1

Paper 1

DIAPHRAGM VALVES FOR CONTROLLING

OPPOSED PAD HYDROSTATIC BEARINGS
W. B. Rowe* J. P. O'Donoghuet

The diaphragm valve described is of very simple construction and may be used to control opposed pad
hydrostatic bearings as an alternative to capillaries, orifices, or other types of control device. Experiments
and analysis for flat pad bearings show good agreement. Data have been computed for a range of operating
characteristics in a form suitable for use by designers. A reference bearing system is proposed for correcting
errors in long slideways in which the reference is a diaphragm controlled hydrostatic bearing. The computed
data provides information necessary to design such a system.

INTRODUCTION pressure is supplied to annular regions on opposite faces

HYDROSTATIC BEARINGS require a control mechanism of a diaphragm. Flow is directed radially inwards between
which ensures a stable relationship between the applied restrictor lands to a central outlet on each side of the
load and the bearing gap. The means of control most diaphragm and supplies the opposite bearing recesses.
commonly employed are capillaries and orifice restrictors. The principle of operation may be understood by con-
For better stability and higher stiffness there are a range sidering the effect of an applied load on the bearing w.
of devices, including pressure-sensing and position- The load is reacted by the increase in the recess pressure
sensing valves. A full description of these devices has been p1 and also by the reduction in the recess pressure pz.
given previously in reference (I)+. These pressures, which also act on the diaphragm, cause
The double diaphragm valve described in this paper an increase in clearance between the diaphragm restrictor
was developed for grinding machine slideways and spindle lands on the highly loaded side while reducing the
bearings. It is a form of pressure-sensing valve designed clearance on the opposite side. Extra flow is forced through
specifically to control equal area opposed pad bearings. the highly loaded bearing pad to maintain the bearing gap,
For practical purposes it is possible with this type of valve while less flow is supplied to the lightly loaded pad. With
to obtain infinite bearing stiffness or even negative stiff- a diaphragm of correct flexibility, bearing stiffness may be
ness. The double diaphragm valve has advantage over so high as to be considered infinite over a useful propor-
other systems for this type of application because of its tion of the load range.
manufacturing simplicity. The principle has also been applied successfully to
Development was carried out and a patent was first hydrostatic journal bearings with four pads, as in Fig. 1.lb.
applied for by Rowe (2). It was also developed by De Gast An advantage of hydrostatic bearings is the high standard
(3) for use with high precision lathes. The reference of accuracy that can be achieved. If the bearing system can
bearing system described in this paper is a development be made load independent (i.e. infinite stiffness) it is
incorporating the double diaphragm valve intended for apparent that the moving bearing will follow a reference
correcting errors in long slideways (4). bar with extreme accuracy. This principle is utilized in a
Fig. 1 . 1 shows
~ a schematic drawing of an opposed pad reference bearing system (Fig. 1.2) which will correct for
hydrostatic bearing with the diaphragm control. Supply errors in a long machine tool slideway.
A large proportion of the load is supported on the main
The M S . of this paper mas received at the Institution on 23rd July bearings by pressurized oil. Since displacement of the
1969 and accepted for publication on 2Ist October 1969. 33
* Principal Lecturer in Production Engineering, Lanchester College main bearings is controlled by the reference bearing, flow
of Technology, Coventry. rate through the main bearing has no significance for
t Principal Lecturer in Mechanical Engineering, Lanchester College bearing performance. It is therefore feasible to incorporate
of Technology, Coventry.
$ References are given in Appendix 1.2. a seal and hence reduce oil collection problems. The
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2 W.B. ROWE AND J. P. O'DONOGHUE

Diophroqm I I
Opposed
main bearing area, and a factor for the relay (if this is not
adjusted to unity).

m
valve beoring
A linearized solution was suggested by de Gast (3) which
.Diophroqm applies only for small variations from the unloaded con-
ditions. It was found that a linearized solution under-
estimates deflections in some cases and overestimates in
P, others. The following results are therefore based on exact
I L I solutions of the basic equations.
Fig. l.la. Schematic arrangement of an opposed pad
hydrostatic bearing controlled by a double diaphragm
valve Notation
A d Effective area of diaphragm valve.
Ae Bearing effective area.

