Advanced Dynamics of Rolling Elements

In any rotating machinery system, the bearing has traditionally been a crit- ical member of the entire system, since it is the component that permits the relative motion between the stationary and moving parts. Depending on the application, a number of different bearing types have been used, such as oil-lubricated hydrodynamic bearings, gas bearings, magnetic suspensions, rolling element bearings, etc. Hydrodynamic bearings can provide any desired load support, but they are limited in stiffness and the associated power loss may be quite large. Gas bearings are used for high-precision applications where the supported loads are relatively light, bearing power losses are very low, and the rotating speeds generally high. For super- precision components where no frictional dissipation or bearing power loss can be tolerated, magnetic suspensions are employed; again, the load support requirements are very low. Rolling element bearings have been widely used for those applications that require greater bearing versatility, due to the requirements for high-load and high-stiffness characteristics, while allowing moderate power loss and permitting variable speeds. A study of the dynamic interaction of rolling elements is, therefore, the subject of this text. Texts covering the analysis and design methodology of rolling elements are very limited. Notable works include Analysis of Stresses and Deflections (Jones, 1946, Vols. I and II), Ball and Roller Bearings, Their Theory, Design and Application (Eschmann, Hasbargen, and Weigand, 1958), Ball and Roller Bearing Engineering (Palmgren, 1959, 3rd ed. ), Advanced Bearing Technology (Bisson and Anderson, 1965), and Rolling Bearing Analysis (Harris, 1966).

Table of Contents:
1. Introduction.- 1.1 Rolling Bearing Elements and Basic Interactions.- 1.2 Types of Analytical Models.- 1.2.1 Quasi-Static Model.- 1.2.2 Dynamic Model.- 1.3 Nomenclature.- 1.3.1 Coordinate Frames.- 1.3.2 Vector Transformations.- 1.3.3 List of Symbols.- 1.4 Summary.- 2. Equations of Motion and Coordinate Transformations.- 2.1 Coordinate Frames and Transformations.- 2.2 Equations of Motion.- 2.2.1 Mass Center Motion.- 2.2.2 Rotational Motion.- 2.3 Moving Coordinate Frames.- 2.4 General Motion Simulation.- 2.5 Summary.- 3. Geometric Interactions in Rolling Bearings.- 3.1 Rolling Element/Race Interactions.- 3.1.1 Ball/Race Interactions.- 3.1.2 Roller/Race Interactions.- 3.1.3 Roller/Race-Flange Interactions.- 3.2 Rolling Element/Cage Interactions.- 3.2.1 Geometric and Kinematic Considerations.- 3.2.2 Hydrodynamic Models.- 3.2.3 Dry Contact Models.- 3.3 Race/Cage Interactions.- 3.3.1 Geometric and Kinematic Considerations.- 3.3.2 Hydrodynamic Models.- 3.3.3 Dry Contact Models.- 3.4 Interactions Between Rolling Elements.- 3.4.1 Ball Bearings.- 3.4.2 Roller Bearings.- 3.5 External System Interactions and Constraints.- 3.5.l Equilibrium Constraint for Ball Bearings.- 3.5.2 Equilibrium Constraint for Roller Bearings.- 3.6 Summary.- 4. Elastohydrodynamic Lubrication.- 4.1 General Consideration in Lubricant Traction Modeling.- 4.1.1 Rolling Element/Race Contact Zone.- 4.1.2 Lubricant Rheology.- 4.1.3 Typical Traction-Slip Behavior.- 4.2 An E1astohydrodynamic Traction Model.- 4.2.1 Film Thickness Computation.- 4.2.2 Computation of Traction.- 4.2.3 Estimation of Lubricant Constitutive Equation.- 4.3 Traction Behavior of Some Lubricants.- 4.3.1 U.S. Specification MIL-L-23699.- 4.3.2 U.S. Specification MIL-L-7808.- 4.3.3 Traction Fluid Santotrac 30.- 4.3.4 Polyphenyl Ether.- 4.3.5 SAE-30-Type Oil.- 4.4 Summary.- 5. Churning and Drag Losses.- 5.l Estimation of Drag Forces.- 5.2 Estimation of Churning Moments.- 5.2.1 Loss on the Cylindrical Surface.- 5.2.2 Loss on the End Surface.- 5.3 Effective Lubricant Viscosity and Density.- 5.4 Summary.- 6. Numerical Integration of the Equations of Motion.- 6.1 Dimensional Organization.- 6.2 Explicit Algorithms.- 6.2.1 Step-Changing Criterion.- 6.3 Implicit Algorithms.- 6.3.1 Predictor Formula.- 6.3.2 Corrector Formula.- 6.3.3 Step-Changing Criterion.- 6.3.4 Change of Order.- 6.3.5 Computational Considerations.- 6.4 Selection of a Method.- 6.5 External Constraints.- 6.5.1 Equilibrium Constraints.- 6.5.2 Fictitious Damping.- 6.6 Summary.- 7. The Computer Program ADORE.- 7.1 Program Overview.- 7.2 Structure of ADORE.- 7.3 ADORE Capabilities.- 7.3.1 Bearing Types.- 7.3.2 Types of Cages.- 7.3.3 Operating Conditions.- 7.3.4 External Constraints.- 7.3.5 Radial Preloads.- 7.3.6 Material Properties.- 7.3.7 Lubricant Traction.- 7.3.8 Churning and Drag.- 7.3.9 Roller Skew.- 7.3.10 Rolling Element Skid.- 7.3.11 Cage Instability.- 7.3.12 Bearing Power Loss.- 7.3.13 Wear.- 7.3.14 Geometric Imperfections.- 7.3.15 Bearing Noise.- 7.3.16 Bearing Life for Arbitrary Load and Speed Cycles.- 7.3.17 Flexibility in Units.- 7.3.18 Graphic Output.- 7.3.19 Integrating Algorithms.- 7.3.20 Restart Capabilities.- 7.4 Input/Output Data.- 7.4.1 Input Data.- 7.4.2 Print Output.- 7.4.3 Plot Output.- 7.4.4 User-Programmable Subroutines.- 7.4.5 Data Management in ADORE.- 7.5 Computer Resource Requirement.- 7.6 Summary.- 8. Some Dynamic Performance Simulations.- 8.1 Numerical Considerations.- 8.2 Vibrational Characteristics.- 8.2.1 Cylindrical Roller Bearings.- 8.2.2 Ball Bearings.- 8.3 General Ball Motion and Skid.- 8.4 Cage Stability.- 8.5 Roller and Cage Motion in Cylindrical Roller Bearings.- 8.5.1 Roller Misalignment and Skew.- 8.5.2 Time-Varying Loads and Speeds.- 8.6 Summary.- 9. Experimental Validation of ADORE.- 9.1 Ball Motion and Skid.- 9.2 Cage Motion.- 9.2.1 Cage Mass Center Orbit.- 9.2.2 Whirl Velocities.- 9.2.3 Coning Motion of Cage.- 9.3 Summary.- 10. Guidelines for Rolling Bearing Design.- 10.1 System Overview.- 10.2 Rotor-Bearing System Interaction.- 10.3 ADORE: A Design and Performance Diagnosis Tool.- 10.4 Summary.- Appendix I: Hertz Point-Contact Solutions.- Appendix II: Shrink Fit and Thermal Expansion of Races.- Appendix III: Fatigue Life Computation.- Appendix IV: Source Listing of ADORE.- Appendix V: Typical Example.- References.- Author Index.