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Externally pressurized gas bearings: A review
Wear, 62 (1980)
299 - 314 0 Elsevier Sequoia &A., Lausanne - Printed in the Netherlands
EXTERNALLY PRESSURIZED GAS BEARINGS: A REVIEW
B, C. MAJUMDAR
Department of mechanical Engineering, indian institute of Technology, Khamgpur (India) (Received
December
l&1978;
in final form August 3,197s)
Summary The literature pertaining to externally pressurized hole admission gas bearings is briefly reviewed. The advantages, ~mi~tions and app~cations of such bearings under various design conditions are discussed.
1. In~oduction To satisfy bearing design requirements sufficient background information is necessary to optimize the design. Design requirements are often met by the proper design of conventional bearing systems, development of new or improved lubricants or development of effective seals to minimize lubricant leakage. However, in many applications even properly designed conventional bearings used with the best available lubricating oils fail to give satisfactory performance. For example solid lubricants such as graphite or molybdenum disulphide are used in some high temperature applications. Gas lubrication, which is discussed in this paper, can also be used and has some special features when compared with oil lubrication. However, gas bearings also have inherent fictions which the designer or user must recognize in order to obtain successful operation. In particular gas bearings are more prone to instability than oil-lubricated bearings. Gas bearings, like liquid- or grease-lubricated bearings, can be (i) selfacting, (ii) squeeze film or (iii) externally pressurized. In a self-acting bearing the load is supported by a wedge film action of the gas caused by relative tangential motion between two surfaces. The squeeze film.bearing supports a load because of the oscillating relative normal motion between the two surfaces. The externally pressurized bearing supports a load because of the pressure in the gas film which is supplied from an external source. External pressurization can alleviate start-up and shutdown wear by permitting jacking of the shaft in the absence of rotation. In many applications large loads (e.g. due to the dead weight of a telescope reflector mirror) or small loads (e.g. due to the mass of inertial navigation gyros) must be supported with virtually no relative motion. Self-acting bearings cannot operate in such conditions and externally pressurized bearings may be required. The
300
coefficient of friction for an externally pressurized bearing is low and is zero when there is no relative motion. If used continuously it has greater stiffness than a self-acting bearing. Thus high load, high stiffness and low coefficient of friction are the attractive features of externally pressurized bearings. However, this type of bearing is not immune to destructive whirl instability. It is necessary to recognize the types of instability which may occur in some possible bearing designs and to ensure that these undesirable phenomena do not occur in the final design.
2. Review It is not possible to credit any one person with the invention of the gas bearing. Patents were obtained by W~tin~ou~ in 1904, De Ferranti in 1906 and Abbot in 1916. The first operational gyro was developed in the U.S.A. by the Sperry Gyroscope Company in 1930. Willis [l] was the first to make an experimental investigation of the radial air flow between two parallel plates. He measured the Bernoulli forces and compared the attractive and repulsive forces in externally pressurized bearings. Many externally pressurized bearings have been designed and m~~a~tured for a variety of practical applications such as machine tools [ 2,3] , spherical bearings for space simulators [ 41 and dental drills [ 5,6] , 2.1. Classification Gas bearings can be roughly classified either by geometry or by the method of compensation. The most common type of extemalfy pressurized journal bearing has supply ports in a row or in several rows [7] . Recesses (Fig. 1) are made at the leading edges of the holes so that a high inlet pressure remains uniform throughout them. Therefore the boundary conditions are recess pressure at the holes and ambient pressure at the ends. Such bearings which are designed to support large loads may develop ~s~bi~ties. Therefore in some bearings gas is fed through plain feed holes without a recess. Gas bearings can also be classified by the method of compensation or restriction. An important component of this bearing is the flow restrictor. The bearing would not function without the restrictor. It would support a load but there would be no change in the load supported with film thickness. In practice it is necessary for the load to vary inversely with the film thickness, and the restrictor performs this function. Capillary, orifice and inher-
CTIVE Afm
= nd,h
Fig. 1. Jet configurations:
(a) plain jet; (b) jet with a circular pocket.
301
aJ[d -OW
CAPILLARY
INHERENT t ORIFICE)
INMERENY WIESYRICYINC
L*wo)
Fig. 2. Various restrictors.
ently compensated restrictors are most commonly used in gas bearings (Fig. 2). Some special types of restrictor also exist. Adams et ~2. [8] introduced a restricting land full journal bearing similar to a step bearing. Mayer and Shaw [9] predicted the characteristics of variable flow restrictors and showed their relative advantages and disadvantages. Newgard and Kiang [lo] used an elastic orifice as a compensating element to provide greater stiffness. 2.2. Design and analysis under steady state conditions The design procedures for gas- and oil-lubricated bearings are similar but gas bearing design is complicated owing to compressibility effects. Externally pressurized bearings also have stability problems. Therefore a compromise is necessary between bearing stiffness and stability. Normally a bearing of higher stiffness is more unstable. It is first necessary to establish a design which has the required stiffness, load and flow characteristics and then to check for stability. This process is repeated until a satisfactory compromise is reached. In an outstanding contribution to externally pressurized gas bearing literature Shires [ll] gives design information for both laminar and turbulent flow. Robinson and Sterry [12] used axial flow theory, i.e. gas flow takes place only along the bearing length, for computing the static load of a hole admission bearing. The pressure distribution of a bearing using the axial flow model is shown in Fig. 3. There is a large discrepancy between the theoretical and the actual load capacity as the effect of circumferential flow is neglected in the idealized bearing. Shires [13] provided a simplified theoretical treatment to compensate for circumferential flow. A semiempirical relation for the load capacity was derived by Robinson and Sterry [12] from a comprehensive series of experiments. Heinrich [14] made a theoretical ,analysis of journal bearings with a single and double row of orifices for small eccentricity ratios. He derived a closed-form solution for the load capacity by assuming a line source. Design curves for the load capacity for various design conditions were also given [ 141. This analysis takes the circumferential flow component into account. Fedor [15] developed a method for the solution of Reynolds equation applied to a full journal bearing, and also gave information [16] concerning
302
Fig. 3. Distribution of pressures in an axial flow model: (a) single central orifice; (b) two orifices in one channel.
