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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Covers the latest developments in modeling elastohydrodynamic lubrication (EHL) problems using the finite element method (FEM) This comprehensive guide introduces readers to a powerful technology being used today in the modeling of elastohydrodynamic lubrication (EHL) problems. It provides a general framework based on the finite element method (FEM) for dealing with multi-physical problems of complex nature (such as the EHL problem) and is accompanied by a website hosting a user-friendly FEM software for the treatment of EHL problems, based on the methodology described in the book. Finite Element Modeling of Elastohydrodynamic Lubrication Problems begins with an introduction to both the EHL and FEM fields. It then covers Standard FEM modeling of EHL problems, before going over more advanced techniques that employ model order reduction to allow significant savings in computational overhead. Finally, the book looks at applications that show how the developed modeling framework could be used to accurately predict the performance of EHL contacts in terms of lubricant film thickness, pressure build-up and friction coefficients under different configurations. Finite Element Modeling of Elastohydrodynamic Lubrication Problems offers in-depth chapter coverage of Elastohydrodynamic Lubrication and its FEM Modeling, under Isothermal Newtonian and Generalized-Newtonian conditions with the inclusion of Thermal Effects; Standard FEM Modeling; Advanced FEM Modeling, including Model Order Reduction techniques; and Applications, including Pressure, Film Thickness and Friction Predictions, and Coated EHL. This book: * Comprehensively covers the latest technology in modeling EHL problems * Focuses on the FEM modeling of EHL problems * Incorporates advanced techniques based on model order reduction * Covers applications of the method to complex EHL problems * Accompanied by a website hosting a user-friendly FEM-based EHL software Finite Element Modeling of Elastohydrodynamic Lubrication Problems is an ideal book for researchers and graduate students in the field of Tribology.
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Table of Contents
Title Page
Copyright
Preface
Nomenclature
About the Companion Website
Part I: Introduction
Chapter 1: Elastohydrodynamic Lubrication (EHL)
1.1 EHL Regime
1.2 Governing Equations in Dimensional Form
1.3 Governing Equations in Dimensionless Form
1.4 Lubricant Constitutive Behavior
1.5 Dimensionless Groups
1.6 Review of EHL Numerical Modeling Techniques
1.7 Conclusion
References
Chapter 2: Finite Element Method (FEM)
2.1 FEM: The Basic Idea
2.2 Model Partial Differential Equation (PDE)
2.3 Steady-State Linear FEM Analysis
2.4 Steady-State Nonlinear FEM Analysis
2.5 Transient FEM Analysis
2.6 Multi-Physical FEM Analysis
2.7 Stabilized FEM Formulations
2.8 Conclusion
References
Part II: Finite Element Modeling Techniques
Chapter 3: Steady-State Isothermal Newtonian Line Contacts
3.1 Contact Configuration
3.2 Geometry, Computational Domains, and Meshing
3.3 Governing Equations and Boundary Conditions
3.4 FEM Model
3.5 Overall Solution Procedure
3.6 Model Calibration and Preliminary Results
3.7 Conclusion
References
Chapter 4: Steady-State Isothermal Newtonian Point Contacts
4.1 Contact Configuration
4.2 Geometry, Computational Domains, and Meshing
4.3 Governing Equations and Boundary Conditions
4.4 FEM Model
4.5 Overall Solution Procedure
4.6 Model Calibration and Preliminary Results
4.7 Conclusion
References
Chapter 5: Steady-State Thermal Non-Newtonian Line Contacts
5.1 Contact Configuration
5.2 Geometry, Computational Domains, and Meshing
5.3 Governing Equations and Boundary Conditions
5.4 FEM Model
5.5 Overall Solution Procedure
5.6 Model Calibration and Preliminary Results
5.7 Conclusion
References
Chapter 6: Steady-State Thermal Non-Newtonian Point Contacts
6.1 Contact Configuration
6.2 Geometry, Computational Domains, and Meshing
6.3 Governing Equations and Boundary Conditions
6.4 FEM Model
6.5 Overall Solution Procedure
6.6 Model Calibration and Preliminary Results
6.7 Conclusion
References
Chapter 7: Transient Effects
7.1 Contact Configuration
7.2 Geometry, Computational Domains, and Meshing
7.3 Governing Equations, Boundary, and Initial Conditions
7.4 FEM Model
7.5 Overall Solution Procedure
7.6 Preliminary Results
7.7 Conclusion
References
Chapter 8: Model Order Reduction (MOR) Techniques
