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Analysis of multirecess conical hydrostatic thrust bearings under rotation
Wear, 89 (1983)
29 - 40
29
ANALYSIS OF MULTIRECESS CONICAL BEARINGS UNDER ROTATION
T. JAYACHANDRA Department (Znd ia)
HYDROSTATIC
THRUST
PRABHU and N. GANESAN
of Applied
(Received July 21,1982;
Mechanics,
Indian
Institute
of Technology,
Madras
600036
in revised form November 15,1982)
Summary
The characteristics of multirecess conical hydrostatic thrust bearings were studied theoretically taking into account the effect of rotational lubricant inertia. Load capacity, flow, stiffness and damping characteristics were computed for the practically useful aspect and resistance ratios for capillary and orifice compensations. The results are presented in a form for the convenience of designers.
1. Introduction
Hydrostatic thrust bearings are widely used in rotating machinery where the trend is for increased speeds. For high speed applications it is necessary to have design data including the effect, of rotation which is neglected in conventional design practice. The effect of rotation is characterized by the rotational and radial inertia terms in the analysis. For lubrication problems with an incompressible fluid the omission of the radial inertia term does not seriously affect the analytical results [ 11. The effect of rotational lubricant inertia on the performance of plane hydrostatic thrust bearings has been studied [ 2 - 41. The effect on annular recess thrust bearings has also been studied [ 51. Salem and Khalil [ 61 studied the inertia effect in conical bearings. Annular recess conical thrust bearings have been studied [ 71 under both static and dyniunic conditions with capillary and orifice restrictors taking into account the-effect of rotational lubricant inertia. The investigations of Dowson [ 41 and Salem and Khalil [6] were of bearings with shallow recesses where other investigations were of deep recess bearings. The conclusions from all studies are that the rotational inertia of the lubricant has a detrimental effect on the load-carrying capacity to a varying degree depending on the aspect and resistance ratios and that the influences on stiffness and damping are insignificant in the practical speed range [ 71. 0043-1648/83/$3.00
0 Elsevier Sequoia/Printed
in The Netherlands
30
Multirecess thrust bearings have more practical importance than the annular recess type because of their capacity to resist any tilting moment. If the bearings are of conical design, they can also take up a small amount of lateral load in addition to axial load to eliminate the need for separate guide bearings in the rotor system. No information is available on the behaviour of multirecess conical thrust bearings. The behaviour of multirecess plane thrust bearings has been reported by Rowe and O’Donoghue [ 81 and it is possible to determine the characteristics of conical bearings from plane thrust bearings by multiplying by a suitable sine function. The effect of rotational lubricant inertia on multirecess bearings has not been studied and damping characteristics are not known. An attempt was made to rectify this by studying multirecess conical hydrostatic thrust bearings taking into account the effect of rotational lubricant inertia. Since the effect of rotation on the stiffness and damping characteristics is insignificant, these characteristics are determined for zero-speed operation.
2. Analysis Figure 1 shows the multirecess conical hydrostatic thrust bearing. The spherical coordinate system given by Constantinescu [9] is used to obtain the Reynolds equation in two dimensions including rotational lubricant inertia for laminar incompressible lubrication:
Fig. 1. Multirecess conical thrust bearing.
31
3
=10
pa2 sin20c a --(h3r2) + 67&! 2 r
ah + 1275
If the dimensionless quantities defined in the nomenclature dimensional Reynolds equation can be written as =A$h a
-32
r)+hr%
ah
are used the non-
+ofz
ah
(2)
This equation is written in the finite difference form and solved by an iterative process using a digital computer to obtain the pressure distribution satisfying the following boundary conditions: (a) jj = 0 at the bearing boundary exposed to the atmospheric pressure; (b) jj = constant = @c)N in the Nth recess; (c) fi = 0 in the region of cavitation. Boundary condition (c) is applied by setting the negative pressures encountered in the solution to zero for the next iteration. In hydrostatic thrust bearings for certain bearing geometries and resistance ratios cavitation may be induced even for parallel films owing to the rotational lubricant inertia [lo]. Hence the cavitation boundary condition is also satisfied and the recess pressures are evaluated by a method due to Heller and Shapiro [ 111. 2.1. Axial load-carrying capacity The load capacity is calculated from the pressure mensionless load-carrying capacity is expressed as
distribution.
The di-
(3) In dimensional
form,
W = P,r4C2W
(4)
2.2. Flow Figure 2 shows the various components of flow out of the recess. The flow out of the bearing is equal to the sum of the flows from the recesses. Qp,
Qr,
e;4,
,a =$t
RECESS
‘C e
CZ? &le
OP,
Qr2
c
QrJ
GG
Q, 0, i 0‘
‘*,x+
Fig. 2. Components of flow out of each recess.
32
The flow across the recess boundary is the algebraic sum of flows due to pressure gradients, rotation and squeeze action. The following are the components of flow out of the recess: (5)
(6)
(7)
Q
2_
= 3(r3
rz2bw
Ae,
(4f
-
4,)
s~2~
(9)
4PsH3 (9, is calculated for dynamic conditions out of each recess is the algebraic sum Qr.9 = G,,,
+ 4, &,I& + Q,,,
only.)