the effects of a single source in a journal bearing. Rothe [17 ] gave a detailed description of the advantages, libations and applications of gas bearings used in U.S. Army ballistic missiles. Laub 1181 presented an analytical treatment developed for a semi-cylindrical bearing with nine orifices for the full journal bearing with 192 radial and 24 axial orifices, He considered only the axial flow component in his analysis. Licht [19] showed that by suitable changes of dependent variables the equation which governs laminar and isothermal flow of the gas lubricant in otherwise geome~c~ly identical bearings could be reduced to the Laplace equation v "p = 0 where p is the gas film pressure. Laub and Norton [ZO] made an analytical and experimental study of spherical gas bearings with orifice compensation. Allen et al. [21] studied the effect of bearing length on the performance of bearings with single and double rows of orifices. This analysis was also based on an idealized flow model. The expe~men~ly determined load capacity was half the value calculated using axial flow theory. It was predicted that circumferential flow could account for the difference. Tang and Gross [22] presented analysis and design charts for the evaluation of the performance of externally pressurized gas bearings. The curves are most effective as when they are properly used the complicated task of the calculation of the do~stream pressure from flow equations is eliminated. The curves can also be used to design bearings either for maximum load or maximum stiffness,
Turnblade [23] proposed a new method of bearing equivalence as a simple technique for design. He developed an equation for rectangular bearings which described the relation between the ratio of effective bearing width to effective length and the ratio of width to length of an arbitrary element. An equivalent rectangular bearing can be conveniently described by estimating the streamline flow pattern within the bearing configuration and solving the equation. The performance of the equivalent bearing is identical to that of the actual bearing. Lemon [24] presented a simplified analysis for determining load, flow and stiffness. He considered the effect of circumferential flow by using an approximate equation and presented a design procedure to obtain optimum stiffness in a journal bearing. A theoretical and experimental analysis of a gas journal bearing with six orifice stations was made by Mori et al. [ 251. The complex potential theory was considered to be the most suitable method for obtaining a point source solution of Reynolds equation. With this theory Mori et al. obtained the load capacity and volume rate of flow and they predicted the actual load and mass flow rate from the incompressible flow solution. Dudgeon and lowe [26] predicted the performance of a static gas journal bearing theoretically by considering axial and circumferential flow. Rieger and Wilcock [27] derived a simplified design method for load, flow and stiffness of thrust and journal bearings. A systematic analysis and simplified procedure for designing journal bearings together with design charts for predicting journal bearing performance have been given [28] . Shires [29] also gave simplified methods for designing thrust and journal bearings. Powell [30] presented design charts of load capacity for lightly loaded bearings. Constantinescu and coworkers [31 - 331 suggested an approximate method of analysis using the line source assumption and claimed that this led to accurate results even when the number of orifice stations was relatively small, e.g. n = 6. Pink [ 341 compared the results of MT1 [ 281, Constantinescu and Salcudean [ 311 and Powell [ 301 with the experimental data of Eusepi and Lewis [ 351 and concluded that there was considerable conflict between the various published design methods in the prediction of static bearing stiffness and load. The above references are restricted in the sense that some do not consider the circumferential flow component and some use a line source assumption. Majumdar [ 36 - 401 carried out theoretical and experimental work on hole admission bearings with four, six and eight orifice stations in two planes at quarter stations. The theory considered both axial and circumferential flow without assuming a line source. When circumferential flow is considered in the analysis the pressure distribution is different from that obtained with the usual line source assumption. Figure 4 shows a typical pressure distribution obtained when the discreteness of the holes is considered. The load capacity and the mass rate of flow of a bearing with six supply holes in two rows at the quarter station are shown in Figs. 5 and 6. The line source assumption is not relevant to the actual behaviour at least at high eccentricity ratios. The optimum design method for maximum load and for maximum stiffness was also indicated.
304
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Fig. 4. (a) Premure distribution in the axial direction. Experimental r~dts: 0.6 = 0.60; =, 6 = 1.10; V, 6 = 2.04;v, 6 = 3.06;& = 4.98 kgf cmb2. Theoretical results (solid CUrVea) calculated using 6 = 46 q(k62T)l” cdA,,IcSpI, and & = pa/pa. (h) Pressure distribution in the circumferential direction. Experimental results: q ,6 = 0.90; ., 6 = 2.12; 0,6 = 2.96; l,6 = 4.10; I, 6 = 5.17;& = 4.96 kgf cmT2. Theoretical results (solid curves) calculated from 6 = 46 v(k61T)lP cdA,/c?p, with n = 6,~=0.3,~~=0.2,i=0.5and&=~,/~,.
oI
3
2
0
4
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00
I I
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6 (c)
Fig. 5. Load capacity of a compensated bearing: (a) n = 6, pa = 0.2; (b) n - 6, pa - 0.5; (c) n = 6,& - 0.33. Solid curves, accurate theoretical resulta; broken curves, approximate theoretical results. hperimental results: 0, e = 0.3; 0, E = 0.6; A, e = 0.9.
The theoretical and experimental analyses of the references quoted so far are directed towards the determination of bearing characteristics at zero speed. The first theoretical investigation considering journal rotation using pressure perturbation theory and a line source assumption is probably that of Lund [41] . He produced design charts for the variation of steady state load capacity and attitude angle with journal speed for various supply pressures and feeding parameters. MT1 [ 281 gave design charts for a.rotating journal and a relation between attitude angle and eccentricity ratio. Castelli and Pirvics [42] proposed a new method for the solution of the generalized Reynolds equation by which the load coefficients and attitude angles of three- and four-pad axial groove cylindrical bearings were found. Castelli and Shapiro [43] obtained a numerical solution of Reynolds equation for hydrodynamic, hydrostatic and hybrid bearings. Bearings with multiple recesses were also studied. Fleming et al. [44] used a pressure perturbation method for an unloaded journal to obtain the steady state load capacity and
306
0
0
I
I
1
f
f
2
3
4
6
faf
(b)
14
I2
IO
b
EC 6
4
z
,
f
I
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2
I 3
I 4
6
(cl Fig. 6. Flow rate of a compensated bearing: (a) R = 6, Is, = 0.5; (b) n = 6,& = 0.33; (c) n = 6, &, = 0.2. Solid curves, accurate theoretical results; broken curves, approximate theoretical results. Experimental results: 0, E = 0.9; 0, E = 0.6; *, 6 = 0.3.
307
attitude angles. Cunningham et al. [45, 461 performed experiments on a rotating journal bearing which had two rows of supply holes with six orifices per row. Their experimental results were compared with the results obtained with linearized p and ph theories [ 43,471. It was shown that the linearized ph perturbation solution was nearer to the experimental results. Powell [48] reported elaborate experimental results for hybrid bearings. Majumdar [49] used a linearized perturbation solution of a similar bearing without the line source assumption. A comparison of the results for the above methods is shown in Figs. 7 and 8.
Fig. 7. Load capacity of a rotating journal: L/D = 1.5; n = 6; 6 = 3.0. Broken curve, line source assumption [ 411; chain curve, line source assumption [47] ; solid curve, discrete source [ 491; 0, experimental results [ 451. Fig. 8. Attitude angle of a rotating journal: L/D = 1.5; n = 6; 6 = 3.0. Broken curve, line source assumption [ 411; solid curve, discrete source [ 411; 0, experimental results [ 46 1.
