Review Articles

J. Tribol. 2010;132(2):020801-020801-9. doi:10.1115/1.4001238.

The Cattaneo–Mindlin concept of interfacial slip in tangentially loaded compliant bodies is revisited and its basic simplifying assumptions are critically examined. It is shown that these assumptions, which, in the absence of modern numerical techniques, were essential in 1949 to enable an elegant quantitative solution of the basic problem of presliding between contacting bodies, may be nonphysical. An alternative approach to the same problem that is based on treating sliding inception as a failure mode involving material plastic yield is discussed. This alternative approach was suggested even before 1949 but for the same lack of modern numerical techniques could only be promoted qualitatively. Some recent theoretical models, which are based on this earlier alternative approach, and in which the simplifying assumptions of the Cattaneo–Mindlin concept were completely relaxed, are described along with their experimental verification. It is shown that the presliding problem between contacting bodies can be accurately solved by these models using realistic physical assumptions and failure criterion.

Commentary by Dr. Valentin Fuster

Research Papers: Contact Mechanics

J. Tribol. 2010;132(2):021401-021401-9. doi:10.1115/1.4000693.

In tribological applications, calculating the contact temperature between contacting surfaces makes it possible to estimate lubricant failure and effectiveness, material failure, and other phenomena. The contact temperature can be divided into two scales: the macroscopic and the microscopic scales. In this article, a semi-analytical transient temperature model is presented, which can be used at both scales. The general theory is presented here and used to calculate the contact temperatures of single micro- and macrocontacts. For the steady state situation, the results obtained are in good agreement with those found in literature. Further, it is shown that the simplification of modeling a microcontact as an equivalent square uniform heat source to simplify the calculation of the maximum temperature is justified in the fully plastic regime. The partition is calculated by setting a continuity condition on the temperature field over the contact. From the results, it can be concluded that at low sliding velocities the steady state assumption, which is often used for microcontacts, is correct. However, at higher sliding velocities, the microcontact is not in the steady state and transient calculation methods are advised.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021402-021402-8. doi:10.1115/1.4000733.

This paper presents an efficient numerical method based on quadratic programming, which may be used to analyze fretting contacts in the presence of wear. The approach provides an alternative to a full finite element analysis, and is much less computationally expensive. Results are presented for wear of a Hertzian contact under full sliding and under partial slip. These are compared with previously published finite element analyses of the same problem. Results are also obtained for the fully worn problem by allowing a large number of wear cycles to accumulate. The predicted traction distributions for this case compare well with the fully worn analytical solution presented in part one of this paper.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021403-021403-12. doi:10.1115/1.4001011.

An elastic contact model for three-dimensional layered or coated materials under coupled normal and tangential loads, with consideration of partial slip effects, has been developed in this paper. The response functions for calculating the displacements and stresses were determined in the frequency domain by using the Papkovich–Neuber potentials. The partial slip contact problem was solved by a numerical procedure based on the conjugate Gradient method and fast Fourier transform technique. The contact pressure, surface shear tractions, stick ratios, rigid body displacements, and subsurface stresses are analyzed under different conditions with variations in the material properties and coating thickness. Results show that stiffer coatings tend to decrease the stick ratios and the rigid ball tangential displacements in comparison to those with compliant coatings under the same contact conditions. For stiffer coatings, the values of the von Mises stress and compressive surface stress increase and the positions of maximum von Mises stress move up to the surface; meanwhile, the distributions of the compressive stress become asymmetric due to the action of the tangential load.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021404-021404-14. doi:10.1115/1.4001012.

It has been widely accepted that the microstructure of bearing materials can significantly affect their rolling contact fatigue (RCF) lives. Hence, microlevel topological features of materials will be of significant importance to RCF investigation. In order to estimate the fatigue lives of bearing elements and account for the effects of topological randomness of the bearing materials, in this work, damage mechanics modeling approach is incorporated into a Voronoi finite element method recently developed by the authors. Contrary to most of the life models existing in the literature for estimating the RCF lives, the current model considers microcrack initiation, coalescence, and propagation stages. The proposed model relates the fatigue life to a damage parameter D, which is a measure of the gradual material degradation under cyclic loading. In this investigation, 40 semi-infinite domains with different microstructural distributions are subjected to a moving Hertzian pressure. Using the fatigue damage model developed, the initiation and total lives of the 40 domains are obtained. Also, the effects of initial material flaws and inhomogeneous material properties (in the form of normal distribution of the elastic modulus) on the fatigue lives are investigated. It is observed that the fatigue lives calculated and their Weibull slopes are in good agreement with previous experimental and analytical results. It is noted that introducing inhomogeneous material properties and initial flaws within the domains decreases the fatigue lives and increases their scatters.

