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TECHNICAL PAPERS

J. Tribol. 2002;124(3):429-437. doi:10.1115/1.1396341.

A model of a thin annular plate sliding against an elastic foundation was developed and used to study thermoelastic instability (TEI) in clutches. The analysis examines the stability of the quasi-steady state solution of the governing equations by considering non-axisymmetric perturbations. The results indicate that above critical values of temperature and sliding speed the response of the plate becomes unstable and exhibits large deformations. Two mechanisms account for this behavior: thermal buckling and bending. It is shown that a conservative approximation of the stability boundaries can be constructed by computing only two points on the stability curve. The boundary between stable and unstable behavior depends on the material properties, geometry, and boundary conditions. The model was used to conduct a parametric study which indicates that stability of the sliding system can be improved by reducing the sliding speed, decreasing the modulus of elasticity of the plate, increasing the thermal conductivity, or increasing the thickness. In addition, for a range of sliding speeds, increasing the stiffness of the friction material improves the stability of the system. For speeds outside this range, increasing the stiffness makes the system less stable.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):438-442. doi:10.1115/1.1467633.

A methodology is presented for the optimization of a tribological contact configuration, namely, a multi-layered elastic structure under normal (frictionless) point contact loading. This work is aimed at developing an algorithm by which the finite element (FE) mesh and the corresponding structure may be generated automatically for each variation in the vector of design variables during optimization iterations. The FE model for contact analysis may be developed in a given commercial solver such as ABAQUS or ANSYS. To do this, a flexible mesh generator, which interfaces with the FE model and the optimizer, was developed. The optimization scheme is implemented using a simulated annealing (SA) algorithm as the optimizer, with an axisymmetric (point contact) FE indentation model in the commercial finite element solver ABAQUS. The results suggest that conventional optimization methods may be employed to examine the design of tribological contact configurations such as multi-layered structures, working seamlessly within the operating system shell (e.g., Unix), and the finite element solver.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):443-447. doi:10.1115/1.1454105.

In ElastoHydrodynamic Lubrication (EHL), transient processes are much more common than stationary ones. Predicting the film thickness under steady state conditions has become straight forward. Using numerical methods, the effect of transient conditions on the film thickness profile can be computed. However, those analyses are very time consuming even using advanced numerical techniques. As such, they are inadequate for industrial applications as design and development. This paper shows that under certain assumptions, an approximate formula of the transient film thickness profile can be derived under transient operating conditions. The variations can occur in the geometry, the load or the hydrodynamic velocity. The theory can handle all variations separately, or even a combination of several parameters varying simultaneously. The analytical approximation obtained is rather good apart from the constriction at the contact edge(s). This approach can be applied to any set of time dependent conditions (load, speed, geometry). As an example an EHL contact is studied in which reversal of the entrainment velocity occurs.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):448-460. doi:10.1115/1.1456455.

All-ceramic ball bearings with silicone nitride balls and silicone nitride rings were tested and the vibration characteristics were compared with those of hybrid ceramic ball bearings and conventional steel ball bearings. The vibration measurement results showed that the overall vibratory velocity levels of the all-ceramic ball bearings are influenced by rotational velocities, and do not change with axial loads. Under a given axial load and rotational velocity, the overall vibratory velocity level of the all-ceramic ball bearing is the lowest, and the hybrid ball bearing the highest. The frequencies of main peaks in the measured vibration spectra of the all-ceramic ball bearing are higher than the frequencies of the corresponding main peaks for the hybrid ceramic ball bearing and the steel ball bearing. To explain the main peaks, modal analysis was done and the relationship between peak and natural vibration was analyzed. The results of the analyses showed that the main peaks are caused by: (1) the mass-type natural vibration of the outer ring in the vertical direction, (2) the bending natural vibration of the outer ring in the radial direction, (3) the moment of inertia-type natural vibration of the outer ring in the angular direction, (4) the mass-type natural vibration of the outer ring in the axial direction, and (5) the bending natural vibration of the outer ring in the axial direction. We also discuss the generating mechanism of the vibration and present the calculation method of the vibration spectra. As a result, it is clear that the vibration spectra of the all-ceramic ball bearing are determined by the amplitude of the waviness of the raceways and ball surface, the mobility, and the non-linear spring constant associated with the contact between the raceways and balls.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):461-467. doi:10.1115/1.1467083.