2.3. Design and analysis under dynamic conditions An important practical consideration in the design of externally pressurized bearings is selfexcited instability; pneumatic instability or air hammer can also be a problem. The evaluation of the selfexcited whirl of externally pressurized bearings has been studied by Mullan and Richardson [50] , Gross [51] and Lemon [ 241. They assumed that selfexcited whirl would occur at a rotational frequency slightly less than double the calculated frequency. Lund [41] used a pressure perturbation method. Licht and Elrod [52] conducted experiments to study air hammer instability in thrust bearings and used distributed parameter analysis to show how the depth of the pool affected the stability. Richardson [53] presented a general stability criterion based on lumped parameter analysis of gas bearings with orifice compensation. Mori [ 541 suggested that the experimentally observed depression of the pressure profile in thrust bearings was caused by the generation of a shock wave in the bearing clearance space. Licht [55] developed a stability analysis for gas journal bearings with externally pressurized pads and Licht and Elrod [56]
308
investigated selfexcited vibration. Stability bounds were established theoretically for circular thrust bearings and verified experimentally. Hirs [57] proposed a new type of bearing in which the stability and load capacity were ensured by directing the flow in the clearance gap through grooves on one of the surfaces towards a zone where both surfaces were plain. Allais [58] studied gas bearing equations and derived relations for maximum stiffness and stable operation. Some approximate formulae were derived for orificecompensated bearings under choked conditions. Speen [ 591 described an integral journal thrust bearing with superior load capacity, stability and stiffness. Majumdar [60] made a theoretical analysis of a bearing under plain vibration of the stationary journal. The results were compared with the simplified solution using a line source assumption. A solution for the hole admission bearing with squeeze action has been made by a quasi-static assumption [ 611 and a pressure perturbation solution [62] . The quasi-static solution overestimates the squeeze load capacity. The stability analysis of rotating externally pressurized bearings was first made by Larson and Richardson [63]. They conducted experiments and showed the variation of stability with bearing clearance. The work on the stability of stationary and rotating journal bearings most generally cited is that of Lund [47] who made a theoretical estimation of the stability of a bearing with a central row of holes. Ono and Tamura [64] derived the theoretical stability characteristics using a quasi-static assumption and verified their results experimentally [65] . Fleming et al. [44] calculated the stability of unloaded journal bearings by using a first-order perturbation theory with steady whirl approximation and found that the effect of the recess volume of an orifice-compensated bearing was to decrease the stability. Elrod and Glanfield [66] developed a very detailed computer method for designing a bearing of similar configuration with a flexibly mounted rotor. Lund [67] predicted the stiffness and damping of gas bearings and the stability behaviour of an elastic rotor on a flexible damped support [68] . Ausman [69] analysed the dynamic behaviour of journal bearings with a sinusoidally timevarying load. Ehod et al. [70] obtained the stability by the response to step jump. Majumdar [60] also determined the stability using a steady whirl approximation and without the line source assumption and compared the results with those of Fleming et al. [44] . Figure 9 shows a detailed comparison of the stability results using the various methods outlined. 2.4. Other effects Mori [ 711 analysed a gas bearing by considering lubricant inertia and turbulence. Mori and Miyamatsu [72] have contributed to the understanding of the characteristics of externally pressurized gas bearings. Various flow models were employed to explain results obtained over a wide range of operating conditions. Their analysis included the entrance effects accompanied by the growth of a boundary layer and the occurrence of a shock wave. Entrance effects were taken into account by McCabe et al. [ 731 and Carfagno and McCabe [74]. Elrod and Glanfield [66] also considered the entrance
309
\
UNSTABLE
UNSTABLE
f STABLE
\
\
loi1
5
Azsrlo
qryd
6
7991)
\
I
(b)
I
I
I
2
3
4
llllll S
676910
A
Fig. 9. Stability characteristics. (a) L/D = 1.0; n = 6; 6 = 3; e, = 0.1; I#JW = 0; solid curve, accurate; chain curve, steady whirl [60] ; broken curve, steady whirl [44]. (b) As for (a) except Qw = 50; 0, experimental results for & = 2; 0, experimental results for ii, = 6.
effect in their stability study. Dayson and Chasman [75] investigated the adequacy of the theoretical model in predicting the unbalanced response of a rotor supported in hydrostatic gas-lubricated journal bearings. The effect of rotor speed and supply pressure on rotor unbalance was also investigated. Mori and Mori [76] showed that the main bearing could give a very high onset speed if an auxiliary bearing was added. They also studied pneumatic phase shifting as a means for stabilizing selfexcited whirl in externally pressurized gas bearings and described constructional details of the stabilizer 1771.
310
A method of increasing stability is to mount the bearing on rubber Orings [78] . Kerr [79] demonstrated a high onset speed in hydrodynamic gas bearings. Powell and Tempest [80] made an experimental investigation of externally pressurized bearings mounted on rubber O-rings. A simplified analysis of the stability of similar bearings has been reported [81] and confirmed Powell and Tempest’s results. 3. Conclusions and recommendations It appears from the survey that the theoretical estimate of static characteristics such as load capacity, maSs flow rate and stiffness of lightly loaded hole admission gas bearings can be predicted with reasonable accuracy from axial flow theory. However, axial flow theory does not appear to be satisfactory for a heavily loaded bearing. The axial flow theory always overestimates load capacity as it does not account for the variation of pressure between the two feeding planes (Fig. 3). The line source assumption for a discrete number of feed holes has been used for a few analyses but such solutions cannot be considered as general solutions. Majumdar’s work (Fig. 5) on similar bearings suggests that the line source assumption is not valid for all hole admission bearings. It is particularly unsuitable if a bearing has less than eight feed holes in a row. Lund [47] used a correction factor of 1.5 to compensate for the feed holes in the calculation of flow but not in the computation of the gas film pressure. Therefore the use of a correction factor with the line source solution is not entirely justified. The advantages and disadvantages of the different methods are summarized in Table 1. Much information on the static behaviour and stability characteristics is available, but little is known about the complicated flow phenomena near the supply holes. The pressure drop at the edge of the supply holes is due to (i) the energy consumption for the development of the boundary layer and (ii) the accelerating energy of the lubricant from the stagnation point. Mori and Miyumatsu [72] gave an approximate expression for pressure drop as 0.7 po(Qo/n dh)2 where h, Q0 and p. are the film thickness, volume flow rate and fluid density respectively. However, this estimate does not consider the effect of the entrance length on the development of boundary layers, the compressibility of the lubricant and the divergence of the flow area. The pressure at the edge of the supply hole can be obtained by subtracting this drop from the predicted pressure downstream of the orifice. Another estimate for the correct pressure at the edge of the supply hole is due to Elrod and Glanfield [66] . Flow settles down some 250 film thicknesses or less from the hole. The pressure tends to p2 = In r for large r where r is the distance measured from the centre of the supply hole. If the pressure curve obtained from the solution of Reynolds equation is extrapolated back to the nozzle curtain area the correct pressure is found. Only the pressure in the neighbourhood of the hole will be incorrect. The correct pressure at the edge is sometimes’5 - 10% less than the pressure predicted by the usual method.
311 TABLE 1 Relative merits of various methods
Method
Advantages
Disadvantages
Considering discrete supply holes
No a priori assumption
Numerical convergence and stability are difficult to ascertain; costly
Line source ~ump~on
Quite simple
Qu~tionable for large e and small number of supply holes
Axial flow theory
Simplest
No a priori justification
As externally pressurized gas journal bearings are becoming more popular it is hoped that further research may resolve some of the outstanding problems and provide more reliable data, design charts, simple design rules and procedures for practising engineers.
