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The effect of rotational inertia on the squeeze film load between porous annular curved plates
235
Wear, 79 (1982) 235 - 240
THE EFFECT OF ROTATIONAL EXERTS ON THE SQUEEZE LOAD BETWEEN POROUS ANNULAR CURVED PLATES
FILM
J. L. GUPTA and K. H. VORA Birla Vishuakarma Mahuvidyalaya (~~gjneering Gujarat (India)
Coliege),
Va~la&h Vidya~agur 388120,
M. V. BHAT Department of Mathematics, Gujarat (India) (Received May 20,198l;
Sardar Pate1 University, Vallabh Vidyanagar 388120,
in revised form November 19, 1981)
Summary
The squeeze film behaviour between rotating annular plates was analysed theoretically for the case when the curved upper plate with a uniform porous facing approached normally the impermeable flat lower plate. Expressions for the pressure and the load capacity were obtained. The load capacity decreased when the speed of rotation of the upper disc increased up to a certain value Be of the curvature parameter B. This trend was reversed for B > Be. The load capacity could be increased without altering the speed of rotation by increasing 8.
I. Introduction
Wu [l] investigated squeeze film behaviour in porous annular discs with both discs rotating. He presented the results in the form of an infinite series involving Bessel functions and showed that the effect of rotation was to reduce pressure, load capacity and response time. Prakash and Vij [Z] extended the analysis of ref. 1 to include the tangential velocity slip at the porous matrix-film interface. Ting 133 simplified the analysis of Wu [I] and obtained results in closed form by considering the rotation of the lower disc only. The discs were assumed to be flat but in practice, owing to elastic, thermal and uneven wear effects, they are usually far from flat. It is therefore necessary to consider the effect of curvature on such configurations. Gupta and Vora [4] considered curvature effects for the squeeze film between non-porous curved annular plates. Gupta et al. [ 51 extended the analysis of ref. 4 by assuming that the curved upper plate had a porous facing of uniform thickness. It was shown that the load capacity could be increased by increasing the curvature parameter with concave plates. How0043-1648/82/0000-0000/$02.75
@ Elsevier Sequoial~nted
in The Netherlands
236
ever, in both of these analyses [4, 5 ] the plates were considered to be nonrotating. Squeeze film behaviour between porous annular plates including the effects of both rotation and curvature is reported in this paper.
2. Analysis The bearing configuration consists of two annular plates, each of inside radius a and outside radius b. The upper plate is curved and has a porous facing of thickness H which is backed by a solid wall. The lower plate is impermeable and flat. The film thickness h [3] is taken as h = ho exp(--Br2)
a
(1)
where r is the radial coordinate, ho is the central film thickness and B is the curvature parameter. In dimensionless form it can be written as h = exp(--BR2)
(2)
where & = Hahn
B =&I2
R = r/a
(3)
The upper plate approaches the lower plate normally with a uniform velocity dh,/dt = ho. The upper and the lower plates rotate with angular velocities GU and 52, respectively. Taking Q;2f= L?,/,QU, and incorporating the MorganCameron approximation, the modified Reynolds equation giving the film pressurep(r) 163 is 3 + lB@H)r $ I
where @is the permeability
and 1-1is the fluid viscosity.
3. Solutions Solving eqn. (4) under the boundary p(r)=0 we obtain
when
r=a
=
and r=b
the dimensionless
1+2tis 12ljJB
L,(R)
conditions (5)
film pressure JYas
+ 6s
(3 + 4s1, + 3Q;))Ls(R)
-
237
1 + 2&S
-
12rjrij
Lx(k) + --$
(3 + 4Rf + 3Stf2)Ln(k)
‘g
(6)
t
where L,(R)
= In
1 + 12$ exp(3B) i 1 + 12$ exp(3BR’)
exp(-3BR2)
L2(Rf = In I
(7)
t
+ 124
exp(-3B)
(8)
+ 12$
I
I(R)=fd” + 12$] I x (exp(-33x2)
(9)
and S = p!Ctu2h03/&
4 = @H/he3 The load-carrying
capacity
k = b/a
(10)
W of the bearing is given by
b W=
_I(I
2xpr dr
which can be expressed
in dimensionless
= 3(1 + 211/5)11 + $3
-
form as
+ 4tir + 3fi:))12 -
(3 + 4fzr + 352;)&(k)
+
1+2$/s 12$
L,(k)
I (11)
where R3 exp(3RR2) 1
I2 =
1 + 121~5exp(3~R2) ’
I
1
dR
R3 exp(-3BR2) exp(-3BR2)
(12)
