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The load-carrying capacity of plain finite partial journal bearings in the turbulent regime
Wear, 58 (1980) 261 - 273 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
261
THE LOAD-CARRYING CAPACITY OF PLAIN FINITE PARTIAL JOURNAL BEARINGS IN THE TURBULENT REGIME
VINAY KUMAR
~~ar~~ent of ~ec~a~icu~ ~ngi~ee~~g, (Haryana), Pin 1321 IS (India) (Received
June 26,1978;in
Region& engineering College, Kur~~she~a
final form February 28,1979)
Serious errors occur in hydrodynamic bearings when the lubricant film becomes turbulent if they are analysed/designed according to laminar theory. Thus there is a need for an accurate description of the load capacity of plain journal bearings in the turbulent regime. It has been shown that the governing ~bulent-lying lubrication equation for plain partial journal bearings of finite length under dynamic operation is a second-order linear non-homogeneous partial differential equation with analytically known variable coefficients which can be split by the classical method of separation of variables. One of the resulting ordinary equations is integrated directly while the other is solved by a direct method. The solution is subject to zeropressure boundary conditions. All the results of interest such as pressure distribution, load-carrying capacity and attitude angle are dimensionless and are obtained in the closed form which is simple and accurate and permits straightforward calculation of numerical data over a wide range of parameters.
1. Introduction High speed rotating machines with hydrodynamic journal bearings may require process fluid lub~~ation. Under adverse conditions it is necessary to overcome shaft sealing problems. Since many process fluids (such as water and liquid metals) have low kinematic viscosities compared with conventional oils, turbulent flow may occur in the bearing films. Therefore it is necessary to extend Reynolds lubrication theory to the turbulent flow regime [ 1 ] ,360” journal bearings have already been analysed [ 2,3], but since bearings are made up of partial arcs with an arbitrary load angle to provide lubrication and to ensure stability, their behaviour under turbulent film conditions is of considerable practical as well as academic interest. In practice partial journal bearings with aspect ratios between 0.25 and 40 are used. Thus the simplifying mathematical assumptions of Sommerfeld or
262
Ocvirk cannot be applied as the actual axial length of the bearing must be considered. Plain partial arc journal bearings of finite length with hydrodynamic turbulent lubrication conditions have been analysed to determine their loadcarrying capacity and attitude angle. Although some data on the static properties [l] existing during turbulent operation of three-dimensional partial journal bearings which have been obtained by numerical computer techniques and verified experimentally are available, they are insufficient for the preparation of complete design charts or for design purposes. Hence an analytical approach yielding closed-form results was adopted for the following reasons. (1) Since turbulence is associated with laminar flow, the analytical results of interest in the laminar regime can be obtained as a special case by applying suitable restrictive conditions. This approach is not possible when numerical methods are used. (2) An insight into the interrelation of the various parameters is obtained which is not possible using numerical methods. (3) Numerical data required for the wide range of parameters involved in the preparation of design charts or tables can be more conveniently and accurately calculated than by a numerical approach. (4) Bearing design is quicker and simpler when the appropriate results are available in closed form because of the iterative procedure generally required. To determine load capacity in a closed form the governing turbulent equation for hydrodynamic pressure [ 1, 21 is made non-dimensional and split into two ordinary linear differential equations with analytically known variable coefficients by using the classical method of separation of variables. The first equation yields the particular solution for the case of a bearing of infinite length while the second is a homogeneous differential equation consisting of a hyperbolic differential equation and a Hill equation. The static characteristics are evaluated from the analytical form of the threedimensional hydrodynamic pressure distribution obtained in the conventional manner which yields dimensionless closed-form results.
2. Assumptions (1) The bearing is aligned so that the circumferential coordinate only. (2) There is no relative axial motion (3) The journal and bearing surfaces (4) Turbulence is isotropic and fully (5) The lubricant is an incompressible (6) The film does not rupture at any
film thickness
is a function
of the
between the surfaces. are smooth, rigid and impermeable. developed. Newtonian liquid. point over the bearing arc.
263
3. The governing hydrodynamic turbulent regime
lubrication
equation in the laminar-
For the journal bearing configuration shown in Fig. 1 the governing pressure equation including turbulence under a whirling and squeezing motion of the journal is given by [ 1,2] _?_(~!~)+!_~~!!$$!!!_I7
(1)
where, as shown by Constantinescu in the discussion of ref. 1, the analytical values of l/G, and l/G, can be closely approximated by 12 + 0.238 X (0.332)Reh0.a and 12 + 0.17 X (0.33)2Re$7 respectively. From Fig. 1 (2) U = Rw -Cc+ cos(0 + Q) + Ci sin(b + (Y) V = -CE$ sin(0 + (u) - C!; cos(0 + (IL)
(3)
Substituting eqns. (2) and (3) in eqn. (1) and non-dimensionalizing the resulting equation by means of the parameters x = R8, h* = h/C, P = pC2/pR2u and z* = z/b gives
(4)
LINE
OF
Fig. 1. Bearing configuration,
264
The terms containing C/2R were neglected since they are extremely small in comparison with the other terms.
