Influence of recess shape on the performance of a capillary compensated circular thrust pad hydrostatic bearing

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Abstract

This work describes a theoretical study concerning the static and dynamic performance of a circular thrust pad hydrostatic bearing having recesses of different geometric shapes. The Finite Element Method has been used to compute the performance characteristics of a circular thrust pad hydrostatic bearing with circular, rectangular, elliptical and annular recesses. The performance has been compared on the basis of the same bearing operating and the same geometric parameters, i.e. the same ratio of bearing to pocket area (Ā) and the same value of restrictor design parameter C̄s2. Further, a comparative study of the various bearing configurations has been carried out vis-à-vis different compensating devices such as capillary, orifice, and constant flow valve restrictors so as to study the combined influence of the geometric shape of recesses and the compensating device on bearing performance. The computed results indicate that to get an improved performance from a hydrostatic circular thrust pad bearing, a proper selection of the geometric shape of the recess in conjunction with the type of restrictor and the value of the restrictor design parameter C̄s2 is essential.

Introduction

Hydrostatic thrust bearing systems have been used in many industrial applications due to their favourable design characteristics. These include high load-carrying capacity, virtual independence of speed, absence of stick–slip characteristics, zero wear of bearing surfaces, very low friction at low or zero speeds, large fluid film stiffness and damping, reduced vibrations and good positional accuracy. Typical industrial applications of hydrostatic thrust bearings are in the machine tool industry, test rigs and dynamometers [26], in lock gates [7], in the circular saw guides used in the forest product industry [5] and in the slippers of axial piston pumps and motors [9]. These have also been used in the analyzing magnet of the nuclear structure facility at SERC Daresbury Laboratory, the Halle optical telescope, the 210-foot diameter tracking antenna at the NASA-jet propulsion laboratory deep space instrumentation facility and Denver’s Mile high stadium [16]. Hydrostatic thrust bearing systems have been extensively investigated by many reasearchers during the last few decades and their research efforts have been focussed on various aspects concerning this class of the bearings. The following review details a few important investigations concerning thrust pad hydrostatic bearings during this time [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].

Ting and Mayer [1] studied the combined influence of centrifugal and thermal effects for a parallel surface stepped thrust bearing and compared the predictions with experimental results. A satisfactory correlation between theory and experiment was observed, particularly for bearings operating at high speed. They also provided design criteria for the selection of the correct bearing geometry and operating conditions. Ghosh and Mujumdar [2] dealt with an analysis of the dynamic behaviour of a compensated hydrostatic step thrust bearing taking into account the fluid inertia and the effect of the compressibility of the fluid in the recess volume. Results were presented for capillary as well as orifice compensated bearings in terms of dimensionless load carrying capacity, oil flow rate, stiffness and damping coefficients.

Generally, hydrostatic thrust pad bearing systems are designed to operate with parallel surfaces. However, the manufacturing, structural and assembly errors and thermal deformations cause tilting of the pad. Many researchers [3], [4], [5], [6], [7], [8], [9] have studied the effect of tilt on the performance characteristics of hydrostatic thrust pad bearings. For hydrostatic thrust pad bearings operating under load, the elastic deformation of the bearing pad alters the fluid film profile and hence the performance characteristics. In recent years many studies, which include bearing pad deformation in the analysis, have been carried out and reported in the literature [10], [11], [12], [13], [14], [15]. These studies indicate that the flexibility of the bearing pad quite significantly affects the performance characteristics and for an optimum design the parameters such as type of restrictor, restrictor design parameter, and deformation coefficient must to be considered.

Addition of solid lubricant to a Newtonian fluid to form a non-Newtonian powder lubricant slurry is the current subject of research activity. A major advantage of the addition of solid lubricants to a traditional Newtonian carrier fluid is the greater stability of mixtures when subjected to extreme temperature. Peterson, Zhenming and Daring [16], [17] studied the non-Newtonian effects of powder lubricant slurries on the performance of hydrostatic thrust pad bearings. Singh et al. [18] obtained closed form expressions for elastohydrostatic load carrying capacity and fluid film pressure distribution for a circular thrust pad bearing operating with power law lubricants. Their results indicated that for a specified value of deformation coefficient, the load carrying capacity decreases with an increase in the value of flow behaviour index for the non-Newtonian lubricants.

Hydrostatic bearings are generally designed on the assumption that they are operated in the fluid film lubrication regime. However, for the case of low speed operations or low viscosity lubricants, these experience mixed lubrication conditions. Accordingly, in recent years many studies showing the effect of surface roughness on the performance of hydrostatic circular thrust pad bearings have been reported in the literature [19], [20].

Yoshimoto et al. [21] studied a rectangular hydrostatic thrust bearing using a self-controlled restrictor employing a floating disk. This type of restrictor is based on the force balance between the upper and lower faces of a floating disk to control the mass flow rate entering the bearing clearance space instead of elastic deformation of the diaphragm. It was reported that a very high fluid film stiffness could be achieved by a hydrostatic thrust bearing when compensated by a self-controlled restrictor employing a floating disk. Younes [22] studied a circular thrust pad hydrostatic bearing with a central step added to a rotating shaft that rested in a deep recess from the point of view of optimal pumping power. Different bearing configurations were considered and it was reported that the addition of the stem enabled this bearing to support radial loads.

Lewis [23] analytically studied the performance of circular thrust pad hydrostatic bearings of various recess shapes, i.e. triangular, square, rectangular etc. A conformal mapping procedure was used to obtain an analytical solution of a simplified Reynolds equation and the computed results were compared with the solutions obtained by an electrical analog technique. Results were presented in terms of load and flow parameters for the different bearings.

