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TECHNICAL PAPERS

J. Tribol. 2001;124(1):1-4. doi:10.1115/1.1400996.

This study investigates fluid absorption behavior of an unirradiated and a heavily crosslinked ultra-high molecular weight polyethylene (UHMWPE) exposed to bovine serum with pressures varying from 0.1 to 10 MPa. The time-response of serum absorption is biphasic, with an initial “soak-in” phase (<1 week) of rapidly increasing mass before settling to a steady-state phase of mass increasing more gradually and linearly with time. Initial soak-in absorption of the unirradiated and crosslinked UHMWPE are very similar, increasing from 0.8 to 4.3 g/m2 as serum pressure is increased from 0.1 to 10 MPa. Steady-state absorption rates of the unirradiated UHMWPE increase from 0.8 to 1.5 (mg/m2 )/hr as serum pressure is increased from 0.1 to 10 MPa, while absorption rates for the crosslinked UHMWPE are lower, increasing from approximately 0.28 to 0.85 (mg/m2 )/hr over the same pressure range. Steady-state rate of absorption is also found to depend on temperature, with values at room temperature being two-fold less than at 37°C. Such fluid absorption characterization is critical to hip and knee simulator testing, in order to separate wear loss and serum absorption gain contributions to total mass change measurements of UHMWPE bearings. In addition to its pertinence in quantification of UHMWPE wear, the extent of fluid film lubrication in total joint replacements as implied by reported serum absorption behavior is also discussed.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):5-13. doi:10.1115/1.1398295.

By applying a closed-form analytical solution Hwu and Fan (1998) for an anisotropic half-plane, the contact characteristics of unidirectional continuous fiber-reinforced plastic (FRP) composites are investigated. The particular condition of a rigid parabolic cylinder in normal sliding contact with the composite is evaluated. The influence of FRP composite matrix material, friction coefficient, fiber material, fiber orientation, and fiber volume fraction on the surface contact pressure are determined and evaluated by comparison to published experimental data and results from the finite element method. From the analytical results, several important trends for the contact characteristics of fiber-reinforced plastics are ascertained and discussed with respect to the wear and design-ability of FRP materials.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):14-19. doi:10.1115/1.1395629.

Hard protective coatings produced by the physical vapor deposition (PVD) processes can potentially be used to prevent rolling contact fatigue (RCF) failures in gears and rolling bearings. Although TiN appears to be an attractive material for such applications, the RCF performance of PVD TiN coatings is limited by their columnar microstructure. One possible solution is to use interlayers interrupting the column growth in TiN to achieve a more equiaxed grain morphology. In this paper, the effect of the coating layer structure on the propagation of a three-dimensional crack through the coating thickness is studied theoretically. Numerical simulations of both planar and kinked three-dimensional cracks under cyclic contact loading are performed using a new numerical approach recently developed by the authors, which is based on a combination of the fast Fourier transform (FFT) technique, the eigenstrain theory, and the conjugate gradient method. The simulation results indicate that high-endurance TiN-based coatings can in principle be produced by alternating relatively thick TiN layers with much thinner interlayers made of another material.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):20-26. doi:10.1115/1.1395630.

The finite element model with the implementation of a robust cyclic plasticity theory was used to simulate the elastic-plastic stresses for the partial slip (stick-slip) line rolling contact. Detailed rolling contact stresses and strains were obtained for up to 40 rolling passes. The partial slip condition greatly affects the residual stress in the rolling direction and the residual shear strain within a thin layer of material near the contact surface. The residual stress in the axial direction was not significantly influenced by the partial slip condition. An increase in friction coefficient drives the location of maximum shear strain to the contact surface. In addition, a comparison was made between the finite element results and the results obtained from an approximate method.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):27-35. doi:10.1115/1.1395628.