Commentary by Dr. Valentin Fuster

Research Papers: Elastohydrodynamic Lubrication

J. Tribol. 2010;132(2):021501-021501-10. doi:10.1115/1.4001104.

This paper presents a study of the spinning influence on film thickness and friction in EHL circular contacts under isothermal and fully flooded conditions. Pressure and film thickness profiles are computed with an original full-system finite element approach. Friction was thereafter investigated with the help of a classical Ree–Eyring model to calculate the longitudinal and transverse shear stresses. An analysis of both the velocity and shear stress distributions at every point of the contact surfaces has allowed explaining the fall of the longitudinal friction coefficient due to the occurrence of opposite shear stresses over the contact area. Moreover in the transverse direction spinning favors large shear stresses of opposite signs, decreasing the fluid viscosity by non-Newtonian effects. These effects have direct and coupled consequences on the friction reduction that is observed in the presence of spinning. They are expected to further decrease friction in real situations due to shear heating.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021502-021502-9. doi:10.1115/1.4001120.

A compliance operator is often utilized to evaluate the elastic displacement of surfaces in the simulation of transient and steady-state elastohydrodynamic lubrication of conformal-contact systems. The values of the compliance operator represent the elastic responses of all nodes when only one node is under a unit load. The accuracy of compliance operator values, computational cost, and storage size are important issues. Our study of steady-state conformal-contact elastohydrodynamic lubrication analyses suggests a method of selective-fine-mesh with selective-storage, as well as a special technique of the combined selective-fine-mesh with selective-storage mapping. These two techniques enable an efficient elasticity procedure for the simulation of steady-state and transient conformal-contact elastohydrodynamic lubrication systems by means of the finite element method.

Commentary by Dr. Valentin Fuster

Research Papers: Friction & Wear

J. Tribol. 2010;132(2):021601-021601-10. doi:10.1115/1.4001449.

A slurry whirling arm erosion test ring was constructed and a series of erosion tests and post-erosion analysis were carried out using a paint erosion indication technique. The pattern of the paint removal presented a highly visual and accelerated map for the erosion process and its behavior. Also, the erosion rate of paint removal was investigated under a number of erosion variables. It was observed that the rebounding of the erodent particles from the sample surface play an important role in developing erosion for this tester. The erosion pattern showed that the effect of the rebound particles depends on the impact velocity and impingement angle. It was also observed that the erosion behavior of paint as a function of impingement angle, impact velocity, and erosion time was similar to that reported in literature for engineering materials. The slurry whirling arm erosion tester seems to be promising for simulating the slurry process in real cases.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021602-021602-5. doi:10.1115/1.4001167.

Erosion wear is a serious problem that constantly accompanies the operation of the system for hydraulic transportation of solid materials. The consequences are the loss of material (steel), loss of work element’s working capacity, great operational expenses, etc. The choice of new materials for working elements, improvements in construction, and optimization of the slurry flow are all various ways of softening the consequences of erosion wear that reflects on the working life of the hydraulic transportation pump’s elements. Data, which would be used for acquiring the above mentioned goals, are mainly acquired through testing in laboratory or semi industrial conditions. It has been proved that numerous influential and time effected values, complex and long term research, etc., make this job an expensive one, and the results are often solely applicable to specific operating conditions. To resolve this problem, the author suggests a combined approach: shorthand experimental researches and mathematical modeling of erosion wear. The suggested model, used for defining the mass loss of the working element affected by solid particles in the slurry, offers the possibility of introducing a greater number of influential values, some of which have a coincidental character. Through the realization of this model, wear intensity data is acquired, which are practically noncoincidental values with a great degree of reliability, and which greatly coincide with the results acquired through measurements conducted on the test facility.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021603-021603-6. doi:10.1115/1.4001170.

