeze-film pressures and forces are also obtained under the sinusoidal squeezing motion of a rigid plate facing a cylindrical porous rubber block. It is found that the difference between the VEHL solutions and the EHL (elastohydrodynamic lubrication) solutions becomes more marked for a higher frequency of squeeze oscillation due to the effect of viscoelasticity of the porous rubber. The VEHL solutions hence correspond better to the measured results and yield a smaller surface deformation of the porous rubber block than the EHL solutions as the frequency of squeeze oscillation increases. The results also show that the fluid inertia effect due to the flow in the squeeze film becomes noticeable for a higher frequency of the squeezing motion as the effect of viscoelasticity of the porous rubber.

Turbulence models of hydrodynamic lubrication

The main theoretical turbulence models for application to hydrodynamic lubrication problems were briefly reviewed, and the course of their development and their fundamentals were explained. Predictions by these models on flow fields in turbulent Couette flows and shear-induced countercurrent flows were compared to existing measurements, and Zhang & Zhang’s combined κ-ɛ model was shown to have surpassingly satisfactory results. The method of application of this combined κ-ɛ model to high speed journal bearings and annular seals was summarized, and the predicted results were shown to be satisfactory by comparisons with existing experiments of journal bearings and annular seals.