8.1 Introduction
8.2 Reduced Solution Space Techniques
8.3 Static Condensation with Splitting (SCS)
8.4 Conclusion
References
Part III: Applications
Chapter 9: Pressure and Film Thickness Predictions
9.1 Introduction
9.2 Qualitative Parametric Analysis
9.3 Quantitative Predictions
9.4 Analytical Film Thickness Predictions
9.5 Conclusion
References
Chapter 10: Friction Predictions
10.1 Introduction
10.2 Quantitative Predictions
10.3 Friction Regimes
10.4 Conclusion
References
Chapter 11: Coated EHL Contacts
11.1 Introduction
11.2 Modeling Subtleties
11.3 Influence of Coating Properties on EHL Contact Performance
11.4 Conclusion
References
Part III: Applications
Appendix A: Numerical Integration
A.1 Line Elements
A.2 Triangular Elements
A.3 Rectangular Elements
A.4 Tetrahedral Elements
A.5 Prism Elements
Appendix B: Sparse Matrix Storage
B.1 Triplet Storage (TS)
B.2 Compressed Row Storage (CRS)
B.3 Compressed Column Storage (CCS)
Appendix C: Shell T9 Lubricant Properties
C.1 Pressure and Temperature Dependence of Density
C.2 Pressure and Temperature Dependence of Viscosity
C.3 Shear Dependence of Viscosity
C.4 Pressure Dependence of Limiting Shear Stress
C.5 Pressure and Temperature Dependence of Thermal Properties
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part I: Introduction
List of Illustrations
Chapter 1: Elastohydrodynamic Lubrication (EHL)
Figure 1.1 Stribeck curve delineating different fluid lubrication regimes.
Figure 1.2 Geometries of (a) conformal and (b) non-conformal contacts.
Figure 1.3 (a) Journal bearing, (b) roller-element bearing, and (c) gears.
Figure 1.4 Geometry of the contacting solids in a general EHL point contact.
Figure 1.5 Kinematics of a general EHL point contact.
Figure 1.6 Free body diagrams in the (a)
x
-direction and (b)
y
-direction for an infinitesimal volume of fluid within the lubricant film.
Figure 1.7 Velocity field within an EHL contact.
Figure 1.8 Mass flow rates into and out of an infinitesimal volume of fluid.
Figure 1.9 EHL film geometry (
xz
-view).
Figure 1.10 Equivalent/reduced geometry of a general EHL point contact.
Figure 1.11 Free body diagram of an infinitesimal volume of solid in the (a)
x
-direction, (b)
y
-direction, and (c)
z
-direction.
Figure 1.12 Elastic deformation of a half-space under the effect of an externally applied point load.
Figure 1.13 Heat interactions over an infinitesimal volume of the solid or fluid domains.
Figure 1.14 Typical double-Newtonian modified Carreau lubricant shear-thinning response at fixed pressure and temperature.
Chapter 2: Finite Element Method (FEM)
Figure 2.1 Approximation of a typical 1D solution of a PDE-governed problem over its domain of application Ω by first-order piecewise polynomials.
Figure 2.2 Typical (a) 1D, (b) 2D, and (c) 3D geometrical domain of application of model PDE with its interior domain
Ω
and boundary .
Figure 2.3 Typical 2D mesh cases: (a) Non-regular non-structured triangular mesh of a random domain, (b) regular structured rectangular mesh of a rectangular domain, and (c) regular structured triangular mesh of a rectangular domain.
Figure 2.4 Typical FEM element shapes: (a) 1D line element, (b) 2D triangular element, (c) 2D rectangular element, (d) 3D tetrahedral element, (e) 3D prism element, and (f) 3D brick element.
Figure 2.5 Overshooting and undershooting problems in high-order interpolation.
Figure 2.6 Use of Pascal's triangle for the definition of terms in 2D Lagrange elements of first order: (a) triangular and (b) rectangular.
Figure 2.7 Reference Lagrange linear elements: (a) 2-node line element, (b) 3-node triangular element, (c) 4-node rectangular element, (d) 4-node tetrahedral element, and (e) 6-node prism element.
Figure 2.8 Use of Pascal's triangle for the definition of terms in 2D second-order Lagrange (a) triangular and (b) rectangular elements.
Figure 2.9 Reference Lagrange quadratic elements: (a) 3-node line element, (b) 6-node triangular element, (c) 9-node rectangular element, (d) 10-node tetrahedral element, and (e) 18-node prism element.
Figure 2.10 Mapping between master and actual elements: (a) line element, (b) triangular element, (c) rectangular element, (d) tetrahedral element, and (e) prism element.
Figure 2.11 Pseudocodes for the assembly process of the global stiffness matrix and source vector.
Figure 2.12 Meshing and node numbering of the solution domain Ω.
Figure 2.13 Solution of the example 1D steady-state linear problem.