Inconel alloys have been used as engineering materials in high temperature and high stress applications due to their excellent mechanical properties. Tribological performances of these alloys, however, have not been conducted extensively. This is because in tribological applications, these materials have not often been utilized in friction and wear-related applications, resulting in a deficiency in the characterization of their tribomechanical properties. In the present research, we investigate the mechanisms of tribological performance of two different Inconel alloys in terms of contact pressures and sliding speeds. We studied their frictional behavior. The wear data were plotted against the pressure×velocity (PV parameter) in order to investigate the changes of surface properties and wear behaviors of the same under the influence of mechanical energy input. It was interesting to find that the wear mechanisms were influenced by the process of tribotesting. There are three competing wear mechanisms found, abrasion, adhesion, and oxidation. Each of those dominates the tribological performance under different conditions.

Topics: Wear , Alloys , Friction
Commentary by Dr. Valentin Fuster

Research Papers: Hydrodynamic Lubrication

J. Tribol. 2010;132(2):021701-021701-9. doi:10.1115/1.4001023.

This paper describes the derivation and numerical solution of a lubrication equation appropriate for high Knudsen number flows and certain types of striated rough surfaces. The derivation begins with the compressible form of the lubrication equation together with the nonlinear series form of the Poiseuille flow reported by Fukui and Kaneko (1990, “A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,” ASME J. Tribol., 112, pp. 78–83.). A multiple-scale analysis is performed on the lubrication equation for a finite-width time-dependent bearing and is limited to either stationary-transverse or longitudinal striated surface roughness of very short length scale. The rough surface averaging that takes place within the multiple-scale analysis includes a fully coupled treatment of the Poiseuille flow. What results is an especially nonlinear lubrication equation with averaged surface roughness effects that is appropriate for high Knudsen number analysis. A rotational transformation is also introduced to provide the roughness averaged lubrication equation in a form that allows analysis of the skewed orientation of a recording head slider with roughness defined relative to the direction of disk motion but with the lubrication equation conveniently expressed in the coordinate system of the slider. A factored-implicit numerical algorithm is described that provides the solution of the roughness averaged lubrication equation. Even though the lubrication equation is highly nonlinear, the numerical scheme is crafted to be fully second-order, time-accurate, and noniterative for tracking the solution in time either to an asymptotic steady-state or in response to a dynamic event. Numerical solutions of several simple geometry bearings are presented that utilize parameters that are typical of the slider-disk interface of current hard disk drives. It is anticipated that the primary benefit of this work may be the ability to accurately and efficiently include the influence of discrete disk data tracks in the air bearing design of very low clearance recording head sliders.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021702-021702-10. doi:10.1115/1.4000721.

The static and dynamic properties of tilting-pad journal bearings with controllable radial oil injection are investigated theoretically. The tilting pads are modeled as flexible structures and their behavior is described using a three-dimensional finite element framework and linear elasticity. The oil film pressure and flow are considered to follow the modified Reynolds equation, which includes the contribution from controllable radial oil injection. The Reynolds equation is solved using a two-dimensional finite element mesh. The rotor is considered to be rigid in terms of shape and size, but lateral movement is permitted. The servovalve flow is governed by a second order ordinary differential equation, where the right hand side is controlled by an electronic input signal. The constitutive flow-pressure relationship of the injection orifices is that of a fully developed laminar velocity profile and the servovalve is introduced into the system of equations by a mass conservation consideration. The Reynolds equation is linearized with respect to displacements and velocities of the nodal degrees of freedom. When all nodal points satisfy static equilibrium, the system of equations is dynamically perturbed and subsequently condensed to a 2×2 system, keeping only the lateral motion of the rotor. As expected, bearing dynamic coefficients are heavily influenced by the control parameters and pad compliance.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021703-021703-11. doi:10.1115/1.4000940.