References
7 8 9 10
R. Willis, On the pressure produced on a flat plate when opposed to a streams of air issuing from an orifice in a plane surface, Tmns. Cambridge Philos. Sot., 4 (1928) 121. H. L. Wunsch, Design of air bearings and their application to measuring instruments and machine tools, Znt. J. Mach. Tool Des. Res., 1 (1961) 198. M. Gram&r and J. Kerr, Air bearings research and applications at National Engineering Laboratory, Scotland, Int. Symp. on Gas-lubricated Bearings, Oct. 1959, U.S. Government Printing Office, Washington, D.C., Oct. 1959. D. F. Wilcock, Design and performance of gas-pressurized spherical space-simulator bearings, J. Basic Eng., 83 (1971) 595. G. L. Green et al., Miniature air-bearing supported air turbine dental hand piece, Rep. 62-90, USAF Aerospace Medical Centre, School of Aerospace Medicine, 1961. J. W. Powell, H. H. Maye and P. R. Dwight, Fundamen~l theory and experiments on hydrostatic air-bearings, Proc. Lubrication and Wear Convention, Bournemouth, May 1963, Inst. Mech. Eng., London, 1963. W. A. Gross, Gas Film Lubrication, Wiley, New York, 1962, p. 279. C. R. Adams, J. Dworski and E. M. Shoemaker, Externally pressurized step journal bearings, Am. Sot. Mech. Eng. Pap. 61-Lub-8, May 1961. J. E. Mayer and M. Shaw, Characteristics of externally pressurized bearings having variable external flow restrictor, J. Basic Eng., 85 (1963) 291. P. M. Newgard and R. L. Kiang, Elastic orifices for pressurized gas bearings, ASLE
Trans., 9 (1966) 311. 11 G. L. Shires, On a type of air lubricated journal bearing, Rep. R61, National Gas Turbine Establishment, 1949. 12 C. H. Robinson and F. Sterry, The static strength of pressure fed gas journal bearings, jet bearings, AERE Rep. R/R 2642, AERE HarweB, Sept. 1958. 13 G. L. Shires, The vented pressure-fed gas journal bearing, Rep. AEEW-R 111, AEE Winfrith, March 1962. 14 G. Heinrich, The theory of the externally pressurized bearing, Zst Znt. Symp. on Gaslubricated Bearings, Rep. ONR 2 Acr-49, U.S. Government Printing Office, Washington, D.C., Oct. 1959.
312 15 J. V. Fedor, Sommerfeld solution for finite bearings with circumferential grooves, J. Basic Eng., 82 (1960) 321. 16 J. V. Fedor, Journal bearings with arbitrary position of source-2, J. Basic Eng., 83 (1961) 572. 17 H. C. Rothe, Externally pressurized gas bearings, Am. Sot. Mech. Eng. Pap. 61 -Lub-4, May 1961. 18 J. H. Laub, Externally pressurized journal gas bearings, ASLE Trans., 5 (1961) 261. 19 L. Licht, Extension of conducting sheet analogy to externally pressurized gas bearings, J, Basic Eng., 83 (1961) 209. 20 J. H. Laub and R. H. Norton, Externally pressurized spherical gas bearings, ASLE Trans., 4 (1961) 172. 21 D. S. Allen, P. J. Stokes and S. Whitley, The performance of externally pressurized bearings using simple orifice restrictors, ASLE Trans., 4 (1961) 181. 22 I. C. Tang and W. A. Gross, Analysis and design of externally pressurized gas bearings, ASLE Trans., 5 (1962) 261. 23 R. C. Turnblade, The design of externally pressurized gas lubricated bearings by the method of bearing equivalence, ASLE Trans., 5 (1962) 385. 24 J. R. Lemon, Analytical and experimental study of externally pressurized air-lubricated journal bearings, J. Basic Eng., 84 (1962) 159. 25 H. Mori, T. Shibayama, I. Shizuma and T. Yamamota, Research on externally pressurized gas bearings, Bull. JSME, 6 (1963) 598. 26 E. H. Dudgeon and I. R. 0. Lowe, A theoretical analysis of hydrostatic gas journal bearings, Me&. Eng. Rep. MT 54, National Research Council of Canada, 1965. 27 N. F. Rieger and D. F. Wilcock, A simplified approach to the design of hydrostatic gas bearings, Mach. Des., 38 (1966) 107. 28 MTI, Design of Gas Bearings, Vol. 1, MTI, Latham, N.Y., 1966. 29 G. L. Shires, The design of pressurized gas bearings, Tribology, 4 (1968) 219. 30 J, W. Powell, Design of Aerostatic Bearings, Machinery Publications, Gt. Britain, 1971. 31 V. N. Constantinescu and M. Salcudean, An approximate method for analysis of externally pressurized journal bearings, 4th Int. Gas Bearing Symp,, Univ. Southampton, March 1967, BHRA, Cranfield, 1967, pap. 1. 32 V. N. Constantinescu, A method for the determination of the pressure distribution in externally pressurized gas-lubricated journal bearings with central row of orifices, Rev. Roum. Sci. Tech., Ser. Met Appl., 12 (1967) 687. 33 V. N. Constantinescu, Pressure distribution in externally pressurized gas-lubricated journal bearings with two rows of orifices, Rev. Roum. Sci. Tech., Ser. Met. Appl., 12 (1967) 1037. 34 E. G. Pink, Investigation into methods for externally pressurized gas journal bearings, 6th Int. Gus Bearing Symp., Univ. Southampton, BHRA, Cranfleld, 1974, pap. A3. 35 M. Eusepi and P. Lewis, Research on gas lubrication at higher temperature and flow rates, 5th Quarterly Rep. MTI 64 TR 53, MTI, Latham, N.Y., Sept. 1964. 36 B. C. Majumdar, Analysis of externally pressurized gas journal bearings, J. Me&. Eng. Sci., 12 (1970) 1. 37 B. C. Majumd~, Theoretieai analysis of externally pressurized air journal bearings, J. Mech. Eng. Sci., I2 (19’70) 123. 38 B. C. Majumdar, Externally pressurized air journal bearings with multiple supply holes. Appl. Sci. Res., 22 (1970) 239. 39 B. C. Majumdar, The numerical solution of aerostatic journal bearings with several supply ports, Wear, 15 (1970) 331. 40 B. C. Majumdar, On the general solution of externally pressurized gas journal bearings, J. Lubr. Technot., 94 (1972) 291. 41 J. W. Lund, The hydrostatic gas bearing with journal rotation and vibration, J. Basic Eng., 84 (1964) 328. 42 V. Castelli and J. Pirvics, Equilibrium characteristics of axial groove gas-lubricated bearings, Am. Sot. Mech. Eng. Pup. 65-Lub-16, 1965.
