Theory of Journal Bearing:

The theoretical background of journal bearings including porous bearings, solid wall bearings with compressible as well as incompressible lubricants lies in References Wu [1] and [2]. As the solid wall bearing can be considered as a special case of the porous bearing, the analysis of the porous bearing can be easily applied to the solid wall bearing. Furthermore, the case of compressible lubricant differentiate that of incompressible lubricant in the non-linearity of the governing Reynolds equation as variable lubricant density opposed to constant lubricant density for the incompressible case. Nevertheless, solution methodologies for both compressible and incompressible cases remain the same in terms of numerical formulations and solver selections. Here, illustration is presented only for the general case of the porous journal bearing with the compressible lubricant.

Utilizing finite difference approximation with Newton-Raphson linearization, the resulted finite-difference equations are solved by the Column Vector method (see Reference Castelli and Pirvics [3]). The details of solution procedures are given in Reference Wu [2]. Once, the porous bearing case is solved, the solid wall bearing is derived as a special case. For the case of incompressible lubricant, the governing equations are reduced to linear partial equations with pressure (P) as the dependent variable. The similar solution techniques can be applied to obtain solutions. For bearing stiffness and damping coefficients, the center of the journal (shaft) is perturbed numerically to compute the variation of the bearing load. Therefore, the stiffness coefficients are calculated by the ratios of the load variations to the deviations of the journal center. The journal center deviation amount can be iterated until asymptotic behavior reveals. The damping coefficients are calculated with the squeeze film term added to the governing equation. Similarly, the numerical perturbation is applied to calculate the damping coefficients.

Theory of Tilting-Pad Bearing:

The governing equation, Reynolds equation, is similar to that for the journal bearing except that the expression for the lubricant film thickness contains additional terms contributed by the pad location, the pad extended angle, the pre-load parameter and the pad tilting angle. To facilitate the applications of the numerical techniques used in the previously illustrated journal bearing case, the pads are located in global coordinates. The tilt angle of each pad is allowed to vary until an equilibrium position is reached and the total carrying load matches the applied load. The steady state position of the shaft center is determined. Then, the numerical perturbation method is employed to calculate both stiffness and damping coefficients.

Theory of Thrust Bearing:

The governing equations for a thrust bearing is slightly different from that for a journal bearing as the bearing length is converted to the bearing pad radial distance. To include the capability of simulating pads with compliant surfaces, the film shapes are versatile to contain linear as well as nonlinear film thickness distributions. In the numerical computations, the finite-difference grids are selected to be variable extents so that the inherent edge boundary layer behavior can be realized to improve the numerical accuracy. Additionally, a inertial factor is included to enable the numerical solutions to correlate with the experimental results of high-speed bearings e.g. foil thrust bearings.

Theory of Rolling Element Bearing:

The dynamic characteristics of rolling element bearings differ from that of fluid film bearings in the following aspects: (1) Due to direct contacts with races, restrained forces on rolling elements are in phase with deflections. Consequently, cross-coupling stiffnesses are insignificant. (2) Damping coefficients are small and negligible. To compute lateral/angular stiffnesses, the relationships between forces/moments and deflections/ inclinations have to be established. Based on these relationships, the stiffnesses are calculated from the differentiations of forces/moments with respect to deflections/inclinations. As the motions of rolling elements are so complex that simplifications are usually made, i.e., treating the dynamic behaviors of rolling elements in quaisy stead state.

Summary:

For the current analysis package, the Reynolds equation is selected for all fluid film bearing formulations. The author however has developed three dimensional Navier-Stokes equation based analytical tools for both high speed journal and thrust bearing with thermoelasticity coupling to deal with air foil bearings. Those developments will be implemented in the future bearing analysis tools. Bearing Analysis Library

References:

[1] Wu, Erh-Rong, “Gas-Lubricated Porous Bearings,”  Dr. Eng. Science Dissertation, Columbia University , 1976
[2] Wu, Erh-Rong, “Gas-Lubricated Porous Bearings of Finite Length, Self-Acting  Journal Bearings,” Journal of Lubrication Technology, ASME Transaction, 1979 
[3] Castelli, V. & Pirivics, J., “Review of Numerical Methods in Gas Bearing Film Analysis,”  Journal of Lubrication Tech., ASME Trans., 1968
[4] Wu, Erh-Rong &  Castelli, V., “Gas-Lubcated Porous Bearings, Infinitely Long Journal Bearings, Steady State Solution,” Journal of Lubrication Tech., ASME  Trans., 1976. 
[5] Wu, Erh-Rong & Castelli, V., “Gas-Lubricated Porous Bearings, Short Journal Bearings, Steady State Solution,” Journal of Lubrication Tech., ASME Trans., 1977. 
[6] Lund , J. W., “Spring and Damping Coefficients for the Tilting-Pad Journal Bearing,” ASLE Trans., 1964 
[7] Lund , J. W., “Calculation of Stiffness and Damping Properties of Gas Bearings,” Journal o Lubrication Tech., ASME Trans., 1968  
[8] Nicholas, J. C., Gunter, E. J. & Allaire, P. E., “Stiffness and Damping Coefficients for the Five Pad Tilting-Pad Bearing,” ASLE Trans. 1977 
[9] Jones, A. B., “A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitray Load and Speed Conditions,” Journal of Basic Engineering, ASME Trans., 1960 
[10] Jones, A. B., “Ball Motion and Sliding Friction in Ball Bearings,” Journal of Basic Eng. , ASME, Trans., 1959 
[11] Elrod, H. G. & McCabe, J. T.,”Theory for Finite-Width High Speed Self-Acting Gas- Lubricated Slider (and Partial-Arc) Bearings,” Journal of Lubrication Tech., ASME Trans., 1969 
[12] Diprima, R. C., “Higher Order Approximations in the Asymptotic Solution of the Reynolds Equation for Slider Bearings at High Bearing Numbers,” Journal of LubricationTech., ASME Trans. 1969 
[13] Gupta, P. K.,  Advanced Dynamics of Rolling Elements, Spring-Verlag, 1984 
[14] Pan, C. H. T., Wu, Erh-Rong & Krauter, A. I., “Rotor Bearing Dynamics Technology Design Guides, Part I, Flexible Rotor Dynamics,” AFAPL-TR-78-6, Air Force Propulsion Lab., Wright-Patterson Air Force Base, Ohio . 1978 
[15] Sundararajan, C. (Editor), Probabilistic Structural Mechanics Handbook, Theory and  Industrial Applications,  Chapman & Hall

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