This theoretical study concentrates on the mathematical analysis of the motion of small (5–50 μm) spherical solid particles in the inlet zone of elastohydrodynamic point contacts, found in various machine elements as, for example, in ball bearings, in order to compute the particle trajectories in the fluid flow. Particles may collide on a moving element of the contact (e.g., a ball in a bearing) or bypass it. For those particles that collide on another element, a fluid and mechanical force analysis reveals if they will be entrapped and, possibly, cause surface damage, or temporarily or finally expelled. Particle rejection is associated with the risk of inlet blockage and fluid starvation, which may further cause film collapse and scuffing. The study gives useful evidence of the probabilities of particle entrapment or rejection, extending the concept to evaluate the probabilities of oil starvation, surface indentation, or both, under various operating conditions.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):468-473. doi:10.1115/1.1467638.

The diagnosis and cause analysis of rolling-element bearing failure have been well studied and established in literature. Failure of bearings due to unforeseen causes were reported as: puncturing of bearings insulation; grease deterioration; grease pipe contacting the motor base frame; unshielded instrumentation cable; the bearing operating under the influence of magnetic flux, etc. These causes lead to the passage of electric current through the bearings of motors and alternators and deteriorate them in due course. But, bearing failure due to localized electrical current between track surfaces of races and rolling-elements has not been hitherto diagnosed and analyzed. This paper reports the cause of generation of localized current in presence of shaft voltage. Also, it brings out the developed theoretical model to determine the value of localized current density depending on dimensional parameters, shaft voltage, contact resistance, frequency of rotation of shaft and rolling-elements of a bearing. Furthermore, failure caused by flow of localized current has been experimentally investigated.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):474-479. doi:10.1115/1.1472457.

The sensitivity of clean and oil-impregnated cotton-phenolic retainers to atmospheric humidity has been measured. The cleaning procedure used here, and vacuum oil-impregnation did not change the dimensions of the retainers. Absorption of water caused the diameter of the retainers to increase by 0.2 to 0.4 percent, depending on the humidity. Oil impregnation of the retainers retarded, but did not stop, the absorption of water. Oil was displaced from the impregnated retainers by water; one-third to one-half of the oil was removed, depending on the atmospheric humidity.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):480-485. doi:10.1115/1.1454101.

A novel hydrodynamic system, called torsional fluid film vibrator (TFFV) is proposed. This device is complementary to the Lanchester’s absorber and presents a classical response of a one-degree of freedom linear system with a periodical self-excitation. The fluid film thickness variation produces a variable viscous drag moment, which drives the elastically supported bush in a torsional oscillatory movement. The TFFV concept is connected with current research to improve the drilling technology of deep holes. The Navier-Stokes equations are solved on the particular geometry of this vibrator and the viscous drag moment is explicitly presented. The theoretical part is continued with the TFFV dynamic simulation and the analysis of the influence of the geometrical parameters on the amplitude of the viscous drag moment. Computed structural friction power and the amplitude of vibration agree reasonably well with the experimental measurements conducted on a TFFV test rig.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):486-493. doi:10.1115/1.1454102.

An improved FEM model was developed to simulate the elastic behavior of a connecting rod bearing, accounting for the displacements caused by the tightening torque applied to the bolts that join the cap and the rod. These initial displacements are added to the pressure induced displacements, to enhance the solution of the elastohydrodynamic bearing lubrication problem. The big end bearing of a marine diesel engine was modeled and analyzed under combustion process loads and inertia loads using the Newton-Raphson method together with the Murty’s algorithm. Some important differences between these results and other results published for the same bearing without the bolt preload are identified and discussed.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):494-505. doi:10.1115/1.1454104.