Figure 2.14 Flowchart of the damped-Newton method.
Figure 2.15 Pseudocodes of the assembly procedure for multi-physical problems.
Figure 2.16 Standard Galerkin solution of the 1D convection-diffusion problem with and , for different values of
β
x
.
Figure 2.17 Solution of the 1D convection-dominated convection-diffusion problem () using stabilized FEM formulations: (a) Isotropic Diffusion (ID), (b) Streamline Upwind Petrov–Galerkin (SUPG), and (c) Galerkin Least Squares (GLS).
Chapter 3: Steady-State Isothermal Newtonian Line Contacts
Figure 3.1 Equivalent / reduced geometry of a line contact.
Figure 3.2 Computational domains of the EHL line contact problem.
Figure 3.3 Meshing of the computational domains of line contacts:
extra coarse
,
normal
, and
extra fine
mesh cases.
Figure 3.4 Pressure solution of a typical highly loaded steady-state isothermal Newtonian EHL line contact using (a) standard Galerkin and (b) stabilized (SUPG or GLS) formulations.
Figure 3.5 (a) Dimensionless Hertzian pressure distribution
P
and (b) corresponding total elastic deformation
z
-component
W
of the contacting solids, over the contact domain
Ω
c
.
Figure 3.6 Dimensionless central and minimum film thickness mesh sensitivity analysis for a typical heavily loaded line contact case.
Figure 3.7 Effect of penalty term on (a) pressure distribution and (b) lubricant film thickness.
Figure 3.8 Lubricant film break-up in the outlet region of the contact.
Figure 3.9 Effect of solid domain size on total elastic deformation of contacting solids.
Figure 3.10 Dimensionless pressure and lubricant film thickness distributions over the contact domain for typical (a) lightly loaded and (b) heavily loaded contacts.
Chapter 4: Steady-State Isothermal Newtonian Point Contacts
Figure 4.1 Equivalent/reduced geometry of a point contact.
Figure 4.2 Computational domains of the EHL point contact problem.
Figure 4.3 Meshing of the computational domains of point contacts:
extra coarse
,
normal
, and
extra fine
mesh cases.
Figure 4.4 Pressure solution of a typical highly loaded steady-state isothermal Newtonian EHL circular contact using (a) standard Galerkin, (b) SUPG, and (c) SUPG + ID formulations.
Figure 4.5
(
a) Dimensionless Hertzian pressure distribution
P
and (b) corresponding total elastic deformation
z
-component
W
of the contacting solids, over the contact domain
Ω
c
, for the circular contact case ().
Figure 4.6 Dimensionless central and minimum film thickness mesh sensitivity analysis for a typical heavily loaded point contact case.
Figure 4.7 Cavitation zone and free cavitation boundary of a typical point contact.
Figure 4.8 Dimensionless pressure and film thickness profiles for test case A.
Figure 4.9 Dimensionless pressure and film thickness profiles for test case B.
Figure 4.10 Dimensionless pressure and film thickness profiles for test case C.
Chapter 5: Steady-State Thermal Non-Newtonian Line Contacts
Figure 5.1 Computational domain of the thermal part.
Figure 5.2
Extra coarse
mesh case for the (a) EHL domain and (b) thermal domain.
Figure 5.3 Shear-thinning response of selected lubricants.
Figure 5.4 Film thickness and temperature convergence with respect to mesh size: (a) moderately loaded case () and (b) highly loaded case ().
Figure 5.5 Flowcharts of the considered weak-coupling strategies.
Figure 5.6 Dimensionless pressure and film thickness distribution over the contact width: (a) moderately loaded case and (b) highly loaded case .
Figure 5.7 Temperature rise over the plane surface , the mid-layer of the lubricant film , and the cylinder surface : (a) moderately loaded case and (b) highly loaded case .
Figure 5.8 Dimensionless lubricant shear stress distribution on the plane surface over the contact width: (a) moderately loaded case and (b) highly loaded case .
Chapter 6: Steady-State Thermal Non-Newtonian Point Contacts
Figure 6.1 Computational domain of the thermal part.
Figure 6.2
Extra coarse
mesh case for the (a) EHL domain and (b) thermal domain.
Figure 6.3 Film thickness and temperature convergence with respect to mesh size: (a) moderately loaded case () and (b) highly loaded case ().
Figure 6.4 Dimensionless pressure profiles over the contact domain: (a) Moderately loaded case and (b) highly loaded case .
Figure 6.5 Dimensionless film thickness profiles over the contact domain: (a) moderately loaded case and (b) highly loaded case .
Figure 6.6 Temperature rise over the mid-layer of the lubricant film : (a) moderately loaded case and (b) highly loaded case .
