Elliptical Contact Research Articles

Analysis of wide EHL rolling contacts with small superimposed transverse speed oscillation

Abstract This paper theoretically investigates the lubricant film build-up in a Newtonian, isothermal wide EHL rolling contact with a small transverse oscillating speed of constant frequency and amplitude. By increasing either elliptical contact width or roller-end gap smoothness, it is found with the aid of perturbation techniques that a) film fluctuations introduced by the transverse speed become smaller in amplitude and limited to the contact edges and, b) the overall minimum film thickness position shifts from the side lobes to the centerline outlet. As a result, the central, average and minimum film thickness values might then all appear to be unaffected by the transverse speed.

Boundary Lubrication in Transient Elliptical Contact: Part 1—Theoretical Formulation and Results

The transient elliptical contacts in boundary lubrication widely exist in modern mechanical systems with high durability. However, little attention has been paid to the squeeze effect of fluid film in this condition. A deterministic model which combines the contact mechanics with the pure squeeze lubrication model has been developed with the aim of understanding the squeeze effect of fluid film under the transient boundary lubrication. The leakage coefficient was introduced to capture the fluid leakage of rough surfaces. The squeeze effect of trapped fluid film was confirmed through comparing the fluid film stiffness in boundary lubrication with that of the elastohydrodynamic lubrication. Additionally, the effects of fluid film entrapment/leakage on the boundary lubrication performance were numerically analyzed during transients. The load capacity of the squeeze films is built up due to the trapped fluid film in the micro-valleys, which can be significantly affected by the interfacial shear coefficient of the boundary films. The simulation results show a good agreement with the experiments and justify the present numerical model is feasible in the boundary lubrication regime.

Contact Area and Shear Stress in Repeated Single-Asperity Sliding of Steel on Polymer

A model for the contact area of a single asperity sliding in a groove after repeated cycles is presented. Based only on the asperity geometry and on data from friction experiments, the model predicts the area of the asymmetric elliptical contact of the asperity sliding in its own groove. It thus allows to determine the shear stress of the steel–polymer couple in the relevant geometry without need for further microscopy of indenter or groove. The model was validated by experiments with an indenter manufactured from slide bearing steel and polyether-ether ketone (PEEK) as substrate. In experiments of 1000 repeated cycles, the contact area was found to vary with varying load and sliding velocity, while the shear stress was 20.5 MPa at a normal pressure of 50–70 MPa, independent of velocity when friction heating is still negligible. Model and experimental confirmation advance single-asperity friction experiments into an efficient method to extract shear stress and contact area for an understanding of sliding friction in metal-polymer contacts.

An experimental study on the contact problem of spherical indentation on the anisotropic elastic half-space based on imaging

The contact problem of spherical indentation on the anisotropic elastic half-space is considered based on an experimental method in this work. The mechanical anisotropy is simulated by uniaxially stretching isotropic material made of silica gel. A special optical system is adopted to capture the contact images during indentation tests. The experimental results further prove the reasonability of the assumption that the contact area of spherical indentation on the anisotropic half-space is elliptical. And the results also show that the direction of strengthening by the uniaxial tension is consistent with the minor axis of the elliptical contact area. Such results validate the associated theory. Besides, the error analysis is made to examine the reliability of the fitting results, where the relative errors are less than 0.8% under the influence of given drawbacks. The indentation modulus can be determined through experimental data. Results show that the indentation modulus monotonically increases with the increase of the prestrain value. In the case of isotropy, the initial elastic modulus is determined under the assumption of incompressibility. The result by indentation test is in a good agreement with that by uniaxial tensile test.

A Model of Mixed Lubrication Based on Non-Normalized Discretization and Its Application for Multilayered Materials

This study presents a generalized model of mixed elastohydrodynamic lubrication, in which the dimensional Reynolds equation is discretized according to a modified differential scheme based on the full analysis of the pressure balance within the lubrication region. The model is capable of a wide range of lubrication regimes from fully hydrodynamic down to boundary lubrication, and both the steady-state and the time-dependent conditions can be considered. A simplified computational procedure is proposed for elliptical contacts without the ellipticity parameters specified. The evolution of lubrication behavior at startup and shutdown conditions is investigated and the transient effect of surface waviness is discussed. The model application is then extended to contacts of multilayered materials, and the effects of the layer stiffness and the fabrication methods on the stress fields and lubrication performance are analyzed. The conclusions may potentially provide some insightful information for the design and analysis of functional materials and their engineering structures.