A new modified Reynolds equation is derived with centrifugal force acting on the hydrodynamic oil film being considered. This equation, together with a cavitation model, is used to obtain the steady-state equilibrium and calculate the rotordynamic coefficients of lightly loaded floating bush journal bearings operating at very high shaft speeds. The bush-to-shaft speed ratio and the linear cross-coupling spring coefficients of the inner oil film is found to decrease with the increase in shaft speed as the axial oil film rupture develops in the inner oil film. The present model can give reasonable explanation to the steady-state behavior and the stability behavior of the bearing observed in actual machines.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):506-514. doi:10.1115/1.1456452.

Full Navier-Stokes equations are solved numerically for a cavity region between two consecutive pads and a parallel lubricating film. Numerical solutions are obtained for a wide range of Reynolds number and various values of a distance between pads. Numerical results show that the inlet pressure build-up is significantly affected by Reynolds number and the distance between two adjacent pads. A new formula is derived of loss coefficient with Reynolds number and a distance factor, for using it in an extended Bernoulli equation, on the basis of numerical results. Experiments are conducted to investigate the validity of the formula of loss coefficient proposed by authors.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):515-523. doi:10.1115/1.1456453.

Some non contacting mechanical face seals are running near the laminar boundary flow limit. A modification of operating conditions leads to a non laminar fluid flow in the seal interface while inertia forces remain negligible. A numerical model has been developed to determine pressure and velocity fields in the sealing dam for laminar to turbulent regime. The turbulent viscosity determination is based on the Elrod and Ng model. Evolutions of seal characteristics (opening force, friction torque, leakage rate[[ellipsis]]) and fluid film dynamic coefficients versus running conditions are presented. Numerical results show that great variations appear in the transition to turbulence.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):524-529. doi:10.1115/1.1456086.

A two-control-volume bulk-flow model is used to predict rotordynamic coefficients for an annular, honeycomb-stator/smooth-rotor gas seal. The bulk-flow model uses Hirs’ turbulent-lubrication model, which requires a friction factor model to define the shear stresses at the rotor and stator wall. Rotordynamic coefficients predictions are compared for the following three variations of the Blasius pipe-friction model: (i) a basic model where the Reynolds number is a linear function of the local clearance, fs=ns Rems (ii) a model where the coefficient is a function of the local clearance, and (iii) a model where both the coefficient and exponent are functions of the local clearance. The latter models are based on data that shows the friction factor increasing with increasing clearances. Rotordynamic-coefficient predictions shows that the friction-factor-model choice is important in predicting the effective-damping coefficients at a lower frequency range (60∼70 Hz) where industrial centrifugal compressors and steam turbines tend to become unstable. At a higher frequency range, irrespective of the friction-factor model, the rotordynamic-coefficient predictions tend to coincide. Blasius-based Models which directly account for the observed increase in stator friction factors with increasing clearance predict significantly lower values for the destabilizing cross-coupled stiffness coefficients.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):530-538. doi:10.1115/1.1456089.

The dynamics of solid third bodies sheared between two rubbing bodies is far from being understood. Yet, this interface plays a prominent role in the velocity accommodation and in the load transmission. In the present paper, a simple model, which uses the Distinct Element Method, is operated in order to understand phenomena occurring in dry contact. In this model, the solid third body is considered as an aggregate of discrete interacting particles. Inter-particle forces are determined by force-displacement law and trajectories are calculated using the Newton’s second law. The global behavior of the simulated contact can be analyzed through the evolution versus time of characteristic parameters calculated by averaging over all the particles. The model is used to study the effect of particle size and inter-particle forces. The influence of particle size is studied in presence of repulsive force (based on Hertz contact model), and in presence of adhesive force (based on JKR contact model). Some promising results are highlighted. In particular, with the boundary conditions chosen in this paper, it is shown that the particle size has a weak influence when inter-particle forces are repulsive but has a dramatic influence when inter-particle adhesion is considered: solid third body goes from a quasi-fluid to a quasi-solid behavior.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):539-546. doi:10.1115/1.1467084.

The effects of lubricant film flow, pressurized and sheared between two parallel sinusoidal wavy surfaces in sliding motion is studied analytically. Results are presented using a flow factor model which provides an average description of the surfaces roughness impact. Two distinct cases are studied in order to compare stationary or time dependent local aperture configurations. Flow factors are computed respectively for each case through spatial or spatio-temporal average, revealing striking differences. The results shed light on the relevance of the composite roughness concept. Special attention is paid to the flow factor analytical behavior when surfaces are near contact.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):547-552. doi:10.1115/1.1467636.