Figure 6.7 Dimensionless lubricant resultant shear stress distribution on the plane surface over the contact domain: (a) moderately loaded case and (b) highly loaded case .
Chapter 7: Transient Effects
Figure 7.1 Shapes of (a) indent and (b) bump surface features placed on one of the contacting surfaces.
Figure 7.2 Dimensionless pressure and film thickness profiles for the indent overrolling simulation at different dimensionless times , corresponding to different locations
X
f
of the surface feature within the contact.
Figure 7.3 Dimensionless pressure and film thickness profiles for the bump overrolling simulation at different dimensionless times corresponding to different locations
X
f
of the surface feature within the contact.
Chapter 8: Model Order Reduction (MOR) Techniques
Figure 8.1 Dimensionless pressure and film thickness profiles obtained using the modal reduction technique for the case , () using (a) and (b) mode shapes.
Figure 8.2 Dimensionless pressure and film thickness profiles obtained using the Ritz-vector-like method for the case , () using (a) and (b) Ritz vectors.
Figure 8.3 Composition of the EHL basis for the low-, medium-, and high-
M
regimes.
Figure 8.4 Dimensionless pressure and film thickness profiles obtained using the EHL-Basis technique for three different line contact test cases: a) , , (low
M
), b) , , (medium
M
) and c) , , (high
M
).
Figure 8.5 Dimensionless pressure and film thickness profiles for the circular contact test case: , , (low
M
).
Figure 8.6 Dimensionless pressure and film thickness profiles for the circular contact test case: , , (medium
M
).
Figure 8.7 Dimensionless pressure and film thickness profiles for the circular contact test case: , , (high
M
).
Figure 8.8 Comparison of dimensionless pressure and film thickness profiles along the central line of the contact in the
x
-direction obtained using the full and reduced models for the three considered circular contact cases: (a) , , (low
M
), (b) , , (medium
M
), and (c) , , (high
M
).
Figure 8.9 Absolute dimensionless pressure deviation and relative dimensionless film thickness deviation between the reduced and full models for the circular contact test case: , .
Figure 8.10 Absolute dimensionless pressure deviation and relative dimensionless film thickness deviation between the reduced and full models for the circular contact test case: , .
Figure 8.11 Absolute dimensionless pressure deviation and relative dimensionless film thickness deviation between the reduced and full models for the circular contact test case: , .
Figure 8.12 Line contact dimensionless pressure and film thickness profiles for a typical (a) lightly loaded contact and (b) highly loaded contact.
Figure 8.13 Circular contact dimensionless pressure and film thickness profiles for a typical (a) lightly loaded contact and (b) highly loaded contact.
Figure 8.14 Decay of normalized coefficients with distance for a considered node in the vicinity of the contact center: (a) line contact and (b) circular contact.
Figure 8.15 Dimensionless elastic deformation of the equivalent solid for the line contact case under unusual loading patterns: (a) triangular and (b) step.
Figure 8.16 Dimensionless elastic deformation of the equivalent solid along the central line of the contact in the
x
-direction for the circular contact case under unusual loading patterns: (a) conical and (b) step.
Figure 8.17 Dimensionless pressure and film thickness profiles for the line contact case , , and with surface features: (a) indent, (b) bump, and (c) waviness.
Figure 8.18 Dimensionless pressure and film thickness profiles along the central line of the contact in the
x
-direction for the circular contact case , , and with surface features: (a) indent, (b) bump, and (c) waviness.
Chapter 9: Pressure and Film Thickness Predictions
Figure 9.1 Influence of external applied load on pressure and film thickness.
Figure 9.2 Influence of mean entrainment speed on pressure and film thickness.
Figure 9.3 Influence of lubricant ambient pressure viscosity on pressure and film thickness.
Figure 9.4 Influence of lubricant viscosity-pressure coefficient on pressure and film thickness.
Figure 9.5 Influence of lubricant compressibility on pressure and film thickness.
Figure 9.6 Influence of slide-to-roll ratio on (a) pressure, (b) film thickness, (c) temperature rise, and (d) shear stress.
Figure 9.7 Influence of thermal and shear-thinning effects on film thickness profile within circular contacts.
Figure 9.8 Film thickness curves for an external applied load ().
Figure 9.9 Film thickness curves for an external applied load .
Figure 9.10 Experimental validation of film thickness correction factors for (a) central and (b) minimum film thickness predictions in circular EHL contacts.
Figure 9.11 Experimental validation of film thickness analytical formulas for (a) central and (b) minimum film thickness predictions in circular EHL contacts.
Chapter 10: Friction Predictions
Figure 10.1 Friction curves obtained from numerical simulations (with variable and constant lubricant thermal properties) and experiments, under different load and mean entrainment speed conditions: (a) and , (b) and , (c) and , and (d) and .
