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8/7/2019 Hydrostatic Bearings-An Introduction
1/15
Design Procedure for
Hydro-Static Bearings (HSB)
Invented by L. D. Girard (Frenchman).
Completely removal of wear and reduction of coefficientof friction to 1/500.
Machines using hydrostatic support show a betterrotational accuracy within 2 micro-inch (0.051 micron)and RMS surface roughness down to 0.25 micro-inch(0.00635 micron)
Externally Pressurized
CoinedbyFuller
8/7/2019 Hydrostatic Bearings-An Introduction
2/15
Features of Hydrostatic support
Surfaces can be separated by full fluid film even atzero speed. No problem with micro roughness and waviness.
Zero friction at zero speed. Useful feature for large size telescopes and radars.
High stiffness Oil film thickness varies as cube root of load.
Why not every bearing is based on Hydrostaticmechanism *K IJ RTGUUWTGUWRRN[4GNK CDK NK V [NK
RTGUUWTGQK NNK PGUCTGCNY C[UK PFQW
3/1Wh
HSB for Large BinocularTelescope Supports for Azimuth and Elevation axes.
Telescope makes a far away object look closer by collecting lightfrom a distant object (objective lens or primary mirror) and bringsthat light (image) to a focus where a second device (eyepiece lens)magnifies the image and brings it to our eye.
A telescope's ability to collect light is directly related to thediameter of the lens or mirror -- the aperture -- that is used togather light. Generally, the larger the aperture, the more light thetelescope collects and brings to focus, and the brighter the finalimage.
Refractors have good resolution, high enough to see details.However, it is difficult to make large objective lenses (greater than4 inches or 10 centimeters) for refractors. Because the aperture islimited, a refractor is less useful for observing faint, deep-skyobjects, like galaxies and nebulae, than reflector types oftelescopes.
8/7/2019 Hydrostatic Bearings-An Introduction
3/15
Thrust Bearings Many loads carried by rotating machinery have
components that act in the direction of theshafts axis of rotation. Bearings supporting
such loads are known as thrust bearings.
Elementary 1-D Analysis
Assume a shaft ofradius Ro is locatedco-axially with a
circular recess ofradius Ri.
Assume all the oil in
recess is at thesupply pressure Ps.
8/7/2019 Hydrostatic Bearings-An Introduction
4/15
Consider a small element of angular extent d ata radius r and radial width dr.
Elemental flow rate:
If flow is symmetrical to the origin, and radialflow rate is constant, then flow rate:
If film thickness is constant, then on integration:
rddr
dphq .
12
3
=
2..12
3
rdr
dphQ =
)(log6
1
3
CrQph
+=
Using two boundary conditions to findunknown values of C1 and Q
Load carrying capacity:
Substituting expression of pand rearranging
i
i
s Rr
RR
rR
pp = 00
0
Rregionin the
log
log
( )drrdpRpWo
i
R
R
is +=
2
0
2.
( )
=
i
o
o
i
os
RR
R
R
RpW
log.2
1
.2
2
2
=
1
2
11
1log.2
1
r
rCW
8/7/2019 Hydrostatic Bearings-An Introduction
5/15
.1 .2 .3 .4 .5 .6 .7 .8 .9
4
6
8
10
12
14
16
18
20
22
load vs ratio
ratio
load
C1 = 10
( )drrdpRpWo
i
R
R
is +=
2
0
2.
)/1log(
1
6 1
30
r
phQ s
=)/1log(
1
1
2r
CQ =
.1 .2 .3 .4 .5 .6 .7 .8 .9
0
20
40
60
80
100
120
140
160
180
200
220
240
flow vs ratio
ratio
flow
C2 = 10
)/1log(1
1
2r
CQ =
Load is not a function of film thickness (h0 ), but flow is a very
strong function of film thickness.
Film thickness is designed based on surface finish & vibration.
8/7/2019 Hydrostatic Bearings-An Introduction
6/15
Power loss
Power consumption in the hydrostatic bearingsystem consists of pumping power and frictionlosses.
2
4
00
4
0 12
.