Study of sliding parameters in engagement of gear hone teeth and processed parts with consideration of elastic deformation

Abstract Gear hone is a special tool used for roll finishing of lateral faces of hardened gear teeth. Over time, penetration of abrasive grains into the metal, wear and breaking in of the hone material, contact and bending deformation of the hone teeth, especially if an elastic binding agent is used, lead to changing the tooth contact geometry, i.e. a point contact turns into an elliptical contact. The gear hone tooth deformation is a function of the contact point height, i.e. for contact points closer to the tooth top, the deformation is many times larger than that occurring for contact points closer to the tooth bottom. It affects the specific sliding ratio between the meshing hone and the workpiece. Meanwhile, slight equalization of the specific sliding along the tooth height occurs. Comparable deformations, errors and metal removal rates applicable to the gear hone and the workpiece would contribute to the effectiveness of the error reduction measures. The elasticity of the tool’s composite material must be duly considered in calculating the geometry, selecting the compound composition and assessing the applicability of gear hones.

Effect of tooth profile modification on the durability of planetary hub gears

Planetary systems offer the advantage of desired speed-torque variation with a lighter, compact and coaxial construction than the traditional gear trains. Frictional losses and Noise, Vibration and Harshness (NVH) refinement are the main concerns. Modification of gear teeth geometry to reduce friction between the mating teeth flanks of vehicular planetary hubs, as well as refining NVH under varying load-speed conditions is one of the remedial actions. However, implementing modifications can result in reduced structural integrity and system durability. Therefore, a contradiction may arise between assuring a high degree of durability and achieving better transmission efficiency, which necessitates detailed system optimisation. An integrated multi-disciplinary analytical approach, including tribology and sub-surface stress analysis is developed. As a preliminary step, Tooth Contact Analysis (TCA) is performed to obtain contact footprint shape of meshing gear teeth pairs, as well as contact kinematics and applied load distribution. Then, an analytical time-efficient Elastohydrodynamic Lubrication (EHL) analysis of elliptical point contact of crowned spur gear tooth is carried out to observe the effect of gear tip relief modification upon planetary hub sub-surface stresses.

Satisfying Boundary Conditions in Homogeneous, Linear-Elastic and Isotropic Half-Spaces Subjected to Loads Perpendicular to the Surface: Distributed Loads on Adjacent Contact Areas

Abstract This work originates from an experimental program on strain distribution near the loaded surface of an airfield concrete pavement,which provided us with results that contrast with the rheological predictions of Boussinesq for a homogeneous, linear-elastic and isotropic half-space. We already reviewed and extended the original work carried out by Boussinesq in previous papers, to provide a closed form second order solution that enabled us to establish a good match between analytical and experimental findings for point-loads. In this paper, we have explained why Boussinesq’s closed form solution for a homogeneous linear-elastic and isotropic half-space subjected to a point-load is not exact, as believed until now, but approximated. Then, we have shown that our second order solution is the actual solution of Boussinesq’s problem. We have also presented the numerical analysis of second order for rectangular and elliptical contact areas, both loaded by uniform and parabolic laws of external pressure. Moreover, we have evaluated the interaction effect provided on the surface of a concrete half-space by the twin wheels of an aircraft landing gear. Extension of the solution to layered systems is also possible, for improving the knowledge of stress propagation into airfield pavements and promoting more effective design standards.

An Isothermal Elastohydrodynamic lubrication of elliptical contact with Multigrid method

ABSTRACT The paper presents steady isothermal elastohydrodynamic lubrication (EHL) of elliptical contact with Roelands pressure–viscosity model over varying rotational speed. The governing Reynolds and film-thickness equation for elliptical contact in roller bearings are analysed by using multigrid method with full approximation scheme. The non-dimensional pressure distribution and film-thickness for elliptical contacts are obtained. The distinct friction models appropriate to lubricants in contact regions are predicted. The result shows that rolling contact weariness does not occur on the surfaces with longitudinal roughness and wear or scraping exists in circular EHL contacts with entrainment along major axis to that of minor axis. The computation predicts that minimum film thickness, central film thickness and elastic deformation increase as the rotational speed increases. The computed results are compared with earlier finding which are in good agreement.