Figure 10.2 Lubricant thermal conductivity variations across the film thickness at different locations along the central line of the contact in the
x
-direction for the case , , and .
Figure 10.3 Lubricant volumetric heat capacity variations across the film thickness at different locations along the central line of the contact in the
x
-direction for the case , , and .
Figure 10.4 Temperature variations across the film thickness at different locations along the central line of the contact in the
x
-direction for the case , , and .
Figure 10.5 Total amount of heat transfer from the lubricant film toward the contacting solids across the two fluid–solid interfaces, for the case and .
Figure 10.6 Typical friction curves under low loading conditions.
Figure 10.7 Typical friction curves under moderate loading conditions.
Figure 10.8 Typical friction curves under high loading conditions.
Figure 10.9 Typical friction curves under very low loading and relatively low mean entrainment speed conditions.
Figure 10.10 Flowchart for qualitative delineation of friction regimes.
Figure 10.11 Measured friction for dry and lubricated (using Shell T9) contacts at very low
SRR
x
, in a crossed-roller instrument.
Figure 10.12 Delimitation of the linear friction regime using the Weissenberg dimensionless number
Wi
.
Figure 10.13 Delimitation of the nonlinear viscous friction regime using the Weissenberg and limiting shear stress (LSS) dimensionless numbers,
Wi
and
Li
, respectively.
Figure 10.14 Delimitation of the plateau friction regime using the limiting shear stress (LSS) and thermoviscous indicator dimensionless numbers,
Li
and
Ti
, respectively.
Figure 10.15 Delimitation of the thermoviscous friction regime using the thermoviscous indicator dimensionless number
Ti
.
Figure 10.16 Flowchart for quantitative delineation of friction regimes.
Chapter 11: Coated EHL Contacts
Figure 11.1 Geometry of a ball-on-disk coated circular contact.
Figure 11.2 Computational domain for the EHL part of the coated TEHL circular contact model.
Figure 11.3 Computational domain for the thermal part of the coated TEHL circular contact model.
Figure 11.4 Influence of coating mechanical properties on dimensionless pressure and film thickness profiles, along the central line of the contact in the
x
-direction, in coated EHL circular contacts, for the case: , , .
Figure 11.5 Influence of coating thermal properties on dimensionless pressure and film thickness profiles, along the central line of the contact in the
x
-direction, in coated EHL circular contacts, for the case: , , .
Figure 11.6 Influence of soft and hard coating thickness on dimensionless pressure and film thickness profiles, along the central line of the contact in the
x
-direction, in coated EHL circular contacts, for the case: , .
Figure 11.7 Influence of coating thermo-mechanical properties on friction in coated EHL circular contacts (friction-reducing configurations with ).
Figure 11.8 Influence of coating thermo-mechanical properties on friction in coated EHL circular contacts (friction-increasing configurations with ).
Figure 11.9 Influence of coating thickness on friction in coated EHL circular contacts ().
Figure 11.10 Influence of coating mechanical properties on dimensionless pressure distribution along the central line of the contact in the
x
-direction for and .
Figure 11.11 Influence of coating mechanical properties on resultant shear stress distribution along the central line of the contact in the
x
-direction, over the mid-layer of the lubricant film, for and .
Figure 11.12 Heat generation and removal within EHL contacts.
Figure 11.13 Influence of coating thermal properties on temperature distribution along the central line of the contact in the
x
-direction over the mid-layer of the lubricant film, for and .
Figure 11.14 Temperature distribution across the solids (substrates coatings) and lubricant film in the
z
-direction at different
X
positions along the central line of the contact in the
x
-direction (, , and ), for the case , , and .
Figure 11.15 Influence of coating thermal inertia on friction up to , for the case ().
Figure 11.16 Influence of coating thickness on dimensionless temperature distribution within the ball in the
z
-direction, along a line passing through the contact center, for and .
Figure 11.17 Influence of coating thickness on dimensionless temperature distribution across the lubricant film thickness in the
z
-direction, along a line passing through the contact center, for and .
Appendix A: Numerical Integration
Figure A.1 Reference elements: (a) 1D line element, (b) 2D triangular element, (c) 2D rectangular element, (d) 3D tetrahedral element, and (e) 3D prism element.
Figure A.2 Mapping between master and actual elements: (a) line element, (b) triangular element, (c) rectangular element, (d) tetrahedral element, and (e) prism element.
Appendix B: Sparse Matrix Storage
Figure B.1 Pseudocodes of the matrix-vector multiplication operation.
