ARTICLE Year : 2011  Volume : 1  Issue : 1  Page : 1015 Effect of Load Orientation on the Stability of a Threelobe Pressure Dam Bearing with Rigid and Flexible Rotors SS Rattan^{1}, NP Mehta^{2}, G Bhushan^{1}, ^{1} Department of Mechanical Engineering, NIT Kurukshetra, Haryana, India ^{2} Director, M.M. Engineering College, Mullana Ambala, Haryana, India Correspondence Address: Sometimes the line of action of the load does not pass through the axis of a bearing and is shifted on either side by a few degrees. The effect of load orientation on the stability of a threelobe pressure dam bearing has been studied in this article. A threelobe pressure dam bearing is produced by incorporating two pressure dams in the upper lobes and a relief track in the lower lobe of an ordinary threelobe bearing. The results show that the stability of a threelobe pressure dam bearing supporting either rigid or flexible rotor is increased for the positive values of load orientation, that is, when the load line is shifted in the opposite direction of rotation.
1. Introduction The analytical dynamic analysis [1],[2] has shown that the cylindrical pressure dam bearings are found to be very stable. The experimental stability analysis of such type of bearings [3] showed that analytical stability analysis provides the general trends in experimental data. The study of noncylindrical pressure dam bearings, such as finite elliptical, half elliptical, offsethalves bearings, threelobe bearings and fourlobe bearings have proved that by incorporation of a pressure dam, the performance of the bearing is improved [4],[5] . A threelobe pressure dam bearing is found to be more stable than an ordinary threelobe bearing [6] . Sometimes the line of action of the load does not coincide with the axis of the bearing and is shifted on either side by a few degrees. The load orientation affects the performance of a bearing. The effect of load orientation on the performance of an ordinary threelobe bearing was studied by Mehta et al. [7] . The effect of load orientation on the performance of a threelobe pressure dam bearing has not yet been reported in the literature. Therefore, the present study was undertaken to investigate the effect of load orientation on the performance of a threelobe pressure dam bearing. The geometry of a threelobe pressure dam bearing is shown in [Figure 1]. A threelobe pressure dam bearing comprises three lobes whose centers of curvatures are not in the geometrical center of the bearing. Thus, although the individual lobes are circular, the geometrical configuration as a whole is not. A rectangular dam or step of depth S d and width L d is cut circumferentially in each of the lobes 1 and 3. The dam starts after the oil hole and subtend arc of qs degrees at the center. Circumferential relieftrack or groove of certain depth and width L t is also cut centrally in the lobe 2 of the bearing. The relieftrack is assumed to be so deep that its hydrodynamic effect is neglected. For concentric position of rotor, there are two reference clearances of the bearing: a major clearance c given by a circle circumscribed by the lobe radius and a minor clearance c m given by an inscribed circle. Thus, the center of each lobe is shifted by a distance e p = (c  c m ) known as ellipticity of the bearing. The various eccentricity and ellipticity are nondimensionalized by dividing the major clearance c.{Figure 1} Ellipticity ratio [INLINE:1] Eccentricity ratio [ε] = e/c [INLINE:2] Lobes 1 and 3 with pressure dams and lobe 2 with a relieftrack are shown in [Figure 2], where 1 1 and 1 2 are circumferential lengths of the bearing before and after the dam.{Figure 2} 2. Analysis [INLINE:3] The nondimensionalized Reynolds equation is as follows: The solution of the equation (1) for pressure distribution in the finite element technique is obtained by minimizing the following variational integral [8] over the individual elements [INLINE:4] where [INSIDE:1]= dimensionless film pressure in the eth element. Each lobe of the bearing is analyzed separately. Since the pressure profiles for the lobes 1 and 3 are symmetrical about the center line of the bearing, only half of the lobe is considered for analysis. Each half is divided into 20 × 4 elements. The mesh size is reduced near the trailing edge where heavy pressures are produced. The resulting matrix in each case is stored in a banded form and is then solved by the Gauss elimination method. The Reynolds equation is an elliptical partial differential equation, and hence must be solved as a boundary value problem. According to McCallion et al. [9] , for a bearing having oil supplied at zero pressure, the largest possible extent of the positive pressure region is given by the boundary conditions that both pressure and pressure gradients are zero at the breakdown and build up boundaries of the oil film. However, it has been shown [10] that even by setting the negative hydrodynamic pressures to zero as they occur in any iteration step, the results tend to satisfy the abovementioned boundary conditions in the limit. The latter approach has been followed in the present analysis. Stiffness and damping coefficients are calculated separately for each lobe and then totaled as described by Mehta