An Engineering Approach to Hertzian Contact Elasticity—Part I

2000 ◽  
Vol 123 (3) ◽  
pp. 582-588 ◽  
Author(s):  
Luc Houpert
Keyword(s):  
Curve Fitting ◽  
Point Contact ◽  
Hertzian Contact ◽  
Smooth Transition ◽  
Analytical Relationship ◽  
Load Parameter ◽  
Minor Effect ◽  
Line Contact ◽  
Outer Race ◽  
Tapered Roller Bearings

Results given in this paper are threefold. In the case of Hertzian line contact, a new load/deformation relationship is derived analytically and use is made of the thickness of the outer race section. A minor effect of the section thickness is shown. The exponent on the deformation is 1.074 (instead of 1.1 usually accepted). Results calculated with the new relationship are successfully compared to results calculated with other published relationships and also are compared successfully to some available experimental results. For the case of point contact, useful relationships, obtained by curve-fitting, are given to calculate easily the load versus deformation, maximum Hertzian pressure and ellipse contact dimension as a function of a dimensionless load parameter and ratio k of equivalent radii (instead of sum of curvatures and elliptical integrals before). A large range of k is covered, from 0.05 (found at roller rib contact) to 13,000 to cover all bearing cases, from ball bearings to spherical and tapered roller bearings. Finally, an important analytical relationship, based on curve-fitting, also is suggested to describe a smooth transition from point contact to line contact as the load increases. It is recommended to define bearing setting and bearing preload with the suggested relationship.

An Engineering Approach to Non-Hertzian Contact Elasticity—Part II

2000 ◽  
Vol 123 (3) ◽  
pp. 589-594 ◽  
Author(s):  
Luc Houpert
Keyword(s):  
Pressure Distribution ◽  
Analytical Solutions ◽  
Curve Fitting ◽  
Numerical Solutions ◽  
Companion Paper ◽  
Hertzian Contact ◽  
Line Contact ◽  
Correction Factors ◽  
Analytical Development ◽  
Single Radius

Roller/race misalignment and deformation are used for calculating analytically the pressure distribution along the roller/race contact and the final roller/race load and moment. Use is made of the surface crowns and race undercuts for calculating contact dimensions with their possible truncations at large misalignment or loads. The pressure distribution is not symmetrical when misalignment occurs. This analytical development was possible by using a slicing technique in which the local roller/race geometrical interference was calculated in each slice of the contact. A mix of point and line contact Hertzian solutions developed in a companion paper “Part I” is used for obtaining the final load per slice. The final analytical solutions (load, moment and pressure) are successfully compared to two numerical solutions described briefly. The analytical model has been slightly fine-tuned using correction factors obtained by curve-fitting for matching the results to the numerical ones. In the curve-fitting, the single radius profile and multi-radius profile are distinguished.

scholarly journals Optical Analysis of Ball Bearing Starvation

1971 ◽  
Vol 93 (3) ◽  
pp. 349-361 ◽  
Author(s):  
L. D. Wedeven ◽  
D. Evans ◽  
A. Cameron
Keyword(s):  
Film Thickness ◽  
Ball Bearing ◽  
Point Contact ◽  
Experimental Measurements ◽  
Line Contact ◽  
Optical Analysis ◽  
Grease Lubrication ◽  
Theoretical Predictions ◽  
Pressure Stress ◽  
Starvation Conditions

Elastohydrodynamic oil film measurements for rolling point contact under starvation conditions are obtained using optical interferometry. The experimental measurements present a reasonably clear picture of the starvation phenomenon and are shown to agree with theoretical predictions. Starvation inhibits the generation of pressure and, therefore, reduces film thickness. It also causes the overall pressure, stress, and elastic deformation to become more Hertzian. Additional experiments using interferometry illustrate: the cavitation pattern, lubricant entrapment, grease lubrication, ball spin, and edge effects in line contact.