The objective of the present work is to study theoretically the influence of wear on the performance of four-pocket capillary-compensated hydrostatic journal bearing operating with micropolar lubricant. In the present study, the lubricant containing additives and contaminants is modeled as micropolar fluid. The modified Reynolds equation for micropolar lubricant is solved using finite element method along with capillary restrictor flow equation as a constraint together with appropriate boundary conditions. The performance characteristics of a capillary-compensated four-pocket worn hydrostatic journal bearing operating with micropolar lubricant have been presented for a wide range of values of nondimensional external load, wear depth parameter, and micropolar parameters. The simulated results have also been presented for two different loading arrangements. In arrangement I, the load line acts through centers of the pockets, whereas in arrangement II, the load line bisects the land between two pockets. The simulated results suggest that a bearing lubricated with lubricant having higher micropolar effect has better static and dynamic performance characteristics as compared with Newtonian lubricant but the bearing lubricated with lubricant having higher micropolar effect is predominantly affected by the wear vis a vis static characteristics parameters as compared with Newtonian lubricant for both loading arrangements. However, in the case of stiffness and damping coefficients, loading arrangement II shows a significant higher enhancement in the value of direct stiffness and damping coefficients in z-direction due to micropolar effect as compared with load arrangement I. And also, the effect of wear on stiffness and damping coefficients in z-direction for bearing operating with micropolar lubricant is of same order as Newtonian lubricant for the loading arrangement II. A similar behavior is observed for the case of stiffness and damping coefficients in x-direction for loading arrangement I.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021704-021704-12. doi:10.1115/1.4001014.

A new thermohydrodynamic analysis model for bump air foil bearings with a detailed thermal model of bump foil structures and rotor is presented. In the developed model, temperatures of lubricating air film, top foil, bump foils, bearing sleeve, and rotor are calculated simultaneously through an iterative process. Reynolds equation and 3D energy equation were applied to the air film, and energy equations were applied to all the other structures around the bearing. Energy and momentum equations were applied to cooling channels to predict spatial temperature distribution along the cooling channels. The thermal growth of the rotor, foil structure, bearing sleeve, and centrifugal growth of the rotor are also considered. For the accuracy of the model, effective heat transfer resistance between the top foil and bearing sleeve was measured for various conditions and implemented into the thermal analysis around the cooling channels. The model was also bench marked with published experimental results for verifications. Using a developed model, parametric studies were performed with different bearing nominal clearances, applied loads, rotating speeds, and cooling conditions through channels.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021705-021705-7. doi:10.1115/1.4000694.

This paper deals with the application of the lattice-Boltzmann method (LBM) to fluid-film lubrication. Compared with the traditional computational approach in lubrication (based on Reynolds equation), LBM does not neglect inertia forces. The implementation of LBM is less demanding than that of the Navier–Stokes solvers for complex geometric configurations. Various wall boundary conditions, as well as the multiple relaxation time model, are discussed. Bearing cavitation is approached in a simplified manner. The LBM solutions for two classic configurations are compared with the corresponding analytic and numeric solutions of the Reynolds or Navier–Stokes equations. The LBM results were satisfactory for the investigated cases.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021706-021706-11. doi:10.1115/1.4001169.

A complete analytical model of bump-type foil bearings taking into consideration the effects of four factors, i.e., the elasticity of bump foil, the interaction forces between bumps, the friction forces at the contact surfaces, and the local deflection of top foil, is presented in this investigation. Each bump is simplified to two rigid links and a horizontally spaced spring, the stiffness of which is determined from Castigliano’s theorem. The interaction forces and the friction forces are coupled with the flexibility of bumps through the horizontal elementary spring. The local deflection of the top foil is described using a finite-element shell model and added to the film thickness to predict the air pressure with Reynolds’ equation. The bump deflections of a strip with ten bumps calculated using the presented model under different load distributions are consistent with the published results. Moreover, the predicted bearing load and film thickness obtained from a foil bearing with a bump circumferential extend of 360 deg also agree very well with the experimental data, especially for predictions with a proper selection of radial clearance (preload of foil structure) and friction coefficients. In addition, the radial clearance and friction force variations in the foil bearing are noted to significantly change the performance of the foil bearing. The predictions demonstrate that the radial clearance of the foil bearing has an optimum value for the maximum load capacity.