4. Splitting of the governing pressure equation Since the right-hand side of eqn, (4) is a function of 0 only, it can be solved by the method of separation of variables and can be rewritten as P(6,z”) = F(B) + g(0)K(z*)
(5)
which for convenience can be expressed as
P=F+gK Substi~ting eqn. (5) in eqn. (4) and updating the variables gives an ordinary non-homogeneous second-order differential equation
(6) and an ordinary homogeneous
second-order differential equation (7)
Dividing eqn. (7) by gK and noting that the sum of the first and second terms, which are functions of f3 and z* respectively, is zero leads to the conclusion that each term must be equal to a constant +n2. This results in further splitting into two ordinary differential equations:
and
(9) Thus the non-homogeneous partial differential equation given by eqn. (4) is equivalent to three ordinary differential equations represented by eqns. (6), (8) and (9). 5. Boundary conditions The Sommerfeld boundary conditions and the choice of the coordinates shown in Fig. 1 yieldsP(0, z*) = 0 = P(A, z) and P(e, f i) = 0. Under these conditions eqn. (5) yields
266
(109 (lob)
F(0) = 0 = F(h)
g(O) = 0 =&f(A) Iq+$ = 0 = Iq-$-)
6. Hydrodynamic
(11)
pressure distribution in the laminar-turbulent
regime
Determination of the hydrodynamic pressure distribution requires the solution of eqns. (6), (8) and (9) under the given boundary conditions. According to ref. 4 Iye, 2) = :
F(e)
m=l
K&)
I
l-
Kn (‘&I
I
Examination of eqn. (12) reveals that only F(8) and K,(z) need be known to determine P(0, z) and thus eqn. (8) need not be solved for g(B). 6.1. ~e~e~i~~~on of F(B) Simplifying and integrating eqn. (6) with respect to 8 gives E
=Ia3 c0s(e + Q) + a4 sin(6 + cu) Geh*3
dt9
Geh*3
+A1
1
Gohe
(13)
where JfI is an integration constant and l/G@ = w
i2+n,{l+ec0s(e+a)}“.8
12 + aI{1 + 0.86 c0s(e + CY)}
Integrating eqn. (13) again with respect to 0 gives
F(e) = ~(12
+ aI)@
+ 0.8faIa31$a + u4(12 + aI)@ + 0.8eaIa,lQ1 +
+ (12 +u1)J&r$O + o&a&I; -&a (14) where J,s is another integration constant and integrals I$“, IF, Ii”, Ii1 and IF are defined in the Appendix. Equation (14) can be simplified to F(e) = (--(6 + O.la,)J,, + a&’
+ asI .,7e+n)(3.6$)
+ (0.8aIJC, + a7)u5
+ 0.8a,a41&o -Je2
+a,,I$e\+
E sin(fI
h*
+ CY)
+ 2ea,Ig0 + (15)
Application of the boundary conditions given by eqn. (10) to eqn. (15) yields
J cl =-
%f7
0.8aJs
+ f8 +a7f6
- (6 + 0.1aI)f7
(16)
266
Jc2 = {as- (6 + o.~~,)J,,l{~,f5(3~5 + fcias
+ (0.8a,Jc,
Thus F(0) as defined (17).
+ %)as(fs
+ f4) + QII=G>+ -f5)
+ %lfi
by eqn. (15) can be determined
+
0.h aqf4
from eqns. (16) and
6.2. Determination of K(z) K(z) is determined by solving eqn. (9) with the boundary given by eqn. (11): K(z) = s
M, cash nbz*/R = 2
n=l
(17)
E
conditions
K,(z*)
(18)
m=l
Substituting eqns. (16) and (17) in eqn. (15) and in turn substituting (15) and (18) in eqn. (12) yieldsP(B, z*) in the closed form.
eqns.
7. Bearing load capacity The bearing load capacity is calculated from the hydrodynamic pressure distribution. For convenience the pressure is resolved into components parallel and perpendicular to the line of centres and summed over the circumference of the journal. Thus from Fig. 1 b/2
W,, = -
A
j-
j- p(e,z)
-b/2
0
b/2
A
wl=-%/2 [ p(e,2) In non-dimensional
WI1
=_
+05
=-.f
are
+0.5 h P(e,2*) cos (e + CY)de d2* r I
-0.5
Wl
Substitution
sin (e + a)R de dz
form these equations
pR=u bR/c2
pR2ubR/C2
cos (0 + a)R de dz
0
A
s
P(e,2*) sin (e + CY) de dz*
--0.5
(1%
0
of eqns. (12), (15) and (18) gives
(20)
+%O%k-
-
a8wl
(0.8a,Jcl
- 2casII$O sin((S+ c) -li”>
+ aT)aB(wl - EI:‘) -
- 0.8ala4{Ih0
sin(0 + a) --I?}
+
WJ. pR2u bR]C2 =
-
2ea8iI--Ii0
cos(6 + a) + Igl} - 0.8ala4{-I&o k
-
Io?,1I--
r&2 cos(6 + 01)
The d~ensionl~ss by
s=
m
c2 pR2bRw
tanh(m b/i%) ~bt~
t
t
cos(8 + a) + Ii”} -
C22f
static load capacity or Sommerfeld number is then given
(WE+ Wf)f’2[;=o=fj = -
c2 w ;+$
pR3 bo
which acts at an angle P = tanl(W,lw,,)
(24)
where WI, and W, are given by eqns. (21) and (22). 8. Sample results and discussion Tables 1 and 2 give results for non-eccentrically loaded half and quarter journal bearings of the type commonly used in practice. The values of ar are chosen to be greater than zero so that no cavitation or negative pressures are encountered during compu~tion at the leading edge which would cause the load-supporting film to rupture and thus modify the boundary conditions and render the mathematical andlysis inapplicable. It is also important that the load angle 180 - cr -0 is confined to the bearing are. To ensure that this condition is satisfied the ratio (180 - a -@)/A which locates the line of action of the external load must be less than unity and positive. For convenience in computation it is advisable to start the calculation with a positive value of Q:such as to ensure that the line of centres passes through the chosen bearing arc. Similarly the maximum vaiues of (Ymust be limited