+ 8, &.,@f + Q.9
The net dimensionless
flow (10)
The flow out of the bearing is a = A@,,
(11)
which in dimensional
form is
2P
Q=-‘H3@
(12)
311
2.3. Frictional loss The frictional loss is computed for the entire land area of the pad irrespective of whether or not cavitation is present. The dimensionless frictional loss is &
= -
1
sin o
J-J iE AL
In dimensional H,=
r)a2rGC4 2H
F3df d@
(13)
form EfL
(14)
33
2.4. Axial stiffness The stiffness and damping coefficients are evaluated numerically. A small change in the parameter e, (2% of the value of 2,) is introduced in the analysis to obtain a change Ams in the load-carrying capacity. The dimensionless stiffness coefficient is (15) In dimensional form this is K, =
Psr4c2 -a, w
(16)
2.5. Damping A small velocity o Aa, is introduced and the change in load-carrying capacity is computed. The dimensionless damping coefficient is (17) In dimensional form,
3. Results and discussion Results pave been obtained for two configurations: one with four recesses and the other with six recesses. The analysis was carried out with the following data for both conditions: bearing outer diameter, 20 cm; speeds of rotation, 0 and 8000 rev min- ‘; frequency of sinusoidal excitation, 62.8 rad s-i; lubricant density, 0.87 X 1K6 kgf s2 cmm4; dynamic viscosity, 20 X 10-s kgf s cmm2; supply pressure, 20 kgf cm -2; design film thickness, 0.005 cm; included angle between the recesses, $J, = 15”; recess ratio, R, = 0.5. Since h was assumed constant under steady state conditions for any speed of rotation, the recess pressure varied. Hence to study the bearing characteristics with respect to speed the resistance ratio is preferred to the pressure ratio p,/P, as for the continuous recess bearing [ 71. For a capillarycompensated bearing (CCB) the resistance ratio is independent of rotation. This is not so with an orifice-compensated bearing (OCB). With rotation, the recess pressure varied and hence orifice resistance also varied. The variation in the orifice resistance is not as significant as that in p,/P,. Resistance ratios are obtained by varying the restrictor resistance and keeping the bearing resistance constant for a particular geometry. For a particular value of y the recess pressure at zero speed is obtained by using the relation PC -=PS
1 I+?
(19)
34
The bearing resistance corresponding to each recess is obtained by dividing the recess pressure by the flow out of the recess at zero speed and is related to the flow out of the bearing by the relation R, =P,IQ,, =p,NlQ
(20)
With Rb known, the restrictor resistance is also known and hence the restrictor can be designed. For minimum frictional loss and maximum load capacity it is preferable to design bearings with radius ratios between 0.4 and 0.6 and the analysis has been carried out for these ratios only. Cavitation did not occur for these ratios within the speed range considered when the resistance ratio was between 0.3 and 3. Hence the following relations can be used to obtain the characteristics of bearings of conical shape from those of a plane pad: IV, = iii,
(21)
0, = QP sin a
(22) (231
The ch~acte~stic constants differ very little for the two confi~rations for all the radius and resistance ratios considered in this study, for both types of compensation, as can be seen from Tables 1 and 2. The characteristics have been plotted for a plane bearing with six recesses . Figure 3 shows the effect of rotation on iii with respect to y. The decrease in ff with speed is higher for higher values of y. The OCB has a larger reduction in @than the CCB. The effect of rotation on Q is shown in Fig. 4. Q increases with speed and is higher for the CCB than the OCB. Figure 5 shows the stiffness characteristics for three practically useful radius ratios. Generally, the OCB has higher ES values than the CCB. The m~imum value of ES occurs at 7 = 1 for the CCB and at y = 0.7 for the OCB. Damping characteristics are shown in Fig. 6. The OCB has a higher cd value than the CCB. Damping increases as +rincreases. Figure 7 shows the effect of semicone angle on the bearing characteristics. g and KS increase with CYwhile cd decreases. The variation in @ and Q with speed is linear as long as the film does not rupture because of cavitation. Hence to arrive at accurate values it is necessary to use Table 1 and to compute the characteristics for any A value between 0 and 0.917.
