Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches

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Features

  • Proposes a robust combination of asymptotic and numerical techniques to solve EHL problems for lightly and heavily loaded line and point contacts
  • Presents a regularization approach that produces stable solutions in heavily loaded EHL contacts
  • Explains the mechanisms taking place in various zones of heavily loaded line and point EHL contacts
  • Shows how to reduce the solution of thermal EHL problems to their isothermal analogs
  • Offers concrete ways to determine important design parameters, including lubrication film thickness and frictional stresses and forces
  • Presents problem solutions in the form of simple analytical formulas, graphs, and tables
  • Contains end-of-chapter exercises to reinforce key skills
  • Includes more than 150 illustrations and a four-page color insert

Summary

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.

Table of Contents

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

Lubricant Degradation in EHL Contacts
Introduction
EHL for Degrading Lubricants
Lubricant Flow Topology
Numerical Method for EHL Problems
Solutions for Lubricants without Degradation
EHL Solutions for Lubricants with Degradation
Lubricant Degradation and Contact Fatigue
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

Non-Steady EHL Problems
Introduction
Properly Formulated Non-Steady EHL Problems
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
Lubricant Degradation
Contact Fatigue
References

Author Bio(s)

Ilya I. Kudish is a professor of mathematics at Kettering University in Flint, Michigan. He is a recipient of all Kettering University research awards—Outstanding New Researcher, Outstanding Applied Researcher, Outstanding Researcher, and Distinguished Researcher Awards—as well as of Rodes and Oswald Professorships. Dr. Kudish is a Fellow of the American Society of Mechanical Engineers (ASME). He has served as a consultant to Caterpillar, Inc. and as a visiting professor at Purdue University (USA), Cardiff University (UK), and INSA (France). Dr. Kudish’s main interests are in the sphere of application mathematical methods to various problems of tribology. Over the years he has made theoretical contributions to the fields of elastohydrodynamic lubrication, stress-induced lubricant degradation, contact problems for coated/rough elastic solids, fracture mechanics, and fracture mechanics-based contact and structural fatigue modeling.

Editorial Reviews

"… 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