A numerical model is presented for computing the static friction coefficient of rough surfaces with a soft thin film. In the calculation, an improved model, based on that due to Derjaguin et al., is used in conjunction with an elastic-plastic contact model for contact with a soft coating. The effects of the film thickness and surface roughness on the static friction coefficient and contact are investigated. The numerical results reflect published experimental observations and show the static friction coefficient depends strongly on surface film thickness, external force and surface roughness. The static friction coefficient (μ) increases with the surface film thickness when the plasticity index ψ≥0.5 whilst μ increases with decreasing film thickness in the very thin film regime when ψ=0.25 and F/AnE<10−4. For real rough surfaces, contact and friction behavior is probably heavily influenced by the existence of such soft, thin surface films, which increase the contact area due to plastic deformation of the film and the contact stiffness of the surface in the case of thin film and light load.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):36-45. doi:10.1115/1.1401017.

The knowledge of contact stresses is critical to the design of a tribological element. It is necessary to keep improving contact models and develop efficient numerical methods for contact studies, particularly for the analysis involving coated bodies with rough surfaces. The fast Fourier Transform technique is likely to play an important role in contact analyses. It has been shown that the accuracy in an algorithm with the fast Fourier Transform is closely related to the convolution theorem employed. The algorithm of the discrete convolution and fast Fourier Transform, named the DC-FFT algorithm includes two routes of problem solving: DC-FFT/Influence coefficients/Green’s function for the cases with known Green’s functions and DC-FFT/Influence coefficient/conversion, if frequency response functions are known. This paper explores the method for the accurate conversion for influence coefficients from frequency response functions, further improves the DC-FFT algorithm, and applies this algorithm to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy. A set of general formulas of the frequency response function for the elastic field is derived and verified. Application examples are presented and discussed.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):46-61. doi:10.1115/1.1401018.

A three-dimensional numerical model is presented to investigate the quasi-static sliding contact behavior of layered elastic/plastic solids with rough surfaces. The model is applicable for both single-asperity contact and multiple-asperity contacts. The surface deformation is obtained based on a variational principle. The surface and subsurface stresses in the layer and the substrate are determined with a Fast Fourier transformation (FFT) based scheme and von Mises and principal tensile stresses are computed accordingly. Contact statistics, such as fractional contact area, maximum pressure/E2 and relative meniscus force are predicted. The results are used to investigate the effect of the contact statistics on friction, stiction, and wear problems such as debris generation, brittle failure, and delamination of layered media. Optimum layer parameters are identified. It allows the specification of layer properties, according to the contact statistics, to reduce friction, stiction, and wear of materials. A normalization procedure is presented to apply the results on various combinations of surface roughness, material properties, and normal load.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):62-71. doi:10.1115/1.1398296.

A non-steady problem for a conformal elastohydrodynamically lubricated (EHL) contact of two infinite cylindrical surfaces with parallel axes is considered. It takes into account the elasticity of cylinders, lubricant viscosity, contact surface velocities, and the applied load. The problem is solved based on the “modified” formulation proposed by Kudish et al. (2000) which is free of such defects as discontinuity of its solution and independence of the solution from some of the initial data. The problem is reduced to a system of nonlinear integro-differential equations. The additional conditions include initial and boundary conditions and Newton’s second law applied to the internal cylinder motion. The main emphasis of the paper is three fold: the analysis of the transient dynamics of the system under constant external conditions and due to abrupt changes in applied load, and the system behavior in the case of a bump/dent presence on the shaft surface. The numerical solutions exhibit damped oscillatory behavior while approaching a steady state. It is observed that in a transient motion the radial displacement of the shaft center may vary by no more than 2.5 percent while the maximum pressure may vary by as much as 350 percent. Moreover, the variations of pressure are greater for stiffer materials.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):72-81. doi:10.1115/1.1399054.