Experimental and numerical study on cavitation under elastohydrodynamic lubrication point contact

2021 ◽  
pp. 135065012110527
Author(s):  
Mingfei Ma ◽  
Wen Wang ◽  
Wenxun Jiang
Keyword(s):  
Contact Area ◽  
Numerical Study ◽  
Ambient Pressure ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Hertzian Contact ◽  
Pressure Area ◽  
Cavitation Pressure ◽  
Experimental Researches ◽  
State 1

As a common phenomenon in elastohydrodynamic lubrication, cavitation has an effect on the completeness of the oil film in the contact area. Many studies have therefore been conducted on cavitation. Experimental researches on cavitation usually rely on optical interference observation, which offers a limited resolution and observation range. In this paper, an infrared thermal camera is used to observe the cavity bubbles on a ball-on-disc setup under sliding/rolling conditions. The results show that the cavity length increases with an increases of the entrainment speed and the viscosity of the lubricants. These observations are explained by a numerical model based on Elrod's algorithm. Effects of entrainment speed and lubricant viscosity on the breakup of cavitation bubbles and the cavitation states are investigated. Both the simulation and experimental results show that a negative pressure area is present behind the Hertzian contact area. The ambient pressure plays a role in maintaining cavitation state 1. The cavitation pressure is close to the vacuum pressure when the entrainment speed is low and to the ambient pressure instead when the entrainment speed is high.

scholarly journals Modeling and Dynamic Analysis of Spherical Roller Bearing with Localized Defects: Analytical Formulation to Calculate Defect Depth and Stiffness

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Behnam Ghalamchi ◽  
Jussi Sopanen ◽  
Aki Mikkola
Keyword(s):  
Dynamic Model ◽  
Contact Stiffness ◽  
Roller Bearing ◽  
Measurement Data ◽  
Contact Forces ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Outer Race ◽  
Rolling Elements ◽  
Localized Defects

Since spherical roller bearings can carry high load in both axial and radial direction, they are increasingly used in industrial machineries and it is becoming important to understand the dynamic behavior of SRBs, especially when they are affected by internal imperfections. This paper introduces a dynamic model for an SRB that includes an inner and outer race surface defect. The proposed model shows the behavior of the bearing as a function of defect location and size. The new dynamic model describes the contact forces between bearing rolling elements and race surfaces as nonlinear Hertzian contact deformations, taking radial clearance into account. Two defect cases were simulated: an elliptical surface on the inner and outer races. In elliptical surface concavity, it is assumed that roller-to-race-surface contact is continuous as each roller passes over the defect. Contact stiffness in the defect area varies as a function of the defect contact geometry. Compared to measurement data, the results obtained using the simulation are highly accurate.

scholarly journals Power Curve-Fitting Control Method with Temperature Compensation and Fast-Response for All-Metal Domestic Induction Heating Systems

Energies ◽  
2019 ◽  
Vol 12 (15) ◽  
pp. 2915 ◽  
Author(s):  
Sang Min Park ◽  
Eunsu Jang ◽  
Dongmyoung Joo ◽  
Byoung Kuk Lee
Keyword(s):  
Induction Heating ◽  
Curve Fitting ◽  
Control Algorithm ◽  
Control Method ◽  
Reference Value ◽  
Fast Response ◽  
Load Parameter ◽  
Power Curve ◽  
Fitting Method ◽  
Curve Fitting Method

Typical domestic induction cooktops can only heat ferromagnetic pots/vessels. However, to increase the availability and marketability of induction heating (IH) cooktop products, heating techniques for all types of metallic pots (i.e., created from metals such as aluminum, copper, and stainless steel) are required. To satisfy the requirements of induction cooktops, this paper proposes the design of an all-metal domestic IH system that can heat any type of metallic pot while considering the temperature variation of the working-coil. A control algorithm using a power curve-fitting method (CFM) is presented to quickly respond to load parameter variations in the IH. In addition, the CFM control algorithm is established to compensate for the power reference value by reflecting the increase in the working-coil temperature during the heating of the non-ferromagnetic pot. To evaluate the performance of the proposed system, the control algorithm strategy and experimental results based on a 3.2 kW all-metal IH cooktop are presented.