1 A m Area of main load-carrying bearing.

B Bearing flow factor (= q77/ph3).
Bd Valve flow factor { = q 7 ) / [ P ( h d f X d ) 3 ] } -
P Dimensionless pressure (= p/P,).
p, Supply pressure.
P Recess pressure.
W Reference bearing load parameter (= w/PJe)-
W Load supported by diaphragm compensated bearing.
X Ratio bearing gap to design gap (= h/h,).
xd Ratio of diaphragm deflection to diaphragm design
L
- X
gap (= X d l h d ) .
Bearing deflection.
P Design pressure ratio (= p,/P,).
h Bearing stiffness.
e x Stiffness parameter (= hho/P,Ae).
hd Diaphragm stiffness.
Ad Diaphragm stiffness parameter (= h d h d / P s A d ) .
I
SuBxes
Fig. 1.lib. An arrangement for diaphragm control of 1 Denotes high-pressure side of bearing and valve.
four-pad hydrostatic journal bearings 2 Denotes low-pressure side of bearing and valve.

reference bearing carries a proportion of the loads but the

proportion must be so small that there is no risk of THEORY
distorting the reference bar. The pressure applied to the The load-deflection relationships are derived in Appendix
main bearings is controlled by the pressure in the reference 1.1. The following basic equations are required for the
bearing by means of a hydraulic relay. Ideally the reference solution:
bar should be mounted so that it is not subject to the
structural distortions of the main bed due to either = 3(~1-~2)(Aa+AeJ - - (1.1)
thermal effects, foundation movements, weight of moving
parts, or cutting forces. However, in the event that the
reference is subject to those effects, it is still possible to
B B
correct for manufacturing inaccuracy. (Ps--Pl)(hd+xd)3 77 = plhI3-
77 . (1.3)
The scheme in Fig. 1 . 2 ~shows the reference bearing
controlling only one main bearing, whereas the scheme in
Fig. 1.26 is for two opposed main bearings. Either scheme
is feasible and there is no reason in the latter arrangement
that the two main bearings should have equal areas. Equation (1.1) relates bearing load w to the recess
From the computed data in Fig. 1.3 it is possible to pressures p1 and p2. Diaphragm deflection may also be
calculate for such arrangements the loads corresponding expressed in terms of the difference in recess pressures,
to a given value of the load parameter. This involves as in equation (1.2), so that it is possible to write dia-
adding, or subtracting as appropriate, the thrust exerted phragm deflection in terms of bearing load, diaphragm
by the main bearings to the load indicated for the reference stiffness, and effective areas. Flow through the diaphragm
bearing. The thrust exerted by a main bearing is calcu- restrictor equals flow from the high-pressure recess, as in
lated from the recess pressure in the diaphragm valve, the equation (1.3). Equation (1.4) is for the low-pressure side.
Proc lnstn Mech Engrs 1969-70 Vol184 Pt 31

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DIAPHRAGM VALVES FOR CONTROLLING OPPOSED PAD HYDROSTATIC BEARINGS 3

/// A M o i n load

a Single main bearing arrangement.

b A hydraulic circuit for a reference bearing system controlling opposed main bearings.
Fig. 1.2. Reference bearing system for correcting errors in long machine tool slideways

In the unloaded condition p , = pPs, x d = 0, h = ho and -+T)2w

hence from equation (1.3) 1-P
P2 = . (1.7)
Bd - -.-P hrJ3
_- . . . (1.5)
(2 - x1)3 +-1-BB
B 1-8 h d 3

Writing parameters in dimensionless form, and manipu-

W = +(Pl-P2) . . . . . . (1.8)
lating equations (1.1)-(lS), leads to the following simul- A computer Program was written to calculate a range of
taneous equations in P,, P,, and X , : values which are presented in Fig. 1.3.
Rearranging equation (1.6) for the design condition
P 2w when P, = p, it is a simple matter to find the value of
diaphragm stiffness which yields infinite bearing stiffness.
PI =
B2w . . (1.6) It is found that
x13+1-8
- (I+&)
j\d(opt) = 6B(1-B) * * *

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4 W.B. ROWE AND J. P. O'DONOGHUE

b
I 1 1 .-
0.1 0.2 0-3 0.4 C ' 0 0.1 02 0.3 0.4 (
W
w=ps (A,I+A,ZJ

a fl = 0.1. b fl = 0.2.