35 TABLE1 Static characteristics(cr = 90";R,= 0.5;@,=15")
CCB and OCB,
CCB, 1\=0.917
OCB,
CCB
A=O.917
and OCB,
A=o 4
6
0.4 1.742
CCB, A = 0.917
OCB,
A = 0.917
A=0
0.3 0.5 0.7 1.0 2.0 3.0
1.506 1.306 1.149 0.98 0.654 0.490
1.475 1.263 0.918 0.573 0.400
1.46 1.24 1.077 0.900 0.558 0.388
5.63 4.95 4.35 3.71 2.48 1.86
5.90
5.98
5.23 3.92 2.62 1.96
5.16 4.53 3.85 2.56 1.92
0.5 1.676
0.3 0.5 0.7 1.0 2.0 3.0
1.338 1.159 1.02 0.87 0.58 0.435
1.32 1.13 0.831 0.529 0.379
1.31 1.12 0.974 0.819 0.520 0.371
7.28 6.24 5.48 4.68 3.12 2.34
7.57 6.56 4.92 3.28 2.46
7.52 6.50 5.7 4.86 3.23 2.42
0.6 1.5
0.3 0.5 0.7 1.0 2.0 3.0
1.136 0.986 0.869 0.741 0.495 0.371
1.125 0.971 0.719 0.467 0.341
1.123 0.966 0.844 0.715 0.463 0.337
9.64 8.36 7.37 6.28 4.20 3.15
9.88 8.58 6.44 4.30 3.23
9.86 8.54 7.5 6.4 4.27 3.2
0.4
1.791
0.3 0.5 0.7 1.0 2.0 3.0
1.50 1.30 1.14 0.973 0.648 0.486
1.46 1.25 0.910 0.566 0.395
1.45 1.23 1.067 0.89 0.551 0.382
5.65 4.83 4.24 3.62 2.41 1.81
5.97 5.17 3.88 2.58 1.94
5.91 5.10 4.47 3.81 2.53 1.89
0.5
1.691
0.3 0.5 0.7 1.0 2.0 3.0
1.33 1.15 1.01 0.862 0.575 0.431
1.31 1.12 0.823 0.524 0.374
1.30 1.11 0.965 0.811 0.515 0.367
7.20 6.16 5.42 4.62 3.08 2.31
7.48 6.49 4.86 3.24 2.43
7.44 6.42 5.63 4.8 3.19 2.39
0.6
1.549
0.3 0.5 0.7 1.0 2.0 3.0
1.12 0.970 0.853 0.728 0.485 0.364
1.11 0.955 0.706 0.457 0.333
1.1 0.948 0.828 0.7 0.452 0.328
9.47 8.11 7.13 6.08 4.06 3.04
9.71 8.41 6.31 4.20 3.15
9.68 8.36 7.34 6.26 4.16 3.12
36 TABLE
2
Stiffness
and damping
Rl
Y
N=4
N=6
E(j(X106)
K, 0.4
0.5
0.6
((w = 90”; R, = 0.5; 4, = 15”)
characteristics
K CCB
OCB CCB __ ___
-. OCB -.
3.18 4.16 4.65
2.37 3.12 3.63 4.14 4.95 5.33
2.06 2.60 2.95 2.64 2.23
3.34 3.91 4.05 3.95 3.19 2.56
1.64 2.18 3.13 4.09 4.56
2.35 3.08 3.58 4.07 4.84 5.20
2.97 3.49 3.62 3.52 2.92 2.30
1.04 1.38 2.0 2.6 2.9
1.49 1.96 2.37 2.57 2.98 3.29
1.82 2.30 -
2.95 3.46 3.59 3.51 2.82 2.27
1.03 1.37 1.96 2.55 2.84
1.35 1.94 2.33 2.53 3.03 3.22
2.5
0.58 0.78 1.09 1.42 1.59
0.84 1.07 1.24 1.43 1.68 1.8
1.54 1.94 -
2.49 2.92 3.03 2.96 2.39 1.92
0.59 0.79 1.09 1.40 1.56
0.85 1.07 1.24 1.40 1.65 1.77
CCB
OCB
CCB
OCB
0.3 0.5 0.7 1.0 2.0 3.0
2.05 2.6 2.96 2.66 2.26
3.33 3.91 4.07 3.98 3.22 2.6
1.63 2.21 -
0.3 0.5 0.7 1.0 2.0 3.0
1.83 2.32 -
0.3 0.5 0.7 1.0 2.0 3.0
1.55 1.95 -
2.63 2.36 2.0
2.23 2.01 1.71
2.94 3.06 3.0 2.44 1.97
Fig. 3. Plots of w us. y for various capillary; 0, orifice; -,A=O;---,A=0.917. Fig. 4. Plots of Gus. bols as for Fig. 3.
l$ (X106)
y for various
-
2.61 2.34 1.98
2.20 1.98 1.67
values of A (a! = 90”;
values
of A ((u = 90”;
RI = 0.4; R, = 0.5; N = 6): C,
RI = 0.4; R, = 0.5; N = 6):.sym-
37 55c 50RI.0
4,'
I
/--
I
/
4.5-
LO-
_ a <
3.5-
lu" 30-
25-
2.0-
15-
lo-
05
0
1
2
3
Y
Fig. 5. Plots of is us. y for various radius ratios ((u= 90”; R, = 0.5; N = 6): -, --, orifice.
capillary;
Fig. 6. Plots of cd us. y for various radius ratios (a! = 90”; R, = 0.5; N = 6): symbols as for Fig. 5.
ISlO-
0
45'
----------I 60' ~(Degrees)
I 75'
90'
Fig. 7. Influence of cone angle on w, 6, ES and & for a CCB (l-21 = 0.4; R, = 0.5; 7 = 1; N = 6): -,A=o;-----,A=o.9l7.