Response to Dr Greenwood\u2019s Comments on \u201cExtending the Double-Hertz Model to Allow Modeling of an Adhesive Elliptical Contact\u201d

© 2018, The Author(s). An adhesive elliptical contact is normally found in microscale applications that involve cylindrical solids, crossing at an angle between 0° and 90°. Currently, only one model is available to describe the elliptical contact’s surface interaction: the approximate Johnson–Kendall–Roberts (JKR) model which is limited to soft materials. In this paper, a new adhesive elliptical model is developed for a wide range of adhesive contacts by extending the double-Hertz theory, where adhesion is modeled by the difference between two Hertzian pressure distributions. Both Hertzian pressures are assumed to have an equivalent shape of contact areas, the only difference being in size. Assuming that the annular adhesive region is obtained by the area difference between the two Hertzian contact areas, the pull-off force curves can be calculated. In the limiting case of an adhesive circular contact, the results are very close to results from the existing models. However, for an adhesive elliptical contact in the JKR domain, lower pull-off forces are predicted when compared to the JKR values. Unlike the developed model, the shape of the JKR contact area varies throughout contact. Results show, particularly for conditions close to the JKR domain, that it is important to take into account that the adhesive region is the result of the two Hertzian contact areas having a non-equivalent shape.

A New Method for Eyring Shear-Thinning Models in Elliptical Contacts Thermal Elastohydrodynamic Lubrication

A new method for solving the shear stress and the effective viscosity of Eyring shear-thinning fluid in thermal elastohydrodynamic lubrication (EHL) was proposed and applied to two models. Model 1 is the thermal EHL model with one-direction velocity, and model 2 is the spinning thermal EHL model in which the velocity varies with coordinates. Comparisons between the new and the existing method were carried out. Results show that only replacing the shear strain rate of model 1 with that of model 2, the shear stress and the effective viscosity of model 2 for Eyring shear-thinning fluid can be obtained. For model 1, results obtained with the two methods are the same. The new method can be qualified and applied into model 2. It is proved that the new method has higher efficiency for shear-thinning fluid than the existing method. Therefore, the new method is more efficient and can be used for spinning Eyring shear-thinning thermal EHL.

Effect of surface topography associated with arbitrary velocity direction on the lubrication film thickness in elliptical contacts

PurposeThe purpose of this study is to describe and observe the effect of surface topography associated with arbitrary directions of rolling and sliding velocities on the performance of lubricating films in elliptical contacts.Design/methodology/approachThe most recently published mixed elastohydrodynamic (EHL) model by Pu and Zhu is used. Three different machined rough surfaces are discussed and the correlated inclined angle of surface velocity varies from 0° to 90° in the analyzed cases. These cases are carried out in a wide range of speeds (five orders of magnitude) while the simulated lubrication condition covers full-film and mixed EHL down to the boundary lubrication.FindingsThe results indicate that the variation of the average film thickness corresponding to different entrainment angles is distinct from those without considering surface roughness. In addition, the surface topography appears to have an immense effect on the lubrication film thickness in the exceptive situation.Originality/valueThis paper has not been published previously. Surface roughness has attracted much attention for many years owing to the significant influence on lubricating property. However, previous studies mainly focus on the counterformal contact with the same direction between surface velocity and principal axis of the contact zone. Little attention has been paid to the specific condition with the arbitrary direction of rolling and sliding velocities found in hypoid gears and worm, and some other components. The purpose of this study is to describe and observe the effect of surface topography associated with arbitrary directions of rolling and sliding velocities on the performance of lubricating films in elliptical contacts based on the most recently published mixed EHL model by Pu and Zhu.

