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Static characteristics of aerostatic porous rectangular thrust bearings
Wear, 77 (1982) 229 - 236
229
STATIC CHARACTERISTICS OF AEROSTATIC RECTANGULAR THRUST BEARINGS
POROUS
K. C. SINGH and N. S. RAO Department
of Mechanical Engineering,
Indian Institute
of Technology,
Kharagpur
(India)
(Received March 5, 1981; in revised form October 7, 1981)
Summary
A theoretical analysis of the steady state performance of aerostatic rectangular thrust bearings with porous pads is presented. The equations governing the gas flow through the porous pad and the bearing clearance are solved simultaneously using a finitedifference method to obtain the pressure distribution in the bearing clearance. The load capacity, the mass rate of flow and the static stiffness are calculated numerically for different bearing dimensions and operating pressures and are presented in graphical form. The load capacity predicted for a square bearing correlates with previous results.
1. Introduction
The advantages of porous gas bearings over conventional recessed bearings have been discussed elsewhere [ 1,2] . Conventional hole-admission recessed bearings with compressible lubricants are more prone to bearing instability in the form of pneumatic hammer. In porous bearings the system instability is damped out, and increased load capacity and stiffness are obtained [ 31. Mori and coworkers [4,5] developed a useful theoretical approach for the circular thrust bearing but the solution was limited to simple boundary conditions. A more versatile solution in chart form was provided by Gargiulo and Gilmour [ 21 for the design of circular thrust bearings. Rectangular thrust bearings have been used successfully in practical applications, particularly for machine-tool slideways. Majumdar and Schmidt [ 61 analysed aerostatic rectangular thrust bearings with porous pads by using a three-dimensional flow model in the porous pad. Rao [ 71 used a onedimensional flow model which is applicable to thin pads only and reported that the load capacity predicted by Majumdar and Schmidt [6] for a square bearing was very high compared with that of a circular thrust bearing, possibly owing to some error in the calculation. It is therefore necessary to recalculate the static characteristics of rectangular thrust bearings with finite thicknesses . 0043-1648/82/0000-0000/$02.75
@ Elsevier Sequoia/Printed in The Netherlands
230
In the present analysis, the porous material is assumed to be isotropic and the flow continuity equation of the porous pad is solved numerically using the Reynolds equation for the bearing clearance as one of the boundary conditions. Central-difference quotients are used to replace the second-order derivatives and the, three-point backward-difference quotient is used to replace the first-order derivative of the governing partial differential equations. The load capacity, the mass rate of flow of the gas and the static stiffness are calculated and presented in graphical form. The predicted load capacity is compared with previous results and is found to be in good agreement with that of a similar circular thrust bearing [ 21.
2. Analysis The bearing configuration shown in Fig. 1 consists porous pad and an impermeable runner. Gas is supplied at ps to the top surface of the pad. The pressure drops to p’ medium and then top in the bearing clearance and finally atmosphere at a pressure pa.
of a rectangular a constant pressure in the porous exhausts to the
Y
E+
Tcc .::,* :,: 1 L--*--i .Y._
I
,_
::-
-‘~
..’
_
x
‘,
.,:.
Fig. 1. Schematic diagram of a porous thrust hearing.
When the usual assumptions continuity equation for the pad, isotropic porous material (i.e. k, equation governing the pressure of can be written as [5,6] a2pf2
a2pf2
a2p’2
-+-+-=
ax2
ay2
for porous gas bearings are made, the flow which is assumed to be composed of = Fz,, = k,), and the modified Reynolds the lubricating film in the bearing clearance
az2
(1)
0
and
(2) When the substitutions i =x/x
Y=ylY
B = PIP,
F’ =P12
are made, eqns. (1) and (2) can be rewritten
8’ = P'IP, in dimensionless
form:
231
(3)
(4) where A=
12k,X2 h3H
The other boundary
conditions
are
fi’=&
at
1=0
and
-l
-l
p’=p
at
Z=l
and
-l
-l
p=l
at
Z=l
and f=l
and -1,
-l
(ambient condition)
ap’ z =o,
-1<35<1,
jr=l,
-1
and 0
ap’ z =o,
-1
f=l,
-1
and 0
p’=l
at -l
1, and -l
1 (sealed end)
for O
end)
ap’ = 0, f = 0, -1 af
=Gp < 1, 0 < Z < 1 (symmetry
condition)
ap’ = 0, p = 0, -1 a7
< 37< 1, 0 d Z < 1 (symmetry
condition)
3. Method
(5)
of solution
The pressure distribution in the lubricating film is determined by eqn. (3) with the boundary conditions given in eqns. (4) and (5). Owing to symmetry it is sufficient to solve for pressure over a quarter of the bearing. Equations (3) and (4) are written in the finite-difference form using centraldifference quotients for second-order derivatives and three-point backwarddifference quotients for the first-order derivative. The finite-difference equations are solved numerically to obtain F by iteration using a successive over-relaxation scheme and imposing the boundary conditions given in eqn. (5). 3.1. Bearing chamc teristics The steady state performance of the bearing is assessed from its load capacity, its stiffness and its mass rate of flow.