38
4. Conclusions The characteristics of multirecess conical hydrostatic thrust bearings under rotation have been studied. Design data for load-carrying capacity and flow are provided for various resistance ratios for both capillary and orifice compensations for a practically useful speed range. Stiffness and damping characteristics are also given because they are useful in studying the response behaviour of bearings under dynamic conditions. The characteristic coefficients do not differ much for both four-recess and six-recess bearings. Hence if a higher tilt resistance is required a six-recess bearing is preferred, but this would entail additional cost and complexities in the hydraulic system.
Acknowledgments The authors thank useful discussions.
Dr. B. S. Prabhu
and Dr.-Ing. B. V. A. Rao for the
References 1 G. R. Symmons and F. B. KIemz, Effect of radial and rotational lubricant inertia on the pressure distribution in externally pressurized thrust bearings, Appl. Sci. Res., 32 (1976) 31 - 44. 2 F. Osterle and W. F. Hughes, The effect of lubricant inertia in hydrostatic thrust bearing lubrication, Wear, 1 (1958) 465 - 471. 3 T. Sasaki, H. Mori and A. Hirai, Theoretical study of hydrostatic thrust bearings, Bull. Jpn. Sot. Mech. Eng., 2 (5) (1959) 75 - 79. 4 D. Dowson, Inertia effects in hydrostatic thrust bearings, J. Basic Eng., 83 (1961) 227 - 234. 5 C. Y. Chow, A non-central feeding hydrostatic thrust bearing, J. Fluid Mech., 72 (1) (1975) 113 - 120. 6 E. Salem and F. Khalil, Thermal and inertia effects in externally pressurized conical oil bearings, Wear, 56 (1979) 251 - 264. 7 T. J. Prabhu and N. Ganesan, Characteristics of conical hydrostatic thrust bearings under rotation, Wear, 73 (1981) 95 - 122. 8 W. B. Rowe and J. P. O’Donoghue, Design Procedures for Hydrostatic Bearings, Machinery Publishing, Brighton, Sussex, 1971. 9 V. N. Constantinescu, Gas Lubrication, American Society of Mechanical Engineers, New York, NY, 1969. 10 J. F. Osterle and W. F. Hughes, Inertia induced cavitation in hydrostatic thrust bearings, Wear, 4 (1961) 228 - 233. 11 S. Heller and W. Shapiro, A numerical solution for the incompressible hybrid journal bearing with cavitation, J. Lubr. Technol., 91 (1969) 508 - 515.
Appendix AL cd
A: nomenclature land area damping constant
(kgf s cm-l)
39
c_d_ Cdc, cdp
e, Ae, AL?, h
Ii H
Q-=
!2-(
r9 rl, r2,r3,r4 P r4c
Rl Rb>Rc, R, R,
dimensionless damping constant dimensionless damping constants for a conical bearing and for a plane bearing film thickness measured axially (cm) small variation in e, (cm) Ae,/W, dimensionless small variation in e, film thickness (cm) h/W, dimensionless film thickness design film thickness (cm) frictional loss (kgf cm 8’) dimensionless frictional loss dimensionless frictional losses for a conical bearing and for a plane bearing stiffness (kgf cm-‘) dimensionless stiffness dimensionless stiffnesses for a conical bearing and for a plane bearing number of recesses pressure (kgf cme2) p/P,, dimensionless pressure recess pressure (kgf cmd2) p,/P,, dimensionless recess pressure supply pressure (kgf cmm2) volumetric flow rate (cm3 s-r) dimensionless volumetric flow rate dimensionless flow rates for a conical bearing and for a plane bearing dimensionless flow rates due to rotation across the recess boundaries at 4 = I$, and at 4 = & dimensionless flow rates due to pressure gradients across the recess boundaries at r = r2 and at r = r3 dimensionless flow rates due to pressure gradients across the recess boundaries at 4 = 4, and at 4 = C& dimensionless flow rates due to centrifugal action (rotation effect) across the recess boundaries at r = r2 and at r = r3 flow rate out of each recess (cm3 s-l) dimensionless flow rate out of each recess dimensionless flow rate due to squeeze action over the recess region slant lengths of the cone (cm) r/r4, dimensionless radius outer radius of the conical bearing r 1/r4, dimensionless radius ratio resistance of the bearing (corresponding to each recess), capillary and orifice (kgf s cme5) r2)/(r4 - rl), recess ratio (r3 -
40
Am,, A&
time (s) load-carrying capacity (kgf) dimensionless load-carrying capacity dimensionless loads for a conical bearing and for a plane bearing variations in dimensionless load-carrying capacities under static and dynamic conditions semicone angle (deg) RJR,, R,/R,, resistance ratio dynamic viscosity (kgf s cmp2) 3nS2r42/2PSH 2, bearing number 2/Ps, rotational lubricant inertia parameter (3/10W2r4, density of the lubricant (kgf s2 cmd4) 3(77~/P,)(r~/H)~, squeeze number wt, dimensionless time included angle between recesses frequency of excitation (rad s-l) angular velocity (rad s-l)
29 - 40
29
ANALYSIS OF MULTIRECESS CONICAL BEARINGS UNDER ROTATION
T. JAYACHANDRA Department (Znd ia)
HYDROSTATIC
THRUST
PRABHU and N. GANESAN
of Applied
(Received July 21,1982;
Mechanics,
Indian
Institute
of Technology,
Madras
600036
in revised form November 15,1982)
Summary
The characteristics of multirecess conical hydrostatic thrust bearings were studied theoretically taking into account the effect of rotational lubricant inertia. Load capacity, flow, stiffness and damping characteristics were computed for the practically useful aspect and resistance ratios for capillary and orifice compensations. The results are presented in a form for the convenience of designers.