The available literature concerning hydrostatic circular thrust pad bearings indicates that the majority of the studies have been carried out for a circular shape of the recess, except the study by Lewis [23]. This may be because of the ease of manufacturing and relative simplicity of these bearings. Owing to rapid technological advancements in manufacturing techniques, the other recess shapes can now be easily manufactured. A recent study concerning hybrid journal bearings by Franchek and Childs [24] demonstrates that the recess shapes may be varied so that the designer has flexibility in selecting an appropriate bearing configuration under given operating conditions. They experimentally studied the five-pocket, high pressure, orifice compensated hybrid journal bearing for cryogenic environments such as that found in the space shuttle main engine (SSME). Bearings having four different recess shapes with square, circular, triangular and a square recess bearing with an angled orifice were studied. Comparisons were made on the basis of the same supply pressure, eccentricity ratio, speed and the same ratio of recess to land area. Their study reported that a square recess bearing with an angled orifice had the best performance among all the bearings studied. Recently the authors [25] analyzed the capillary compensated four recess hydrostatic/hybrid journal bearing with different geometric shapes of recess. It was reported that from the fluid film stiffness point of view, the following criterion may be useful for the designer :S̄circular>S̄square>S̄elliptical>S̄triangular

It was further reported that a square recess bearing is better from the view point of minimum fluid film thickness. The available reported study [23] for the case of a circular thrust pad bearing has certain limitations. This study has been carried out for uncompensated bearings. However, it is rather impractical to use a bearing without any restrictor because the performance of the bearing is considerably affected by the kind of restrictor used in the system. Further, the study [23] has been carried out using a simplified form of Reynolds equation and the results have been computed for the static characteristics only by using less accurate solution techniques. From the design point of view, a study which deals with the static and dynamic characteristics is more desirable. The available studies in the case of hydrostatic/hybrid journal bearings [24], [25] clearly indicate that bearing designers can make better use of the recess shapes during the design process. Thus, it is expected that in the case of circular thrust pad hydrostatic bearings too, a designer can make use of this aspect to advantage. To the best of the authors knowledge no study has yet been reported in the literature that analyses the influence of geometric shape of the recess on the static and dynamic performance of circular thrust pad compensated hydrostatic bearings. Therefore, the objective of this study was to bridge this gap and to study the effect of the geometric shape of the recess on the static and dynamic performance of hydrostatic circular thrust pad compensated bearings. The various geometric shapes of the recesses studied in this work are circular, rectangular, elliptical and annular. The bearing performance has been compared on the basis of the same ratio of bearing to pocket area. This is because the chosen recess shapes used in the present study do not have a common dimensional parameter such as landwidth, recess diameter etc. Therefore, the ratio of area of bearing to recess area (Ā) seems to be one of the most logical parameters for comparison as used in earlier studies [24], [25]. Further, a comparative study has been carried out of all the bearing configurations vis-à-vis different flow control devices so as to study the combined influence of geometric shapes of the recesses and the compensating device on the bearing performance. The results presented in this study are expected to be useful for bearing designers.

Section snippets

Governing equations

A hydrostatic circular thrust pad bearing with a central recess is shown in Fig. 1. The lubricant from an externally pressurized source is fed to the bearing through a restrictor. The flow of an incompressible lubricant in the clearance space of a circular thrust pad hydrostatic bearing, disregarding hydrodynamic effects, is governed by the following non-dimensional Reynolds equation [14]:∂α h̄36p̄∂α+∂β h̄36p̄∂β=2 h̄∂τwhere α=Xro; β=Yro

Using a finite element formulation, Galerkin’s

Restrictor flow equation

The equation for the flow of lubricant (Q̄R) through the various restrictors is expressed in non-dimensional form as [15]:Q̄R=C̄s11−p̄cmwhere m=1 for the capillary restrictor, m=0.5 for the orifice restrictor.

For a constant flow valve restrictorQ̄R=Q̄spwhere Q̄sp is the specified flow for a constant flow valve restrictor.

Boundary conditions

The boundary conditions used for the solution of lubricant flow field are described as:

  • (i)

    At the external boundary, pressures are zero.

  • (ii)

    The pressures for nodes on the pocket boundary are equal.

  • (iii)

    The flow of lubricant through the restrictor is equal to the bearing input flow.

Performance characteristics

The performance characteristics of a hydrostatic circular thrust pad compensated bearing are obtained by using the following expressions [10]:F̄o=e=1nt−1+1−1+1i=1nl=8p̄oiNiJ̄dξ dη+j=1npĀcj p̄ocjS̄=−F̄oh̄=e=1nt−1+1−1+1i=1nl=8p̄oiNih̄J̄dξ dη+j=1npĀcjp̄ocjh̄C̄=−F̄oḣ̄=e=1nt−1+1−1+1i=1nl=8p̄oiNiḣ̄J̄dξ dη+j=1npĀcjp̄ocjḣ̄

Results, discussion and conclusion

The Reynolds equation (1), governing the flow of lubricant in the clearance space between runner and thrust pad, is solved using the finite element method to obtain the nodal pressures. To compute the nodal pressures, the system fluidity matrix and the corresponding column vectors on the right hand side of the system equation (2) are required to be generated. After incorporating the necessary modifications in the system equation of the discretized lubricant flow field for the continuity of flow

Conclusions

On the basis of the results and the foregoing discussion, the general conclusions drawn are presented below:

  • The static and dynamic performance of a circular thrust pad bearing change appreciably as the geometric shape of the recess changes.

  • The lubricant flow requirement of a capillary compensated annular recessed bearing is nearly 87% more than the circular recessed bearing for the same value of restrictor design parameter (C̄s2).

  • It has been observed that the value of load carrying capacity (F̄o

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