scholarly journals Microscopic and mechanical properties of undistributed and remoulded red clay from Guiyang, China

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Yanzhao Zhang ◽  
Shaoyun Pu ◽  
Rita Yi Man Li ◽  
Jing Zhang
Keyword(s):  
Point Contact ◽  
Morphological Properties ◽  
Line Contact ◽  
Red Clay ◽  
Undisturbed Soil ◽  
Pore Area ◽  
Distribution Index ◽  
Curved Slice ◽  
Clay Aggregates ◽  
Scanning Electron

Abstract Unconsolidated-undrained (UU) tests were conducted to investigate the mechanical and morphological properties of undisturbed and remoulded red clay, with the microscopic characteristics determined by scanning electron microscopy (SEM). The microanalysis showed that the red clay aggregate was granular, curved-slice and thin layered and flower-shaped ellipsoid, with X and Y-type cracks and pores in the undisturbed red clay. Moreover, the contact modes of red clay aggregates were point contact, line contact, surface contact and mosaic contact. In addition, the main microstructure red clay was flocculation, honeycomb and pseudosphere structures. The pores in undisturbed soil were arranged in one direction, with no obvious directionality in remoulded red clay. The pore area, perimeter and maximum length of undisturbed red clay were smaller than those of remoulded red clay, with a larger probability entropy, probability distribution index and fractal dimension of pore distribution of undisturbed red clay than remoulded red clay. UU tests showed that the shear strength of undisturbed red clay was higher than that of remoulded red clay.

Hertzian Contact Stress Modeling in Railway Bearings for Assorted Load Conditions and Geometries

2014 ◽  
Author(s):  
Michael A. Mason ◽  
Charles P. Cartin ◽  
Parham Shahidi ◽  
Mark W. Fetty ◽  
Brent M. Wilson
Keyword(s):  
Finite Element ◽  
Contact Stress ◽  
Rolling Contact ◽  
Hertzian Contact ◽  
Modeling Tools ◽  
Heat Treat ◽  
Roller Bearings ◽  
Tapered Roller ◽  
Tapered Roller Bearings ◽  
Load Conditions

Increasing freight car loads demand higher performance tapered roller bearings. As the stress state on railway bearing applications continues to increase, further advancement in the modeling tools and methods used for subsurface contact stress evaluations are needed. Heat treat specifications and contact geometries for railway bearings were originally developed for ideal load conditions. However, in railroad applications, tapered roller bearings are exposed to a vast range of load conditions that are seldom perfect. Moreover, when comparing global rail markets, there are often differences in bearing loads, railcar wear conditions, maintenance practices, and reliability versus utilization expectations. Advanced modeling techniques need to be developed by bearing designers in order to meet the specific needs of each individual rail market. Prior research has shown that subsurface stresses, resulting from rolling contact, are the primary factor in the development of fatigue cracks in railway bearings. In addition, finite element modeling software has previously been used to analyze Hertzian contact stresses under rolling contact. Recent advancements in the technology and computational power of finite element methods can be used to numerically analyze more detailed simulations of complex geometries and biased load conditions in railway bearings. These improvements in the tapered roller bearing modeling methodology are necessary to determine the material, heat treat specifications, and geometry required to meet the demands of specific railway bearing applications. Furthermore, the specific risks associated with some common railway bearing design and modeling assumptions will be evaluated. An exploratory list of these assumptions include: line versus point contact, load deflection factor, zero contact angle, rigid body assumptions, linear material behavior, neglect for overload, and uniform loading on the bearing. Emphasis will be placed on potential improvements in the theoretical and finite element prediction of surface and subsurface stresses in railway bearings under rolling contact with a review of prior research on the subject.