313 43 V. Ca&,elli and W. Shapiro, Improved method for numerical solutions of general incompressible fluid lubrication problems, Am. SOC. Mech. Eng. hp. 66-Lb4, 1%X3. 44 D. P. Fleming, R. E. Cunningham and W. J. Anderson, Zero-load stability of rotating externally pressurized gas-lubricated journal bearings, J. Lubr. Technol., 92 (1970) 325. 45 R. E. Cunningham, D. P. Fleming and W. J. Anderson, Experiments on rotating externally pressurized journal bearings, I - Load and stiffness, NASA Tech. Note TN D-5191, 1969. 46 R. E. Cunningham, D. P. Fleming and W. J. Anderson, Experiments on rotating externally pressurized journal bearings, II - Attitude angle and air flow, NASA Tech. Note TN D-51 92, 1969. 47 J. W. Lund, A theoretical analysis of whirl instability and pneumatic hammer for a rigid rotor in pressurized gas bearings, J. Lubr. Technol., 89 (1967) 154. 48 J. W. Powell, Experiments on hybrid air journal bearing, Am. Sot. Mech. Eng. Pap. 64-WA/Lub-11, 1964. 49 B. C. Majumdar, Analysis of externally pressurized gas bearings with journal rotation, Wear, 24 (1973) 15. 50 P. J. Mullan and H. H. Richardson, Plane vibration of inherently compensated gas journal bearing analysis and comparison with experiments, ASLE Trans., 7 (1964) 277. 51 W. A. Gross, Investigation of whirl in externally pressurized gas journal bearings, J. Basic Eng., 84 (1962) 132. 52 L. Licht and E. G. Elrod, A study of the stability of externally pressurized gas bearings, J. Appl. Mech., 82 (1960) 250. 53 H. H. Richardson, Static and dynamic characteristics of compensated gas bearings, Tmns. ASME, 80 (1958) 1503. 54 H. Mori, A theoretical investigation of pressure depression in externally pressurized gas-lubricated circular thrust bearings, J. Basic Eng., 83 (1961) 201. 55 L. Licht, Air-hammer instability in pressurized journal gas bearings, J. Basic Eng., 85 (1961) 235. 56 L. Licht and E. G. Elrod, Experimental study of stability of externally pressurized gas-lubricated thrust bearings, J. Lubr. Technol., 85 (1963) 25. 57 G. G. Hirs, Partly-grooved, externally pressurized bearings, Proc. Inst. Mech. Eng., London, 180 (3) (1965 - 66). 58 D. C. Allais, The design of externally pressurized gas thrust bearings for maximum stiffness and stability, ASLE Trans., 5 (1962) 254. 59 G. B. Speen, Pressure and flow studies of externally pressurized gas-lubricated bearings, Trans. ASME, 5 (1962) 242. 60 B. C. Majumdar, Dynamic characteristics of externally pressurized gas journal bearings with multiple supply holes, Wear, 34 (1975) 289. 61 B. C. Majumdar, Multirecess aerostatic journal bearings with cyclic squeeze, Wear, 21 (1972) 9. 62 B. C. Majumdar, Effects of cyclic squeeze on externally pressurized gas journal bearings, J. Lubr. Technol., 97 (1975) 69. 63 R. H. Larson and H. H. Richardson, A preliminary study of whirl instability of pressurized gas journal bearings, J. Basic Eng., 84 (1962) 511. 64 K. Ono and A. Tamura, Whirl instability of externally pressurized gas journal bearings, Bull. JSME, 11 (1968) 707. 65 0. Tanaguchi, A. Tamura and K. Ono, Experimental study of whirl instability of externally pressurized air journal bearings, Bull. JSME, 11 (1968) 173. 66 H. G. Elrod and G. A. Glanfield, Computer procedure for the design of flexibly mounted externally pressurized gas-lubricated journal bearings, 5th Int. Gas Bearing Symp., Univ. Southampton, March 1971, BHRA, Cranfield, 1971, pap. 22. 67 J. W. Lund, Calculation of stiffness and damping properties of gas bearings, J. Lubr. Technol, 90 (1968) 693. 68 J. W. Lund, The stability of an elastic rotor on journal bearings with flexible, damped supports, J. Appl. Mech., 87 (1965) 911.
314 69 J. Ausman, On the behaviour of gas-lubricated journal bearings subjected to sinusoidally time varying loads, J. Basic Eng., 87 (1965) 589. 70 H. G. Elrod, J. T. McCabe and T. Y. Chu, Determination of gas bearing stability to a step-jump, Am. Sot. Mech. Eng. Pap. 67-Lub-5, 1967. 71 H. Mori, A theory of externally pressurized circular thrust gas bearings with consideration of effect of lubricant inertia, J. Basic Eng., 85 (1963) 304. 72 H. Mori and Y. Miyumatsu, Theoretical flow models for externally pressurized gas bearings, J. Lubr. Technol., 93 (1969) 181. 73 J. T. McCabe, H. G. Ehod, S. P. Carfagno and R. Colster, Entrance effects in gas bearings, Rep. I-C 2429-1, Franklin Institute Laboratory, Nov. 1969. 74 S. P. Carfagno and J. T. McCabe, Summary of investigation of entrance effects in circular thrust bearing, Interim Rep. Z 2049-24, Franklin Institute Laboratory, Aug. 1965. 75 R. D. Dayson and M. R. Chasman, Experimental rotor unbalance response using hydrostatic gas lubrication, J. Lubr. Technol., 95 (1973) 469. 76 H. Mori and A. Mori, A stabilizing method of the externally pressurized gas journal bearings, 4th Znt. Gas Bearing Symp., Univ. Southampton, March 1967, BHRA, Cranfield, 1967. 77 A. Mori and H. Mori, An application of pneumatic phase shifting to stabilization of externally pressurized journal gas bearings, J. Lubr. Technol,, 95 (1973) 33. 78 A. G. Montgomery and F. Sterry, A simple air bearing rotor for high rotational speed, AERE Rep. ED/R 1617, AERE Harwell, 1955. 79 J. Kerr, The onset and cessation of half-speed whirl in air lubricated self-pressurized journal bearings, NEL Rep. 237, National Engineering Laboratory, East Kilbride, 1966. 80 J. W. Powell and M. C. Tempest, A study of high speed machine with rubber stabilized air bearings, J. Lubr. Technol., 90 (1968) 701. 31 B. C. Majumdar, Stability characteristics of gas-lubricated bearings supported on rubber 0 rings, Tribal. Znt., 8 (1975) 127.
299 - 314 0 Elsevier Sequoia &A., Lausanne - Printed in the Netherlands
EXTERNALLY PRESSURIZED GAS BEARINGS: A REVIEW
B, C. MAJUMDAR
Department of mechanical Engineering, indian institute of Technology, Khamgpur (India) (Received
December
l&1978;
in final form August 3,197s)
Summary The literature pertaining to externally pressurized hole admission gas bearings is briefly reviewed. The advantages, ~mi~tions and app~cations of such bearings under various design conditions are discussed.