=
=
+=
R
R
h
RP
PQP
PPP
if
sh
fht
2
4
00
4
0
2
2
0
3
0
112
1
)/log(6
11
+= R
R
h
R
PRR
h
Pi
s
i
t
==
=
=
03
0
2
0
0
2
)(
R
iR
ff drrh
PrFP
rAh
rF
h
UAF
Petroff equation
Example: W = 1000 N, =5000 rpm, R0=100
mm, Ri=50 mm, =0.01 Pa.s, 1=0.6, 2=0.9.Optimize minimum film thickness for minimumpower loss
2
4
00
4
0
2
2
0
3
0
1
12
1
)/log(6
11
+=
R
R
h
RP
RR
hP is
i
t
( )
=
io
o
i
os
RR
R
R
RpW
log.2
1
.2
2
20
23
01 hC
hCPt +=
( )Pa824,58
5.01
)2log(.2
1.0*
1000
rad/s6.52360
5000*2
22=
=
==
ss PP
sC
C
/N.m448.0
)N/(s.m10*35.4
2
2
211
1
=
=
8/7/2019 Hydrostatic Bearings-An Introduction
7/15
.0004 .0006 .0008 .001 .0012 .0014 .0016 .0018 .002
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
Power loss vs film thickness
Film thickness
Powerloss
Example: W = 1000 N, =5000 rpm, R0=100 mm,=0.01 Pa.s, 1=0.6, 2=0.9, h0=1mm. Optimize ratio(Ri/R0) for minimum power loss
( )( )4
221*7.478
1
)/1log(5.353 r
r
rPt +
=
.1 .2 .3 .4 .5 .6 .7 .8 .9
500
550
600
650
700
750
800
850
Power loss vs ratio
ratio
Powerloss
8/7/2019 Hydrostatic Bearings-An Introduction
8/15
.1 .2 .3 .4 .5 .6 .7 .8 .9
280
300
320
340
360
380
400
420
440
460
480
500
Power loss vs ratio
ratio
Powe
rloss
speed = 2500 rpm
.1 .2 .3 .4 .5 .6 .7 .8 .9
200
225
250
275
300
325
350
375
400
425
450
475Power loss vs ratio
ratio
Powerloss
Speed = 1250 rpm
Step hydrostatic bearing to support vertical turbo-charger inpower plant.
8/7/2019 Hydrostatic Bearings-An Introduction
9/15
Restrictor In earlier slides, it was assumed that recess pressure
was equal to supply pressure.
This means change in load requires change inperformance of pump.
Pump performance can be regulated:
Manually
Automatically
To automat the pump performance one needs
sensor, amplifier, controller, etc. To reduce cost, often self regulating called restrictor
is used.
( )
=
i
o
o
i
os
RR
R
R
RpW
log.2
1
.2
2
2
Restrictor Constant flow restrictor
If flow is constant, recess pressure and film thicknessare related.
Increase in load, is balanced by increase in recesspressure and corresponding decrease in filmthickness.
Constant supply pressure restrictor Recess pressure is kept lower than supply pressure
Drop in pressure, from supply pressure to recesspressure, depends is controlled by the fixed restrictorplaced between supply manifold and the bearing.
Increase in load, reduces the flow by decreasing filmthickness, recess pressure increases and equilibriumis restored.
)/1log(
1
6 1
3
0
r
phQ s
=
8/7/2019 Hydrostatic Bearings-An Introduction
10/15
Constant supply pressurerestrictors
Most commonly used restrictorsare capillary and orifice. Capillary is relatively long and
narrow opposed to and orifice whichis short in the direction of flow.
In a capillary, flow occurs due toshearing and is dependent onviscosity of fluid, whereas flow inorifice is due to inertia and dependson density.
Flow in capillary is directlyproportional to pressure differenceand that in an orifice is dependent onsquare root of pressure difference.
Although the pumping power lossesare higher for these types ofcompensation devices, the initialcost is much less.
c
cc
l
RPQ
8
4=
/.24
2
PCd
Q Do
O =
A smaller flow
produces a
smaller pressure
drop.
Hydrostatic Bearing Film Stiffnesswith Constant Feed Rate
)/log(
1
6
3
0
io
s
RR
phQ
=
( )
=
i
o
o
i
os
RR
R
R
RpW
log.2
1
.2
2
2
3
22
3
0
2
22
*)5.01(*)1.0(*01.0*31.3oo
io
h
Q
h
Q
R
RRW
=
00
13StiffnesshW
dhdWK ==
If W is doubled, and Q is kept constant, what will be
relative change in film thickness?