01 02 03 04 0;

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DIAPHRAGM VALVES FOR CONTROLLING OPPOSED PAD HYDROSTATIC BEARINGS 5

I I I I 1
01 02 03 04 05

16

1-4

10
-=-I<"
II
%-
n
5 0.8
Fig. 1.3. Theoretical relationships between load, recess
pressure, and clearance for a diaphragm controlled <I e
bearing
'-II
0 6

0.4

0.2

0
5

g = 0.7.

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6 W. B. ROWE AND J. P. O’DONOGHUE

The charts are given for seven values of the design pressure
ratio /3 and for various values of diaphragm stiffness of
which one value is calculated for optimum bearing stiffness.
0.1 10 x wo 0.18 1-8 X WO
0.2 5XWo 0.34 1.7 x wo
Design of diaphragm controlled opposed pad 0.3 3.33 x wo 0.42 1.4 x wo
bearings 0.4 2.5 x w0 0.54 1.35 X W O
0.5 2xwo 065 1.3 X W O
The design calculations are extremely simple. It is possible 0.6 1.67 X w0 0.79 1.32 X W O
to calculate the optimum diaphragm stiffness, but this has 0.7 1.43 x wo 0.81 1.15 x wo
1
I I
not been found to be very successful. I n practice, the
optimum thickness has been found by experiment. The
final adjustment can be made by varying the supply Opposed main bearing arrangement
pressure. This can be seen from the form of diaphragm Assuming, as before, a unity relay factor for pressure the
stiffness parameter. The required condition is supported load expression has now an additional term for
the low-pressure main bearing :
w = PsA,W+PsA,,P,-PsAm~P~
It is also known that P2 = P1-2W by definition, and
It follows that an increase in P, is equivalent to a reduction therefore
in h d .
w = P s A e W + P s A m ~ P ~ - P s A m ~ ( P ~ -(1.12)
2W)
Example: A bearing is to carry 1000 lbf with negligible
deflection. The supply pressure is to be 500 lbf/in2. At the design condition P1 = P2 = /3, w = W,,, and
Selecting /3 = 0.5, the maximum value of load para- W = 0, so that equation (1.12) reduces to a form useful
meter for negligible deflection is W = 0.15. to a designer:

wmax = PsAe W
W O = Ps(Aml-Am2) .
. (1.13)
The difference Am,-Am2 must be adjusted so that the
1000 4 bearing thrust just equals the design load condition.
= - in2
Hence
= 500~0.15 3 +
However, the total area Aml Am2may be made as large
as desired, and in this way it is possible to achieve a
and therefore each pad should have an effective area of maximum load many times greater than the design load.
5 in2. The remainder of the calculations for flow rate,
pad dimensions, etc., follow design procedure 2 from Kinematic design
reference (5).
In machine tool slideways it is usual to provide at least
two slides with support along the length of the supported
Design of the reference bearing system table. Sometimes there is a bearing at each corner of the
Single main bearing arrangement table; that is, four bearings, of which two or all are opposed
pad bearings. From considerations of kinematics these
One configuration of the system involves one main bearing
arrangements are not ideal and lead to some structural
in which the pressure is controlled from the high-
deflections. With diaphragm controlled bearings designed
pressure side of a diaphragm controlled reference bearing.
for infinite stiffness, the structure must deflect until each
The load supported, assuming unity relay factor, is
bearing is at its operating gap. If the structure is too rigid
w = PsAeW+PsAmP, . . (1.10) to allow this, the diaphragm valves will compensate to the
Since it is desirable that forces on the reference bearing limit and the resultant bearing stiffness will be low.
should be as small as possible, it may be arranged that A better arrangement is to make three of the four bearings
W = 0 at the design condition. Writing equation (1.10) diaphragm controlled and the remaining bearing con-
for the design condition when P1 = /3: trolled by capillary. Where the machine arrangement will
allow a three-point support, each formed by a diaphragm
w0 = PsAm/3 . . . (1.11) controlled bearing, this arrangement should be used.
The value P,A,/3 must be arranged to equal the design
load. EXPERIMENTS
It is not immediately apparent whether there is an Details of the test rig are shown in Fig. 1.4. The pockets
advantage in making /3 low or not. Table 1.1 shows that were machined in a 1-in square section bar so that the
the low value of /3 has an improved load range over which bearings were symmetrically arranged with respect to the
deflections are negligible. I n this case w,,, is the thrust loading point. Three displacement indicator positions are
exerted by the main bearing pad only. shown which allowed the bar to be set up for equal bearing
If insufficient load range is unobtainable from this gaps under no load.
arrangement, it is necessary to go to an opposed main By making the diaphragm valve adjustable it was
bearing arrangement. possible to vary the design pressure ratio /3. The recess
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DIAPHRAGM VALVES FOR CONTROLLING OPPOSED PAD HYDROSTATIC BEARINGS 7
W
10