Appendix C: Shell T9 Lubricant Properties
Figure C.1 Pressure and temperature dependence of Shell T9 density.
Figure C.2 Pressure and temperature dependence of Shell T9 viscosity.
Figure C.3 Shear dependence of Shell T9 viscosity.
Figure C.4 Traction curve employed in determining the value of the limiting-stress pressure coefficient
Λ
.
Figure C.5 Pressure and temperature dependence of Shell T9 thermal conductivity.
Figure C.6 Pressure and temperature dependence of Shell T9 volumetric heat capacity.
List of Tables
Chapter 2: Finite Element Method (FEM)
Table 2.1 Mathematical formulation of the shape functions of Lagrange linear elements
Table 2.2 Mathematical formulation of the shape functions of Lagrange quadratic elements
Chapter 3: Steady-State Isothermal Newtonian Line Contacts
Table 3.1 Properties of the different considered mesh cases
Table 3.2 Effect of the penalty term on the pressure and film thickness solutions
Chapter 4: Steady-State Isothermal Newtonian Point Contacts
Table 4.1 Properties of the different considered mesh cases
Table 4.2 Effect of the penalty term on the pressure and film thickness solutions
Table 4.3 Operating conditions for test cases A, B, and C
Chapter 5: Steady-State Thermal Non-Newtonian Line Contacts
Table 5.1 Mesh specifications in terms of numbers of elements and degrees of freedom
Table 5.2 Lubricant properties, solid material properties, and operating conditions
Table 5.3 Comparison of numbers of iterations required for convergence of full- and weak-coupling strategies
Table 5.4 Comparison of execution times of full and weak-coupling strategies
Chapter 6: Steady-State Thermal Non-Newtonian Point Contacts
Table 6.1 Mesh specifications in terms of numbers of elements and degrees of freedom
Table 6.2 Lubricant properties, solid material properties, and operating conditions
Chapter 8: Model Order Reduction (MOR) Techniques
Table 8.1 Modified WLF parameters for the CPRI, CPRP, and PENNZ lubricants
Table 8.2 Line contact error behavior: comparison between the full and reduced models
Table 8.3 Line contact performance analysis: comparison between the full and reduced models
Table 8.4 Circular contact error behavior: comparison between the full and reduced models
Table 8.5 Circular contact performance analysis: comparison between the full and reduced models
Table 8.6 Properties of the different mesh cases considered for line and circular contacts
Table 8.7 Number of nonzero entries of [
T
ee
] and (before and after splitting) for the five different line and circular contact mesh cases
Table 8.8 Choice of stopping criterion for splitting algorithm within SCS
Table 8.9 Choice of relaxation factor for splitting algorithm within SCS
Table 8.10 Offline phase computational overhead of the SCS technique
Table 8.11 Relative dimensionless film thickness deviations between the full and SCS models for the line contact case
Table 8.12 Performance analysis for the line contact case: comparison between the full and SCS models
Table 8.13 Relative dimensionless film thickness deviations between the full and SCS models for the circular contact case
Table 8.14 Performance analysis for the circular contact case: comparison between the full and SCS models
Chapter 9: Pressure and Film Thickness Predictions
Table 9.1 Lubricant properties, solid materialproperties, and operating conditions
Table 9.2 Tait–Doolittle parameters of the model strong and fragile liquids
Table 9.3 Combinations of double-Newtonian modified Carreau parameters employed in numerical experiments
Table 9.4 Parameters and standard deviations for the correction factors
Table 9.5 Parameters and standard deviations for the complete film thickness formulas
Chapter 11: Coated EHL Contacts
Table 11.1 Operating conditions and solid material properties of numerical experiments
Appendix A: Numerical Integration
Table A.1 Gauss point locations and weights for the 1D reference line element
Table A.2 Gauss point locations and weights for the 2D reference triangular element
Table A.3 Gauss point locations and weights for the 2D reference rectangular element
Table A.4 Gauss point locations and weights for the 3D reference tetrahedral element
Table A.5 Gauss point locations and weights for the 3D reference prism element
Finite Element Modelling of Elastohydrodynamic Lubrication Problems
Wassim Habchi
Lebanese American University
Lebanon
This edition first published 2018
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Library of Congress Cataloging-in-Publication Data
Names: Habchi, Wassim, 1982- author.
Title: Finite element modeling of elastohydrodynamic lubrication problems /
by Wassim Habchi.
Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes
bibliographical references and index. |
Identifiers: LCCN 2017050348 (print) | LCCN 2018000724 (ebook) | ISBN
9781119225157 (pdf) | ISBN 9781119225140 (epub) | ISBN 9781119225126
(cloth)
Subjects: LCSH: Elastohydrodynamic lubrication. | Finite element method.