1. In~oduction To satisfy bearing design requirements sufficient background information is necessary to optimize the design. Design requirements are often met by the proper design of conventional bearing systems, development of new or improved lubricants or development of effective seals to minimize lubricant leakage. However, in many applications even properly designed conventional bearings used with the best available lubricating oils fail to give satisfactory performance. For example solid lubricants such as graphite or molybdenum disulphide are used in some high temperature applications. Gas lubrication, which is discussed in this paper, can also be used and has some special features when compared with oil lubrication. However, gas bearings also have inherent fictions which the designer or user must recognize in order to obtain successful operation. In particular gas bearings are more prone to instability than oil-lubricated bearings. Gas bearings, like liquid- or grease-lubricated bearings, can be (i) selfacting, (ii) squeeze film or (iii) externally pressurized. In a self-acting bearing the load is supported by a wedge film action of the gas caused by relative tangential motion between two surfaces. The squeeze film.bearing supports a load because of the oscillating relative normal motion between the two surfaces. The externally pressurized bearing supports a load because of the pressure in the gas film which is supplied from an external source. External pressurization can alleviate start-up and shutdown wear by permitting jacking of the shaft in the absence of rotation. In many applications large loads (e.g. due to the dead weight of a telescope reflector mirror) or small loads (e.g. due to the mass of inertial navigation gyros) must be supported with virtually no relative motion. Self-acting bearings cannot operate in such conditions and externally pressurized bearings may be required. The
300
coefficient of friction for an externally pressurized bearing is low and is zero when there is no relative motion. If used continuously it has greater stiffness than a self-acting bearing. Thus high load, high stiffness and low coefficient of friction are the attractive features of externally pressurized bearings. However, this type of bearing is not immune to destructive whirl instability. It is necessary to recognize the types of instability which may occur in some possible bearing designs and to ensure that these undesirable phenomena do not occur in the final design.
2. Review It is not possible to credit any one person with the invention of the gas bearing. Patents were obtained by W~tin~ou~ in 1904, De Ferranti in 1906 and Abbot in 1916. The first operational gyro was developed in the U.S.A. by the Sperry Gyroscope Company in 1930. Willis [l] was the first to make an experimental investigation of the radial air flow between two parallel plates. He measured the Bernoulli forces and compared the attractive and repulsive forces in externally pressurized bearings. Many externally pressurized bearings have been designed and m~~a~tured for a variety of practical applications such as machine tools [ 2,3] , spherical bearings for space simulators [ 41 and dental drills [ 5,6] , 2.1. Classification Gas bearings can be roughly classified either by geometry or by the method of compensation. The most common type of extemalfy pressurized journal bearing has supply ports in a row or in several rows [7] . Recesses (Fig. 1) are made at the leading edges of the holes so that a high inlet pressure remains uniform throughout them. Therefore the boundary conditions are recess pressure at the holes and ambient pressure at the ends. Such bearings which are designed to support large loads may develop ~s~bi~ties. Therefore in some bearings gas is fed through plain feed holes without a recess. Gas bearings can also be classified by the method of compensation or restriction. An important component of this bearing is the flow restrictor. The bearing would not function without the restrictor. It would support a load but there would be no change in the load supported with film thickness. In practice it is necessary for the load to vary inversely with the film thickness, and the restrictor performs this function. Capillary, orifice and inher-
CTIVE Afm
= nd,h
Fig. 1. Jet configurations:
(a) plain jet; (b) jet with a circular pocket.
301
aJ[d -OW
CAPILLARY
INHERENT t ORIFICE)
INMERENY WIESYRICYINC
L*wo)
Fig. 2. Various restrictors.
ently compensated restrictors are most commonly used in gas bearings (Fig. 2). Some special types of restrictor also exist. Adams et ~2. [8] introduced a restricting land full journal bearing similar to a step bearing. Mayer and Shaw [9] predicted the characteristics of variable flow restrictors and showed their relative advantages and disadvantages. Newgard and Kiang [lo] used an elastic orifice as a compensating element to provide greater stiffness. 2.2. Design and analysis under steady state conditions The design procedures for gas- and oil-lubricated bearings are similar but gas bearing design is complicated owing to compressibility effects. Externally pressurized bearings also have stability problems. Therefore a compromise is necessary between bearing stiffness and stability. Normally a bearing of higher stiffness is more unstable. It is first necessary to establish a design which has the required stiffness, load and flow characteristics and then to check for stability. This process is repeated until a satisfactory compromise is reached. In an outstanding contribution to externally pressurized gas bearing literature Shires [ll] gives design information for both laminar and turbulent flow. Robinson and Sterry [12] used axial flow theory, i.e. gas flow takes place only along the bearing length, for computing the static load of a hole admission bearing. The pressure distribution of a bearing using the axial flow model is shown in Fig. 3. There is a large discrepancy between the theoretical and the actual load capacity as the effect of circumferential flow is neglected in the idealized bearing. Shires [13] provided a simplified theoretical treatment to compensate for circumferential flow. A semiempirical relation for the load capacity was derived by Robinson and Sterry [12] from a comprehensive series of experiments. Heinrich [14] made a theoretical ,analysis of journal bearings with a single and double row of orifices for small eccentricity ratios. He derived a closed-form solution for the load capacity by assuming a line source. Design curves for the load capacity for various design conditions were also given [ 141. This analysis takes the circumferential flow component into account. Fedor [15] developed a method for the solution of Reynolds equation applied to a full journal bearing, and also gave information [16] concerning
302
Fig. 3. Distribution of pressures in an axial flow model: (a) single central orifice; (b) two orifices in one channel.
