Design of multirecess hydrostatic oil journal bearings M.K Ghosh* and B.C. Majumdart

This paper presents computer generated design data in terms of load capacity and oil f low for multirecess hydrostatic journal bearings. The Reynolds equation for a finite bearing was solved on a high speed digital computer satisfying appropriate boundary conditions and using the finite difference method. Results for various L/D ratios, recess to bearing area ratios, number of recesses etc are presented for capillary and orifice compensated bearings

Multirecess hydrostatic journal bearings have found applica- tion in machine tools and the aerospace industries due to their high stiffness and low friction. The bearing configura- tion can either have axial drain grooves in between recesses or can be without drain grooves. The bearings without axial drain grooves have better characteristics due to the inter- action of flow between the recesses and have become popular. Capillary and orifice restrictors are most common- ly employed.

Raimondi and Boyd I first presented a complete theoretical analysis for a multirecess hydrostatic journal bearing based on a one-dimensional flow model. Rippel 2 subsequently presented detailed analysis, design procedure and design data for various configurations of externally pressurised oil bearings including a rlaultirecess hydrostatic journal bearing. Cowley and Kher a presented both theoretical and experimental results for a four-recess hydrostatic journal bearing. Davies 4 presented a general theoretical analysis for multirecess hydrostatic journal bearings, taking into account the effect of shaft rotation.

O'Donoghue and Rowe s'6 developed a design method for externally pressurised bearings based on the pad load and flow coefficients which depend only on the geometrical shape and proportions of the bearing and are independent of the control device used. Design data in terms of pad load and flow coefficients were presented for various geometric configurations based on short land theory.

O'Donoghue et al 7,s discussed the procedure for the optimisation of externally pressurised bearings on the basis of minimum power consumption and obtained opti- mum sill width ratio (0.25) and optimum speed parameter. O'Donoghue and Rowe 9 also gave the exact computer solution of Reynolds equation for a finite multirecess hydrostatic journal bearing using the f'mite difference method. Decker and Shapiro a° reported on the computer aided design procedure for hydrostatic bearings. However, adequate design data were not presented. Ghosh ~1'12 obtained a computer solution of Reynolds equation for bearings with large circumferential sills and showed that an optimum sill angle existed at which load capacity of the bearing was a maximum. Experimental verification of the theoretical results was also reported and finite element solutions have recently been recorded t3 .

*Institute of Technology, Banaras Hindu University, Varanasi 221005, India tIndian Institute of Technology, Kharagpur 721302, India

The above review reveals that most of the theoretical analysis and design data available in the literature are for bearings with short axial and circumferential lands. Results available for bearings with large axial and circum- ferential sills are inadequate and the effect of various parameters eg LID ratio, recess to bearing area ratio, number of recesses etc are nbt depicted clearly in-the literature.

This paper presents computer generated design data in terms of recess pressures, load capacity and oil flow for multirecess hydrostatic journal bearings. The Reynolds equation for a finite bearing was solved on a high speed digital computer satisfying proper boundary conditions. Results for various LID ratios, recess [o bearing area ratios, number of recesses etc are presented for capillary and orifice compensated bearings.

Analysis

A multirecess hydrostatic bearing system is shown in Fig 1. The off is supplied from a constant pressure source at a pressure Ps and it enters the bearing through a restrictor (either capillary or orifice). The recess pressures get adjusted in such a way as to carry the net load on the bearing.