The basic lubrication equations are deduced from the original second-order fluid constitutive equations. Two examples of lubrication, a plane inclined slider and a journal bearing, are calculated respectively. The Reynolds boundary conditions are used in the calculation of the journal bearing. In this calculation, it is found that the load carrying capacities of the slider and the journal bearing are of different tendencies with the increase of the Deborah number. Furthermore, the results show that with the decrease of the film thickness, the increase of the normal stress of second-order fluid is greater than that of Newtonian fluid. Finally, it is found that the distribution of the normal stress changes significantly at a certain thickness.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):553-561. doi:10.1115/1.1467637.

This paper studies the nonlinear dynamic analysis of a flexible rotor supported by externally pressurized porous gas journal bearings. A time-dependent mathematical model for externally pressurized porous gas journal bearings is presented. The finite difference method and the Successive Over Relation (S.O.R.) method are employed to solve the modified Reynolds’ equation. The system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal center in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal center. This paper shows how the dynamic behavior of this type of system varies with changes in rotor mass and bearing number. The results of this study contribute to a further understanding of the nonlinear dynamics of gas-lubricated, externally pressurized, porous rotor-bearing systems.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):562-567. doi:10.1115/1.1456454.

When the spacing between the slider and the disk is smaller than 10 nm, the effect of the intermolecular forces between the two solid surfaces can no longer be ignored. This effect on the flying attitude of practical slider designs is investigated here numerically. The three-dimensional slider surface is discretized into non-overlapping unstructured triangles. The intermolecular forces between each triangular cell of the slider and the disk surface are formulated, and their contributions to the total vertical force, as well as the pitch and roll moments, are included in a previously developed steady state air bearing design code based on a multi-grid finite volume method with unstructured triangular grids [3–5]. It is found that the van der Waals force has significant influence on the flying height and has non-negligible effect on the pitch angle for both positive pressure sliders and negative pressure sliders, when the flying height is below 5 nm. When the flying height is below 0.5 nm, the repulsive portion of the intermolecular force becomes important and also has to be included.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):568-574. doi:10.1115/1.1456084.

This paper describes the air film dynamics for micro-textured flying head slider bearings in magnetic hard disk drives and it discusses their ability to increase the air film damping. In order to study the effects of micro-textures on air film dynamic characteristics of flying head slider bearings, two-dimensional micro-textures are modeled as deterministic rectangular models in the transverse and longitudinal directions. Dynamic analysis of these micro-textured flying head slider bearings is carried out by computing the impulse responses of the sliders and applying the modal analysis method to obtain their modal frequencies, damping ratios, and mode shapes. It is found that micro-textures on air bearing surfaces in the transverse direction have very significant effects and increase the air film damping of the slider’s three vibration modes but do not affect the air film stiffness. The same is not true for micro-texture in the longitudinal direction. The effects of transverse micro-textures’ spacing and depth on the air film dynamics are also numerically investigated in detail. The optimum values for high air film damping ratios are obtained. It is found that transverse micro-textures having small spacing are not very effective, being equivalent to the surface roughness effect.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):575-583. doi:10.1115/1.1456085.

Spreading characteristics of molecularly thin lubricant on a grooved surface have been studied numerically by Monte Carlo simulations and compared with measurements obtained by perfluoropolyether (PFPE) thin film spreading on a solid surface with minute grooves. In the simulations, by incorporating the interactions between molecules and the side surfaces of a groove, the Monte Carlo method based on the Ising model was extended to the case of a surface with grooves and applied to simulate the spreading of non-polar lubricant inside a groove. Compared with the spreading on a smooth surface, lubricant spreads rapidly inside a groove, indicating an acceleration of the spreading along the groove. In the experiments, lubricant spreading on a surface with groove-shaped textures was measured by making use of the phenomenon in which diffracted light decreases or vanishes in the lubricant-covered region. Based on the results showing lubricant spreading predominantly along the groove, the accelerating effect obtained in the simulations is well confirmed by the measurements.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):584-594. doi:10.1115/1.1456087.