dR
+ 12Q
4. Results and discussion Various special cases that can be deduced considered.
from the present
analysis are
238
(1) When neither plate rotates, Sz, = fl, = 0. Then eqns. (6) and (11) reduce to those obtained by Gupta et al. [ 51 for non-rotating porous annular curved plates. (2) With rl, -+ 0 and setting s2, = 0, = 0, we obtain the results for nonrotating and non-porous annular curved plates [ 41, (3) When both plates are impermeable and rotating, the results can be obtained by requiring $ -+ 0 in eqns. (6) and (11). (4) When the upper plate does not rotate (fin, = 0), eqns. (6) and (11) yield 1
-
p1 - -L,(R) 123/B
+
1 ----=L,(fi) AL, - 1 12$/B
+
1l.Rf $+2(k)1 -I(h 1
and
where
(5) When the lower plate does not rotate, results can be obtained from eqns. (6) and (11) by setting Q-2f= 0. (6) When both plates rotate with the same velocity (!AU = ai), results can be obtained from eqns. (6) and (11) by setting Qzf = 1. (‘7) With fi + 0, the results for rotating parallel plates are 3 1+23/s+ $(3+4s&+3n,“) x p=-1+1211/ i i
x
R2-1-(ha-l)% I
1 @,3
IL
4 1+12$J
1+24ilS+
$(3+4Q,+3$Zf2)
k*-l-
(k2 - 1)2
In k
These results reduce to those of Ting [ 3] if we set 52, = 0, as a consequence of which 1+2$/s+
;(3+4.R,+312ta)=l+~
Wu [l] considered this case without making the assumption that H is small. The effects on the load capacity due to variations in the speed parameter s2 f, the inertia parameter S or S1 and the curvature parameter a are given in Tables 1 - 4. The following observations can be made. Table 1 shows that w increases when S numerically decreases up to a certain value & of B, where 0.125 B &, < 0.25. This trend is reversed when B 2 Be. When S is sufficiently large and the plate is convex, p becomes
239
TABLE1 Valuesof~forvariousvalues ofg andS Li
S
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250
-2.19
-1.00
-0.05
-1.634 -0.831 1.722 3.724 10.695 56.719 113.865
-0.437 0.125 2.127 3.823 10.197 55.367 112.406
0.518 0.889 2.450 3.902 9.800 54.288 111.244
12j,= O.OOl;& = 2;k = 2,
negative and the film ruptures. w increases sharply when B > 0.25 with a concave plate. Table 2 shows that when Q;2tis numerically large, i@becomes negative and the film ruptures with a convex plate for sufficie_ntly large values of B. W decreases when a f increases numerically for B < BO. This trend is reversed when .6 2 &. The entries in the column for which fif = 0 are the values of w when the lower plate does not rotate (CL1= 0) and those in the column for which IRf = 1 are the values of w when both plates rotate at the same angular velocity (Q2, = R,). Thus the load capacity of the bearing with concave plates when the lower plate does not rotate is greater than that when both plates rotate with the same angular velocity for B < &,. This trend is reversed for B> Be. TABLE2 Valuesof Wforvariousvalues ofE and fi;tf
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250
-10
-2
0
I
2
10
-0.006 0.470 2.272 3.859 10.019 54.885 111.892
0.553 0.917 2.461 3.905 9.786 54.247 111.201
0.562262 0.924178 2.464584 3.906097 9.782265 54.237800 111.191400
0.597 0,912 2.459 3.905 9.788 54.255 111.209
0.518 0.889 2.450 3.902 9.800 54.288 111.244
-0.181 0.330 2.213 3.845 10.092 55.084 112.109
123,= 0.001;s=-0.05; k = 2.