With the usual basic assumptions for an isoviscous, incom- pressible lubricant the generalised Reynolds equation for a nonrotating journal (neglecting the time dependent terms) can be written as

~-~(h3aP~ a ap ~--, + ~ (h 3 ~ - ) = 0 (1)

Using the following substitutions

-fi = PIPs,-h = h/C, 0 ."- x /R , 2 = z / (L/2)

Equation (1) becomes

a (~s a~ +(D/L)2 a ap --a0 bff ) ~-~ (h-3 ~-~ ) = 0 (2)

where h = 1 + eo cos 0

Equation (2) is written in a finite difference form and solved by iteration using a high speed digital computer satisfying the following appropriate boundary conditions:

ff (0, + 1)=0

a__~_ff (0, 0) = 0 for symmetry

0301-679X/80/020073-06 $02.00 © 1980 IPC Business Press TRIBOLOGY international April 1980 73

Ghosh and Majumdar - Design of multirecess hydrostatic oil journal bearing

' P = Pr at the rth recess

(0, g) = ff (0 +27r, g ) fo r periodicity

The recess pressures, Pr, are evaluated using the method described by Majumdar 14 .

Static load capacity and oil f low of the bearing

The static load capacity of the compensated bearing is calculated as:

L/2 27r W c = - 2 f f p c o s 0 R d 0 d z

O O

The oil flow rate of the compensated bearing is:

2C 3 2r~ -~p ( l+eoCOS0)aRd0 Q c - 12/a o f [ ~-zlz=L/2

The static load capacity and oil flow rate are expressed in dimensionless form as

Wc = lye Q c - 12/aQc LDp-~s' CaPs

Results and discussion

Effect of concentric pressure ratio/3

Dimensionless load capacity Wc vs concentric pressure ratio/3 at different eccentricity ratios is shown in Fig 2 for both capillary and orifice compensated bearings. It is observed that the load capacity of the compensated bearing has an optimum value with respect to/3. The orifice compensated bearing is seen to exhibit higher load carrying capacity.

Variation of dimensionless oil flow Qc vs concentric pressure ratio/3 is shown in Figs 6 and 7 for capillary and orifice compensated bearings, respectively. Oil flow is found to increase with increase in/3. It is also seen that up to/3 ~< 0.6 (approximately) the oil flow increase is nearly linear with respect to/3 and beyond/3 > 0.6 oil flow increase is large and nonlinear with respect to/3. Since large amount of oil flow would mean more power required

to pump the fluid into the bearing clearance, it would be advantageous to operate at a value of/3 ~< 0.6. An optimum value of/3 with respect to load capacity is also observed to lie between 0.3 and 0.6. Therefore, while designing the bearing an optimum value of/3 should be chosen for load capacity and this also reduces the oil flow. For the known values of/3, dimensionless capillary and orifice design parameters can be calculated from the concentric flow parameter.

Effect of LID ratio

Dimensionless load capacity I# c vs LID ratio is shown in Fig 3. It is seen that the dimensionless load capacity of the bearing decreases with increase in L/D ratio. However, this does not mean that the actual load capacity of the bearing is also reduced, since increase in length of the bearing results in increase in the projected area of the bearing for the same diameter.

Variation of dimensionless oil flow Qc with L/D ratio for capillary and orifice compensation is shown in Figs 8 and 9, respectively. It is observed that oil flow decreases with increase in L/D ratio. Therefore, for a particular load to be carried by the bearing, a higher value of LID ratio must be chosen.

Effect of recess to bearing area ratio

Variation of dimensionless load capacity We with recess to bearing area ratio is shown in Fig 4. Since large recesses mean large areas of constant pressure, the load capacity of the bearing increases with increase in recess to bearing area ratio.

Variation of dimensionless off flow Qc with recess to bearing area ratio is shown in Fig 10. Oil flow increases almost linearly with E and b for ~ and b ~< 0.6 approximately, beyond which increase in oil flow is quite high. Therefore, there seems to be an optimum value of ~ and b for which a balance can be obtained between load capacity of the bearing and the correspond-

