1 Nonlinear Time dependent Simulations of High speed Gas Lubricated Bearings An object oriented model Balaji Sankar Post Graduate Research Trainee, National Aerospace Laboratories Bangalore, India balajis_dd@nal.res.in Sadanand Kulkarni Scientist, Propulsion Division National Aerospace Laboratories Bangalore, India sadanandsk@nal.res.in Abstract—The non-linear time dependent solution of the compressible Reynolds’ equation with squeeze damping is presented in this paper. The equations are discretized using a Finite Volume technique and the non-linear system is solved using under relaxation techniques. The difficulties faced at high shaft speeds and flexible foundations are discussed. The results are validated with existing literature. The advantages and disadvantages of using an object oriented programming methodology for the simulation are also highlighted. Keywords:Reynolds’ Equation, Gas Lubricated Bearings, Simulation Gas bearings are in use from 1960 for supporting rotors. The obvious advantage is that they do not need a separate supply of tailored lubricant and can run on air. The surface of the journal and the bush are separated by a layer of air characterized by a very low viscosity (compared to oil). They also have the capacity to tolerate high speeds as they do not face the shear thinning effects and high temperature effects that plague oil bearings. The main reason is that the heat generated in gas lubricated bearings is quickly dissipated and they operate in an approximately isothermal condition. The major disadvantage of gas lubricated foil bearings is that the viscosity and hence damping offered by the lubricating medium are low compared to oil lubricated bearings and hence the bearings are subjected to self-excited whirl, which limits their range of operating parameters.Under unstable operating conditions the amplitude of oscillation of the journal inside the bush exceeds the clearance and strikes the bush leading to the bearing failure. This stability problem in gas lubricated bearings has been analyzed by a number of researchers. Castelli and Elrod [1]solved the incompressible nonlinear Reynolds’ equation to obtain the pressure distribution around the journal. In turn pressure is integrated to obtain the forces acting on the journal. Determined forces were used to obtain the acceleration of the shaft. The co-ordinates of the shaft and its velocities were considered as the state variables. These state variables were integrated in time to get the orbit of the shaft. Stability of the journal-bearing system was analyzed from these orbit plots. This method of analyzing the stability of the shaft was described as a computational experimental rig [2]. “This technique uses the computer as an accurate experimental test rig; it operates exactly in accordance with assumed governing equations”. In these basic studies, the bearing was assumed to be rigid and did not consider the flexible bump foil arrangement in analysis. As a next step, gas lubricated bearings with flexible foundation were analyzed by Heshmat [3]. In his study Reynolds’ equation was discretized using Finite Differences and the resulting system was solved using multi-dimensional Newton’s method. In this study squeeze damping term in the Reynolds’ equation was not considered while solving for the pressure distribution. Della Corte [4] estimated the load carrying capacity and stability of gas foil bearings experimentally. The effect of radial clearance on the load carrying capacity of the foil bearings was studied experimentally by Radil [5]. There have been many works concentrating on the structural part of the gas lubricated bearings with flexible foundation. References [4][5] [6]provide modeling techniques and equations to estimate the structural flexibility and damping coefficients of the flexible foundation of the gas lubricated bearing. A more recent work by Peng and Khonsari concentrates on the solution of compressible Reynolds’ equation with flexible foundation [7]. But, in this also the effect of squeeze damping term was not considered and only the effect of bearing number was considered. The solution presented in this paper considers the unsteady equations for compliant foil bearings with compressible gas flow under iso-thermal conditions. I. THEORY A. Compliant foil bearings A very concise introduction to compliant foil bearings can be found in [7,8]. A brief description of its structure and operation is presented here. The configuration of a typical foil bearing is shown in Fig. 1. It comprised of a cylindrical shell (sleeve) lined with corrugated bumps (bump foil). The corrugated bump foil forms the flexible foundation. The encircling foil (supported by the bump foil) encircles the shaft. In practice, the bearing diameter is often designed to have very small clearance. When the shaft rotates, it drags air into the space between encircling foil and the shaft. This leads to pressure being developed in the converging portion. Upon reaching a certain speed of journal, called ‘‘lift-off speed’’, pressure developed will be sufficient to form a layer