TABLE1 180'journal bearings withoffset toads (b/D = 1)
Re,
E
Q
(“1 2000
0.4
0.6
0.8
3000
0.4
0.6
0.8
35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80
180-CY---p
P
s
88.73 88.428 88.71 89.414 87.69 86.69 87.23 88.746 87.24 84.386 85.247 87.865 88.86 88.60 88.85 89,478 87.92 87.05 87.537 88.88 87.458 85.008 85.7688 88.097
24.82 25.353 25.828 26.122 15.11 16.00 16.753 17.204 9.666 11.04 12.112 12.72 34.578 35.233 35.817 36.179 21.278 22.37 23.3 23.856 13.884 15.58 16.91 17.662
h 0.3126 0.231 0.1461 0.0588 0.3184 0.2406 0.1543 0.0625 0.321 0.2534 0.165 0.0674 0.3114 0.230 0.1453 0.0585 0.3171 0.2386 0.1526 0.0618 0.3197 0.25 0.1624 0.0661
to where the ratio of the load angle to the bearing arc angle becomes negative or greater than unity. With centrally loaded partial arc bearings the ratio (180 - a - P)/X is 0.5. The value of CY that satisfies this condition is obtained by trial and error. When Q is known it locates the line of centres with respect to the bearing arc boundaries. During the calculation m = 1 can be adopted in eqns. (21) and (22) as the series is found to be convergent. Although the data are given for b/D = 1, it can be readily validated for any other desired value of the aspect ratio. The following inferences can be drawn from the examination of data contained in Tables 1 and 2. (1) For given values of Re, and e there is more than one value of load capacity and attitude angle because S depends on 0 which is itself a multivalued function of E, (2) As IXincreases for a given Re, and E, the value of S increases while the ratio (180 - (Y- p)/X decreases. However, the trend of p is inconclusive. (3) For any particular value of 01and E the value of S increases with increasing Re, while fi shows a downward trend.
269
TABLE2 Eccentrically
loaded 90" journal bearings( b/D=
Re,
f
a
(") 2000
0.4
0.6
0.8
3000
0.4
0.6
0.8
5000
0.4
0.6
0.8
1)
180-CY---p
S
P
x 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60
0.783 0.451 0.1221 0.7626 0.424 0.0886 0.7421 0.3932 0.044 0.784 0.4521 0.1233 0.764 0.4258 0.0906 0.744 0.3957 0.0469 0.7851 0.4534 0.1246 0.7657 0.4278 0.0928 0.7462 0.3985 0.0503
79.532
94.41 109.015 81.37 96.842 112.028 83.21 99.608 116.04 79.44 94.3 108.9 81.23 96.68 111.85 83.0356 99.3846 115.776 79.338 94.19 108.78 81.085 96.5 111.6477 82.84 99.133 115.47
0.3356 0.4232 0.504 0.481 0.6765 0.9178 0.6245 0.99667 1.612 0.41385 0.52 0.616 0.5944 0.83 1.117 0.772 1.22 1.954 0.556 0.695 0.819 0.8 1.11 1.48 1.042 1.63 2.574
(4) The values of S and 0 decrease with increasing E for given values of Re, andcr. Plain journal bearings have been analysed considering (1) the axial length or side leakage, (2) the partial bearing arc, (3) the arbitrary load angle, (4) hydrodynamic lubrication with Newtonian lubricant and (5) superlaminar-turbulent operation to provide information regarding their performance characteristics. Numerical techniques have been applied with turbulent lubrication theory in previous determinations of these characteristics for design purposes. This approach entailed extended computer running time with the result that only a limited amount of data could be computed. To overcome this problem we have used the analytical approach of splitting the dynamic turbulent lubrication equation by the use of the familiar separation of variables technique and solving directly for the constituent hydrodynamic pressures under the chosen boundary conditions in the closed form. To reduce the number of variables and aid in the efficiency of
270
calculation non-dimensional parameters have been used throughout, The load capacity and attitude angle were also determined in the analytical form without any approximation. Thus all the results of interest are available in the closed form, which is simple and exact and hence readily permits accurate determination of numerical data over a wide range of parameters. Extensive design data based on this work will be presented in subsequent publications. The analytical results presented can also be used to study the elastohy~odyn~ic lubrication of journal bearings in the turbulent regime under both steady state and dynamic conditions. They can also be used for the determination of stiffness and damping coefficients and this application will be discussed in a future paper.