Motion of a thin elliptic plate under symmetric and asymmetric orthotropic friction forces

The anisotropy of a friction force is proved to be an important factor in various contact problems. We study the dynamical behavior of thin plates with respect to symmetric and asymmetric orthotropic friction. The terminal motion of plates with circular and elliptic contact areas is analyzed. The evaluation of friction forces for both symmetric and asymmetric orthotropic cases is shown using an analytic approach. Regular pressure distribution is considered. Differential equations are formulated and solved numerically for a number of initial conditions. The method used gives more accurate results compared to the previous study. Examples show the significant influence of friction force asymmetry on motion.

Elliptic surfaces and contact conics for a 3-nodal quartic

Let ${\mathcal Q}$ be an irreducible $3$-nodal quartic and let ${\mathcal C}$ be a smooth conic such that ${\mathcal C} \cap {\mathcal Q}$ does not contain any node of ${\mathcal Q}$ and the intersection multiplicity at $z \in {\mathcal C} \cap {\mathcal Q}$ is even for each $z$. In this paper, we study geometry of ${\mathcal C} + {\mathcal Q}$ through that of integral sections of a rational elliptic surface which canonically arises from ${\mathcal Q}$ and $z \in {\mathcal C} \cap {\mathcal Q}$. As an application, we construct Zariski pairs $({\mathcal C}_1 + {\mathcal Q}, {\mathcal C}_2 + {\mathcal Q})$, where ${\mathcal C}_i$ $(i = 1, 2)$ are smooth conics tangent to ${\mathcal Q}$ at four distinct points.

Extending the Double-Hertz Model to Allow Modeling of an Adhesive Elliptical Contact

An adhesive elliptical contact is normally found in microscale applications that involve cylindrical solids, crossing at an angle between 0° and 90°. Currently, only one model is available to describe the elliptical contact’s surface interaction: the approximate Johnson–Kendall–Roberts (JKR) model which is limited to soft materials. In this paper, a new adhesive elliptical model is developed for a wide range of adhesive contacts by extending the double-Hertz theory, where adhesion is modeled by the difference between two Hertzian pressure distributions. Both Hertzian pressures are assumed to have an equivalent shape of contact areas, the only difference being in size. Assuming that the annular adhesive region is obtained by the area difference between the two Hertzian contact areas, the pull-off force curves can be calculated. In the limiting case of an adhesive circular contact, the results are very close to results from the existing models. However, for an adhesive elliptical contact in the JKR domain, lower pull-off forces are predicted when compared to the JKR values. Unlike the developed model, the shape of the JKR contact area varies throughout contact. Results show, particularly for conditions close to the JKR domain, that it is important to take into account that the adhesive region is the result of the two Hertzian contact areas having a non-equivalent shape.

Hertz Contact Stress of Deep Groove Ball Bearing

Calculation of hertz contact stress between the deep groove ball bearing and inner race using analytical and MESYS tool analysis. The calculation procedure consists of calculation of maximum contact pressure at different loads using analytical formulas and comparison of results with MESYS tool to predict the contact pressure between inner race and ball of single row deep groove ball bearing and the contact between inner race and ball is considered as an elliptical contact.

Wheel-rail Interactions in High Speed Railway Networks during Rapid Train Transit

Implementation of high speed railway networks in India requires a thorough investigation of the contact pressure and frictional stresses induced in wheel-rail interfaces during rapid train transit. In this study, 3-D numerical moving load simulations of wheel-rail interactions in broad gauge railway networks in India were performed for fast and heavy train operations, and the trends in variation of contact pressure and frictional stresses with increase in speed of the train were quantified. Simulations were performed in ANSYS, with a hybrid model incorporating a flexible wheel-set running over a couple of rigid rails, which aided in the estimation of frictional stresses at the interface of wheel-rail contact during train run, while maintaining a reduced number of elements associated with lengthy railway track sections. Analyses were done for train operations ranging from maximum speeds of 160 to 200 km/h and an axle load of 32.5 tonnes, as recommended by Indian Railways for operation of freight carriers in the existing broad gauge tracks. It was noted that with increase in speeds of trains, there was a reduction in the area of wheel-rail contact. The elliptical Hertzian contact patch of the static train wheels were found to transform into a more rounded patch during rapid train transit, causing concentration of stresses at the centre and effectively increasing the maximum stresses at the points of contact. Contact pressure and frictional stresses were found to increase quadratically, with an increase in the speed of the train. It was ascertained that the rate of wheel and rail wear during high speed train operations would be significant, and that high speed networks would require new wheel and rail sections, and the adoption of materials with very high frictional wear capacity.