Analysis of Microelastohydrodynamic and Mixed Lubrication Effects in Gear Tooth Contacts

2003 ◽  
Author(s):  
M. J. A. Holmes ◽  
H. P. Evans ◽  
R. W. Snidle
Keyword(s):  
Contact Area ◽  
Transient Analysis ◽  
Gear Tooth ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Contact Analysis ◽  
Line Contact ◽  
Hydrodynamic Equations ◽  
Analysis Technique ◽  
Film Thinning

The paper presents results obtained using a transient analysis technique for point contact elastohydrodynamic lubrication (EHL) problems based on a formulation that couples the elastic and hydrodynamic equations. Results are presented for transverse ground surfaces in elliptical point contact that show severe film thinning at the transverse limits of the contact area. This thinning is caused by transverse leakage of the lubricant from the contact in the remaining deep valley features. A comparison is also made between the point contact results on the entrainment centre line and the equivalent line contact analysis.

scholarly journals A Pocket-Textured Surface for Improving the Tribological Properties of Point Contact under Starved Lubrication

Materials ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 1789
Author(s):  
Wei Wang ◽  
Wenhan Zhao ◽  
Yang Liu ◽  
Hui Zhang ◽  
Meng Hua ◽  
...  
Keyword(s):  
Tribological Properties ◽  
Equivalent Stress ◽  
Reduction Rate ◽  
Point Contact ◽  
Operating Conditions ◽  
Experimental Results ◽  
Hertzian Contact ◽  
Textured Surface ◽  
Oil Storage ◽  
Starved Lubrication

This paper reports a novel pocket-textured surface for improving the tribological properties of point contact under starved lubrication by possibly storing and releasing oil, and homogenizing the surface contact pressure. The ball-on-disk experimental results confirmed the coefficient of friction (COF) and wear reduction effect of such pocket-texturing. The maximum reduction rate was 40% compared with a flat surface under the same operating conditions. Analyses on experimental results attributed the oil storage effect and enhanced the secondary lubrication effect within the starved lubrication state, to become the main mechanism. In addition, the plate elasticity and the Hertzian contact principles were employed to estimate the pressure and the load acting on the surface. The experimental results and numerical analysis substantiated the design of pocket-textured surface, making it likely to enlarge about 50% of contact surface and to reduce 90% of equivalent stress in comparison to those of conventional surfaces.

Estimation of Cavity Length in EHL Rolling Point Contact

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
K. Stadler ◽  
N. Izumi ◽  
T. Morita ◽  
J. Sugimura ◽  
B. Piccigallo
Keyword(s):  
Cavity Length ◽  
Point Contact ◽  
Hertzian Contact ◽  
Careful Examination ◽  
Contact Radius ◽  
Cavity Pressure ◽  
Pressure Index ◽  
Lubricated Contacts ◽  
Transient Simulations ◽  
Good Agreement

Reciprocating lubricated contacts sometimes suffer from oil starvation due to cavitation at the reversal of motion. However, the behavior of cavities is not well understood such that starvation can be theoretically predicted. In this study, the length of cavity in a steady state elastohydrodynamic lubricated point contact was calculated. For numerical simulation, a modified Elrod algorithm was used. An equation was obtained for the cavity enclosed in the oil meniscus. The equation was constructed with Moes dimensionless parameters M and L, assumed pressure of cavity, and viscosity pressure index of the lubricant. The dimensionless cavity length (or the ratio of cavity length by Hertzian contact radius) is proportional to the product of M−a and Lb. Careful examination of the equation elucidated that the cavity length is dominated by the viscosity, sum velocity, cavity pressure, and geometry of the contact. Experimental measurements with a ball-on-disk apparatus have shown good agreement. The validity of the equation suggests that the algorithm is applicable for complete transient simulations. In practice, the estimated cavity length can be a parameter for starvation.

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