Nomenclature a b c
F(e) &T(e) GX
constants defined in the Appendix axial width of the journal bearing radial clearance function of 6 defining the hydrodynamic pressure in a journal bearing without end flow another function of 0 known as the coupling hydrodynamic pressure coefficient defined in ref. 1 = 12 + 0.238(0.33)2Ret8,
coefficient defined in ref. 1 = 12 + 0.17(0.33)2Re~7, C{l + E cos(8 + a)), hydrodynamic film thickness at any point integrals defined in the Appendix IT function of z defining the hydrodynamic pressure in the z direction K(z) n2 positive integers k&m hydrodynamic pressure P(V) inner radius of the bearing shell R Reynotds number based on clearance Re, Reynolds number based on film thickness Reh Sommerfeld number or dimensionless static load-carrying capacity S t time tangential velocity at any point on the journal surface u radial velocity of the journal centres V load-carrying capacity W rectilinear coordinates X,Y,Z attitude angle of the partial bearing attitude angle of the load ; eccentricity ratio angle measured in the direction of journal rotation ; bearing arc span x absolute viscosity of the Newtonian lubricant P kinematic viscosity of the Newtonian lubricant Y density of the lubricant P angle between the centre of the bearing arc and the line of the centres Q, angular speed of the journal w A point above any quantity denotes its time derivative.
GZ h
271
References 1 C. W. Ng and C. H. ‘I’. Pan, A linearized turbulent lubrication tbeory, J. Basic Eng., 87 (1965) 675 - 688. 2 Kr. Vinay, Theoretical pressure distribution in self-acting hydrodynamic journal bearings without end flow considering turbulence, J. Inst. Eng. (India), 49 (Jan. 1969) Kr. Vinay, Analytical load carrying capacity of finite length hydrodynamic journal bearings in turbulent regime, J. Inst. Eng. ~Z~dia), 50 (1970) P. C. Warner, Static and dynamic properties of partial journal bearings, J. Basic Eng., (June 1963) 247 - 256. Kr. Vinay, Theoretical pressure in moderately loaded self-acting journal bearing with incompressible turbulent film considering end leakage, J. Inst. Eng. (Zndia), 50 (March 1970). 6 F. K. Orcutt, Investigation of a partial arc pad bearing in superlaminar speed flow regime, J. Basic Eng., 87 (1965) 145 - 152. 7 M. I. Smith and D. D. Fuller, Journal bearing operation at superlaminar speeds, Trans. AS’ME, 78 (1956) 469. 8 E. B. Arvas, B. Starnlicht and R. J. Wernick, Analysis of plain cylindrical journal bearings in turbulent regime, J. Basis Eng., 86 (June 1964) 387 - 395. 9 A. Szeri and D. Powers, Full journal bearings in turbulent and laminar regimes, J. Meek Eng. Sci., 9 (3) (1967) 167 - 176. 10 F. K. Orcutt and E. B. Arwas, The steady-state and dynamic characteristics of a full circular bearing and a partial arc bearing in the laminar and turbulent flow regimes, J. Lubr. Technol., 89 (April 1967). 11 H. F. Black, Empirical treatment of hydrodynamic journal bearing performance in the superlaminar regime, J. Me&. Eng. Sci., 12 (2) (1970). 12 J. F. Booker, A table of the journal bearing integral, J. Baeic Eng., (June 1965) 533 535.
Appendix tzl= 0.238(0.332Re,.)0*8 = 0.040383 Re$8 a2 = 0.17(0.332Re~)o*7 = 0,036Re~7 as= E(O.5 -&w) a4 = k/w a5 =
l/(1 -9)
as = as(6 + O.la,)/e a7 = as( 12 - 0.6aI)/e as = 0.8aI a8/& as = ~(6 + O.la,)/e aI0 fi
= ad1 = l/(1
-
3~)
+ E cos
f2=fiEsin~
a)
1 f4 =
1 +
cos(h + a)
E
fs = f4E sin(h + a) 2
f6 = (1 _
E2p2
tar-’
f, = a,{f5(3a5
+ f4)
f3
= %(f6
-f3)
f9
=a5(f6
-f5
wl
= Iy
-
f2t3%
+ 0m8a1a4 E -f3
(f4
+ fl))
+ %O(f6
-fl)
+ a9(fz
-f3)
-f9,
+f2)
sin(8 + a) -I:”
w2 = I’i” -ry
cos(0 + a)
From refs. 1 and 2 ,,,, _ Ik
sin’(0 + a) coP(O
- J
18” =
I?