8/7/2019 Hydrostatic Bearings-An Introduction
11/15
.001 .002 .003 .004 .005 .006 .007 .008 .009 .010
250
500
750
1000
1250
1500
1750
2000
2250
Load vs film thickness
film thickness, mm
load,
N
Blue line --> 0.0001 flow
Red line --> 0.001 flow
Green line --> 0.01 flow
Capillary Compensation
174B--)/log(
1
6
33
0 =>== roio
r PhBQRR
PhQ
( )
cc
rscc
c
crsc
rlmm
PPkQl
RPPQ
50,15.0r
0.01assume)(8
c
4
==
=>=
=
( )
039.0A
log.2
1
.
eff3
2
2
2
=+
=
=
co
cseff
reff
i
o
o
i
or
kBh
kPAW
PA
RR
R
R
RPW
co
c
s
r
c
kBh
k
P
P
+
=
=
3
co
o
oo
co
o
co
cseff
o
kBh
Bh
h
W
h
W
kBh
Bh
kBh
kPAh
W
+=
++=
3
3
3
2
3
3or,
3
Lower value of kc
increases stiffness
8/7/2019 Hydrostatic Bearings-An Introduction
12/15
.001 .002 .003 .004 .005 .006 .007 .008 .0090
.2
.4
.6
.8
1
1.2
1.4
1.6
W vs h
film thickness, m
Load,
N
Lower value of kc reduces load
Carrying capacity
.001 .002 .003 .004 .005 .006 .007 .008 .009
0
200
400
600
800
1000
1200
1400
W vs h
film thickness, m
Load,
N
8/7/2019 Hydrostatic Bearings-An Introduction
13/15
Orifice Compensation
)(/).(24
2
rsoOrsDoO PPkQPPCd
Q ==
ro
io
r PhBQRR
PhQ 3
3
0
)/log(
1
6==
( )
++=
=
62
6222
2
2
2
2
4
log.2
1
.
o
soooo
eff
reff
i
o
o
i
or
hB
PhBkkkAW
PA
RR
R
R
RPW
262
2
oro
o
s
r
o
kPhB
k
P
P
+=
=
Hydrostatic Lift Useful to avoid metal to metal
contact under heavy static load
conditions. Ex. Synchronous
condenser, rolling mills, etc.
How to estimate load capacity ?
Trial and error method
Good for first of its kind.
Numerical modeling and simulation
Assume a shaft of radius r being floated in a bearing of radius R by
oil pumped through a slot at pressure PS
cosCh
.12
rateflowElemental
r
3
e
brd
dph
q
=
=
8/7/2019 Hydrostatic Bearings-An Introduction
14/15
Hydrostatic lift..
rCee
/)cos1(ChcosCh rr
===
( )b
rd
dpCq r .
12
cos1.
rateflowElemental
33
=
( )331
cos1
12
risepressureElemental
=
d
bC
qrdp
r
( )( ) ( ) ( )
+
++
= D
bC
qrP
r
cos1
coscos
1.2
2
cos1.1.2
cos3-4.sin.12
solutionGeneral
1
5.22
2
222
2
3
1
( )( ) ( )
( )
++
=
==
1
5.22
2
22
2
0
cos1.2
2
1.2
-4.
90at0PusingevaluatedbecanD,n,integratioofconstant
D
( )( ) ( )
( )
++
=
==
1
5.22
2
22
2
3
1
0
ss
cos1.2
2
1.2
-4.120atPPusingdeterminedbecanPpressuresupply
r
sbC
qrP
-1 -.8 -.6 -.4 -.2 0 .2 .4 .60
20
40
60
80
100
120
140
160
180
200
220Pressure versus eccentricity ratio
eccentricity ratio
Supplypressure
Negative value of eccentricity ratio, describe the journal
position when it is above the bearing center.
8/7/2019 Hydrostatic Bearings-An Introduction
15/15
Load Carrying Capacity
=2
0
.dpcosbr2W
Wloadappliedthebalancewill
.b.cosprdforceofcomponentverticaland.brdareaonactspPressure
( )( ) ( ) ( )
dC
qrW
r
.coscos1
coscos
1.2
2
cos1.1.2
cos3-4.sin.24 15.22
2
222
22
03
1
2
++
=
( )
=
23
1
2
1
212
rC
qrW
-1 -.8 -.6 -.4 -.2 0 .2 .4 .60
20
40
60
80
100
120
140
160
180Load versus eccentricity ratio
eccentricity ratio
Load