-._2 8
-
0

z
111 6
!I
z
--
3

, 4
0
c
0
w
_I

0.037in

LOAD-lbf

Fig. 1.5. Experimental results for diaphragm controlled

hydrostatic bearings (corrected for structural
deflections)

450 lbf compared with 425 lbf from theory, which is an

Oil outlet to bearing
_.-.L
error in the same direction as the capillary results.
The optimum diaphragm thickness was found to be
0.037 in, and hence this thickness corresponds closely to
Oil
/-- A, = 1.5 on Fig. 1.3e. A diaphragm thickness of 0.049 in
inlet- __ is 2.3 times stiffer, and therefore this characteristic
Oil o u t l e t t o b e a r i n q
.-.-A
corresponds closely with A, = 3.37 also on Fig. 1.3e, and
shows good agreement.
It can be seen that negative stiffness can actually occur
and it was possible to obtain the shape of the deflection
curve. I n severe cases this condition leads to static
b instability. The bearing collapses towards the load,
Fig. 1.4. Test rig for opposed pad hydrostatic bearing reversing its direction of travel in a limited cycle oscilla-
w i t h diaphragm control : ( a ) bearing arrangement, tion. This condition was apparent by a hammering
( b ) adjustable diaphragm valve vibration in the rig. The vibration in some circumstances
may be quite inconspicuous, but leads to fatigue failure of
pressures were monitored with standard pressure gauges. the diaphragm if allowed to continue. An example of this
For the results given, the supply pressure was maintained is shown in Fig. 1.6 from a diaphragm used to control a
constant at 400 lbf/in2. journal bearing. From the circular markings on the
The diaphragm valve details were : restrictor lands, diaphragm it can be seen that there has been hammering
+
3 in outside diameter, in inside diameter; oil annulus against the restrictor lands.
surrounding plunger, 2 in diameter. The experimental results show that the diaphragm valve
The capillary compensated load-bearing capacity may has greater ultimate load capacity than a capillary con-
be calculated from the formulae and data of reference (5). trolled bearing, although the load range over which
For p , = 200 lbf/in2 the theoretical collapse load is deflections can be considered negligible is only a propor-
375 lbf, while for p o = 180 lbf/in2 the value is increased tion of the ultimate capacity. The stiffness is higher and
to 388 lbf. The values obtained from experiments were deflections lower for loads up to at least half the ultimate
400 and 410 lbf. The agreement between theory and load.
experiment to 64 per cent is within reasonable limits of This is true even if the diaphragm is 100 per cent stiffer
errors in standard pressure gauges and other experimental than the optimum value. Small changes in the design
errors. The results are shown in Fig. 1.5. The readings pressure ratio make apparent differences to the load-
have not been shown because the original results were deflection graph. However, the tuning of the system is
averaged and corrected for structural deflections. Correc- more sensitive to variations in supply pressure.
tion for standard deflections is almost always necessary,
particularly near the optimum condition. CONCLUSIONS
At optimum stiffness the measured load capacity was The analysis gives excellent correlation with the measured
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8 W. B. ROWE AND J. P. O'DONOGHUE