1. Introduction
Hydrostatic thrust bearings are widely used in rotating machinery where the trend is for increased speeds. For high speed applications it is necessary to have design data including the effect, of rotation which is neglected in conventional design practice. The effect of rotation is characterized by the rotational and radial inertia terms in the analysis. For lubrication problems with an incompressible fluid the omission of the radial inertia term does not seriously affect the analytical results [ 11. The effect of rotational lubricant inertia on the performance of plane hydrostatic thrust bearings has been studied [ 2 - 41. The effect on annular recess thrust bearings has also been studied [ 51. Salem and Khalil [ 61 studied the inertia effect in conical bearings. Annular recess conical thrust bearings have been studied [ 71 under both static and dyniunic conditions with capillary and orifice restrictors taking into account the-effect of rotational lubricant inertia. The investigations of Dowson [ 41 and Salem and Khalil [6] were of bearings with shallow recesses where other investigations were of deep recess bearings. The conclusions from all studies are that the rotational inertia of the lubricant has a detrimental effect on the load-carrying capacity to a varying degree depending on the aspect and resistance ratios and that the influences on stiffness and damping are insignificant in the practical speed range [ 71. 0043-1648/83/$3.00
0 Elsevier Sequoia/Printed
in The Netherlands
30
Multirecess thrust bearings have more practical importance than the annular recess type because of their capacity to resist any tilting moment. If the bearings are of conical design, they can also take up a small amount of lateral load in addition to axial load to eliminate the need for separate guide bearings in the rotor system. No information is available on the behaviour of multirecess conical thrust bearings. The behaviour of multirecess plane thrust bearings has been reported by Rowe and O’Donoghue [ 81 and it is possible to determine the characteristics of conical bearings from plane thrust bearings by multiplying by a suitable sine function. The effect of rotational lubricant inertia on multirecess bearings has not been studied and damping characteristics are not known. An attempt was made to rectify this by studying multirecess conical hydrostatic thrust bearings taking into account the effect of rotational lubricant inertia. Since the effect of rotation on the stiffness and damping characteristics is insignificant, these characteristics are determined for zero-speed operation.
2. Analysis Figure 1 shows the multirecess conical hydrostatic thrust bearing. The spherical coordinate system given by Constantinescu [9] is used to obtain the Reynolds equation in two dimensions including rotational lubricant inertia for laminar incompressible lubrication:
Fig. 1. Multirecess conical thrust bearing.
31
3
=10
pa2 sin20c a --(h3r2) + 67&! 2 r
ah + 1275
If the dimensionless quantities defined in the nomenclature dimensional Reynolds equation can be written as =A$h a
-32
r)+hr%
ah
are used the non-
+ofz
ah
(2)
This equation is written in the finite difference form and solved by an iterative process using a digital computer to obtain the pressure distribution satisfying the following boundary conditions: (a) jj = 0 at the bearing boundary exposed to the atmospheric pressure; (b) jj = constant = @c)N in the Nth recess; (c) fi = 0 in the region of cavitation. Boundary condition (c) is applied by setting the negative pressures encountered in the solution to zero for the next iteration. In hydrostatic thrust bearings for certain bearing geometries and resistance ratios cavitation may be induced even for parallel films owing to the rotational lubricant inertia [lo]. Hence the cavitation boundary condition is also satisfied and the recess pressures are evaluated by a method due to Heller and Shapiro [ 111. 2.1. Axial load-carrying capacity The load capacity is calculated from the pressure mensionless load-carrying capacity is expressed as
distribution.
The di-
(3) In dimensional
form,
W = P,r4C2W
(4)
2.2. Flow Figure 2 shows the various components of flow out of the recess. The flow out of the bearing is equal to the sum of the flows from the recesses. Qp,
Qr,
e;4,
,a =$t
RECESS
‘C e
CZ? &le
OP,
Qr2
c
QrJ
GG
Q, 0, i 0‘
‘*,x+
Fig. 2. Components of flow out of each recess.
32
The flow across the recess boundary is the algebraic sum of flows due to pressure gradients, rotation and squeeze action. The following are the components of flow out of the recess: (5)
(6)
(7)
Q
2_
= 3(r3
rz2bw
Ae,
(4f
-
4,)
s~2~
(9)
4PsH3 (9, is calculated for dynamic conditions out of each recess is the algebraic sum Qr.9 = G,,,
+ 4, &,I& + Q,,,
only.)