Nomenclature

a Axial length of recess

~- Dimensionless axial recess width, a l l b Circumferential length of recess

Dimensionless circumferential recess length, Nb/TrD

C Radial clearance

Cd Discharge coefficient of orifice D Diameter of journal

d o Orifice diameter

e o Eccentricity

h Film thickness

Dimensionless fi lm thickness, h/C

L Length of bearing

N Number of recesses

p Film pressure

p- Dimensionless f i lm pressure, P/Ps Pr Recess pressure at rth recess

Pr Dimensionless recess pressure at rth recess, Pr/Ps Ps Supply pressure

Qc Oil flow rate

Qc Dimensionless flow rate, 12/JOe/Caps R Journal radius

W c Load capacity

H/c Dimensionless load capacity, Well DPs x, z Coordinates

/3 Concentric position recess to supply pressure ratio 6 o Orifice restrictor coefficient,

37T/J Cd do 2 2 ½ ( ) C 3 P Ps

6o Eccentricity ratio, eo/C 0 Angular coordinate, x/R 1~ Absolute viscosity of the lubricant

p Density of the lubricant

74 TRIBOLOGY international April 1980

Ghosh and Majumdar - Design o f multirecess hydrostatic oi l journal bearing

~e Pressure

]nt supply pressure J ~ - - Pressure

<>F,,., ( ~ . Motor

-.() .

gouge

control valve

Fig 1 Externally pressurised multi-recess ]ournal bearing system

0.6

0.5

04

8 as

._E 0.2 , , - , ,

0.1

0.2 0.4 0,6 0.8 Concentric pressure ratio, ,8

i.o

Fig 2 Load capacity of compensated bearing vs concentric pressure ratio

0.6

0.5

I~. 0.4 >,

8

=_o

0.2

0.1

. . . . Orifice compensated bearing Capillary compensated bearing

J 0.5 1.0 1.5 2.0 2.5

L I D

Fig 3 Load capacity of compensated bearing vs L/D ratio

T R I B O L O G Y in ternat iona l Apr i l 1980 75

Ghosh and Majumdar - Design o f multirecess hydrostatic oil journal bearing

ing increase in the pumping power required to force the liquid into the bearing. The opt imum value for ~ and lies between 0.4 and 0.6. Optirnisation by O'Donoghue et al 8 shows that this value is approximately equal to 0.5.

0.6

0.5

N=4 /3 = 0.6 L=I.O

i~ 0.4

'8

~ 0.3

g

E O.2 i:5

~ = b =

0.8

0.6

0.5

0.4

- - 0.2

0.I

0 0,2 0.4 0,6 0.8 1.0 Eccentricity ratio,, o

Fig 4 Load capacity of capillary compensated bearing vs bccentricity ratio for various recess bearing area ratios

I~ 04

o. 8

g "N

-~ 0.2

0,6 | /3 = 0.6 I 0=~=0.6

~=1.0

05

N= 6

3

0,1

I I I I

0 0.2 0.4 0.6 0.8

Eccentricity ratio,, o

i.o

Effect of number of recesses The variation of dimensionless load capacity Wc and dimensionless oil flow rate Qc with number of recesses is shown in Figs 5 and 1 1, respectively. It is quite clear that the number of recesses has very little effect on either load capacity or oil flow of the bearing. However, a symmetrical number of recesses (either 4 or 6) is usually selected in the design.

The above analysis does not consider the effect of journal speed (or hydrodynamic effects). It has been found 15 that the effect of journal rotation on the performance of a bearing having a large sill area is not significant up to a

6/a03 bearing number A equal to 1, where A Ps (C/R) 2

and 03 is the angular speed of the journal. This analysis Is accounts for the cavitation effect of hybrid multirecess journal bearings. Therefore the present data can be used for a hybrid journal bearing if the bearing number does not exceed 1.

Design procedure The following parameters must be determined:

Concentric pressure ratio/3 Concentric oil flow rate ~)c Length to diameter ratio LID Dimensionless restrictor parameters/i c or 6o

In order to design the bearings to carry a particular load W, the following steps may be followed:

Usually LID ratio is taken as 1 but a higher value may be chosen.