the effects of a single source in a journal bearing. Rothe [17 ] gave a detailed description of the advantages, libations and applications of gas bearings used in U.S. Army ballistic missiles. Laub 1181 presented an analytical treatment developed for a semi-cylindrical bearing with nine orifices for the full journal bearing with 192 radial and 24 axial orifices, He considered only the axial flow component in his analysis. Licht [19] showed that by suitable changes of dependent variables the equation which governs laminar and isothermal flow of the gas lubricant in otherwise geome~c~ly identical bearings could be reduced to the Laplace equation v "p = 0 where p is the gas film pressure. Laub and Norton [ZO] made an analytical and experimental study of spherical gas bearings with orifice compensation. Allen et al. [21] studied the effect of bearing length on the performance of bearings with single and double rows of orifices. This analysis was also based on an idealized flow model. The expe~men~ly determined load capacity was half the value calculated using axial flow theory. It was predicted that circumferential flow could account for the difference. Tang and Gross [22] presented analysis and design charts for the evaluation of the performance of externally pressurized gas bearings. The curves are most effective as when they are properly used the complicated task of the calculation of the do~stream pressure from flow equations is eliminated. The curves can also be used to design bearings either for maximum load or maximum stiffness,
Turnblade [23] proposed a new method of bearing equivalence as a simple technique for design. He developed an equation for rectangular bearings which described the relation between the ratio of effective bearing width to effective length and the ratio of width to length of an arbitrary element. An equivalent rectangular bearing can be conveniently described by estimating the streamline flow pattern within the bearing configuration and solving the equation. The performance of the equivalent bearing is identical to that of the actual bearing. Lemon [24] presented a simplified analysis for determining load, flow and stiffness. He considered the effect of circumferential flow by using an approximate equation and presented a design procedure to obtain optimum stiffness in a journal bearing. A theoretical and experimental analysis of a gas journal bearing with six orifice stations was made by Mori et al. [ 251. The complex potential theory was considered to be the most suitable method for obtaining a point source solution of Reynolds equation. With this theory Mori et al. obtained the load capacity and volume rate of flow and they predicted the actual load and mass flow rate from the incompressible flow solution. Dudgeon and lowe [26] predicted the performance of a static gas journal bearing theoretically by considering axial and circumferential flow. Rieger and Wilcock [27] derived a simplified design method for load, flow and stiffness of thrust and journal bearings. A systematic analysis and simplified procedure for designing journal bearings together with design charts for predicting journal bearing performance have been given [28] . Shires [29] also gave simplified methods for designing thrust and journal bearings. Powell [30] presented design charts of load capacity for lightly loaded bearings. Constantinescu and coworkers [31 - 331 suggested an approximate method of analysis using the line source assumption and claimed that this led to accurate results even when the number of orifice stations was relatively small, e.g. n = 6. Pink [ 341 compared the results of MT1 [ 281, Constantinescu and Salcudean [ 311 and Powell [ 301 with the experimental data of Eusepi and Lewis [ 351 and concluded that there was considerable conflict between the various published design methods in the prediction of static bearing stiffness and load. The above references are restricted in the sense that some do not consider the circumferential flow component and some use a line source assumption. Majumdar [ 36 - 401 carried out theoretical and experimental work on hole admission bearings with four, six and eight orifice stations in two planes at quarter stations. The theory considered both axial and circumferential flow without assuming a line source. When circumferential flow is considered in the analysis the pressure distribution is different from that obtained with the usual line source assumption. Figure 4 shows a typical pressure distribution obtained when the discreteness of the holes is considered. The load capacity and the mass rate of flow of a bearing with six supply holes in two rows at the quarter station are shown in Figs. 5 and 6. The line source assumption is not relevant to the actual behaviour at least at high eccentricity ratios. The optimum design method for maximum load and for maximum stiffness was also indicated.
304
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1
I.0
n-e
I
1.0
I
nl.3 E SO.3
0.8 -
n=e
n-6
e -0.3
01
0
I
0.5 T'
I 1.0
I
0
0.5 x
I I.0
(a)
(b)
Fig. 4. (a) Premure distribution in the axial direction. Experimental r~dts: 0.6 = 0.60; =, 6 = 1.10; V, 6 = 2.04;v, 6 = 3.06;& = 4.98 kgf cmb2. Theoretical results (solid CUrVea) calculated using 6 = 46 q(k62T)l” cdA,,IcSpI, and & = pa/pa. (h) Pressure distribution in the circumferential direction. Experimental results: q ,6 = 0.90; ., 6 = 2.12; 0,6 = 2.96; l,6 = 4.10; I, 6 = 5.17;& = 4.96 kgf cmT2. Theoretical results (solid curves) calculated from 6 = 46 v(k61T)lP cdA,/c?p, with n = 6,~=0.3,~~=0.2,i=0.5and&=~,/~,.
oI
3
2
0
4
C’ P*
3
6 PI
(4
--
00
I I
I
I 3
*
I 4
I
6 (c)
Fig. 5. Load capacity of a compensated bearing: (a) n = 6, pa = 0.2; (b) n - 6, pa - 0.5; (c) n = 6,& - 0.33. Solid curves, accurate theoretical resulta; broken curves, approximate theoretical results. hperimental results: 0, e = 0.3; 0, E = 0.6; A, e = 0.9.
The theoretical and experimental analyses of the references quoted so far are directed towards the determination of bearing characteristics at zero speed. The first theoretical investigation considering journal rotation using pressure perturbation theory and a line source assumption is probably that of Lund [41] . He produced design charts for the variation of steady state load capacity and attitude angle with journal speed for various supply pressures and feeding parameters. MT1 [ 281 gave design charts for a.rotating journal and a relation between attitude angle and eccentricity ratio. Castelli and Pirvics [42] proposed a new method for the solution of the generalized Reynolds equation by which the load coefficients and attitude angles of three- and four-pad axial groove cylindrical bearings were found. Castelli and Shapiro [43] obtained a numerical solution of Reynolds equation for hydrodynamic, hydrostatic and hybrid bearings. Bearings with multiple recesses were also studied. Fleming et al. [44] used a pressure perturbation method for an unloaded journal to obtain the steady state load capacity and
306
0
0
I
I
1
f
f
2
3
4
6
faf
(b)
14
I2
IO
b
EC 6
4
z
,
f
I
I
2
I 3
I 4
6
(cl Fig. 6. Flow rate of a compensated bearing: (a) R = 6, Is, = 0.5; (b) n = 6,& = 0.33; (c) n = 6, &, = 0.2. Solid curves, accurate theoretical results; broken curves, approximate theoretical results. Experimental results: 0, E = 0.9; 0, E = 0.6; *, 6 = 0.3.
307
attitude angles. Cunningham et al. [45, 461 performed experiments on a rotating journal bearing which had two rows of supply holes with six orifices per row. Their experimental results were compared with the results obtained with linearized p and ph theories [ 43,471. It was shown that the linearized ph perturbation solution was nearer to the experimental results. Powell [48] reported elaborate experimental results for hybrid bearings. Majumdar [49] used a linearized perturbation solution of a similar bearing without the line source assumption. A comparison of the results for the above methods is shown in Figs. 7 and 8.
Fig. 7. Load capacity of a rotating journal: L/D = 1.5; n = 6; 6 = 3.0. Broken curve, line source assumption [ 411; chain curve, line source assumption [47] ; solid curve, discrete source [ 491; 0, experimental results [ 451. Fig. 8. Attitude angle of a rotating journal: L/D = 1.5; n = 6; 6 = 3.0. Broken curve, line source assumption [ 411; solid curve, discrete source [ 411; 0, experimental results [ 46 1.