232
Load capacity The load capacity is given by
3.1.1.
x
Y
w=4
@ -P,)
ss 0 0
which can be rewritten
b
dy
(6)
in dimensionless
form as
W
W= 4xuP,
-PaI (7)
3.1.2. Mass rate of flow The mass rate of flow is given by
which can be rewritten G=
in dimensionless
form as
24qATG hatis
- P,~)
(9)
3.1.3. Static stiffness The static stiffness is defined thickness : sc-$
= _dW
as the rate of change of load with film
z
(10)
dA dh which can be rewritten
in dimensionless
form as
Sh
g=
4XYo7, -PA = 3A g Hence s at any value of A can be calculated A versus W.
(11) from the slope of the curve of
233
4. Results and discussion The variation in the load capacity w and the mass flow rate G with the feeding parameter A for various bearing supply pressures and dimension? are shown in Figs. 2 - 6. Figure 7 shows the variation in the static stiffness S with A for different supply pressures. The load capacities calculated in this 0.6
r
0.5 0.L \N 0.3
0.2
0.1
0
1
3
7
5
9
A
Fig. 2. Variation in the load with the feeding parameter for different values of & for open end () and sealed end (- - -) conditions (X/Y = 1.0; X/H = 5.0). @6
0.6
r r
A
A
Fig. 3. Variation in the load with the feeding parameter for different values of the pad shape ratio (jr* = 5.0;X/H = 5.0). Fig. 4. Variation in the load with the feeding parameter for different values of the thickness parameter (& = 5.0; X/Y = 1.0).
234 20-
20-
16-
16-
12e
c
A
A
Fig. 5. Variation in the flow with the feeding parameter for different values of the pad shape ratio @s = 5.0; X/H = 5.0). Fig. 6. Variation in the flow with the feeding parameter for different values of the thickness parameter (& = 5.0; X/Y = 1.0).
0.6-
0.65
w
0
1 1
I 3
I 5 A
I 7
I 9
0
1
2
3
5
A
Fig. 7. Variation in the static stiffness with the feeding parameter for different values of the supply pressure (X/Y = 1.0; X/H = 5.0). Fig. 8. Comparison of the load capacities of square [6] and circular [ 21 thrust bearings with the present results (ps = 3.0; X/H = R/H = 5.0): -O-, square thrust bearing; -, circular thrust bearing; - - -, present results.
work are compared with those calculated elsewhere for circular [ 21 and square [ 61 thrust bearings in Fig. 8. Figure 2 shows that the load capacity w increases steadily with supply pressure. Bearings with sealed ends show an appreciable improvement in load capacity compared with those with open ends for the same supply conditions. Figures 3 and 5 show the variation in w and in the mass flow rate d with the pad shape ratio X/Y. Both the load and the flow decrease with increasing X/Y. For a given bearing area an increase in the pad shape ratio (X/Y > 1)
235
causes
of the to be to ambient Thus a thrust bearing more efficiently a rectangular Figures 4 6 show both the and th_e increase with thickness parameter However, the in W higher values - 10) of X/H is not pronounced as the bearing tends to be thin. The strength of the bearing should be considered when the value of X/H is selected. A value of X/H of 5 appears suitable. Figure 7 shows the variation in static stiffness for different supply pressures. s decreases slightly with increasing supply pressure. The maximum static stiffness is attained at lower values of the feeding parameter (2 < A < 3). There is little variation in the stiffness in the upper range of the feeding parameter (A > 7), and hence the bearing should be operated in this range.
4.1. Comparison with previous results The load capacity of a square thrust bearing predicted by the present analysis is compared with the results of Majumdar and Schmidt [6] in Fig. 8. The load capacity of a circular thrust bearing as predicted by Gargiulo and Gilmour [2] is also shown. The results taken from ref. 2 correspond to a bearing with R/H = X/H = 5.0. The feeding parameter of a circular thrust bearing is defined as A = 12k,R 2/Hh3. Gargiulo and Gilmour [ 21 assumed a three-dimensional flow through the porous pad and obtained good correlation with experimental results, thereby confirming the validity of the assumption. Slightly smaller values of w were obtained from the present solution than from the earlier work [ 21. These are justified because the circular thrust bearing provides an optimum shape for a given surface area and operates more efficiently than a square bearing. The results predicted by Majumdar [6] are much higher than those of Gargiulo and Gilmour [2] and of the present work. This may be due to computational errors.