+ 8, &.,@f + Q.9
The net dimensionless
flow (10)
The flow out of the bearing is a = A@,,
(11)
which in dimensional
form is
2P
Q=-‘H3@
(12)
311
2.3. Frictional loss The frictional loss is computed for the entire land area of the pad irrespective of whether or not cavitation is present. The dimensionless frictional loss is &
= -
1
sin o
J-J iE AL
In dimensional H,=
r)a2rGC4 2H
F3df d@
(13)
form EfL
(14)
33
2.4. Axial stiffness The stiffness and damping coefficients are evaluated numerically. A small change in the parameter e, (2% of the value of 2,) is introduced in the analysis to obtain a change Ams in the load-carrying capacity. The dimensionless stiffness coefficient is (15) In dimensional form this is K, =
Psr4c2 -a, w
(16)
2.5. Damping A small velocity o Aa, is introduced and the change in load-carrying capacity is computed. The dimensionless damping coefficient is (17) In dimensional form,
3. Results and discussion Results pave been obtained for two configurations: one with four recesses and the other with six recesses. The analysis was carried out with the following data for both conditions: bearing outer diameter, 20 cm; speeds of rotation, 0 and 8000 rev min- ‘; frequency of sinusoidal excitation, 62.8 rad s-i; lubricant density, 0.87 X 1K6 kgf s2 cmm4; dynamic viscosity, 20 X 10-s kgf s cmm2; supply pressure, 20 kgf cm -2; design film thickness, 0.005 cm; included angle between the recesses, $J, = 15”; recess ratio, R, = 0.5. Since h was assumed constant under steady state conditions for any speed of rotation, the recess pressure varied. Hence to study the bearing characteristics with respect to speed the resistance ratio is preferred to the pressure ratio p,/P, as for the continuous recess bearing [ 71. For a capillarycompensated bearing (CCB) the resistance ratio is independent of rotation. This is not so with an orifice-compensated bearing (OCB). With rotation, the recess pressure varied and hence orifice resistance also varied. The variation in the orifice resistance is not as significant as that in p,/P,. Resistance ratios are obtained by varying the restrictor resistance and keeping the bearing resistance constant for a particular geometry. For a particular value of y the recess pressure at zero speed is obtained by using the relation PC -=PS
1 I+?
(19)
34
The bearing resistance corresponding to each recess is obtained by dividing the recess pressure by the flow out of the recess at zero speed and is related to the flow out of the bearing by the relation R, =P,IQ,, =p,NlQ
(20)
With Rb known, the restrictor resistance is also known and hence the restrictor can be designed. For minimum frictional loss and maximum load capacity it is preferable to design bearings with radius ratios between 0.4 and 0.6 and the analysis has been carried out for these ratios only. Cavitation did not occur for these ratios within the speed range considered when the resistance ratio was between 0.3 and 3. Hence the following relations can be used to obtain the characteristics of bearings of conical shape from those of a plane pad: IV, = iii,
(21)
0, = QP sin a
(22) (231
The ch~acte~stic constants differ very little for the two confi~rations for all the radius and resistance ratios considered in this study, for both types of compensation, as can be seen from Tables 1 and 2. The characteristics have been plotted for a plane bearing with six recesses . Figure 3 shows the effect of rotation on iii with respect to y. The decrease in ff with speed is higher for higher values of y. The OCB has a larger reduction in @than the CCB. The effect of rotation on Q is shown in Fig. 4. Q increases with speed and is higher for the CCB than the OCB. Figure 5 shows the stiffness characteristics for three practically useful radius ratios. Generally, the OCB has higher ES values than the CCB. The m~imum value of ES occurs at 7 = 1 for the CCB and at y = 0.7 for the OCB. Damping characteristics are shown in Fig. 6. The OCB has a higher cd value than the CCB. Damping increases as +rincreases. Figure 7 shows the effect of semicone angle on the bearing characteristics. g and KS increase with CYwhile cd decreases. The variation in @ and Q with speed is linear as long as the film does not rupture because of cavitation. Hence to arrive at accurate values it is necessary to use Table 1 and to compute the characteristics for any A value between 0 and 0.917.
