# Elastohydrodynamic Lubrication for Line and Point Contacts

## Asymptotic and Numerical Approaches

710 pages | 153 B/W Illus.

Hardback: 9781466583894
pub: 2013-06-17
US Dollars\$206.95
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Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches describes a coherent asymptotic approach to the analysis of lubrication problems for heavily loaded line and point contacts. This approach leads to unified asymptotic equations for line and point contacts as well as stable numerical algorithms for the solution of these elastohydrodynamic lubrication (EHL) problems.

A Unique Approach to Analyzing Lubrication Problems for Heavily Loaded Line and Point Contacts

The book presents a robust combination of asymptotic and numerical techniques to solve EHL problems for lightly and heavily loaded line and point contacts. It also proposes a reasonably simple and naturally based regularization approach that produces stable solutions in heavily loaded EHL contacts. The book offers a clear understanding of the processes taking place in heavily loaded line and point EHL contacts as well as of the proper solution structure for EHL problems. It outlines concrete ways to determine important design parameters such as lubrication film thickness and frictional stresses and forces. The book establishes a close link between EHL problems for heavily loaded point and line contacts.

Fine Tune Your Methods for Solving Elastohydrodynamic Lubrication Problems

In most cases, the equations in the book are derived from first principles. The author describes each of the asymptotic and numerical methods in detail, making it easier for readers to apply them to various problems. The problem solutions are presented in the form of simple analytical formulas, graphs, and tables. Almost all the chapters include exercises that highlight key points and skills. Suitable for engineering and applied mathematics students, this is also a unique resource for researchers and practitioners who want to fine tune their solution methods and design better numerical methods to tackle elastohydrodynamic lubrication problems.

### Reviews

"Elastohydrodynamic lubrication (EHL) is a relatively new area in the development of lubrication theory and practice. This monograph is devoted to selected performances of liquid lubricants, particularly mineral oils and synthetic fluids. … combines asymptotic and numerical techniques to solve EHL problems for line and point contacts. Most chapters include exercises with different levels of difficulty."

Mathematical Reviews, January 2015

"… the present work uses completely different mathematical tools represented by asymptotic methods. … they enable to obtain very elegant and useful limiting solutions that accompany numerical methods or gave answers to stiff problems where these later techniques converge with difficulty or fail. The book is very well written, with an excellent mathematical background and bridges in a successful manner the gap between mathematics and technology. The reviewer strongly recommends the book to all lubrication scientists from academy or industry."

—Mihai ARGHIR, Université de Poitiers, France

"This book offers a comprehensive approach to the solution of the elastohydrodynamic lubrication (EHL) problem under a wide variety of operating conditions/regimes using a combination of asymptotic and numerical techniques. It also offers insight into physical aspects and phenomena governing the EHL problem such as lubricant rheology (Newtonian and non-Newtonian), solid properties, lubricant degradation, kinematic considerations, soft solids, grease lubrication, non-steady operation, starvation, [and] mixed friction … In addition, a variety of problems are offered at the end of each chapter, helping readers reach a higher level of understanding of the complex nature of the EHL problem."

—Wassim Habchi, Lebanese American University, Byblos, Lebanon

"This book introduces to the tribology community the application of asymptotic methods to solve an important and difficult class of problems in lubrication. These analytical techniques overcome some limitations of traditional numerical schemes for determining EHL film thickness and, moreover, aid in the development of insight as to the physical mechanisms at play. …like the comprehensive coverage of EHL regimes: line and point contacts, lightly and heavily loaded conditions, Newtonian and non-Newtonian rheology, isothermal and thermal behavior, steady-state and transient conditions. I also like that the author provides a detailed introduction to lubricant rheology. …I am confident that the work is both technically sound and practically relevant. The author appears to take care in providing a systematic and rigorous treatment and conveys a good appreciation for the engineering implications of his work. …This book provides a welcome contribution to the topic of elastohydrodynamic lubrication (EHL), which is critically relevant to the operation of gears, rolling element bearings and several other machine components. The author demonstrates pitfalls associated with traditional numerical approaches and effectively applies analytical asymptotic methods to develop solutions for various EHL regimes, including those with thermal, non-Newtonian and transient effects. …This book introduces researchers in tribology to a powerful mathematical technique that promises to advance lubrication theory."