The traction developed between a thin flexible web, wrapped around a non-vented, rotating cylindrical roller is studied experimentally and theoretically. A series of eight webs representing a wide range of surface roughness characteristics are traction-tested against the same roller over a wide speed range. A mathematical model that couples air film pressure, web deflection, and asperity deformations is used to model the web/roller interface. An optimization technique is used to estimate the asperity compliance function parameters based on the experimental results and the mathematical model. A new model for determining the asperity engagement height, for surfaces with non-Gaussian peak height distribution, is proposed when the roughness of both surfaces is taken into account. Results are presented that indicate the viability and utility of the new methods.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):595-599. doi:10.1115/1.1467081.

It is today possible to manufacture so smooth surfaces that they can elastically conform totally to each other over the whole Hertzian contact area. For pure rolling lubrication such surfaces only need an oil film of molecular dimensions to get total separation. When the rolling motion is combined with sliding, the pressure fluctuations inside the Hertzian contact redistribute the oil and make metal-to-metal contact possible. The redistribution velocity is a function of the slip rate S and the number of asperities N from the inlet to the outlet of the Hertzian contact area. The asperity top oil film thickness decreases with a factor of the order 2(NS) going from the inlet to the outlet of the Hertzian contact.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):600-606. doi:10.1115/1.1467089.

We numerically investigated the tracking ability, the dynamic contact and friction forces of a 2-DOF model of a tripad slider over a random wavy disk surface with 1 nm rms value in the near-contact regime. The air bearing was modeled as a lumped spring and dashpot in order to consider a general design methodology of the flying slider in the near-contact regime. The nominal flying height was changed from the contact regime to the near-contact regime. We studied the effects of the front and rear air bearing stiffnesses, the nominal flying height and the friction coefficient on the tracking ability and contact force. As a result, we found that the spacing variation is caused not only by the slider dynamics but also by the micro-waviness of the disk surface and the distance of the contact pad (head-gap) position from the rear air bearing center. We also derived the closed form frequency response functions of the spacing variation to the disk surface waviness. The approximation agreed with the numerical simulation. The effect of the friction coefficient on the tracking ability can be neglected when the flying height is more than 1 nm.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):607-616. doi:10.1115/1.1467086.

This paper presents the results of a feasibility study on air-riding seal aeroelasticity for large-diameter aero-engines. A literature survey of previous seal studies revealed a significant amount of experimental work but numerical modeling using CFD techniques was relatively scarce. Indeed, most existing theoretical studies either deal with the structural behavior, or use simplified flow modeling. The aeroelasticity stability of a simplified air-riding seal geometry, devised for this particular feasibility study, was analyzed in three dimensions for typical engine operating conditions. Both the unsteady flow and structural vibration aspects were considered in the investigation. The boundary conditions and the seal gap were varied to explore the capabilities and limitations of a state-of-the-art unsteady flow and aeroelasticity code. The methodology was based on integrating the fluid and structural domains in a time-accurate fashion by exchanging boundary condition information at each time step. The predicted characteristics, namely lift and flow leakage as a function of pressure and seal gap, were found to be in agreement with the expected behavior. Operating seal gaps were determined from the actual time histories of the seal motion under the effect of the aerodynamic and the restoring spring forces. Both stable and unstable cases were considered. It was concluded that, in principle, the existing numerical tools could be used for the flow and aeroelasticity analyses of hydrostatic seals. However, due to large Mach number variations, the solution convergence rate was relatively slow and it was recognized that a preconditioner was needed to handle seal flows. For small gaps of about 10 microns, typical of spiral groved seals, the flow has a high Knudsen number, indicating that the Navier-Stokes formulations may no longer be valid. Such cases require a totally different treatment for the modeling of steady and unsteady aerodynamics, either by modifying the transport parameters of the Navier-Stokes equations or by considering rarefied gas dynamics.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):617-626. doi:10.1115/1.1398546.

