Simulation and Modeling of the Hydrodynamic, Thermal, and Structural Behavior of Foil Thrust Bearings by Robert Jack Bruckner Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisor: Dr. Joseph M. Prahl Department of Mechanical and Aerospace Engineering CASE WESTERN RESERVE UNIVERSITY August, 2004 Dedications All of the work leading to and included in these pages is dedicated to the loving support of my family, Lisa, Eric, and Elisabeth Table of Contents CHAPTER 1 Introduction to Foil Bearings .........................................................................................17 1.1 Historical Context of Hydrodynamics.............................................................17 1.2 Foil Bearing State of the Art...........................................................................19 1.3 Aviation Turbofan Engine Application...........................................................22 1.4 Typical Geometries and Characteristics of Foil Thrust Bearings.....................23 CHAPTER 2 Development of the Governing Equations......................................................................29 2.1 The Generalized Foil Bearing Problem...........................................................29 2.2 Reynolds Equation .........................................................................................29 2.2.1 Development from Mass and Momentum Conservation..............................29 2.2.3 Cylindrical (Thrust Pad) Form of Reynolds Equation .................................40 2.3 Density Models ..............................................................................................43 2.3.1 Power Law Models.....................................................................................43 2.3.2 The Energy Equation..................................................................................44 CHAPTER 3 Solution Technique and Numerical Method ...................................................................54 3.0 Overview .......................................................................................................54 3.1 Derivative Expansions and Coefficient Functions...........................................55 3.2 Finite Differencing Scheme............................................................................56 3.3 Boundary Conditions......................................................................................57 3.4 Iteration Strategy............................................................................................59 1 3.5 Mathematica Code .........................................................................................60 CHAPTER 4 One-Dimensional Bearing Analysis ...............................................................................65 4.1 Analytic Solutions..........................................................................................65 4.1.1 Rigid Straight Taper Bearing......................................................................66 4.1.2 Rayleigh Step Bearing................................................................................68 4.1.3 Double Taper Bearing ................................................................................72 4.1.4 Isobaric Channel Flow with Viscous Heat Generation ................................77 4.2 Numerical Solutions.......................................................................................80 4.2.1 Parallel Channel Flow with Viscous Heat Generation.................................80 4.2.2 Power Law Density Models........................................................................82 4.2.3 Compliant Structure ...................................................................................84 4.2.4 Viscous Heat Generation............................................................................85 4.3 General Observation.......................................................................................88 CHAPTER 5 Cylindrical (Thrust Pad) Bearing Analysis.....................................................................90 5.1 Impacts of Side Leakage for Zero Preload ......................................................90 5.2 Impacts of Side Leakage for Highly Loaded Bearings ..................................110 5.2.1 Incompressible, Constant Property Lubricant ...........................................111 5.2.2 Impacts of Power Law Density Models ....................................................115 5.2.3 Impacts of Compliant Foundation.............................................................119 5.5 Impacts of Viscous Heat Generation.............................................................124 5.6 General Observation.....................................................................................130 2 CHAPTER 6 Comparison of Simulations and Experiments...............................................................131 6.1. Low Load Wear ...........................................................................................131 6.2 Foil Thermocouple Test ...............................................................................135 6.2 Torque versus Load Data..............................................................................140 CHAPTER 7 Summary and Conclusion............................................................................................142 Appendix A.................................................................................................................146 Summary of Governing Equations ...............................................................................146 Appendix B Mathematica Iterator Loop ..........................................................................................150 Appendix C Closed form solution to the Rayleigh step bearing .......................................................151 Appendix D Closed form solution to the double taper bearing .........................................................155 References...................................................................................................................159 3 List of Tables Table 1. Generation I foil thrust bearing parameters......................................................28 Table 2. Table of characteristic squeeze number and modified Reynolds number..........33 4 List of Figures Figure 1. Figure of Osborne Reynolds original hydrodynamic analysis. ........................25 Figure 2. Typical foil journal bearing............................................................................26 Figure 3. Figure of foil thrust bearing having complex structure. ..................................26 Figure 4. Figure of the Generation I foil thrust bearing .................................................27 Figure 5. Typical top foil shape for generation I foil thrust bearings..............................27 Figure 6. Typical load deflection curve for an 8 pad generation I foil thrust bearing......28 Figure 7. Representation of a rectilinear foil bearing.....................................................52 Figure 8. Representation of a thrust foil bearing...........................................................52 Figure 9. Diagram of the energy equation control volume.............................................53 Figure 10. Diagram of the division of the energy equation domain................................53 Figure 11. Typical numerical grid in thrust pad geometry. ............................................60 Figure 12. Program flow chart. .....................................................................................61 Figure 13. Sample of program convergence on load......................................................62 Figure 14. Mathematica code to solve Reynolds equation in cylindrical coordinates. ....63 Figure 15. Mathematica code for solving the structural model in cylindrical coordinates. ..............................................................................................................................64 Figure 16. Load capacity versus runner drag and power loss for a rigid, straight taper bearing with an incompressible, constant property lubricant...................................67 Figure 17. Operating characteristics of the straight taper bearing having contraction ratios, k, of 10, 5, 2.2, and 1.5. ...............................................................................68 Figure 18. Load capacity contours for the Rayleigh step bearing as a function of contraction ratio, k, and change location, cl............................................................70 5 Figure 19. Load capacity versus power loss for the Rayleigh step bearing.....................71 Figure 20. Operating characteristics of the Rayleigh step bearing having change locations, cl, of 0.9, 0.718, 0.5, and 0.2. .................................................................72 Figure 21. Load capacity plot for the double taper bearing as a function of inlet contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.25. ..............................................................................................................................74 Figure 22.Load capacity plot for the double taper bearing as a function of inlet contraction ratio, ka, and exit contraction ratio, kb, for a change location, cl, or 0.50. ..............................................................................................................................74 Figure 23. Load capacity plot for the double taper bearing as a function