et al. [11] . The values of these stiffness and damping coefficients, shaft flexibility, and dimensionless speed are then used to evaluate the coefficients of the characteristic equation [12] , which is a polynomial of the 6th order for flexible rotors. [INLINE:5] For a rigid rotor, the value of F is taken as zero. The system is considered as stable if the real part of all roots is negative. For a particular bearing geometry and eccentricity ratio, the values of dimensionless speed are increased until the system becomes unstable. The maximum value of speed for which the bearing is stable is then adopted as the dimensionless threshold speed. The stability threshold curve divides any figure into major zones. The zone above this curve is unstable, whereas the zone below this curve is stable. The minimum value of this curve is termed the minimum threshold speed. Mostly the curve has a vertical line, toward the left of which the bearing is stable at all speeds. This portion is called the zone of infinite stability. With an increase in the minimum threshold speed, the stability curve shifts upward thus increasing the stable zone. Similarly, with an increase in the zone of infinite stability, the stability threshold curve shifts toward the right increasing the stable zone. Thus with an increase in the minimum threshold speed or zone of infinite stability or both, the stability increases. The present analysis has been done for the bearing with the following parameters, which have been optimized for the best stability [6] . [INLINE:6] The value of ellipticity ratio δ = 0.5 is selected for the present study. To study the effect of load orientation on the performance of the bearing, the value of g is varied from 15° to +15°. The stability curves are then drawn for different values of γ and the performance of the bearing is predicted. The load orientation is taken to be positive if it is oriented in the opposite direction to that of rotation of the shaft and negative if it is oriented in the same direction. 3. Results and Discussion The effect of load orientation on the stability of a threelobe pressure dam bearing is shown in [Figure 3],[Figure 4],[Figure 5],[Figure 6],[Figure 7]. [Figure 3] shows the effect of load orientation on the stability of a threelobe pressure dam bearing supporting a rigid rotor (F = 0). It is observed from the figure that negative values (−5°, −15°) of load orientation adversely affect the stability of the bearing. The plots show that both the minimum threshold speed and the zone of infinite stability decrease with the increase in negative values of load orientation angle. The values of minimum threshold speed at load orientation angles 0°, −5°, and −15° are 10.8, 9.8, and 8.4, respectively. The zone of infinite stability decreases at a load orientation of −15° (up to S = 0.39) from the value at 0° (up to S = 0.51). The positive values of the load orientation have favorable effect on the stability of the bearing in the considered range. There is an increase in the value of minimum threshold speed (from 10.8 to 16.3) and the zone of infinite stability (up to S = 0.51 to 0.9) when the angle of load orientation increases from 0° to +15°. The values of minimum threshold speed and zone of infinite stability at the various load orientation angles for a threelobe pressure dam bearing supporting a rigid rotor are shown in [Table 1].{Figure 3}{Figure 4}{Figure 5}{Figure 6}{Figure 7}{Table 1} [Figure 4],[Figure 5],[Figure 6],[Figure 7] show the effect of load orientation on the stability of a threelobe pressure dam bearing supporting a flexible rotor. The effect of flexibility of a rotor on the stability of the bearing is considered by using nonzero values of F in the characteristic equation. The increase in the value of F indicates more flexibility of a rotor. The value of dimensionless flexibility F of most of the practical rotors may vary from 0.5 to 4 and the same range has been considered. The same effects are observed for the bearing supporting a flexible rotor as for a rigid rotor. The negative values of load orientation adversely affect the stability of the threelobe pressure dam bearing, while there is improvement in the stability for positive values of load orientation up to +15°. Another observation from [Figure 4],[Figure 5],[Figure 6],[Figure 7] is that for a particular load orientation angle, with increase in the flexibility of the rotor the zone of infinite stability remains unchanged, but there is a decrease in the value of minimum threshold speed, thus reducing the stability of the bearing. [Table 2] depicts the values of minimum threshold speed and zone of infinite stability of a threelobe pressure dam bearing supporting a flexible rotor at various values of F for a particular angle of load orientation.{Table 2} 4. Conclusions The stability of a threelobe pressure dam bearing is affected by the load orientation. The negative values of the load orientation adversely affect the stability, while there is improvement in the stability for the positive values of load orientation. The increase in the flexibility of the rotor decreases the stability of a threelobe pressure dam bearing for a particular value of load orientation. 5. Notation [INLINE:7] References