Three synthetic oils used in spacecraft mechanisms (Pennzane SHF X2000, Nye 186 A, and Fomblin Z25) have been studied. Rheological tests were performed in order to characterize the behavior of each fluid versus pressure, temperature and shear rate. That includes the determination of the WLF viscosity model parameters. Tribological tests were carried out to measure the traction coefficient for operating conditions representative of spacecraft applications. Experimental results were compared with theoretical ones obtained using a non-newtonian thermal model. The modeling of traction in EHL contact is based on the Johnson and Tevaarwerk’s model modified to account for the shear heating of the fluid. The variations of the lubricant thermal conductivity with respect to temperature and pressure are also considered. In this model, input data comes directly from rheological experiments or are derived from other measurements through physical relationships. These rheological and tribological characterizations are essential to better describe the behavior of a lubricated rolling bearing.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):82-90. doi:10.1115/1.1398289.

This research presents a nonlinear model to analyze the ball bearing vibration due to the waviness in a rigid rotor supported by two or more ball bearings. The waviness of a ball and each races is modeled by the superposition of sinusoidal function, and the position vectors of inner and outer groove radius center are defined with respect to the mass center of the rotor in order to consider five degrees of freedom of a general rotor-bearing system. The waviness of a ball bearing is introduced to these position vectors to use the Hertzian contact theory in order to calculate the elastic deflection and nonlinear contact force resulting from the waviness while the rotor has translational and angular motion. They can be determined by solving the nonlinear equations of motion with five degrees of freedom by using the Runge-Kutta-Fehlberg algorithm. Numerical results of this research are validated with those of prior researchers. The proposed model can calculate the translational displacement as well as the angular displacement of the rotor supported by two or more ball bearings with waviness. It also characterizes the vibration frequencies resulting from the various kinds of waviness in rolling elements, the harmonic frequencies resulting from the nonlinear load-deflection characteristics of ball bearing, and the sideband frequencies resulting from nonlinearity of the waviness interaction.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):91-102. doi:10.1115/1.1398549.

The lubrication with oil-in-water emulsions is usually supposed to be governed by the oil phase being concentrated in a reservoir that supplies the contact. In this paper, another lubrication process in an elastohydrodynamic ball/disc contact is presented. It is found that above a critical entrainment speed a thick film grows in the contact with time and reaches a limiting thickness. Viscous adherent boundary layers are formed and observed on both the surfaces. Pressure and speed are required for the film build-up. An additional sliding speed at constant entrainment speed induces a shearing at the interface between the boundary layers and the substrate that depends on the nature of the contacting surface. It is shown that these high viscosity boundary films ensure a starved lubrication. The modelling of the starvation process allows us to evaluate the viscosity of these boundary layers and their mean supply rate to the contact. A fine analysis of this latter parameter shows that the lubricating film results from the equilibrium between the flow rate of lubricating particles in the inlet zone and the amount of particles that cannot stay in the contact between two passages which strongly depends on the entrainment speed. A classical adsorption process does not seem responsible for the anchorage of the boundary films to the surfaces. The adherence of the films is explained by an approach based on electric interactions between ionic surfactants and the oxidized metallic surfaces according to the position of their isoelectric point compared to the pH of the emulsion.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):103-108. doi:10.1115/1.1398287.

A simple experimental technique is developed that allows the pressures and stresses predicted by EHL analyses to be checked. A soft, rough steel disc is run, under controlled conditions, against a harder, smooth counterface. The pressures generated cause the soft disc to plastically deform. Once deformation ceases the residual profile may be measured and used as input to an EHL solver. The calculated pressures are then used to determine the stress distributions as the rough surface passes through the conjunction. After allowing for the build up of residual stress the maximum von Mises’ stress should be equal to the yield strength of the disc. This provides an accurate, quantitative check on the theoretical values.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):109-113. doi:10.1115/1.1398288.

The inverse approach, described in detail in a companion paper, is applied to two contacts. The first is a line contact with transverse roughness; the second a point contact with an ellipticity ratio of four containing an isolated transverse surface feature. In each case the surface profile was monitored as the operating conditions became more severe. These profiles were used to define the surface in a multi-level EHL solver and the pressures and subsurface stresses calculated. After allowing for the build up of residual stress, the maximum subsurface stress was compared with the yield strength of the rough surface. Good agreement was obtained indicating, first, that EHL theory is accurate for the rolling case examined and, second, that the Hooke-Venner hypothesis of equivalence between roughness effects in line and point contacts is valid.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):114-120. doi:10.1115/1.1401016.