––Jeffrey L. Streator, G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, USA

I Basic Properties of Solids and Fluids Involved in Lubricated Contacts

Basic Properties and Rheology of Lubricating Oils

Introduction

Rheology Relationships for Lubricating Oils

Polymer Thickening and Shear Stability

Closure

Exercises and Problems

References

Basic Properties of Elastic Solids

Introduction

Material Density

Material Elastic Properties

Material Thermal Diffusivity and Specific Heat Capacity

Closure

References

II Asymptotic Methods and Relationships Relevant to Elastohydrodynamic Lubrication Theory

Basics of Asymptotic Expansions and Methods

Introduction

Ordering, Order Sequences, and Asymptotic Expansions

Asymptotic Sequences and Expansions

Asymptotic Methods

Closure

Exercises and Problems

References

Basics of the Theory of Elastohydrodynamically Lubricated (EHL) Contacts

Introduction

Simplified Navier-Stokes and Energy Equations

Some Classic Results for Smooth Elastic Solids

Closure

Exercises and Problems

References

III EHL Problems for Lightly Loaded Line and Point Contacts

Lightly Loaded Lubrication Regimes for Line and Point Contacts

Introduction

Lightly Loaded Lubrication Regimes for Line Contacts

Asymptotic Approach to Lightly Loaded Point EHL Problems

Closure

Exercises and Problems

References

IV Isothermal EHL Problems for Heavily Loaded Line Contacts with Newtonian Lubricant

Asymptotic Approaches to Heavily Loaded Lubricated Line Contacts

Introduction

Numerical Solution for EHL Contacts

Pre-Critical Lubrication Regimes

Practical Use of the Asymptotic Solutions

Compressible Fluids in Heavily Loaded Contacts

Over-Critical Lubrication Regimes

Choosing Pre- or Over-Critical Lubrication Regimes

Analysis of the Ertel-Grubin Method

Numerical Solution of Asymptotic Equations

Numerical Validation of the Asymptotic Analysis

Numerical Precision, Grid Size, and Stability Considerations

Closure

Exercises and Problems

References

V Isothermal and Thermal EHL Problems for Line Contacts and Lubricants with Newtonian and Non-Newtonian Rheologies

Thermal EHL Problems for Line Contacts

Introduction

Formulation of TEHL Problem

Analytical Approximations for Newtonian Fluids in TEHL Contacts

Numerical Solutions of Asymptotic TEHL Problems for Newtonian Lubricants

Closure

Exercises and Problems

References

Regularization of the Isothermal EHL Problems

Introduction

Regularized Solution of Asymptotic Problems

Numerical Method for the Isothermal EHL Problem

Some Numerical Solutions of the Regularized Isothermal EHL Problem

Closure

Exercises and Problems

References

Approximations for Non-Newtonian Fluids in Line Contacts

Introduction

Formulation of Isothermal Line EHL Problem for Non-Newtonian Fluids

Isothermal Line EHL Problem for Pure Rolling. Pre- and Over-Critical Lubrication Regimes. Some Numerical Examples

Isothermal Line EHL Problem for Relatively Large Sliding. Pre- and Over-Critical Lubrication Regimes

Choosing Pre- and Over-critical Lubrication Regimes for Non-Newtonian Lubricants

Closure

Exercises and Problems

References

TEHL Problems for Non-Newtonian Lubricants in Line Contacts

Introduction

TEHL Problem Formulation for Non-Newtonian Fluids

Asymptotic Analysis of the Problem for Heavily Loaded Line Contacts

Heat Transfer in the Contact Solids

Regularization for Non-Newtonian Fluids

Friction in Heavily Loaded Lubricated Contacts

Closure

Exercises and Problems

References

VI Stress-Induced Lubricant Degradation in Line EHL Contacts

Introduction

Lubricant Flow Topology

Numerical Method for EHL Problems

EHL Solutions for Lubricants with Degradation

Closure

Exercises and Problems

References

VII Isothermal and Thermal EHL Problems for Point Contacts and Lubricants with Different Rheologies