Commentary by Dr. Valentin Fuster

Research Papers: Magnetic Storage

J. Tribol. 2010;132(2):021901-021901-10. doi:10.1115/1.4000734.

The critical conditions (critical stress and critical temperature) for the demagnetization of perpendicular magnetic recording disks were investigated. A tribo-demagnetization test of a perpendicular magnetic recording disk with a low load ball-on-disk system and the scan of the disk with the magnetic head were sequentially carried out to evaluate the critical force and sliding velocity for the occurrence of demagnetization, and the relationship between the two critical factors. Then, a finite element model using thermomechanical coupling was developed to calculate the critical stress and temperature of the magnetic disk based on the critical force and sliding velocity of the experiment result. Finally, a method based on the tribo-demagnetization test in combination with finite element analysis to calculate the critical conditions for the demagnetization of the perpendicular magnetic recording disk under sliding contact was proposed.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):021902-021902-11. doi:10.1115/1.4000848.

Scratch-related magnetic signal degradation can occur during magnetic storage hard disk drive operations when the read-write heads contact the spinning multilayer disks. To investigate this phenomenon, controlled nanoscratch experiments were performed on perpendicular magnetic recording media using various indenters of different radii of curvature. Various loading conditions were used to cause permanent scratches that were measured using atomic force microscopy. The nanoscratch experiments were simulated using finite element analysis (FEA) that included the detailed nanometer scale thin-film multilayer mechanical properties. The permanently deformed field in the subsurface magnetic recording layer was extracted from the FEA results. The residual scratch widths measured on the surface of the magnetic storage disk were directly compared with the residual subsurface widths of the region on the magnetic recording layer, where extensive permanent lateral deformation was present. It was found that the subsurface widths of the deformed regions were significantly larger than the surface scratch widths. Thus, subsurface thin-film layers, such as the magnetic recording layer, could be damaged without observable damage to the protective top surface carbon overcoat. The exact location and extent of damage to the magnetic recording layer depends on the scratch load, size of scratch tip, and the friction at the interface. Such permanent deformation in magnetic recording layer could lead to demagnetization, which has been reported in the literature.

Commentary by Dr. Valentin Fuster

Research Papers: Micro-Nano Tribology

J. Tribol. 2010;132(2):022001-022001-6. doi:10.1115/1.4001105.

Over the past few years, the importance of nanoscale technology in industries, such as data storage, micro-electro-mechanical systems (MEMs), and conventional sliding and rolling element bearings, has increased significantly. This is due to increased performance criteria and emerging technologies at smaller scales. One way to increase tribological performance of such applications is through nanoscale surface texturing. These textures will allow for precise control of the performance of lubricated surfaces with very thin films. This work examines how the behavior of the lubricant changes as the geometry of the texture is decreased toward the nanoscale. This work uses existing scale dependent lubrication theories to model the hydrodynamic lubrication of textured surfaces in attempt to predict how nanoscale textures will perform. The theoretical results show that the scale effects of a lubricant between textured surfaces can decrease the load carrying capacity while also decreasing the friction force. Overall, the friction force decreases more than the load carrying capacity and so the effective friction coefficient is decreased. It should be noted that relative to larger scale textured surfaces, the load support can also decrease with the decreasing scale of the texture.

Commentary by Dr. Valentin Fuster

Research Papers: Other (Seals, Manufacturing)

J. Tribol. 2010;132(2):022201-022201-7. doi:10.1115/1.3204774.

A numerical study of a labyrinth-type turbine seal flutter in a large turbofan engine is described. The flutter analysis was conducted using a coupled fluid-structure interaction code, which was originally developed for turbomachinery blade applications. The flow model is based on an unstructured, implicit Reynolds-averaged Navier–Stokes solver. The solver is coupled to a modal model for the structure obtained from a standard structural finite element code. During the aeroelasticity computations, the aerodynamic grid is moved at each time step to follow the structural motion, which is due to unsteady aerodynamic forces applied onto the structure by the fluid. Such an integrated time-domain approach allows the direct computation of aeroelastic time histories from which the aerodynamic damping, and hence, the flutter stability, can be determined. Two different configurations of a large-diameter aeroengine labyrinth seal were studied. The first configuration is the initial design with four fins, which exhibited flutter instability during testing. The second configuration is a modified design with three fins and a stiffened ring. The steady-state flow was computed for both configurations, and good agreement was reached with available reference data. An aeroelasticity analysis was conducted next for both configurations, and the model was able to predict the observed flutter behavior in both cases. A flutter mechanism is proposed, based on the matching of the structural frequencies to the frequencies of waves traveling in the fluid, in the interfin cavities and in the high- and low-pressure cavities.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):022202-022202-7. doi:10.1115/1.4000838.