Microgeometrical tooth profile modification influencing efficiency of planetary hub gears

Planetary hub systems offer desired speed and torque variation with a lighter, compact and coaxial construction than the traditional gear trains. Generated friction between the mating teeth flanks of vehicular planetary hubs under varying load-speed conditions is one of the main sources of power loss. Modification of gear tooth geometry as well as controlling the contacting surface topography is the remedial action. The paper studies the effect of tooth crowning and tip relief upon system efficiency. It includes an analytical elastohydrodynamic analysis of elliptical point contact of crowned spur gear teeth. The analysis also includes the effect of direct contact of asperities on the opposing meshing surfaces. Tooth contact analysis (TCA) is used to obtain the contact footprint shape as well as contact kinematics and load distribution. A parametric study is carried out to observe the effect of gear teeth crowning and tip relief with different levels of surface finish upon the planetary hubs' power loss.

A Vibration Model of Ball Bearings with a Localized Defect Based on the Hertzian Contact Stress Distribution

To study the vibration mechanism of ball bearings with localized defects, a vibration model of a ball bearing based on the Hertzian contact stress distribution is proposed to predict the contact force and vibration response caused by a localized defect. The calculation of the ball‐raceway contact force when the ball passes over the defect is key to establishing a defect vibration model. Hertzian contact theory indicates that the contact area between the ball and the raceway is an elliptical contact surface; therefore, a new approach is used to calculate the ball‐raceway contact force in the defect area based on the stress distribution and the contact area. The relative motion between the inner ring, the outer ring, and the balls is considered in the proposed model, and the Runge‐Kutta algorithm is used to solve the vibration equations. In addition, vibration experiments of a bearing with an outer ring defect under different loads are performed. The numerical signals and experimental signals are compared in the time and frequency domains, and good correspondence between the numerical and experimental results is observed. Comparisons between the traditional model and the proposed model reveal that the proposed model provides more reasonable results.

About the impact of a viscoelastic on a fixed semi-space

The impact interaction of a viscoelastic body bounded by a quadratic paraboloid in the contact zone with an absolutely rigid fixed half-space having a flat boundary surface is considered. We have introduced the assumption that the force of resistance to the deformation of a body that hits depends not only on its elastic characteristics, but also on the square of the velocity of its center of mass. Additionally, the well-known dependences of G. Hertz on the distribution of contact deformations and dynamic pressure were used. Under these assumptions, the first integral of a non-linear differential equation of motion of the second order is expressed in elementary functions. In the same functions, the expression for the maxima was obtained: dynamic compression, impact force, dimensions of the elliptical contact area, pressure at the center of this area. An extremum study has established that with a viscoelastic impact, the maxima of the impact force and contact pressure can be reached not at the end of the body compression phase, but somewhat earlier during this process. A condition is derived when the maxima of the forces of viscoelastic and elastic impacts are the same. It is related to the coefficient of viscosity and the square of the velocity of the collision of bodies. It is shown that at the end of the compression stage, the impact force and the contact pressure during viscoelastic impact can be significantly less than the maximum, as well as those that the theory of impact of perfectly elastic bodies leads to. A compact formula for calculating the coefficient of speed recovery with a direct central impact is derived. It is shown that it depends on the coefficients of viscosity for compression and decompression and on the square of the velocity of collision of bodies. With the increase of this value, the speed recovery coefficient decreases. The duration in time of the stages of compression and expansion are represented by improper convergent integrals of the second kind, which are not expressed analytically in terms of known functions. The integrands in them have an algebraic singularity in the series 1/2. Therefore, it is recommended to single out the singular part in quadratures and integrate it analytically, and calculate the regular part on a computer. Examples of calculations are given and a comparative analysis of numerical results is carried out.