1 + E cos(fY + cY)}k +p
-2IP E2
Ip = -
I$? +Iy E (1 -Es)@
gL__ I;’
{
Ii” + I;” E 1
I;” =
26{1 + E c03(e
+ a)}2
1
E sin(8 + a)
2(1 - E2) [ - (1 + E c03(e
Ip= = -
+ 01))~
+ 3I!$’ -IF
1
~(1 + E c03(e
13
+ 21p -I? E2
=-
Ip =
+ a) d8
e 3h(e
+ ~1) + c~)/(i
+ E c06(e
1 -E2
+ a))
+ IF
1
273
I’p = IF
2 (1
_
e2)1,2
g2 qf+y
tan-lI(
(1 -Es)@
=-
+ 21p - 8 e2
I;” + 1’1”
Ii’=-
E
p
(1
=-
-
E2)1y
+ e -e
sin(e
E2 ry
I;‘=-
-
cos(fJ
+ a)
E
p
=--
1 log{1 E
I?1 =-
I?++8 E
+ f c0s(e
+ (w)}
+ a)
261
THE LOAD-CARRYING CAPACITY OF PLAIN FINITE PARTIAL JOURNAL BEARINGS IN THE TURBULENT REGIME
VINAY KUMAR
~~ar~~ent of ~ec~a~icu~ ~ngi~ee~~g, (Haryana), Pin 1321 IS (India) (Received
June 26,1978;in
Region& engineering College, Kur~~she~a
final form February 28,1979)
Serious errors occur in hydrodynamic bearings when the lubricant film becomes turbulent if they are analysed/designed according to laminar theory. Thus there is a need for an accurate description of the load capacity of plain journal bearings in the turbulent regime. It has been shown that the governing ~bulent-lying lubrication equation for plain partial journal bearings of finite length under dynamic operation is a second-order linear non-homogeneous partial differential equation with analytically known variable coefficients which can be split by the classical method of separation of variables. One of the resulting ordinary equations is integrated directly while the other is solved by a direct method. The solution is subject to zeropressure boundary conditions. All the results of interest such as pressure distribution, load-carrying capacity and attitude angle are dimensionless and are obtained in the closed form which is simple and accurate and permits straightforward calculation of numerical data over a wide range of parameters.
1. Introduction High speed rotating machines with hydrodynamic journal bearings may require process fluid lub~~ation. Under adverse conditions it is necessary to overcome shaft sealing problems. Since many process fluids (such as water and liquid metals) have low kinematic viscosities compared with conventional oils, turbulent flow may occur in the bearing films. Therefore it is necessary to extend Reynolds lubrication theory to the turbulent flow regime [ 1 ] ,360” journal bearings have already been analysed [ 2,3], but since bearings are made up of partial arcs with an arbitrary load angle to provide lubrication and to ensure stability, their behaviour under turbulent film conditions is of considerable practical as well as academic interest. In practice partial journal bearings with aspect ratios between 0.25 and 40 are used. Thus the simplifying mathematical assumptions of Sommerfeld or
262
Ocvirk cannot be applied as the actual axial length of the bearing must be considered. Plain partial arc journal bearings of finite length with hydrodynamic turbulent lubrication conditions have been analysed to determine their loadcarrying capacity and attitude angle. Although some data on the static properties [l] existing during turbulent operation of three-dimensional partial journal bearings which have been obtained by numerical computer techniques and verified experimentally are available, they are insufficient for the preparation of complete design charts or for design purposes. Hence an analytical approach yielding closed-form results was adopted for the following reasons. (1) Since turbulence is associated with laminar flow, the analytical results of interest in the laminar regime can be obtained as a special case by applying suitable restrictive conditions. This approach is not possible when numerical methods are used. (2) An insight into the interrelation of the various parameters is obtained which is not possible using numerical methods. (3) Numerical data required for the wide range of parameters involved in the preparation of design charts or tables can be more conveniently and accurately calculated than by a numerical approach. (4) Bearing design is quicker and simpler when the appropriate results are available in closed form because of the iterative procedure generally required. To determine load capacity in a closed form the governing turbulent equation for hydrodynamic pressure [ 1, 21 is made non-dimensional and split into two ordinary linear differential equations with analytically known variable coefficients by using the classical method of separation of variables. The first equation yields the particular solution for the case of a bearing of infinite length while the second is a homogeneous differential equation consisting of a hyperbolic differential equation and a Hill equation. The static characteristics are evaluated from the analytical form of the threedimensional hydrodynamic pressure distribution obtained in the conventional manner which yields dimensionless closed-form results.
2. Assumptions (1) The bearing is aligned so that the circumferential coordinate only. (2) There is no relative axial motion (3) The journal and bearing surfaces (4) Turbulence is isotropic and fully (5) The lubricant is an incompressible (6) The film does not rupture at any
film thickness
is a function
of the
between the surfaces. are smooth, rigid and impermeable. developed. Newtonian liquid. point over the bearing arc.
263
3. The governing hydrodynamic turbulent regime
lubrication
equation in the laminar-
For the journal bearing configuration shown in Fig. 1 the governing pressure equation including turbulence under a whirling and squeezing motion of the journal is given by [ 1,2] _?_(~!~)+!_~~!!$$!!!_I7
(1)
where, as shown by Constantinescu in the discussion of ref. 1, the analytical values of l/G, and l/G, can be closely approximated by 12 + 0.238 X (0.332)Reh0.a and 12 + 0.17 X (0.33)2Re$7 respectively. From Fig. 1 (2) U = Rw -Cc+ cos(0 + Q) + Ci sin(b + (Y) V = -CE$ sin(0 + (u) - C!; cos(0 + (IL)
(3)
Substituting eqns. (2) and (3) in eqn. (1) and non-dimensionalizing the resulting equation by means of the parameters x = R8, h* = h/C, P = pC2/pR2u and z* = z/b gives
(4)
LINE
OF
Fig. 1. Bearing configuration,
264
The terms containing C/2R were neglected since they are extremely small in comparison with the other terms.