The physical conditions (pressure, shear stress, temperature) generated in an elastohydrodynamic (EHD) contact are the main parameters governing the evolution of the physico-chemical properties of the lubricant within the contact. This paper presents in-situ measurements of pressure in an EHD point contact using Raman microspectroscopy. Two model lubricants are studied: a pure polyphenyl ether (5P4E) oil and a mixture of this oil with a liquid crystal. The influence of load and speed on the pressure distributions is investigated. The Petrusevich pressure spike is measured and its evolution with the operating parameters is in good agreement with the theoretical predictions. The addition of the liquid crystal to the base oil leads to an important diminution of the mixture viscosity, and consequently of the film thickness in the contact. The different rheological properties of the lubricants induce significant variations of the pressure profiles.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):121-128. doi:10.1115/1.1402181.

A new approach to the life assessment for linear bearings using three-parameter Weibull distribution is discussed and theoretical basic dynamic load ratings are calculated for linear motion ball guides (LMBG) and ball bushes. Life test experiments are also carried out for these two types of linear bearings with varying design features for each type. The concept of a minimum life prior to which there is no failure, is introduced in the analysis of the life test data. The effects of introducing ball cages in LMBGs and rounding of the edges in the case of ball bushes on their lives, basic dynamic load ratings, and the λbm factors are examined. It is observed that compared with a two-parameter Weibull distribution, an analysis based on a three-parameter Weibull distribution clearly reveals the differences in the lives of linear bearings for changes in their designs.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):129-136. doi:10.1115/1.1402179.

A mechanical model of cold rolling of foil is coupled with a sophisticated tribological model. The tribological model treats the “mixed” lubrication regime of practical interest, in which there is “real” contact between the roll and strip as well as pressurized oil between the surfaces. The variation of oil film thickness and contact ratio in the bite is found by considering flattening of asperities on the foil and the build-up of hydrodynamic pressure through the bite. The boundary friction coefficient for the contact areas is taken from strip drawing tests under similar tribological conditions. Theoretical results confirm that roll load and forward slip decrease with increasing rolling speed due to the decrease in contact ratio and friction. The predictions of the model are verified using mill trials under industrial conditions. For both thin strip and foil, the load predicted by the model has reasonable agreement with the measurements. For rolling of foil, forward slip is overestimated. This is greatly improved if a variation of friction through the bite is considered.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):137-143. doi:10.1115/1.1398297.

Measured rotordynamic impedances are presented for two hole-pattern-stator seals and one smooth bore seal. These measured results are compared to predictions from a two-control-volume model and realized in the code ISOTSEAL (constant-temperature seal code). The hole-pattern seals have cell depths of 2.03 mm and 3.18 mm with a cell diameter of 1.59 mm. The hole-area density factor for both hole-pattern seals is 43 percent. The seal diameter is 114.71 mm with an L/D ratio of 0.75. Measured results for radial impedances and leakage were obtained. Test conditions involved three speeds out to 20,200 rpm, three inlet pressures out to 17.2 bar, and two exit-to-inlet pressure ratios of 40 percent and 54 percent. As predicted, the hole-pattern seals exhibit frequency-dependent rotordynamic coefficients K(Ω), k(Ω), C(Ω), c(Ω). Results of the tests show that the 3.18 mm hole-pattern seal has the highest average effective stiffness and lowest effective damping. Direct and effective stiffness were under-predicted in all cases; however, measured direct and effective damping are reasonably well predicted. Impedance predictions improve with increasing pressure ratio. Comparisons of leakage correlate extremely well with predictions; worse case deviations never exceed 10 percent. Results show that leakage decreases as cell depth increases. Results also show that the exit temperature increases substantially with increasing rotational speed.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):144-150. doi:10.1115/1.1398294.