Table 3 showsJhat for a given j the numerical increase in S implies the steady increase in W. In contrast, when Rf numerically increases, w also increases. For the same numerical value of R,, the load capacity is greater
240 TABLE 3 Values of w for various values of S and & S
fif
-2.19 -1.00 -0.05
-10
-2
0
1
2
10
20.284 14.576 10.019
10.055 9.905 9.286
9.896 9.832 9.782
10.175 9.960 9.788
10.695 10.197 9.800
23.461 16.036 10.092
-
_
124!’ = 0.001; g = 0.25; k = 2.
when the plates rotate in the same direction than when they rotate in opposite directions. Table 4 gives the values of the load capacity @, when the upper disc does not rotate. For the values of I? considered here, RI does not become negative and there is no rupture of the film. WI increases when S1 numerically decreases for B < &. This trend is reversed for B 2 & . TABLE 4 Values of %I for various values of g and 5’1 E
Sl
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250 12+=
-2.19
-1.00
-0.05
0.281 0.700 2.370 3.883 9.899 54.558 111.539
0.437 0.825 2.422 3.896 9.834 54.380 111.347
0.562266 0.924185 2.464599 3.906120 9.782325 54.238400 111.193350
0.001; k = 2.
Tables 1 and 4 show that WI > w for a given S or S1 and 8 < &,. This trend is reversed when B 2 &,. References H. Wu, The squeeze film between rotating porous annular disks, Wear, 18 (1971) 461 - 470. J. Prakash and S. K. Vij, Effect of velocity slip on the squeeze film between rotating porous annular discs, Wear, 38 (1976) 73 - 85. L. L. Ting, A mathematical analog for determination of porous annular disc squeeze film behaviour including the fluid inertia effect, .I. &sic Eng., 94 (1972) 417 - 421. J. L. Gupta and K. H. Vora, Analysis of squeeze films between curved annular plates, J. Lubr. Technol., 102 (1980) 48 - 50. J. L. Gupta, K. C. Pate1 and J. V. Hingu, An analysis of the squeeze film between porous annular curved plates, J. Math. Phys. Sci. (India), 14 (1980) 611 - 618. K. H. Vora and M. V. Bhat, The load capacity of a squeeze film between curved porous rotating circular plates, Wear, 65 (1980) 39 - 46.
Wear, 79 (1982) 235 - 240
THE EFFECT OF ROTATIONAL EXERTS ON THE SQUEEZE LOAD BETWEEN POROUS ANNULAR CURVED PLATES
FILM
J. L. GUPTA and K. H. VORA Birla Vishuakarma Mahuvidyalaya (~~gjneering Gujarat (India)
Coliege),
Va~la&h Vidya~agur 388120,
M. V. BHAT Department of Mathematics, Gujarat (India) (Received May 20,198l;
Sardar Pate1 University, Vallabh Vidyanagar 388120,
in revised form November 19, 1981)
Summary
The squeeze film behaviour between rotating annular plates was analysed theoretically for the case when the curved upper plate with a uniform porous facing approached normally the impermeable flat lower plate. Expressions for the pressure and the load capacity were obtained. The load capacity decreased when the speed of rotation of the upper disc increased up to a certain value Be of the curvature parameter B. This trend was reversed for B > Be. The load capacity could be increased without altering the speed of rotation by increasing 8.
I. Introduction
Wu [l] investigated squeeze film behaviour in porous annular discs with both discs rotating. He presented the results in the form of an infinite series involving Bessel functions and showed that the effect of rotation was to reduce pressure, load capacity and response time. Prakash and Vij [Z] extended the analysis of ref. 1 to include the tangential velocity slip at the porous matrix-film interface. Ting 133 simplified the analysis of Wu [I] and obtained results in closed form by considering the rotation of the lower disc only. The discs were assumed to be flat but in practice, owing to elastic, thermal and uneven wear effects, they are usually far from flat. It is therefore necessary to consider the effect of curvature on such configurations. Gupta and Vora [4] considered curvature effects for the squeeze film between non-porous curved annular plates. Gupta et al. [ 51 extended the analysis of ref. 4 by assuming that the curved upper plate had a porous facing of uniform thickness. It was shown that the load capacity could be increased by increasing the curvature parameter with concave plates. How0043-1648/82/0000-0000/$02.75
@ Elsevier Sequoial~nted
in The Netherlands
236
ever, in both of these analyses [4, 5 ] the plates were considered to be nonrotating. Squeeze film behaviour between porous annular plates including the effects of both rotation and curvature is reported in this paper.