Opt imum value of the concentric pressure ratio parameter/3 is to be determined from Fig 2 for a particular eccentricity ratio. Usually operating eccentricity ratio is kept less than or equal to 0.5.

45.0

40.0

35.0

50.0

25.0

.~ 20.0

i:5 15.0

10.0

5.0

t N=4 5=~=0.6 L=I.O

0.8

- - 0.6

_ 0.4

0.0

f I I I I I 0.2 0.4 0.6 0.8 1.0

Concentric pressure ratio,/3

Fig 5 Load capacity of capillary compensated bearing vs eccentricity ratio for various number of recesses

Fig 6 Oil flow of capillary compensated bearing vs pressure ratio

76 TRIBOLOGY international April 1980

Qeo - 12 p Qeo C3 Ps

Oil flow rate Qeo = 6.1 x 10 -6 m3/s

An equation can be derived to get a relationship between 80, ~ and Qco:

Since orifice compensated bearings exhibit higher load capacity, orifice compensation may be chosen.

For the optimum value of the parameter/3 obtained, concentric oil flow rate Qe is found from either Fig 6 or 7 depending on the type of compensation chosen.

Using the concentric oil flow rate Qe and the concentric pressure ratio/3 dimensionless restrictor parameter 8 c or 8 o is calculated.

Design example

A multirecess hydrostatic journal bearing has the following specifications:

Operating eccentricity ratio, eo = 0.4 Diameter of the journal, D = 100 x 10 -3 m Diametral clearance, 2C = 100 x 10 -6 m Viscosity of oil,/a = 24 x 10 -3 Ns/m 2 Density of oil, p = 0.9 x 103 kg/m 3 Supply pressure, Ps = 106 N/m2

To design the bearing for maximum load capacity choose LID _-_ 1.0,~ = 0.6 and b = 0.6, N = 4, Orifice compensation Therefore L = 100 x 10 -3 m, a = 60 x 10 -3 m, b = 47.1 x 10 -a m

From Fig 2, the optimum value of concentric pressure ratio # for eo = 0.4 is obtained as/ /= 0.55, and the corresponding dimensionless load capacity ~/e = 0.28.

Concentric oil flow parameter Qeo as obtained from Fig 7 = 14.00 (approximately)

LDPs '

thus load capacity of the bearing

Wc = ~ LDps We = 2800 N

45.0

40.0 -

35.0 -

3QO

i 25,0

i 2QO

15.0

IQO

5.0

L=0 .6

-- 0.8 0.6

0.4 - % = 0.0,0. 2

I 0.2 0.4 0.6 0.8 1.0

Concentric pressure rotio,,8

Fig 7 Oil flow of oroqce compensated bearing vs pressure ratio

45.0

40,0

3~LO

300

25.0

0

2Q0

i l 5 . 0

IO0

5.0

45.0

B = a 6

Ghosh and Majumdar - Design o f multirecess hydrostatic oil journal bearing

I J I L 05 1.0 1.5 2.0 2.5

L / D

Fig 80i l f low of capillary compensated bearing vs L/D ratio

40.0

35.0

30.0

,%= 0.0

0.4,0.6

~ 25.0

j ~oo

15.0

IQ0

5.0

J J L i - ~ 0 0.5 1.0 1.5 2.0 2.5

L I D

Fig 90i l f low of orifice compensated bearing vs L/D ratio

TRIBOLOGY international April 1980 77

Ghosh and Majumdar - Design of multirecess hydrostatic oil journal bearing

~o 5 o - sv ( i - ~)1/~

Substituting values of Qco, N and/3

8o = 5.2

Now taking a value of Ca = 0.8, the orifice diameter d o = 4 . 8 x 1 0 -s m

This design will also be valid for a rotating journal if the journal speed is limited to about 7 rad/s. With C/R = 0.001, M = 24 × 10 -3 Ns/m = ,Ps = 106 N/m=, the angular velocity of the journal is found to be approximately 7 rad/s.