An experimental study of the chip-tool interface and its evolution in the low speed cutting of metals has been carried out. Specially prepared transparent glass and sapphire tools have been used to cut commercially pure metals such as lead, aluminum and copper. The chip-tool interface has been observed in situ using optical microscopy and recorded on film and video tape. By observing the motion of inhomogeneities in the chip, and profilometry of the chip and tool surfaces, it has been established that there is intimate sliding contact between the chip and the tool at and near the cutting edge. Farther away from the cutting edge and close to the end of the chip-tool contact, metal transfer and sticking are observed between the chip and tool surfaces. It has been shown that metal deposition on the rake face initially occurs near the point at which the chip curls out of contact with the tool and progressively extends outward and away from the cutting edge in conjunction with an increase in the length of contact as cutting progresses. The sticking and sliding zones are unchanged when these pure metals are machined with tungsten carbide tools.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):627-636. doi:10.1115/1.1454103.

The effect of a deposited soft thin metallic film on friction properties of a hardened steel substrate has been investigated experimentally and theoretically. The dependency of the film thickness and contact load on the static friction coefficient is presented. The experimental observations show that deformation of the film in contact was plastic, thereby confirming the assumption of the theoretical calculation. The effect of the film thickness on the contact area has been analyzed. A model for calculating the static friction coefficient of contacting rough surfaces in the presence of a soft thin film has been used. Results from the numerical calculations have been compared with the present static friction measurements performed on a pin on plate reciprocating apparatus. The rise in friction that occurs with increasing thickness for very thin films is discussed in detail. The calculated results, which predict the correct trend of the friction behavior from the present experiment, cover an extremely large range of F/AnE from 10−12 to 10−2, where three different dependencies of F/AnE on the static friction coefficient can be identified. An investigation into the discrepancy between the calculated and experimental values for the static friction coefficient μ suggests that an accurate prediction of the magnitude of μ depends to a great extent on the level of accuracy in measuring the value of the constant ζ, the effective hardness of the film.

Commentary by Dr. Valentin Fuster
J. Tribol. 2002;124(3):637-644. doi:10.1115/1.1473144.

The influence of material anisotropy and friction on ring deformation has been examined in relation to the distribution of normal pressure and frictional shear stress, deformed ring shapes, and estimated errors in the coefficient of friction. Based on the flow rule associated with von Mises’ and Hill’s yield criteria, the analyses have been carried out with the finite element method (FEM) for three cases, namely, (1) an anisotropic ring oriented 90 deg to the axis of rotational symmetrical anisotropy under uniform coefficient of friction; (2) an isotropic ring under frictional anisotropy condition; and (3) an anisotropic ring oriented 0 deg to the axis of rotational symmetrical anisotropy under uniform coefficient of friction. In the first two cases, the results show that the influence of anisotropy on ring deformation is quite similar to that obtained by changing the frictional condition. Therefore, in the third case, if the anisotropic behavior is mistakenly attributed to friction, the possible estimated error for the coefficient of friction can be as high as 80 percent for a pronounced anisotropic material. Deformed ring shapes have been verified in experiments using the extruded annealed aluminum alloy AA6082 (Al-Si1Mg0.9Mn0.1).

Commentary by Dr. Valentin Fuster

TECHNICAL NOTE

J. Tribol. 2002;124(3):645-646. doi:10.1115/1.1467596.

The analysis of the one-dimensional journal bearing leads to an interesting integral that is continuous but has an analytic singularity involving the inverse tangent at π/2. This difficulty was resolved by a clever and non-intuitive transformation attributed to Sommerfeld. In this technical brief we show that the transformation has its origin in the geometry of the ellipse and Kepler’s equation that is based upon his observations of the planets in the Solar system. The derivation of the transformation is a problem or exercise in Sommerfeld’s monograph, Mechanics. The transformation is the relation between the two angles that characterize the ellipse, the closed orbit of a body in a central inverse square force field. The angle measured about the focus is the true anomaly (angle) and the angle measured about the center is the eccentric anomaly (angle). We establish the analogy between the orbital radius in terms of the eccentric anomaly and the film thickness of the journal bearing in terms of its central angle.

Commentary by Dr. Valentin Fuster

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