Classification: LCC TJ1077.5.E43 (ebook) | LCC TJ1077.5.E43 H33 2018 (print)
| DDC 621.8/9--dc23
LC record available at https://lccn.loc.gov/2017050348
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Cover Images: Front cover background © malija/iStockphoto;
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Preface
This book is intended for graduate students and/or researchers interested in modeling the elastohydrodynamic lubrication (EHL) problem using the finite element method (FEM). The level of details provided would allow readers to build their own in-house FEM-based EHL codes from scratch or to use any of the variety of available commercial FEM software to implement them. This latter option is probably the most attractive advantage of FEM modeling of the EHL problem. In fact, though FEM has grown over the years to become the dominant methodology used in scientific computing, for the EHL problem, the most widespread techniques today are based on the finite difference method. The lack of available commercial software allowing the implementation of finite difference codes is a major obstacle. The wide availability of commercial FEM software and handbooks is a clear testimony to the widespread use of the methodology in a variety of scientific areas. It is also a clear sign of maturity of the technology, which has been developed and carefully improved over the years to meet the complex requirements imposed by different fields of science and industries. Another advantage of FEM is its object-oriented nature, which allows a significant flexibility in extending models to include new features. Also, features such as non-structured meshing, high-order elements, and model order reduction (MOR) offer FEM a major advantage in terms of computational performance (memory and speed). The methodology detailed in this book also enables a straightforward incorporation of effects that are difficult (if not impossible) to include in traditional modeling approaches, for example, non-homogenous and/or anisotropic and/or nonlinear solid material properties, and plastic deformations.
EHL is a lubrication regime in which contacting surfaces in relative motion are fully separated by a lubricant film. It is a sub-field of tribology, the science of interacting surfaces in relative motion. Within EHL films, hydrodynamic pressures of several gigapascals may develop. This leads to an elastic deformation of the contacting solids, thus the name “elastohydrodynamic”. Also, film thicknesses may be as low as a few nanometers, and shear stresses within the lubricant film may reach hundreds of megapascals. Under such conditions, the Newtonian limit of most lubricants is exceeded, and significant heat generation by shear may occur within EHL conjunctions. Lubricant temperature rise within the film may exceed 100°C in some extreme cases. This makes EHL modeling a rather complex process involving some strong coupling between various physics: hydrodynamics, linear elasticity, heat transfer, rheology, and so on.
The author has gained extensive experience in FEM modeling of the EHL problem over the last decade. He started working on this topic as a PhD student at Institut National des Sciences Appliqées (INSA) de Lyon (France), back in 2005. Immediately after his PhD, he moved back to his home country, Lebanon, where he held an Assistant Professor of Mechanical Engineering position at the Lebanese American University (LAU), School of Engineering, between 2008 and 2014. In 2014, he became an Associate Professor of Mechanical Engineering at LAU and still holds this position to date. In his positions at LAU, the author continued his work on FEM modeling of the EHL problem, extending the methodology to incorporate some advanced and EHL-specific MOR techniques, but also using it to study a variety of EHL configurations: coated surfaces, complex rheology, and so on. The author also used the developed tools to help advance the fundamental understanding of friction generation and film-forming capability in EHL contacts. With this book, the author wishes to share his experience with graduate students and researchers interested in the topic. The author also hopes it will assist readers in building their own EHL solvers and using them to further advance the field of EHL.
The book is divided into three distinct parts. The first part is an introductory one in which both the EHL and FEM fields are introduced. Chapter 1 offers a complete and general overview of the EHL problem and lays down the equations governing the different coupled physics that are involved. In Chapter 2, the FEM method is introduced with enough details for non-familiar readers to be able to grasp the different modeling techniques introduced in subsequent chapters. Chapter 2 should not be viewed as a comprehensive coverage of the FEM method, but rather as a “just enough” coverage for the book to be complete and for readers to be able to go through it, without the need for further readings. Obviously, a full coverage of the FEM method would require several handbooks and is beyond the scope of the current book. Readers who are interested in a deeper treatment of FEM and its mathematical foundations may refer to the wide variety of available handbooks on the topic.
In the second part, the FEM techniques used to model the EHL problem are described in detail, under a variety of configurations. These techniques are implemented with in-house codes written in the standard C++ programming language. In Chapter 3, the modeling of the steady-state isothermal Newtonian line contact problem is described. Chapter 4 offers an extension to the more general case of a point contact. Chapters 5 and 6 describe the incorporation of non-Newtonian and thermal effects into the previously described models for line and point contacts, respectively. Chapter 7 details the incorporation of transient effects into the modeling of the EHL problem. Finally, Chapter 8 describes some advanced MOR techniques, specifically developed for the EHL problem to boost the computational performance of its corresponding FEM models in terms of both computational speed and memory requirements.