Isothermal EHL Problems for Heavily Loaded Point Contacts with Newtonian Lubricants

Introduction

Problem Formulation

Pre-Critical Lubrication Conditions

Over-Critical Lubrication Conditions

Discussion and Validation of Results

Closure

Exercises and Problems

References

Isothermal EHL Point Contacts with Skewed Direction of Entrained Lubricant

Introduction

Problem Formulation

Pre-Critical Lubrication Conditions

Over-Critical Lubrication Conditions

Closure

Exercises and Problems

References

Lubricated Heavily Loaded Rolling and Spinning Ball in a Grooved Raceway

Introduction

Problem Formulation

Case of Pure Spinning

Pre-Critical Lubrication Conditions

Over-Critical Lubrication Conditions

Closure

Exercises and Problems

References

Thermal EHL Problems for Heavily Loaded Point Contacts with Newtonian Lubricants

Introduction

Problem Formulation

Asymptotic Solution for the Lubricant Temperature T. Case of Not Spinning EHL Contact

Approximations of Sliding Frictional Stresses. No Spinning EHL Contact Case

Asymptotic Solution for the Lubricant Temperature T. Case of EHL Contact with Spinning

Regularization Approaches to Solution of Isothermal Point EHL Problems

Closure

Exercises and Problems

References

Isothermal EHL Problems for Heavily Loaded Point Contacts. Non-Newtonian Lubricants

Introduction

Problem Formulation

Case of Pure Rolling

Case of High Slide-to-Roll Ratio. Approximation of the Reynolds Equation

Closure

Exercises and Problems

References

Thermal EHL Problems for Heavily Loaded Point Contacts. Non-Newtonian Lubricants

Introduction

Problem Formulation

Solutions for fx and fy. High Slide-to-Roll Ratio

Solution for Lubricant Temperature T. High Slide-to-Roll Ratio

Approximate Reynolds Equation. High Slide-to-Roll Ratio

Straight Lubricant Entrainment. High Slide-to-Roll Ratio

Skewed Lubricant Entrainment without Spinning. High Slide-to-Roll Ratio

Skewed Lubricant Entrainment with Spinning. High Slide-to-Roll Ratio

Regularization Approach

Closure

Exercises and Problems

References

VIII Some Other Topics in Elastohydrodynamic Lubrication

Analysis of EHL Contacts for Soft Solids

Introduction

Formulation of an EHL Problem for Soft Solids

Qualitative Analysis of the EHL Problem

Surface Velocities for Soft Solids

Dimensionless Variables and Numerical Method

Numerical Results and Discussion

Closure

Exercises and Problems

References

Non-Newtonian Lubricants and Scale Effects

Introduction

Results of Perturbation Analysis

Application of Analytical Result to an EHL Experiment

Numerical Solution

Application of Numerical Results to an EHL Experiment

Closure

Exercises and Problems

References

Lubrication of Line Contacts by Greases

Introduction

Formulation of the EHL Problems for Greases

Properties of the Problem Solution for Greases

Greases in a Contact of Rigid Solids

Regimes of Grease Lubrication without Cores

Closure

Exercises and Problems

References

Introduction

Non-Steady Lubrication of a Journal Bearing

Closure

Exercises and Problems

References

Lubricant Starvation and Mixed Friction Problems for Point Contacts

Introduction

Starved Lubrication and Lubricant Meniscus

Formulation and Analysis of a Mixed Lubrication Problem

Dry Narrow Contact of Elastic Solids

Closure

Exercises and Problems

References

Final Remarks

Introduction

EHL Contacts for Rough Surfaces

Contact Fatigue

References