2. Analysis The bearing configuration consists of two annular plates, each of inside radius a and outside radius b. The upper plate is curved and has a porous facing of thickness H which is backed by a solid wall. The lower plate is impermeable and flat. The film thickness h [3] is taken as h = ho exp(--Br2)
a
(1)
where r is the radial coordinate, ho is the central film thickness and B is the curvature parameter. In dimensionless form it can be written as h = exp(--BR2)
(2)
where & = Hahn
B =&I2
R = r/a
(3)
The upper plate approaches the lower plate normally with a uniform velocity dh,/dt = ho. The upper and the lower plates rotate with angular velocities GU and 52, respectively. Taking Q;2f= L?,/,QU, and incorporating the MorganCameron approximation, the modified Reynolds equation giving the film pressurep(r) 163 is 3 + lB@H)r $ I
where @is the permeability
and 1-1is the fluid viscosity.
3. Solutions Solving eqn. (4) under the boundary p(r)=0 we obtain
when
r=a
=
and r=b
the dimensionless
1+2tis 12ljJB
L,(R)
conditions (5)
film pressure JYas
+ 6s
(3 + 4s1, + 3Q;))Ls(R)
-
237
1 + 2&S
-
12rjrij
Lx(k) + --$
(3 + 4Rf + 3Stf2)Ln(k)
‘g
(6)
t
where L,(R)
= In
1 + 12$ exp(3B) i 1 + 12$ exp(3BR’)
exp(-3BR2)
L2(Rf = In I
(7)
t
+ 124
exp(-3B)
(8)
+ 12$
I
I(R)=fd” + 12$] I x (exp(-33x2)
(9)
and S = p!Ctu2h03/&
4 = @H/he3 The load-carrying
capacity
k = b/a
(10)
W of the bearing is given by
b W=
_I(I
2xpr dr
which can be expressed
in dimensionless
= 3(1 + 211/5)11 + $3
-
form as
+ 4tir + 3fi:))12 -
(3 + 4fzr + 352;)&(k)
+
1+2$/s 12$
L,(k)
I (11)
where R3 exp(3RR2) 1
I2 =
1 + 121~5exp(3~R2) ’
I
1
dR
R3 exp(-3BR2) exp(-3BR2)
(12)
dR
+ 12Q
4. Results and discussion Various special cases that can be deduced considered.
from the present
analysis are
238
(1) When neither plate rotates, Sz, = fl, = 0. Then eqns. (6) and (11) reduce to those obtained by Gupta et al. [ 51 for non-rotating porous annular curved plates. (2) With rl, -+ 0 and setting s2, = 0, = 0, we obtain the results for nonrotating and non-porous annular curved plates [ 41, (3) When both plates are impermeable and rotating, the results can be obtained by requiring $ -+ 0 in eqns. (6) and (11). (4) When the upper plate does not rotate (fin, = 0), eqns. (6) and (11) yield 1
-
p1 - -L,(R) 123/B
+
1 ----=L,(fi) AL, - 1 12$/B
+
1l.Rf $+2(k)1 -I(h 1
and
where
(5) When the lower plate does not rotate, results can be obtained from eqns. (6) and (11) by setting Q-2f= 0. (6) When both plates rotate with the same velocity (!AU = ai), results can be obtained from eqns. (6) and (11) by setting Qzf = 1. (‘7) With fi + 0, the results for rotating parallel plates are 3 1+23/s+ $(3+4s&+3n,“) x p=-1+1211/ i i
x
R2-1-(ha-l)% I
1 @,3
IL
4 1+12$J
1+24ilS+
$(3+4Q,+3$Zf2)
k*-l-
(k2 - 1)2
In k
These results reduce to those of Ting [ 3] if we set 52, = 0, as a consequence of which 1+2$/s+
;(3+4.R,+312ta)=l+~
Wu [l] considered this case without making the assumption that H is small. The effects on the load capacity due to variations in the speed parameter s2 f, the inertia parameter S or S1 and the curvature parameter a are given in Tables 1 - 4. The following observations can be made. Table 1 shows that w increases when S numerically decreases up to a certain value & of B, where 0.125 B &, < 0.25. This trend is reversed when B 2 Be. When S is sufficiently large and the plate is convex, p becomes
239
TABLE1 Valuesof~forvariousvalues ofg andS Li
S
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250