35 TABLE1 Static characteristics(cr = 90";R,= 0.5;@,=15")
CCB and OCB,
CCB, 1\=0.917
OCB,
CCB
A=O.917
and OCB,
A=o 4
6
0.4 1.742
CCB, A = 0.917
OCB,
A = 0.917
A=0
0.3 0.5 0.7 1.0 2.0 3.0
1.506 1.306 1.149 0.98 0.654 0.490
1.475 1.263 0.918 0.573 0.400
1.46 1.24 1.077 0.900 0.558 0.388
5.63 4.95 4.35 3.71 2.48 1.86
5.90
5.98
5.23 3.92 2.62 1.96
5.16 4.53 3.85 2.56 1.92
0.5 1.676
0.3 0.5 0.7 1.0 2.0 3.0
1.338 1.159 1.02 0.87 0.58 0.435
1.32 1.13 0.831 0.529 0.379
1.31 1.12 0.974 0.819 0.520 0.371
7.28 6.24 5.48 4.68 3.12 2.34
7.57 6.56 4.92 3.28 2.46
7.52 6.50 5.7 4.86 3.23 2.42
0.6 1.5
0.3 0.5 0.7 1.0 2.0 3.0
1.136 0.986 0.869 0.741 0.495 0.371
1.125 0.971 0.719 0.467 0.341
1.123 0.966 0.844 0.715 0.463 0.337
9.64 8.36 7.37 6.28 4.20 3.15
9.88 8.58 6.44 4.30 3.23
9.86 8.54 7.5 6.4 4.27 3.2
0.4
1.791
0.3 0.5 0.7 1.0 2.0 3.0
1.50 1.30 1.14 0.973 0.648 0.486
1.46 1.25 0.910 0.566 0.395
1.45 1.23 1.067 0.89 0.551 0.382
5.65 4.83 4.24 3.62 2.41 1.81
5.97 5.17 3.88 2.58 1.94
5.91 5.10 4.47 3.81 2.53 1.89
0.5
1.691
0.3 0.5 0.7 1.0 2.0 3.0
1.33 1.15 1.01 0.862 0.575 0.431
1.31 1.12 0.823 0.524 0.374
1.30 1.11 0.965 0.811 0.515 0.367
7.20 6.16 5.42 4.62 3.08 2.31
7.48 6.49 4.86 3.24 2.43
7.44 6.42 5.63 4.8 3.19 2.39
0.6
1.549
0.3 0.5 0.7 1.0 2.0 3.0
1.12 0.970 0.853 0.728 0.485 0.364
1.11 0.955 0.706 0.457 0.333
1.1 0.948 0.828 0.7 0.452 0.328
9.47 8.11 7.13 6.08 4.06 3.04
9.71 8.41 6.31 4.20 3.15
9.68 8.36 7.34 6.26 4.16 3.12
36 TABLE
2
Stiffness
and damping
Rl
Y
N=4
N=6
E(j(X106)
K, 0.4
0.5
0.6
((w = 90”; R, = 0.5; 4, = 15”)
characteristics
K CCB
OCB CCB __ ___
-. OCB -.
3.18 4.16 4.65
2.37 3.12 3.63 4.14 4.95 5.33
2.06 2.60 2.95 2.64 2.23
3.34 3.91 4.05 3.95 3.19 2.56
1.64 2.18 3.13 4.09 4.56
2.35 3.08 3.58 4.07 4.84 5.20
2.97 3.49 3.62 3.52 2.92 2.30
1.04 1.38 2.0 2.6 2.9
1.49 1.96 2.37 2.57 2.98 3.29
1.82 2.30 -
2.95 3.46 3.59 3.51 2.82 2.27
1.03 1.37 1.96 2.55 2.84
1.35 1.94 2.33 2.53 3.03 3.22
2.5
0.58 0.78 1.09 1.42 1.59
0.84 1.07 1.24 1.43 1.68 1.8
1.54 1.94 -
2.49 2.92 3.03 2.96 2.39 1.92
0.59 0.79 1.09 1.40 1.56
0.85 1.07 1.24 1.40 1.65 1.77
CCB
OCB
CCB
OCB
0.3 0.5 0.7 1.0 2.0 3.0
2.05 2.6 2.96 2.66 2.26
3.33 3.91 4.07 3.98 3.22 2.6
1.63 2.21 -
0.3 0.5 0.7 1.0 2.0 3.0
1.83 2.32 -
0.3 0.5 0.7 1.0 2.0 3.0
1.55 1.95 -
2.63 2.36 2.0
2.23 2.01 1.71
2.94 3.06 3.0 2.44 1.97
Fig. 3. Plots of w us. y for various capillary; 0, orifice; -,A=O;---,A=0.917. Fig. 4. Plots of Gus. bols as for Fig. 3.
l$ (X106)
y for various
-
2.61 2.34 1.98
2.20 1.98 1.67
values of A (a! = 90”;
values
of A ((u = 90”;
RI = 0.4; R, = 0.5; N = 6): C,
RI = 0.4; R, = 0.5; N = 6):.sym-
37 55c 50RI.0
4,'
I
/--
I
/
4.5-
LO-
_ a <
3.5-
lu" 30-
25-
2.0-
15-
lo-
05
0
1
2
3
Y
Fig. 5. Plots of is us. y for various radius ratios ((u= 90”; R, = 0.5; N = 6): -, --, orifice.
capillary;
Fig. 6. Plots of cd us. y for various radius ratios (a! = 90”; R, = 0.5; N = 6): symbols as for Fig. 5.
ISlO-
0
45'
----------I 60' ~(Degrees)
I 75'
90'
Fig. 7. Influence of cone angle on w, 6, ES and & for a CCB (l-21 = 0.4; R, = 0.5; 7 = 1; N = 6): -,A=o;-----,A=o.9l7.