4. Splitting of the governing pressure equation Since the right-hand side of eqn, (4) is a function of 0 only, it can be solved by the method of separation of variables and can be rewritten as P(6,z”) = F(B) + g(0)K(z*)
(5)
which for convenience can be expressed as
P=F+gK Substi~ting eqn. (5) in eqn. (4) and updating the variables gives an ordinary non-homogeneous second-order differential equation
(6) and an ordinary homogeneous
second-order differential equation (7)
Dividing eqn. (7) by gK and noting that the sum of the first and second terms, which are functions of f3 and z* respectively, is zero leads to the conclusion that each term must be equal to a constant +n2. This results in further splitting into two ordinary differential equations:
and
(9) Thus the non-homogeneous partial differential equation given by eqn. (4) is equivalent to three ordinary differential equations represented by eqns. (6), (8) and (9). 5. Boundary conditions The Sommerfeld boundary conditions and the choice of the coordinates shown in Fig. 1 yieldsP(0, z*) = 0 = P(A, z) and P(e, f i) = 0. Under these conditions eqn. (5) yields
266
(109 (lob)
F(0) = 0 = F(h)
g(O) = 0 =&f(A) Iq+$ = 0 = Iq-$-)
6. Hydrodynamic
(11)
pressure distribution in the laminar-turbulent
regime
Determination of the hydrodynamic pressure distribution requires the solution of eqns. (6), (8) and (9) under the given boundary conditions. According to ref. 4 Iye, 2) = :
F(e)
m=l
K&)
I
l-
Kn (‘&I
I
Examination of eqn. (12) reveals that only F(8) and K,(z) need be known to determine P(0, z) and thus eqn. (8) need not be solved for g(B). 6.1. ~e~e~i~~~on of F(B) Simplifying and integrating eqn. (6) with respect to 8 gives E
=Ia3 c0s(e + Q) + a4 sin(6 + cu) Geh*3
dt9
Geh*3
+A1
1
Gohe
(13)
where JfI is an integration constant and l/G@ = w
i2+n,{l+ec0s(e+a)}“.8
12 + aI{1 + 0.86 c0s(e + CY)}
Integrating eqn. (13) again with respect to 0 gives
F(e) = ~(12
+ aI)@
+ 0.8faIa31$a + u4(12 + aI)@ + 0.8eaIa,lQ1 +
+ (12 +u1)J&r$O + o&a&I; -&a (14) where J,s is another integration constant and integrals I$“, IF, Ii”, Ii1 and IF are defined in the Appendix. Equation (14) can be simplified to F(e) = (--(6 + O.la,)J,, + a&’
+ asI .,7e+n)(3.6$)
+ (0.8aIJC, + a7)u5
+ 0.8a,a41&o -Je2
+a,,I$e\+
E sin(fI
h*
+ CY)
+ 2ea,Ig0 + (15)
Application of the boundary conditions given by eqn. (10) to eqn. (15) yields
J cl =-
%f7
0.8aJs
+ f8 +a7f6
- (6 + 0.1aI)f7
(16)
266
Jc2 = {as- (6 + o.~~,)J,,l{~,f5(3~5 + fcias
+ (0.8a,Jc,
Thus F(0) as defined (17).
+ %)as(fs
+ f4) + QII=G>+ -f5)
+ %lfi
by eqn. (15) can be determined
+
0.h aqf4
from eqns. (16) and
6.2. Determination of K(z) K(z) is determined by solving eqn. (9) with the boundary given by eqn. (11): K(z) = s
M, cash nbz*/R = 2
n=l
(17)
E
conditions
K,(z*)
(18)
m=l
Substituting eqns. (16) and (17) in eqn. (15) and in turn substituting (15) and (18) in eqn. (12) yieldsP(B, z*) in the closed form.
eqns.
7. Bearing load capacity The bearing load capacity is calculated from the hydrodynamic pressure distribution. For convenience the pressure is resolved into components parallel and perpendicular to the line of centres and summed over the circumference of the journal. Thus from Fig. 1 b/2
W,, = -
A
j-
j- p(e,z)
-b/2
0
b/2
A
wl=-%/2 [ p(e,2) In non-dimensional
WI1
=_
+05
=-.f
are
+0.5 h P(e,2*) cos (e + CY)de d2* r I
-0.5
Wl
Substitution
sin (e + a)R de dz
form these equations
pR=u bR/c2
pR2ubR/C2
cos (0 + a)R de dz
0
A
s
P(e,2*) sin (e + CY) de dz*
--0.5
(1%
0
of eqns. (12), (15) and (18) gives
(20)
+%O%k-
-
a8wl
(0.8a,Jcl
- 2casII$O sin((S+ c) -li”>
+ aT)aB(wl - EI:‘) -
- 0.8ala4{Ih0
sin(0 + a) --I?}
+
WJ. pR2u bR]C2 =
-
2ea8iI--Ii0
cos(6 + a) + Igl} - 0.8ala4{-I&o k
-
Io?,1I--
r&2 cos(6 + 01)
The d~ensionl~ss by
s=
m
c2 pR2bRw
tanh(m b/i%) ~bt~
t
t
cos(8 + a) + Ii”} -
C22f
static load capacity or Sommerfeld number is then given
(WE+ Wf)f’2[;=o=fj = -
c2 w ;+$
pR3 bo
which acts at an angle P = tanl(W,lw,,)
(24)
where WI, and W, are given by eqns. (21) and (22). 8. Sample results and discussion Tables 1 and 2 give results for non-eccentrically loaded half and quarter journal bearings of the type commonly used in practice. The values of ar are chosen to be greater than zero so that no cavitation or negative pressures are encountered during compu~tion at the leading edge which would cause the load-supporting film to rupture and thus modify the boundary conditions and render the mathematical andlysis inapplicable. It is also important that the load angle 180 - cr -0 is confined to the bearing are. To ensure that this condition is satisfied the ratio (180 - a -@)/A which locates the line of action of the external load must be less than unity and positive. For convenience in computation it is advisable to start the calculation with a positive value of Q:such as to ensure that the line of centres passes through the chosen bearing arc. Similarly the maximum vaiues of (Ymust be limited