5. Conclusions (1) (2) directly (3) stiffness
The analysis presented is valid for pads of any thickness. As the results are presented in dimensionless form, they can be used in bearing design. A compromise is required between the load capacity and the static of the bearing to decide the operating value of the feeding parameter.
Nomenclature mass rate of flow 24q&TG/h3@,2
- pa2),
dimensionless mass rate of flow
236 h H k,, k,, k, t ;I pa pS pS P P’ bi S 9 T W w x, Y, 2 X, y 77 A
film thickness thickness of the porous pad permeability coefficients in the x, y and z directions pressure (absolute) in the bearing film p/pa, dimensionless pressure (absolute) in the bearing film pressure (absolute) in the porous medium p’/p,, dimensionless pressure (absolute) in the porous medium ambient pressure (absolute) supply pressure (absolute) $pa, dimensionless supply pressure (absolute) p , modified pressure (absolute) ratio p12, modified pressure (absolute) ratio gas constant static stiffness Sh/4XY@, - p,), dimensionless static stiffness absolute temperature load capacity W/BXY(p, - p,), dimensionless load capacity coordinates dimensions of the bearing (Fig. 1) coefficient of absolute viscosity of the gas 12k,X2/h3H, dimensionless feeding parameter
References 1 2 3 4 5 6 7
H. J. Sneck, J. Lubr. Technol., 90 (4) (1968) 804 - 809. E. P. Gargiulo and P. W. Gilmour, J. Lubr. Technol., 90 (4) (1968) 810 - 817. S. A. Sheinbergand V. G. Shuster,Mach. Tool. (U.S.S.R.), 31 (11) (1960) 24. H. Mori, H. Yabe and T. Ono, J. Basic Eng., 87 (3) (1965) 613 - 621. H. Mori, H. Yabe and T. Shibayama, J. Basic Eng., 87 (3) (1965) 622 - 630. B. C. Majumdar and J. Schmidt, Wear, 32 (1975) 1 - 8. N. S. Rao,Znt. J. Mach. Tool Des. Res., 19 (1979) 87 - 93.
229
STATIC CHARACTERISTICS OF AEROSTATIC RECTANGULAR THRUST BEARINGS
POROUS
K. C. SINGH and N. S. RAO Department
of Mechanical Engineering,
Indian Institute
of Technology,
Kharagpur
(India)
(Received March 5, 1981; in revised form October 7, 1981)
Summary
A theoretical analysis of the steady state performance of aerostatic rectangular thrust bearings with porous pads is presented. The equations governing the gas flow through the porous pad and the bearing clearance are solved simultaneously using a finitedifference method to obtain the pressure distribution in the bearing clearance. The load capacity, the mass rate of flow and the static stiffness are calculated numerically for different bearing dimensions and operating pressures and are presented in graphical form. The load capacity predicted for a square bearing correlates with previous results.
1. Introduction
The advantages of porous gas bearings over conventional recessed bearings have been discussed elsewhere [ 1,2] . Conventional hole-admission recessed bearings with compressible lubricants are more prone to bearing instability in the form of pneumatic hammer. In porous bearings the system instability is damped out, and increased load capacity and stiffness are obtained [ 31. Mori and coworkers [4,5] developed a useful theoretical approach for the circular thrust bearing but the solution was limited to simple boundary conditions. A more versatile solution in chart form was provided by Gargiulo and Gilmour [ 21 for the design of circular thrust bearings. Rectangular thrust bearings have been used successfully in practical applications, particularly for machine-tool slideways. Majumdar and Schmidt [ 61 analysed aerostatic rectangular thrust bearings with porous pads by using a three-dimensional flow model in the porous pad. Rao [ 71 used a onedimensional flow model which is applicable to thin pads only and reported that the load capacity predicted by Majumdar and Schmidt [6] for a square bearing was very high compared with that of a circular thrust bearing, possibly owing to some error in the calculation. It is therefore necessary to recalculate the static characteristics of rectangular thrust bearings with finite thicknesses . 0043-1648/82/0000-0000/$02.75
@ Elsevier Sequoia/Printed in The Netherlands
230
In the present analysis, the porous material is assumed to be isotropic and the flow continuity equation of the porous pad is solved numerically using the Reynolds equation for the bearing clearance as one of the boundary conditions. Central-difference quotients are used to replace the second-order derivatives and the, three-point backward-difference quotient is used to replace the first-order derivative of the governing partial differential equations. The load capacity, the mass rate of flow of the gas and the static stiffness are calculated and presented in graphical form. The predicted load capacity is compared with previous results and is found to be in good agreement with that of a similar circular thrust bearing [ 21.