-2.19
-1.00
-0.05
-1.634 -0.831 1.722 3.724 10.695 56.719 113.865
-0.437 0.125 2.127 3.823 10.197 55.367 112.406
0.518 0.889 2.450 3.902 9.800 54.288 111.244
12j,= O.OOl;& = 2;k = 2,
negative and the film ruptures. w increases sharply when B > 0.25 with a concave plate. Table 2 shows that when Q;2tis numerically large, i@becomes negative and the film ruptures with a convex plate for sufficie_ntly large values of B. W decreases when a f increases numerically for B < BO. This trend is reversed when .6 2 &. The entries in the column for which fif = 0 are the values of w when the lower plate does not rotate (CL1= 0) and those in the column for which IRf = 1 are the values of w when both plates rotate at the same angular velocity (Q2, = R,). Thus the load capacity of the bearing with concave plates when the lower plate does not rotate is greater than that when both plates rotate with the same angular velocity for B < &,. This trend is reversed for B> Be. TABLE2 Valuesof Wforvariousvalues ofE and fi;tf
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250
-10
-2
0
I
2
10
-0.006 0.470 2.272 3.859 10.019 54.885 111.892
0.553 0.917 2.461 3.905 9.786 54.247 111.201
0.562262 0.924178 2.464584 3.906097 9.782265 54.237800 111.191400
0.597 0,912 2.459 3.905 9.788 54.255 111.209
0.518 0.889 2.450 3.902 9.800 54.288 111.244
-0.181 0.330 2.213 3.845 10.092 55.084 112.109
123,= 0.001;s=-0.05; k = 2.
Table 3 showsJhat for a given j the numerical increase in S implies the steady increase in W. In contrast, when Rf numerically increases, w also increases. For the same numerical value of R,, the load capacity is greater
240 TABLE 3 Values of w for various values of S and & S
fif
-2.19 -1.00 -0.05
-10
-2
0
1
2
10
20.284 14.576 10.019
10.055 9.905 9.286
9.896 9.832 9.782
10.175 9.960 9.788
10.695 10.197 9.800
23.461 16.036 10.092
-
_
124!’ = 0.001; g = 0.25; k = 2.
when the plates rotate in the same direction than when they rotate in opposite directions. Table 4 gives the values of the load capacity @, when the upper disc does not rotate. For the values of I? considered here, RI does not become negative and there is no rupture of the film. WI increases when S1 numerically decreases for B < &. This trend is reversed for B 2 & . TABLE 4 Values of %I for various values of g and 5’1 E
Sl
-0.1250 -0.0625 0.0625 0.1250 0.2500 0.5000 0.6250 12+=
-2.19
-1.00
-0.05
0.281 0.700 2.370 3.883 9.899 54.558 111.539
0.437 0.825 2.422 3.896 9.834 54.380 111.347
0.562266 0.924185 2.464599 3.906120 9.782325 54.238400 111.193350
0.001; k = 2.
Tables 1 and 4 show that WI > w for a given S or S1 and 8 < &,. This trend is reversed when B 2 &,. References H. Wu, The squeeze film between rotating porous annular disks, Wear, 18 (1971) 461 - 470. J. Prakash and S. K. Vij, Effect of velocity slip on the squeeze film between rotating porous annular discs, Wear, 38 (1976) 73 - 85. L. L. Ting, A mathematical analog for determination of porous annular disc squeeze film behaviour including the fluid inertia effect, .I. &sic Eng., 94 (1972) 417 - 421. J. L. Gupta and K. H. Vora, Analysis of squeeze films between curved annular plates, J. Lubr. Technol., 102 (1980) 48 - 50. J. L. Gupta, K. C. Pate1 and J. V. Hingu, An analysis of the squeeze film between porous annular curved plates, J. Math. Phys. Sci. (India), 14 (1980) 611 - 618. K. H. Vora and M. V. Bhat, The load capacity of a squeeze film between curved porous rotating circular plates, Wear, 65 (1980) 39 - 46.