Analysis of multi lobe journal bearings with surface roughness using finite difference method

Multi lobe journal bearings are used for high operating speeds and high loads in machines. In this paper symmetrical multi lobe journal bearings are analyzed to find out the effect of surface roughnessduring non linear loading. Using the fourth order RungeKutta method, time transient analysis was performed to calculate and plot the journal centre trajectories. Flow factor method is used to evaluate the roughness and the finite difference method (FDM) is used to predict the pressure distribution over the bearing surface. The Transient analysis is done on the multi lobe journal bearings for threedifferent surface roughness orientations. Longitudinal surface roughness is more effective when compared with isotopic and traverse surface roughness.


Introduction
Hydrodynamic journal bearings are used for carrying high loads in different machines. In these fluid film bearings there is a relative motion between the bearing and journal surfaces, when the journal rotates. The fluid is dragged into the wedge shape gap due to which hydrodynamic pressures are generated and able to support an externally applied load. The importance of roughness in predicting bearing performance has gained considerable attention in Tribology. Due to the operation of hydro dynamic bearings at high speed, the problems of instability is encountered. Instability of the journal bearings will ruin the bearings and machine itself. Multi lobe journal bearings maintain the stability of the bearings at high speeds and different loading conditions.

Fundamentals of multi lobe bearings
The Multi-lobe bearings have lobes depending on the number of lobes they are classified as two lobe, three lobe and four lobes. In two lobe theThey accommodates clearance, preload, offset and provision for lubricants to take away the heat during running conditions. Due to this load carrying capacity is increased and in turn produces a stabilizing effect on the shaft. The multi lobe bearings are more stable in major considerations for dynamic characteristics. Non-circular bearing geometry enhances shaft stability under proper working conditions; they reduce power losses and increase oil flow rate, due to which bearing temperature is reduced. Non-circular journal bearings are used in high speed machinery.
The above equation (1) is a second-order, non-homogenous, partial differential equation.
Where is the pressure at any point , is the film thickness at any point , , , and are the pressures at the four adjacent points of .
The three types of surface roughness used in this analysis are Isotropic surface roughness (γ=1), Transverse surface roughness (γ=0.3) and Longitudinal surface roughness (γ=3). Generalized Reynolds equation of steady state is solved in the finite difference method with over relaxation factor to obtain the non-dimensional pressure distribution in each lobe.
where t is time, x and z are the Cartesian coordinates, μis the absolute viscosity of the lubricant, U is the peripherical velocity of the rotor and h is the film thickness, according to the bearing geometry the hydrodynamic pressure is calculated by using the above expressions for transient state.Now the Reynolds equation is solved numerically for pressure by finite difference method by satisfying the Reynolds boundary conditions. The problem of presenting eccentricity as a function of the Sommerfeld number presents some difficulty. The bearing eccentricity has advantage of simplicity. It is a physical dimension easily visualized in that it tells how far the center of the shaft is away from the center of the bearing.

Results
The variation of Non Dimensional Load and Non dimensional frictional force of multi lobe bearing for various surface roughnesses's are determined and plotted. There is a deviation in results due to the surface roughness profiles.    Figures 5 to 7represents the effect of load carrying capacity at different eccentricity ratios for two lobe, three lobe and four lobe journal bearings. It is observed from the figures that three lobe journal bearing and four lobe journal bearing are having maximum load carrying capacity when compared to two lobe journal bearing. The longitudinal surface roughness has more effect on maximum load carrying capacity when compared to transverse and isotropic surface roughness.  10 represents the effect of frictional force at different eccentricity ratios for two lobe, three lobe and four lobe journal bearings. It is observed from the figures that two lobe journal bearing is having maximum frictional force when compared to three lobe and four lobe journal bearing. The longitudinal and transverse surface roughness has more effect on friction force when compared to isotropic surface roughness. From figure 11, it has been observed that the three lobe journal bearings are more stable when compared to two lobe and four lobe journal bearings. The path trajectories are plotted at eccentricity ratio 0.6, with longitudinal surface roughness.

Conclusions
The analysis is done on the multi lobe journal bearings, with longitudinal, transverse and isotropic surface roughness orientations. The Longitudinal surface roughness has more effect onMulti lobe journal bearings. Pressure is maximum exactly midway between the lobes. Due to this, the rotor will be well balanced while rotating and vibrations due to bearing will be minimum. From the results it indicates that the surface roughness plays a major role on the performance of static characteristics of the multi lobe journal bearings. From the graphs it is observed that the three lobe and four lobe journal bearing performance is almost close when compared with two lobe journal bearings. Longitudinal surface roughness is more effective when compared with isotopic and traverse roughness.