A non-axisymmetric steady state numerical model has been developed for use in the design of mechanical seals for down-hole tools. It is applicable to other types of mechanical seals in which friction from the static secondary seal is significant, as well. This model predicts such quantities as leakage rate, external fluid incursion rate, film thickness distribution, fluid pressure and contact pressure distributions, interface temperature and face deformation. Model results show that for a given set of operating conditions, there is a range of equilibrium states of the seal, corresponding to a range of inclinations of the non-rotating seal face relative to the rotating face. Depending on the inclination, it is possible to have negative leakage due to the asymmetry of the film thickness distribution. Even with positive leakage, under some circumstances incursion of the external fluid can occur as a result of the asymmetry.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):151-157. doi:10.1115/1.1401015.

A properly designed mechanical face seal must satisfy two requirements: (1) the seal must be stable, and (2) the seal forced response must be such that the stator tracks the misaligned rotor with the smallest clearance possible, with the smallest relative tilt, and with the largest minimum film thickness. The stability issue was investigated in a previous paper. Here a numerical solution is presented for the transient response of a noncontacting gas lubricated face seal that is subjected to stator and rotor forcing misalignments. The seal dynamic response is obtained in axial and angular modes of motion in a coupled analysis where the Reynolds equation and the equations of motion are solved simultaneously. The steady-state response is first identified for a reference case. Subsequently a parametric study is performed to gauge the influence of the various seal effects, such as speeds, inner to outer radii ratios, face coning heights, pressure drops, support stiffness and damping, and forcing misalignments. The transient responses to static stator misalignment and rotor runout are given, showing that properly designed coned face seals can operate in a stable mode with the stator tracking dynamically a misaligned rotor.

Commentary by Dr. Valentin Fuster
J. Tribol. 2000;124(1):158-165. doi:10.1115/1.1332398.

This paper outlines a five degrees of freedom model of a rotating spindle supported by a pair of lubricated angular contact ball bearings. The ball to raceway contacts are simulated by non-linear contact springs, representing the elastic deformation of the mating rolling members and nonlinear spring/dampers, corresponding to the contact elastohydrodynamic oil film thickness. A regression formula is used to model the latter and includes the damping contributed by the squeeze film effect caused by the mutual convergence of bearing rings. Some results of simulation studies with the model are also presented, in both the time and frequency domains. They include the overall system response, when subjected to varying spindle mass or the number of balls in the support bearings. Furthermore, comparisons are made between the simulated response of the dry and lubricated models. The overall contribution to damping of the elastohydrodynamic oil films between the rolling elements and their raceways is shown to be slight.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):166-177. doi:10.1115/1.1399053.

A numerical model was developed to study the thermal effects on the lubrication mechanism of radially-grooved thrust washers. A mass-conserving transient thermohydrodynamic (THD) analysis was performed by solving the modified Reynolds and energy equations for the lubricant pressure and temperature distributions. The heat transfer equations were also solved simultaneously to obtain the temperature fields of the solids (thrust washers). Due to different thermal time responses of the lubricant film and the solids, heat transfer equations of the runner and the thrust washer pad were treated as in quasi-steady state at each time step in the transient solution. Elrod cavitation algorithm was implemented to include lubricant cavitation. The results show that thermal effects do not only reduce load carrying capacity and the frictional torque but also increase side flow rate. Moreover, the numerical model also demonstrates that the thermal effects have greater influence on the load support when the groove depth and groove numbers increase. Furthermore, the analytical results also show that there exists certain operating conditions before thermal effects become the dominating factors in influencing the thrust washer performance.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):178-185. doi:10.1115/1.1396342.

A transient non-Newtonian TEHD analysis for dynamically loaded journal bearings in mixed lubrication is developed for direct problem. A mass conserving cavitation is included and viscoelasticity subsuming shear thinning is characterized by the upper converted Maxwell fluid and the power law fluid models. The Reynolds equation, the transient two-dimensional energy equation in the film, the quasi-steady-state two-dimensional heat conduction equation in the bushing and heat balance in the journal are solved using finite difference formulations. The thermal and elastic deformation equations are solved using the finite element method. The relation between the average contact pressure and the average gap is numerically determined in micro scale. Effects of roughness texture, degree of asperity contact, shear index, relaxation time, thermal and elastic deformations on bearing behavior are investigated.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):186-195. doi:10.1115/1.1398291.

