ELASTOHYDRODYNAMIC ANALYSIS OF CIRCULAR AND NON CIRCULAR JOURNAL BEARINGS WITH NON NEWTONIAN LUBRICANTS

Abstract

Hydrodynamic journal bearings are generally used in high speed rotating machines and are more often required to carry heavy loads. In such applications, high temperature and pressure develop in the hydrodynamic fluid-film. Due to high pressure developed in the fluid-film, the bearing surfaces deform. The deformations may quite often have magnitudes of the order of the film thickness, thus altering the performance characteristics of the journal bearings significantly. The classical theory of elastohydrodynamic lubrication assumes that the lubricant behaves as a Newtonian viscous fluid. However, the characteristics of the lubricants, in order to meet the specific requirements of many engineering applications, are often controlled by adding additives in the lubricating oils. The polymer thickened oils behave as pseudo plastic or dilatant fluids. The constitutive equations of these lubricants become nonlinear. Therefore, for computing the performance characteristics of the oil lubricated circular and non circular journal bearings, more realistic constitutive equations (non-Newtonian model) of lubricating oils, including pressure and temperature (piezo-thermal) dependent viscosity and flexibility of the bearing shell should be considered. Survey of the available literature reveals that the studies are limited either to the flexible bearings with Newtonian lubricants or to the rigid bearings with non-Newtonian lubricants. Recently, some EHD studies on line contacts with non-Newtonian lubricants are reported. Even, the literature on the performance characteristics of rigid circular journal bearings with non-Newtonian lubricants considering piezo-thermal effects is quite scant and only few static performance characteristics are reported. To the best of author's knowledge, no literature is available on the static and dynamic performance characteristics of flexible circular and non circular journal bearings with non-Newtonian lubricants. Also, the transient motion trajectories (linear and nonlinear) are reported for rigid and flexible bearings with Newtonian lubricants only. Therefore, the present work was undertaken to bridge the gap as above. In the present work, the three dimensional momentum and continuity equations for Newtonian fluids in cylindrical coordinates and cubic shear stress law as constitutive equation for non-Newtonian lubricant have been used to obtain pressure and velocity components in the lubricant flow field for elastohydrodynamic (END) studies. Three dimensional elasticity equations are solved to get the deformations in the bearing shell using the pressure developed in the fluid film as external load. For viscosity dependent on pressure and temperature (piezo-thermal), three dimensional energy equation in the flow field and one dimensional heat conduction equation for the bush are taken. Generally, the heat conduction in the bush in axial and circumferential direction is small as compared to heat conduction in radial direction. Therefore, in the bearing body heat conduction in the radial direction only has been considered to save the computer time. The viscosity variation with temperature for the non-Newtonian lubricants is reported in the literature using only power law model. Hence, the same has been used in the present work for elastothermohydrodynamic (ETHD) studies. But the analysis and solution algorithm presented here are general and can be used with any suitable model for non-Newtonian lubricants with the suitable relation for pressure and temperature dependent viscosity. The solutions of lubricant flow, elasticity and thermal equations are obtained using the finite element method and direct iteration scheme. For a journal bearing having non-Newtonian lubricant with pressure and temperature dependent viscosity, the ETHD analysis is computationally very expensive as nested iterations are required in establishing the pressure, velocity, deformation and temperature fields and the static equilibrium position of the journal centre for a given vertical load. Each time, the extent of positive pressure zone satisfying the Reynolds boundary condition has to be established. As such, the ETHD studies are carried out only for circular journal bearings. The transient motion trajectories of rigid and flexible journal bearings with non-Newtonian lubricants are obtained using linearized as well as the more realistic nonlinear equations of the disturbed motion of the journal. The trajectories are obtained by numerically integrating the governing equations of motion for a specified initial disturbance using fourth order Runge-Kutta method. For the nonlinear motion trajectories, each point on the trajectory is obtained by computing the film force components four times, which needs the establishment of converged solutions including the determination of extent of film in each iteration. The static and dynamic performance characteristics of the bearings including stability analysis using Routh's criterion and transient motion trajectories are obtained for the following cases: 1. Elastohydrodynamic (EHD) lubrication of circular and non circular journal bearings with non-Newtonian lubricants. 2 Elastothermohydrodynamic (ETHD) lubrication of circular journal bearings with non-Newtonian lubricants. 3. Linearized and nonlinear transient motion trajectories of rigid and flexible circular, two-lobe and three-lobe journal bearings with non-Newtonian lubricants. The performance characteristics of rigid and flexible circular and non circular journal bearings with Newtonian and non-Newtonian lubricants are reported as function of load capacity. The static performance characteristics in terms of eccentricity ratio, attitude angle, minimum fluid-film thickness, side flow and power loss and dynamic performance characteristics in terms of four stiffness coefficients, four damping coefficients, threshold speed and whirl frequency ratio are obtained. The EHD studies with non-Newtonian lubricants are given for various values of deformation coefficient (C d) ranging from 0.0 to 1.0 in the case of circular journal bearing and 0.0 to 0.5 in the case of non circular journal bearings. For ETHD analysis, the study is made for dd=0.1 as it is computationally very expensive. The linear and nonlinear motion trajectories are obtained for circular and non circular journal bearings with Newtonian and non-Newtonian lubricants for two representative loads (W=2.0,4.0)and three values of K (K=0.0, 0.1 and 1.0) and for selected values of journal mass so as to have nonlinear trajectories in stable and unstable zones. The trajectories for flexible bearings are obtained for a fixed value of deformation coefficient (Cdz0.1), nonlinearity factor (K=1.0) and external load (W=4.0) and one transient motion trajectory is obtained for each type of bearing because this is computationally very expensive.. The static and dynamic performance characteristics of the hydrodynamic journal bearings presently studied are appreciably affected in higher ranges of the values of three design parameters; i.e., load (W), deformation coefficient (d ) for the bearing shell and the nonlinearity factor (K) of the lubricant constitutive equation. The consideration of piezo-thermal effects also, alter the performance characteristics considerably. For a fixed value of minimum fluid-film thickness, the load carrying capacity is reduced when piezo-thermal effects are considered and flexibility of the bearing shell further reduces it. The linearized equations of motion give limit cycle when journal mass equals the critical mass for all types of bearings with Newtonian and non-Newtonian lubricants. The nonlinear equations of motion predict a lower value of stability margin than predicted by linearized equations. The stability margin is further reduced at higher values of nonlinearity factor (R). However, inclusion of flexibility in the analysis, considerably improves the stability margin. The EHD studies indicate that the effects of W, eci and K on the performance characteristics are significant for each type of bearing configuration studied (circular, two-lobe and three-lobe). The studies presented here also indicate that for thermal analysis (piezo-thermal effects), the inclusion of bearing flexibility becomes necessary to obtain more accurate bearing performance characteristics. From the stability view point, studies on nonlinear motion trajectories are more informative, particularly with respect to the interactions among ea and K. The analysis, solution algorithm and the results presented here would help the bearing designers in establishing more accurate bearing designs.