Conclusion Effect of various design parameters eg concentric recess to supply pressure ratio, LID ratio, recess to bearing area ratio, number of recesses etc on the load capacity and oil flow rate of multirecess hydrostat ic journal bearings is discussed. The design procedure presented utilises computer generated data and shows that an op t imum design for the maximum load capacity is possible by proper selection of various design parameters.

45.0

"40 .0 -

3 5 . 0 -

3 0 . 0 - -

25.0 --

"5

20.0 g

~ t5.0 -" i:5

I0.0 -

5.0 --

References 1. Raimondi A.A. and Boyd J. An Analysis of Orifice and

Capillary Compensated Hydrostatic Journal Bearings. ASME-ASCE Lub. Conf. Baltimore USA, Oct. 1954

2. Rippel H.C. Design of Hydrostatic Bearings. Machine Design, August 1963 to December 1963, pts. 1 to 10

3. Cowley A. and Khet A.K. The Design and Performance Characteristics of a Capillary Compensated Hydrostatic Journal Bearing. Proc. 8th Int. MTDR Conf., September 1967, Pt. 1,397

45.0

4 Q O

35.0

3QO

o

25.0 g

2 0 . 0 Q

15.0

IQO

5.0

/lli- "f°'°

I L 1 _ _ I _ _ ~ 0.2 0.4 0.6 0.8 I.O

Fig I 0 Oil f low o f capillary compensated bearing vs recess to bearing area ratio

/3=0.6

6=/3=O6

L=I.0

N = 5 , 4 , 5 , 6

I I I I

0 0.2 0.4 0.6 0.8

Eccentricity ratio,%

Fig 11 Oil f l ow o f capillary compensated bearing vs eccentricity ratio f o r various number o f recesses

i.O

4, Davies P.B. A General Analysis of Muitkecess Hydrostatic Journal Bearings. Proc. L Mech. E. Lond., 184, Pt. 1, No, 43, 1968-70

5, O'Donoghue J.P. and Rowe W.B. Hydrostatic Bearing Design. Tribology International, 1969, 2(1), 25- 71

6, O'Donoghue J.P. and Rowe W.B. Design of Hydrostatic Journal Bearings. Mach. and Prod. Engg., 1968, 11, 1284

7. O'Donoghue J.P. et al. Optimisation of Externally Pressurised Bearing for Minimum Power and Low Temperature Rise. Tribology International, 1970, 3,153

8. O'Donoghue J.P. et al. Design of Hydrostatic Bearing Using an Operating Parameter. Wear, 1969, 14, 355

9. O'Donoghue J.P. and Rowe W.B. Hydrostatic Journal Bearings (Exact Procedure). Tribology International, 1970, 3, 230

10.Decker O. and Shapiro W. Computer Aided Design of Hydro- static Bearings for Machine Tool Applications. Proc. 9th Int. MTDR Conf. 1968 (Sept.) Pt. 2, PP 797-834

11.Ghosh B. An Exact Analysis of a Hydrostatic Journal Bearing with a Large Circumferential Sill. Wear, 1972, 21, 367-376

12. Ghosh B. Load and Flow Characteristics of a Capillary- Compensated Hydrostatic Journal Bearing. Wear, March 1973, 23(3) 377-386

13.Singh D.V., Sinhasan R. and Ghai R,C. Finite Element Analysis of Orifice Compensated Hydrostatic Journal Bearings. Tribology International, 1976, 9(6), 281-284

14. Majumdar B.C. The Numerical Solution of Hydrostatic Oil Journal Bearings with Several Supply Ports. Wear, 1969, 14, 389- 396

15. Gho~ M.K. Dynamic Behaviour of Externally Pressurised Multirecess Oil Journal Bearings, Ph.D. thesis, Mechanical Engineering Department, HT, Kharagpur, 1977

78 T R I B O L O G Y i n t e r n a t i o n a l A p r i l 1980