The last part covers some areas of application of the numerical tools developed in the second part and showcases how these could be used to establish a proper quantitative and fundamental understanding of the EHL problem. The author has used these tools over the years for a variety of applications. A complete coverage would be beyond the scope of this book. However, the author has selected the applications he thought would be most representative and interesting for readers willing to gain a deeper insight into the EHL problem. In Chapter 9, the developed tools are used to accurately predict pressure and film thickness in EHL contacts. Also, a proper understanding of the physical mechanisms behind film-forming capability is established. Chapter 10 covers the accurate prediction of friction in EHL contacts, which is a far more complex goal to achieve. Further, an interesting discussion on the delineation of EHL friction regimes using dimensionless groups is offered. Finally, Chapter 11 describes how surface coatings can be incorporated in the FEM analysis of EHL contacts, in a rather straightforward manner. It also offers an interesting discussion on how surface coatings may be selected on the basis of their thermo-mechanical properties to significantly enhance the frictional response of EHL contacts, without affecting the fatigue life of their corresponding machine components.
To conclude, the author wishes to express his extreme gratitude to Philippe Vergne from INSA de Lyon (France), his PhD advisor, for introducing him to the field of EHL, and to his co-advisor, Dominique Eyheramendy from Ecole Centrale de Marseille (France), for sharing his FEM knowledge and expertise. It was their scientific and social skills as well as their trust and insightful vision that allowed the author to start his research journey in the field of EHL on solid ground. The author is extremely grateful to his colleague and long-term friend Jimmy Issa for his careful reading of the second chapter of the book. The author also wishes to thank his many collaborators with whom he has had the pleasure of working and exchanging ideas on the EHL problem over the years. A special appreciation goes to Scott Bair from the Georgia Institute of Technology (United States), who has been more than a collaborator, an inspiration, and a “research soul mate”. Finally, the author wishes to dedicate this book to his family, his wife Maya and his baby girl Leah, for their endless love and support.
About the Companion Website
Don't forget to visit the companion website for this book:
www.wiley.com/go/habchi/Modeling-of-EHD-Lubrication-Problems
There you will find valuable material designed to enhance your learning, including:
FEM-based EHL software with user-friendly GUI
The password to download companion website material: methodology
Scan this QR code to visit the companion website.
Part IIntroduction
Chapter 1Elastohydrodynamic Lubrication (EHL)
1.1 EHL Regime
Fluid film lubrication is an essential mechanism for the safe operation of many machine elements/components in relative motion, for example, gears and roller-element bearings. It consists in separating contacting components in relative motion by inserting a high-viscosity fluid, known as a lubricant, between their corresponding surfaces. In a lubricated contact, the lubricant generally serves two distinct purposes. Primarily, it separates the contacting surfaces (partially or fully) and prevents direct solid-to-solid contact between surface asperities. On the one hand, this prevents wear in the corresponding components, providing a longer fatigue life. On the other hand, it leads to reduced friction and energy dissipation. A secondary purpose is that of cooling the lubricated components. In fact, the lubricant separating the contacting surfaces acts as an energy carrier. It enters the contact, extracts much of the heat generated by the relative motion of the surfaces, and carries it away from the contact. This prevents overheating and thermal damage of the contacting solids.
In general, three fluid lubrication regimes are defined, which are distinguishable by their range of friction coefficients on a Stribeck [1] curve, as illustrated in Figure 1.1. These are as follows:
Boundary lubrication
: A major part of the contact load is supported by the direct contact of the surface asperities. This regime is characterized by high friction coefficients, governed by the properties of the contacting solids.
Mixed lubrication
: The contact load is supported by both the direct contact of the surface asperities and the lubricant film. Friction coefficients for this regime are lower than for boundary lubrication and are governed by the properties of the solids as well as those of the lubricant.
Full film lubrication
: Contacting surfaces are fully separated by a lubricant film. Friction coefficients are relatively low and are governed by lubricant properties.
Figure 1.1 Stribeck curve delineating different fluid lubrication regimes.
Under full film lubrication, two sub-regimes may be distinguished:
Hydrodynamic lubrication
(HL): Pressures generated within the lubricating film are relatively low and do not induce any significant elastic deformation of the contacting solids. This is typical of conformal contacts, for which centers of curvature of the two contacting surfaces are located on the same side of the contact, as shown in
Figure 1.2
a. Such contacts are characterized by large contact areas and therefore low pressures. Journal bearings (see
Figure 1.3
a) are typical mechanical devices subject to hydrodynamic lubrication.
Elastohydrodynamic lubrication