2. Analysis The bearing configuration shown in Fig. 1 consists porous pad and an impermeable runner. Gas is supplied at ps to the top surface of the pad. The pressure drops to p’ medium and then top in the bearing clearance and finally atmosphere at a pressure pa.
of a rectangular a constant pressure in the porous exhausts to the
Y
E+
Tcc .::,* :,: 1 L--*--i .Y._
I
,_
::-
-‘~
..’
_
x
‘,
.,:.
Fig. 1. Schematic diagram of a porous thrust hearing.
When the usual assumptions continuity equation for the pad, isotropic porous material (i.e. k, equation governing the pressure of can be written as [5,6] a2pf2
a2pf2
a2p’2
-+-+-=
ax2
ay2
for porous gas bearings are made, the flow which is assumed to be composed of = Fz,, = k,), and the modified Reynolds the lubricating film in the bearing clearance
az2
(1)
0
and
(2) When the substitutions i =x/x
Y=ylY
B = PIP,
F’ =P12
are made, eqns. (1) and (2) can be rewritten
8’ = P'IP, in dimensionless
form:
231
(3)
(4) where A=
12k,X2 h3H
The other boundary
conditions
are
fi’=&
at
1=0
and
-l
-l
p’=p
at
Z=l
and
-l
-l
p=l
at
Z=l
and f=l
and -1,
-l
(ambient condition)
ap’ z =o,
-1<35<1,
jr=l,
-1
and 0
ap’ z =o,
-1
f=l,
-1
and 0
p’=l
at -l
1, and -l
1 (sealed end)
for O
end)
ap’ = 0, f = 0, -1 af
=Gp < 1, 0 < Z < 1 (symmetry
condition)
ap’ = 0, p = 0, -1 a7
< 37< 1, 0 d Z < 1 (symmetry
condition)
3. Method
(5)
of solution
The pressure distribution in the lubricating film is determined by eqn. (3) with the boundary conditions given in eqns. (4) and (5). Owing to symmetry it is sufficient to solve for pressure over a quarter of the bearing. Equations (3) and (4) are written in the finite-difference form using centraldifference quotients for second-order derivatives and three-point backwarddifference quotients for the first-order derivative. The finite-difference equations are solved numerically to obtain F by iteration using a successive over-relaxation scheme and imposing the boundary conditions given in eqn. (5). 3.1. Bearing chamc teristics The steady state performance of the bearing is assessed from its load capacity, its stiffness and its mass rate of flow.
232
Load capacity The load capacity is given by
3.1.1.
x
Y
w=4
@ -P,)
ss 0 0
which can be rewritten
b
dy
(6)
in dimensionless
form as
W
W= 4xuP,
-PaI (7)
3.1.2. Mass rate of flow The mass rate of flow is given by
which can be rewritten G=
in dimensionless
form as
24qATG hatis
- P,~)
(9)
3.1.3. Static stiffness The static stiffness is defined thickness : sc-$
= _dW
as the rate of change of load with film
z
(10)
dA dh which can be rewritten
in dimensionless
form as
Sh
g=
4XYo7, -PA = 3A g Hence s at any value of A can be calculated A versus W.
(11) from the slope of the curve of
233
4. Results and discussion The variation in the load capacity w and the mass flow rate G with the feeding parameter A for various bearing supply pressures and dimension? are shown in Figs. 2 - 6. Figure 7 shows the variation in the static stiffness S with A for different supply pressures. The load capacities calculated in this 0.6
r
0.5 0.L \N 0.3
0.2
0.1
0
1
3
7
5
9
A
Fig. 2. Variation in the load with the feeding parameter for different values of & for open end () and sealed end (- - -) conditions (X/Y = 1.0; X/H = 5.0). @6
0.6
r r
A
A
Fig. 3. Variation in the load with the feeding parameter for different values of the pad shape ratio (jr* = 5.0;X/H = 5.0). Fig. 4. Variation in the load with the feeding parameter for different values of the thickness parameter (& = 5.0; X/Y = 1.0).
234 20-
20-
16-
16-
12e
c
A
A
Fig. 5. Variation in the flow with the feeding parameter for different values of the pad shape ratio @s = 5.0; X/H = 5.0). Fig. 6. Variation in the flow with the feeding parameter for different values of the thickness parameter (& = 5.0; X/Y = 1.0).