A numerical model is developed for the dynamic analysis of gas lubricated spiral groove face seals. Effects of the rotor runout, misalignment, face contact, as well as the stiffness and damping of the secondary seal are considered. Seal axial and angular motions and key parameters such as torque, power loss, and flow rate are obtained by a global time integration scheme, which traces the time history of the dynamic sealing behaviors. The analysis of gas film lubrication, face contact, and the seal dynamics is cast as an inverse problem and the solution is obtained iteratively at each time step. Dynamic tracking motion and key sealing characteristics of a representative spiral groove gas seal undergoing transient operations are presented.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):196-202. doi:10.1115/1.1398292.

The gas film stiffness and damping coefficients for a non-contacting gas face seal are obtained from the unsteady nonlinear Reynolds equation using the perturbation method. The seal assembly is converted to an equivalent spring-damper-mass system. The stator tracking motion is treated as a forced vibration caused by the rotor motion due to its runout and misalignment. The seal steady-state dynamic responses are solved semianalytically. Results for a typical spiral groove gas face seal agree well with that from a full numerical simulation. Stability of the seal axial pulsating and conical whirl are examined using the frequency dependent dynamic force and moment coefficients.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):203-211. doi:10.1115/1.1396343.

Principles of a continuously adjustable hydrodynamic bearing are described together with an analysis model for studying its theoretical performance. The model included an expanded form of the governing Reynolds equation which took account of non-uniform variations in the fluid film thickness. A solution procedure was devised whereby for a given set of adjustment conditions, simultaneously converged fields of fluid film thickness, temperature, viscosity and pressure would result, together with oil film forces. A wide range of operating characteristics were studied with results predicting advantages and benefits over conventional hydrodynamic bearings.

Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):212-219. doi:10.1115/1.1400997.

An extended computational bulk-flow analysis for prediction of performance in angled injection, orifice-compensated hydrostatic/hydrodynamic thrust bearings is presented. The fluid motion within the thin film lands is governed by mass conservation and momentum transport equations. Mass flow conservation and a simple model for momentum transport within the hydrostatic bearing recesses are also accounted for. A perturbation analysis for small amplitude shaft axial motions and angulations leads to zeroth and first-order equations describing the equilibrium and perturbed fluid flows. The computational procedure predicts the bearing flow rate, thrust load and restoring moments, drag torque, and 27 force and moment coefficients. The effects of misalignment on the dynamic performance of a refrigerant fluid-hybrid thrust bearing are evaluated at an optimal operating condition. The axial force/displacement stiffness coefficient and the direct moment/angle stiffness coefficients show a maximum for a certain recess pressure ratio, while the damping coefficient steadily increases with the applied load. As the misalignment angle increases, both moment and force coefficients also increase. Most operating conditions show a whirl frequency ratio equal to 0.50. Thus, thrust hybrid bearings offer the same limited stability characteristics as hydrodynamic thrust bearings when undergoing self-excited shaft angular motions.

Commentary by Dr. Valentin Fuster

TECHNICAL NOTES

J. Tribol. 2001;124(1):220-223. doi:10.1115/1.1396345.

Using a dry point contact model, a parametric study of the peak pressure due to a dent was performed. The parameters varied were the dent depth and diameter. The numerical analysis ws complemented by an analytical analysis. Using the elastic body hypothesis, the study of the dented contact shows that various configurations can occur. Introducing the parameters SC1, a dimensionless pressure parameter and SC2, a dimensionless dent geometry parameter, the pressure peak in all these configurations can be represented in a single graph.

Topics: Pressure , Geometry
Commentary by Dr. Valentin Fuster
J. Tribol. 2001;124(1):223-226. doi:10.1115/1.1398545.

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