2.3. Design and analysis under dynamic conditions An important practical consideration in the design of externally pressurized bearings is selfexcited instability; pneumatic instability or air hammer can also be a problem. The evaluation of the selfexcited whirl of externally pressurized bearings has been studied by Mullan and Richardson [50] , Gross [51] and Lemon [ 241. They assumed that selfexcited whirl would occur at a rotational frequency slightly less than double the calculated frequency. Lund [41] used a pressure perturbation method. Licht and Elrod [52] conducted experiments to study air hammer instability in thrust bearings and used distributed parameter analysis to show how the depth of the pool affected the stability. Richardson [53] presented a general stability criterion based on lumped parameter analysis of gas bearings with orifice compensation. Mori [ 541 suggested that the experimentally observed depression of the pressure profile in thrust bearings was caused by the generation of a shock wave in the bearing clearance space. Licht [55] developed a stability analysis for gas journal bearings with externally pressurized pads and Licht and Elrod [56]
308
investigated selfexcited vibration. Stability bounds were established theoretically for circular thrust bearings and verified experimentally. Hirs [57] proposed a new type of bearing in which the stability and load capacity were ensured by directing the flow in the clearance gap through grooves on one of the surfaces towards a zone where both surfaces were plain. Allais [58] studied gas bearing equations and derived relations for maximum stiffness and stable operation. Some approximate formulae were derived for orificecompensated bearings under choked conditions. Speen [ 591 described an integral journal thrust bearing with superior load capacity, stability and stiffness. Majumdar [60] made a theoretical analysis of a bearing under plain vibration of the stationary journal. The results were compared with the simplified solution using a line source assumption. A solution for the hole admission bearing with squeeze action has been made by a quasi-static assumption [ 611 and a pressure perturbation solution [62] . The quasi-static solution overestimates the squeeze load capacity. The stability analysis of rotating externally pressurized bearings was first made by Larson and Richardson [63]. They conducted experiments and showed the variation of stability with bearing clearance. The work on the stability of stationary and rotating journal bearings most generally cited is that of Lund [47] who made a theoretical estimation of the stability of a bearing with a central row of holes. Ono and Tamura [64] derived the theoretical stability characteristics using a quasi-static assumption and verified their results experimentally [65] . Fleming et al. [44] calculated the stability of unloaded journal bearings by using a first-order perturbation theory with steady whirl approximation and found that the effect of the recess volume of an orifice-compensated bearing was to decrease the stability. Elrod and Glanfield [66] developed a very detailed computer method for designing a bearing of similar configuration with a flexibly mounted rotor. Lund [67] predicted the stiffness and damping of gas bearings and the stability behaviour of an elastic rotor on a flexible damped support [68] . Ausman [69] analysed the dynamic behaviour of journal bearings with a sinusoidally timevarying load. Ehod et al. [70] obtained the stability by the response to step jump. Majumdar [60] also determined the stability using a steady whirl approximation and without the line source assumption and compared the results with those of Fleming et al. [44] . Figure 9 shows a detailed comparison of the stability results using the various methods outlined. 2.4. Other effects Mori [ 711 analysed a gas bearing by considering lubricant inertia and turbulence. Mori and Miyamatsu [72] have contributed to the understanding of the characteristics of externally pressurized gas bearings. Various flow models were employed to explain results obtained over a wide range of operating conditions. Their analysis included the entrance effects accompanied by the growth of a boundary layer and the occurrence of a shock wave. Entrance effects were taken into account by McCabe et al. [ 731 and Carfagno and McCabe [74]. Elrod and Glanfield [66] also considered the entrance
309
\
UNSTABLE
UNSTABLE
f STABLE
\
\
loi1
5
Azsrlo
qryd
6
7991)
\
I
(b)
I
I
I
2
3
4
llllll S
676910
A
Fig. 9. Stability characteristics. (a) L/D = 1.0; n = 6; 6 = 3; e, = 0.1; I#JW = 0; solid curve, accurate; chain curve, steady whirl [60] ; broken curve, steady whirl [44]. (b) As for (a) except Qw = 50; 0, experimental results for & = 2; 0, experimental results for ii, = 6.
effect in their stability study. Dayson and Chasman [75] investigated the adequacy of the theoretical model in predicting the unbalanced response of a rotor supported in hydrostatic gas-lubricated journal bearings. The effect of rotor speed and supply pressure on rotor unbalance was also investigated. Mori and Mori [76] showed that the main bearing could give a very high onset speed if an auxiliary bearing was added. They also studied pneumatic phase shifting as a means for stabilizing selfexcited whirl in externally pressurized gas bearings and described constructional details of the stabilizer 1771.
310
A method of increasing stability is to mount the bearing on rubber Orings [78] . Kerr [79] demonstrated a high onset speed in hydrodynamic gas bearings. Powell and Tempest [80] made an experimental investigation of externally pressurized bearings mounted on rubber O-rings. A simplified analysis of the stability of similar bearings has been reported [81] and confirmed Powell and Tempest’s results. 3. Conclusions and recommendations It appears from the survey that the theoretical estimate of static characteristics such as load capacity, maSs flow rate and stiffness of lightly loaded hole admission gas bearings can be predicted with reasonable accuracy from axial flow theory. However, axial flow theory does not appear to be satisfactory for a heavily loaded bearing. The axial flow theory always overestimates load capacity as it does not account for the variation of pressure between the two feeding planes (Fig. 3). The line source assumption for a discrete number of feed holes has been used for a few analyses but such solutions cannot be considered as general solutions. Majumdar’s work (Fig. 5) on similar bearings suggests that the line source assumption is not valid for all hole admission bearings. It is particularly unsuitable if a bearing has less than eight feed holes in a row. Lund [47] used a correction factor of 1.5 to compensate for the feed holes in the calculation of flow but not in the computation of the gas film pressure. Therefore the use of a correction factor with the line source solution is not entirely justified. The advantages and disadvantages of the different methods are summarized in Table 1. Much information on the static behaviour and stability characteristics is available, but little is known about the complicated flow phenomena near the supply holes. The pressure drop at the edge of the supply holes is due to (i) the energy consumption for the development of the boundary layer and (ii) the accelerating energy of the lubricant from the stagnation point. Mori and Miyumatsu [72] gave an approximate expression for pressure drop as 0.7 po(Qo/n dh)2 where h, Q0 and p. are the film thickness, volume flow rate and fluid density respectively. However, this estimate does not consider the effect of the entrance length on the development of boundary layers, the compressibility of the lubricant and the divergence of the flow area. The pressure at the edge of the supply hole can be obtained by subtracting this drop from the predicted pressure downstream of the orifice. Another estimate for the correct pressure at the edge of the supply hole is due to Elrod and Glanfield [66] . Flow settles down some 250 film thicknesses or less from the hole. The pressure tends to p2 = In r for large r where r is the distance measured from the centre of the supply hole. If the pressure curve obtained from the solution of Reynolds equation is extrapolated back to the nozzle curtain area the correct pressure is found. Only the pressure in the neighbourhood of the hole will be incorrect. The correct pressure at the edge is sometimes’5 - 10% less than the pressure predicted by the usual method.
311 TABLE 1 Relative merits of various methods
Method
Advantages
Disadvantages
Considering discrete supply holes
No a priori assumption
Numerical convergence and stability are difficult to ascertain; costly
Line source ~ump~on
Quite simple
Qu~tionable for large e and small number of supply holes
Axial flow theory
Simplest
No a priori justification
As externally pressurized gas journal bearings are becoming more popular it is hoped that further research may resolve some of the outstanding problems and provide more reliable data, design charts, simple design rules and procedures for practising engineers.
References
7 8 9 10
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