38
4. Conclusions The characteristics of multirecess conical hydrostatic thrust bearings under rotation have been studied. Design data for load-carrying capacity and flow are provided for various resistance ratios for both capillary and orifice compensations for a practically useful speed range. Stiffness and damping characteristics are also given because they are useful in studying the response behaviour of bearings under dynamic conditions. The characteristic coefficients do not differ much for both four-recess and six-recess bearings. Hence if a higher tilt resistance is required a six-recess bearing is preferred, but this would entail additional cost and complexities in the hydraulic system.
Acknowledgments The authors thank useful discussions.
Dr. B. S. Prabhu
and Dr.-Ing. B. V. A. Rao for the
References 1 G. R. Symmons and F. B. KIemz, Effect of radial and rotational lubricant inertia on the pressure distribution in externally pressurized thrust bearings, Appl. Sci. Res., 32 (1976) 31 - 44. 2 F. Osterle and W. F. Hughes, The effect of lubricant inertia in hydrostatic thrust bearing lubrication, Wear, 1 (1958) 465 - 471. 3 T. Sasaki, H. Mori and A. Hirai, Theoretical study of hydrostatic thrust bearings, Bull. Jpn. Sot. Mech. Eng., 2 (5) (1959) 75 - 79. 4 D. Dowson, Inertia effects in hydrostatic thrust bearings, J. Basic Eng., 83 (1961) 227 - 234. 5 C. Y. Chow, A non-central feeding hydrostatic thrust bearing, J. Fluid Mech., 72 (1) (1975) 113 - 120. 6 E. Salem and F. Khalil, Thermal and inertia effects in externally pressurized conical oil bearings, Wear, 56 (1979) 251 - 264. 7 T. J. Prabhu and N. Ganesan, Characteristics of conical hydrostatic thrust bearings under rotation, Wear, 73 (1981) 95 - 122. 8 W. B. Rowe and J. P. O’Donoghue, Design Procedures for Hydrostatic Bearings, Machinery Publishing, Brighton, Sussex, 1971. 9 V. N. Constantinescu, Gas Lubrication, American Society of Mechanical Engineers, New York, NY, 1969. 10 J. F. Osterle and W. F. Hughes, Inertia induced cavitation in hydrostatic thrust bearings, Wear, 4 (1961) 228 - 233. 11 S. Heller and W. Shapiro, A numerical solution for the incompressible hybrid journal bearing with cavitation, J. Lubr. Technol., 91 (1969) 508 - 515.
Appendix AL cd
A: nomenclature land area damping constant
(kgf s cm-l)
39
c_d_ Cdc, cdp
e, Ae, AL?, h
Ii H
Q-=
!2-(
r9 rl, r2,r3,r4 P r4c
Rl Rb>Rc, R, R,
dimensionless damping constant dimensionless damping constants for a conical bearing and for a plane bearing film thickness measured axially (cm) small variation in e, (cm) Ae,/W, dimensionless small variation in e, film thickness (cm) h/W, dimensionless film thickness design film thickness (cm) frictional loss (kgf cm 8’) dimensionless frictional loss dimensionless frictional losses for a conical bearing and for a plane bearing stiffness (kgf cm-‘) dimensionless stiffness dimensionless stiffnesses for a conical bearing and for a plane bearing number of recesses pressure (kgf cme2) p/P,, dimensionless pressure recess pressure (kgf cmd2) p,/P,, dimensionless recess pressure supply pressure (kgf cmm2) volumetric flow rate (cm3 s-r) dimensionless volumetric flow rate dimensionless flow rates for a conical bearing and for a plane bearing dimensionless flow rates due to rotation across the recess boundaries at 4 = I$, and at 4 = & dimensionless flow rates due to pressure gradients across the recess boundaries at r = r2 and at r = r3 dimensionless flow rates due to pressure gradients across the recess boundaries at 4 = 4, and at 4 = C& dimensionless flow rates due to centrifugal action (rotation effect) across the recess boundaries at r = r2 and at r = r3 flow rate out of each recess (cm3 s-l) dimensionless flow rate out of each recess dimensionless flow rate due to squeeze action over the recess region slant lengths of the cone (cm) r/r4, dimensionless radius outer radius of the conical bearing r 1/r4, dimensionless radius ratio resistance of the bearing (corresponding to each recess), capillary and orifice (kgf s cme5) r2)/(r4 - rl), recess ratio (r3 -
40
Am,, A&
time (s) load-carrying capacity (kgf) dimensionless load-carrying capacity dimensionless loads for a conical bearing and for a plane bearing variations in dimensionless load-carrying capacities under static and dynamic conditions semicone angle (deg) RJR,, R,/R,, resistance ratio dynamic viscosity (kgf s cmp2) 3nS2r42/2PSH 2, bearing number 2/Ps, rotational lubricant inertia parameter (3/10W2r4, density of the lubricant (kgf s2 cmd4) 3(77~/P,)(r~/H)~, squeeze number wt, dimensionless time included angle between recesses frequency of excitation (rad s-l) angular velocity (rad s-l)