TABLE1 180'journal bearings withoffset toads (b/D = 1)
Re,
E
Q
(“1 2000
0.4
0.6
0.8
3000
0.4
0.6
0.8
35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80 35 50 65 80
180-CY---p
P
s
88.73 88.428 88.71 89.414 87.69 86.69 87.23 88.746 87.24 84.386 85.247 87.865 88.86 88.60 88.85 89,478 87.92 87.05 87.537 88.88 87.458 85.008 85.7688 88.097
24.82 25.353 25.828 26.122 15.11 16.00 16.753 17.204 9.666 11.04 12.112 12.72 34.578 35.233 35.817 36.179 21.278 22.37 23.3 23.856 13.884 15.58 16.91 17.662
h 0.3126 0.231 0.1461 0.0588 0.3184 0.2406 0.1543 0.0625 0.321 0.2534 0.165 0.0674 0.3114 0.230 0.1453 0.0585 0.3171 0.2386 0.1526 0.0618 0.3197 0.25 0.1624 0.0661
to where the ratio of the load angle to the bearing arc angle becomes negative or greater than unity. With centrally loaded partial arc bearings the ratio (180 - a - P)/X is 0.5. The value of CY that satisfies this condition is obtained by trial and error. When Q is known it locates the line of centres with respect to the bearing arc boundaries. During the calculation m = 1 can be adopted in eqns. (21) and (22) as the series is found to be convergent. Although the data are given for b/D = 1, it can be readily validated for any other desired value of the aspect ratio. The following inferences can be drawn from the examination of data contained in Tables 1 and 2. (1) For given values of Re, and e there is more than one value of load capacity and attitude angle because S depends on 0 which is itself a multivalued function of E, (2) As IXincreases for a given Re, and E, the value of S increases while the ratio (180 - (Y- p)/X decreases. However, the trend of p is inconclusive. (3) For any particular value of 01and E the value of S increases with increasing Re, while fi shows a downward trend.
269
TABLE2 Eccentrically
loaded 90" journal bearings( b/D=
Re,
f
a
(") 2000
0.4
0.6
0.8
3000
0.4
0.6
0.8
5000
0.4
0.6
0.8
1)
180-CY---p
S
P
x 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60 30 45 60
0.783 0.451 0.1221 0.7626 0.424 0.0886 0.7421 0.3932 0.044 0.784 0.4521 0.1233 0.764 0.4258 0.0906 0.744 0.3957 0.0469 0.7851 0.4534 0.1246 0.7657 0.4278 0.0928 0.7462 0.3985 0.0503
79.532
94.41 109.015 81.37 96.842 112.028 83.21 99.608 116.04 79.44 94.3 108.9 81.23 96.68 111.85 83.0356 99.3846 115.776 79.338 94.19 108.78 81.085 96.5 111.6477 82.84 99.133 115.47
0.3356 0.4232 0.504 0.481 0.6765 0.9178 0.6245 0.99667 1.612 0.41385 0.52 0.616 0.5944 0.83 1.117 0.772 1.22 1.954 0.556 0.695 0.819 0.8 1.11 1.48 1.042 1.63 2.574
(4) The values of S and 0 decrease with increasing E for given values of Re, andcr. Plain journal bearings have been analysed considering (1) the axial length or side leakage, (2) the partial bearing arc, (3) the arbitrary load angle, (4) hydrodynamic lubrication with Newtonian lubricant and (5) superlaminar-turbulent operation to provide information regarding their performance characteristics. Numerical techniques have been applied with turbulent lubrication theory in previous determinations of these characteristics for design purposes. This approach entailed extended computer running time with the result that only a limited amount of data could be computed. To overcome this problem we have used the analytical approach of splitting the dynamic turbulent lubrication equation by the use of the familiar separation of variables technique and solving directly for the constituent hydrodynamic pressures under the chosen boundary conditions in the closed form. To reduce the number of variables and aid in the efficiency of
270
calculation non-dimensional parameters have been used throughout, The load capacity and attitude angle were also determined in the analytical form without any approximation. Thus all the results of interest are available in the closed form, which is simple and exact and hence readily permits accurate determination of numerical data over a wide range of parameters. Extensive design data based on this work will be presented in subsequent publications. The analytical results presented can also be used to study the elastohy~odyn~ic lubrication of journal bearings in the turbulent regime under both steady state and dynamic conditions. They can also be used for the determination of stiffness and damping coefficients and this application will be discussed in a future paper.