Fig. 1.6. Fatigue failure of diaphragm from journal bearing rig

characteristics, although it has not been found possible to APPENDIX 1.1

calculate optimum diaphragm stiffness with sufficient D E R I V A T I O N O F BASIC E Q U A T I O N S F O R
accuracy. D I A P H R A G M C O N T R O L L E D O P P O S E D PAD B E A R I N G
From the analysis it is possible to see the effects on The thrust exerted by the bearing may be written in terms
performance of the various parameters such as supply of the bearing areas and the differential pressures:
pressure, design pressure ratio, and diaphragm stiffness.
Computed data have been given from which a designer w = +h-Pz)(Ae,+AeJ (1.1) -
can determine required areas, pressure ratios, etc. for a Both effective areas are included in equation (1.1) to avoid
reference bearing system. confusion. Thus A , = A,,+&. Dividing equation (1.1)
on each side by the term P,A, yields the dimensionless
form which is used in Fig. 1.4:
ACKNOWLEDGEMENTS W = +(Pl-P2) (l.la) . . .
The authors extend their gratitude to Wickman Ltd for The diaphragm deflection under the action Of the
permission reproduce the experimental and in differential pressures will follow a relationship of the form
particular to Mr M. Lambert of Wickman Ltd for his Ad
valuable assistance. #d = (PI-PZ) A, * * (la2)

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DIAPHRAGM VALVES FOR CONTROLLING OPPOSED PAD HYDROSTATIC BEARINGS 9

This equation is made dimensionless by writing The expression for Pz is derived similarly, starting from
the equation of flow rates in the low-pressure side:
xd = &
xd

B
P1-Pz = Pl
- -P, (P~-Pz)(hd-Xx,)~ ;
Bd
= p2hZ3-
77
. (1.4)
PS
leading to
hdhd
Ad =-
PsAd P 2w
Hence
P, =
1-8(1-x)
(1.7)
p1- PZ B (I-?)~
x d =- . . . (1.2a) (2-~,)3+- ’
Ad
Combining (1. la) and (1.2a), where 2-X, = X,.
2w Substituting for Pz in equation (1.7) from equation
. . . . (1.2b) (l.la>,
Considering the high-pressure side of the bearing and 2w
diaphragm valve it is possible to equate the expressions
P, = 1-P
for flow rate through each, which leads to equation (1.3) : +2W (1.14)
(2 - X d 3 +-1-PB (I+?)

At the design condition p , = PPS;x d = 0 ; and h, = ho; Equations (1.6) and (1.14) were solved as two simultaneous
hence from equation (1.3), equations in P, and X , with the restriction that x d
cannot be greater than 1. The restriction was accommo-
EL-.-P ho3 . . . (1.5) dated by an instruction of the form
B 1-P h d 3
2w 2w
Using equation (1.5) to eliminate Bd and B from if - > 1 then write =1
equation (1.3), *d /‘d

and writing P, = p,/P,, x d = X d / h d , X I = h,/ho leads APPENDIX 1.2

to a dimensionless expression for P,:
REFERENCES

(I) O’DONOGHUE,J. P. and ROWE, W. B. ‘Compensation

methods for externally pressurized bearings’, Engineers’
Digest 1969 30 (No.4), 49.
(2) ROWE,W. B. ‘Hydrostatic bearing’, Patent Application No.
22072166, 1966.
Substituting for x d from equation (1.2b) yields the form
(3) DE GAST,J. G. C. ‘A new type of controlled restrictor’,
used in the computer program: Advances in Machine Tool Design and Research, Proc. 7rh
Int. M.T.D.R. Conf. 1966, 273.
(4) ROWE,W. B. ‘Hydrostatic bearing’, Patent Application No.
51919167, 1967.
( 5 ) O’DONOGHUE, J. P. and ROWE,W. B. ‘Hydrostatic bearing
design’, Tribology 1969 2 (No.I), 25.

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