To clarify the sealing characteristics of metal gasket seals, leakage rates of gas and the real contact area of the seal surfaces were measured under several closing loads. The static seal consisted of a ring-shaped copper gasket and the two steel flanges that held the gasket in place. Gasket widths in the radial direction were 2 mm, 3 mm, and 5 mm. The contact surfaces of the flanges were finished by lathe turning. To determine the leakage flow paths between the gasket and the flanges, the real contact situation between them was observed using a thin polymer film 1μm in thickness. The results indicated the leakage flow paths on the gasket surface were the radial direction perpendicular to the lathe-turned groove and the circumferential direction along the groove. As the closing loads increased, the leakage flow in the radial direction ceased and only that in the circumferential direction remained. Therefore, the cross-section of the aperture for the leakage flow in the circumferential direction was evaluated from the measured real contact area, and the leakage rates were estimated by assumption of laminar flow. The results agreed well with the measured leakage rates.

Commentary by Dr. Valentin Fuster

Technical Briefs

J. Tribol. 2010;132(2):024501-024501-3. doi:10.1115/1.4000735.

To analyze the contact between two spherical bodies with different radii of curvature, the three-dimensional (3D) Hertz theory for elliptical contact is typically used. When the two contacting bodies have high conformity, such as the case for ball-in-groove, the Hertz theory may break down. In this research, finite element analysis (FEA) was used to assess the validity of 3D Hertz theory as found in the roller-housing contact of constant velocity joints. The contact area, normal approach, and contact pressure results show that Hertz agrees with FEA predictions for low compressive loads, where the contact ellipse is within the geometrical contact dimensions. At higher loads the contact ellipse extends beyond the contacting geometrical dimensions and the simplified analytical Hertz results diverge from the FEA results.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):024502-024502-4. doi:10.1115/1.4001024.

Flying clearance distribution with thermal flying height control (or thermomechanical actuation) is characterized. Especially, factors contributing to variation in the flying clearance are identified based on the flying height change profiles taken from the burn-in process of hard disk drives and Gage R&R (repeatability and reproducibility) test of touch down repeatability. In addition, the effect of static temperature compensation scheme on the flying clearance distribution is investigated, and the disadvantage of static adaptation to temperature change is identified. In order to avoid early catastrophic head-disk interface failures due to poor static temperature compensation, dynamic clearance adjustment is necessary whenever environmental condition changes. Otherwise, static temperature compensation using the individual temperature sensitivity values of each head needs to be applied.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):024503-024503-7. doi:10.1115/1.4000941.

This paper presents an analytical model for the basic design calculations of plain journal bearings. The model yields reasonable accuracy as compared with published numerical solutions under the same conditions. The principles and procedures of the formulations are presented along with accuracy analyses.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):024504-024504-8. doi:10.1115/1.4001013.

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.

Commentary by Dr. Valentin Fuster
J. Tribol. 2010;132(2):024505-024505-4. doi:10.1115/1.4001195.

The average flow model proposed by Patir and Cheng offers a great convenience for the analysis of rough surfaces in lubrication. The contact factor introduced by Wu and Zheng helps to solve a difficulty in local film evaluation using the average flow model. This paper reports a simple method to calculate the contact factor. Method validation is demonstrated by the comparison of the contact factors for Gaussian surfaces obtained with the present method and the fitting formula of Wu and Zheng. The proposed method cannot only easily compute the contact factor values for Gaussian surfaces; it can also be used for those of non-Gaussian and measured surfaces, especially those with unknown probability density distribution of the roughness height.

Commentary by Dr. Valentin Fuster

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