Nomenclature a b c
F(e) &T(e) GX
constants defined in the Appendix axial width of the journal bearing radial clearance function of 6 defining the hydrodynamic pressure in a journal bearing without end flow another function of 0 known as the coupling hydrodynamic pressure coefficient defined in ref. 1 = 12 + 0.238(0.33)2Ret8,
coefficient defined in ref. 1 = 12 + 0.17(0.33)2Re~7, C{l + E cos(8 + a)), hydrodynamic film thickness at any point integrals defined in the Appendix IT function of z defining the hydrodynamic pressure in the z direction K(z) n2 positive integers k&m hydrodynamic pressure P(V) inner radius of the bearing shell R Reynotds number based on clearance Re, Reynolds number based on film thickness Reh Sommerfeld number or dimensionless static load-carrying capacity S t time tangential velocity at any point on the journal surface u radial velocity of the journal centres V load-carrying capacity W rectilinear coordinates X,Y,Z attitude angle of the partial bearing attitude angle of the load ; eccentricity ratio angle measured in the direction of journal rotation ; bearing arc span x absolute viscosity of the Newtonian lubricant P kinematic viscosity of the Newtonian lubricant Y density of the lubricant P angle between the centre of the bearing arc and the line of the centres Q, angular speed of the journal w A point above any quantity denotes its time derivative.
GZ h
271
References 1 C. W. Ng and C. H. ‘I’. Pan, A linearized turbulent lubrication tbeory, J. Basic Eng., 87 (1965) 675 - 688. 2 Kr. Vinay, Theoretical pressure distribution in self-acting hydrodynamic journal bearings without end flow considering turbulence, J. Inst. Eng. (India), 49 (Jan. 1969) Kr. Vinay, Analytical load carrying capacity of finite length hydrodynamic journal bearings in turbulent regime, J. Inst. Eng. ~Z~dia), 50 (1970) P. C. Warner, Static and dynamic properties of partial journal bearings, J. Basic Eng., (June 1963) 247 - 256. Kr. Vinay, Theoretical pressure in moderately loaded self-acting journal bearing with incompressible turbulent film considering end leakage, J. Inst. Eng. (Zndia), 50 (March 1970). 6 F. K. Orcutt, Investigation of a partial arc pad bearing in superlaminar speed flow regime, J. Basic Eng., 87 (1965) 145 - 152. 7 M. I. Smith and D. D. Fuller, Journal bearing operation at superlaminar speeds, Trans. AS’ME, 78 (1956) 469. 8 E. B. Arvas, B. Starnlicht and R. J. Wernick, Analysis of plain cylindrical journal bearings in turbulent regime, J. Basis Eng., 86 (June 1964) 387 - 395. 9 A. Szeri and D. Powers, Full journal bearings in turbulent and laminar regimes, J. Meek Eng. Sci., 9 (3) (1967) 167 - 176. 10 F. K. Orcutt and E. B. Arwas, The steady-state and dynamic characteristics of a full circular bearing and a partial arc bearing in the laminar and turbulent flow regimes, J. Lubr. Technol., 89 (April 1967). 11 H. F. Black, Empirical treatment of hydrodynamic journal bearing performance in the superlaminar regime, J. Me&. Eng. Sci., 12 (2) (1970). 12 J. F. Booker, A table of the journal bearing integral, J. Baeic Eng., (June 1965) 533 535.
Appendix tzl= 0.238(0.332Re,.)0*8 = 0.040383 Re$8 a2 = 0.17(0.332Re~)o*7 = 0,036Re~7 as= E(O.5 -&w) a4 = k/w a5 =
l/(1 -9)
as = as(6 + O.la,)/e a7 = as( 12 - 0.6aI)/e as = 0.8aI a8/& as = ~(6 + O.la,)/e aI0 fi
= ad1 = l/(1
-
3~)
+ E cos
f2=fiEsin~
a)
1 f4 =
1 +
cos(h + a)
E
fs = f4E sin(h + a) 2
f6 = (1 _
E2p2
tar-’
f, = a,{f5(3a5
+ f4)
f3
= %(f6
-f3)
f9
=a5(f6
-f5
wl
= Iy
-
f2t3%
+ 0m8a1a4 E -f3
(f4
+ fl))
+ %O(f6
-fl)
+ a9(fz
-f3)
-f9,
+f2)
sin(8 + a) -I:”
w2 = I’i” -ry
cos(0 + a)
From refs. 1 and 2 ,,,, _ Ik
sin’(0 + a) coP(O
- J
18” =
I?
1 + E cos(fY + cY)}k +p
-2IP E2
Ip = -
I$? +Iy E (1 -Es)@
gL__ I;’
{
Ii” + I;” E 1
I;” =
26{1 + E c03(e
+ a)}2
1
E sin(8 + a)
2(1 - E2) [ - (1 + E c03(e
Ip= = -
+ 01))~
+ 3I!$’ -IF
1
~(1 + E c03(e
13
+ 21p -I? E2
=-
Ip =
+ a) d8
e 3h(e
+ ~1) + c~)/(i
+ E c06(e
1 -E2
+ a))
+ IF
1
273
I’p = IF
2 (1
_
e2)1,2
g2 qf+y
tan-lI(
(1 -Es)@
=-
+ 21p - 8 e2
I;” + 1’1”
Ii’=-
E
p
(1
=-
-
E2)1y
+ e -e
sin(e
E2 ry
I;‘=-
-
cos(fJ
+ a)
E
p
=--
1 log{1 E
I?1 =-
I?++8 E
+ f c0s(e
+ (w)}
+ a)