0.6-
0.65
w
0
1 1
I 3
I 5 A
I 7
I 9
0
1
2
3
5
A
Fig. 7. Variation in the static stiffness with the feeding parameter for different values of the supply pressure (X/Y = 1.0; X/H = 5.0). Fig. 8. Comparison of the load capacities of square [6] and circular [ 21 thrust bearings with the present results (ps = 3.0; X/H = R/H = 5.0): -O-, square thrust bearing; -, circular thrust bearing; - - -, present results.
work are compared with those calculated elsewhere for circular [ 21 and square [ 61 thrust bearings in Fig. 8. Figure 2 shows that the load capacity w increases steadily with supply pressure. Bearings with sealed ends show an appreciable improvement in load capacity compared with those with open ends for the same supply conditions. Figures 3 and 5 show the variation in w and in the mass flow rate d with the pad shape ratio X/Y. Both the load and the flow decrease with increasing X/Y. For a given bearing area an increase in the pad shape ratio (X/Y > 1)
235
causes
of the to be to ambient Thus a thrust bearing more efficiently a rectangular Figures 4 6 show both the and th_e increase with thickness parameter However, the in W higher values - 10) of X/H is not pronounced as the bearing tends to be thin. The strength of the bearing should be considered when the value of X/H is selected. A value of X/H of 5 appears suitable. Figure 7 shows the variation in static stiffness for different supply pressures. s decreases slightly with increasing supply pressure. The maximum static stiffness is attained at lower values of the feeding parameter (2 < A < 3). There is little variation in the stiffness in the upper range of the feeding parameter (A > 7), and hence the bearing should be operated in this range.
4.1. Comparison with previous results The load capacity of a square thrust bearing predicted by the present analysis is compared with the results of Majumdar and Schmidt [6] in Fig. 8. The load capacity of a circular thrust bearing as predicted by Gargiulo and Gilmour [2] is also shown. The results taken from ref. 2 correspond to a bearing with R/H = X/H = 5.0. The feeding parameter of a circular thrust bearing is defined as A = 12k,R 2/Hh3. Gargiulo and Gilmour [ 21 assumed a three-dimensional flow through the porous pad and obtained good correlation with experimental results, thereby confirming the validity of the assumption. Slightly smaller values of w were obtained from the present solution than from the earlier work [ 21. These are justified because the circular thrust bearing provides an optimum shape for a given surface area and operates more efficiently than a square bearing. The results predicted by Majumdar [6] are much higher than those of Gargiulo and Gilmour [2] and of the present work. This may be due to computational errors.
5. Conclusions (1) (2) directly (3) stiffness
The analysis presented is valid for pads of any thickness. As the results are presented in dimensionless form, they can be used in bearing design. A compromise is required between the load capacity and the static of the bearing to decide the operating value of the feeding parameter.
Nomenclature mass rate of flow 24q&TG/h3@,2
- pa2),
dimensionless mass rate of flow
236 h H k,, k,, k, t ;I pa pS pS P P’ bi S 9 T W w x, Y, 2 X, y 77 A
film thickness thickness of the porous pad permeability coefficients in the x, y and z directions pressure (absolute) in the bearing film p/pa, dimensionless pressure (absolute) in the bearing film pressure (absolute) in the porous medium p’/p,, dimensionless pressure (absolute) in the porous medium ambient pressure (absolute) supply pressure (absolute) $pa, dimensionless supply pressure (absolute) p , modified pressure (absolute) ratio p12, modified pressure (absolute) ratio gas constant static stiffness Sh/4XY@, - p,), dimensionless static stiffness absolute temperature load capacity W/BXY(p, - p,), dimensionless load capacity coordinates dimensions of the bearing (Fig. 1) coefficient of absolute viscosity of the gas 12k,X2/h3H, dimensionless feeding parameter
References 1 2 3 4 5 6 7
H. J. Sneck, J. Lubr. Technol., 90 (4) (1968) 804 - 809. E. P. Gargiulo and P. W. Gilmour, J. Lubr. Technol., 90 (4) (1968) 810 - 817. S. A. Sheinbergand V. G. Shuster,Mach. Tool. (U.S.S.R.), 31 (11) (1960) 24. H. Mori, H. Yabe and T. Ono, J. Basic Eng., 87 (3) (1965) 613 - 621. H. Mori, H. Yabe and T. Shibayama, J. Basic Eng., 87 (3) (1965) 622 - 630. B. C. Majumdar and J. Schmidt, Wear, 32 (1975) 1 - 8. N. S. Rao,Znt. J. Mach. Tool Des. Res., 19 (1979) 87 - 93.