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Stability of submerged oil journal bearings, considering thermal effects
Wear,
157 (1992)
19
19-29
Stability of submerged thermal effects
oil journal bearings, considering
R. Pai and B. C. Majumdar Deparhnent (Received
of Mechanical
Engineering,
Indian
July 24, 1991; revised and accepted
Institute January
of Technology,
Kharagpur
(India)
14, 1992)
Abstract
The stability of submerged oil journal bearings was investigated considering thermal effects. The generalized Reynolds equation was integrated using the cavitation model proposed by Kicinski along the lines of Jakobsson-Floberg-Olsson theory. Heat transfer between the film and both the bush and the shaft is takdn into account. The viscosity is assumed to vary with temperature according to an exponential law. The results are compared with those obtained when temperature effects were neglected.
1. Introduction
Most studies on bearing performance routinely assume the oil film to be isothermal. This greatly simplifies the problem at the cost of accuracy. For a more accurate prediction of bearing performance, it is important to carry out a thermodynamic (THD) analysis. Experimental studies [l, 21 indicate that thermal effects should be considered in lubrication because of the strong dependence of viscosity on temperature. Although thermal effects were recognized by early researchers, Cope [3] was probably the first to derive the governing equations systematically for viscous flow in bearings. Assuming that oil carried away the entire frictional heat, he derived a simplified energy equation in which the variation of viscosity across the film thickness was neglected. Dowson [4] modified the classical Reynolds equation in order to take into account variations in lubricant viscosity and density both along and across the film. He obtained a new equation called the generalized Reynolds equation, which when associated with the energy equation describes properly the thermal phenomenon in hydrodynamic lubrication. Dowson and March [5] numerically simulated an infinitely long journal bearing. Comparing results with their experiments, they found a similar trend for the oil temperature distribution. McCallion ef al. [6] reported a solution technique which has received much attention owing to its simplicity. Majumdar [7] solved the energy and Reynolds equations numerically. He showed that the extent and load capacity of the film decrease if one includes the variation in density and viscosity in the lubricant as a function of temperature. Ezzat and Rohde [S] examined the thermal transient problem in finite slider bearings. Using an explicit scheme, they solved the energy equation by marching forward in time. Boncompain and coworkers [g-11] performed a rigorous THD analysis.
0043-1648/92/$5.00
0 1992 - Elsevier
Sequoia.
All rights reserved
20
Ott and Paradissiadis [12] developed a method of integrating the energy equation to take care of cavitation areas and the reverse flow at the film inlet. Submerged bearings were first investigated by Jakobsson and Floberg [13]. They proposed a cavitation model for stationary cavitation and evaluated bearing performance characteristics. Olsson 1141 and Kicinski [15] extended this cavitation model to nonstationary cavitation and to account for film history effects. More recently, we [16] have performed a stability analysis of submerged bearings under dynamic load without including thermal effects. The novelty of the submerged journal bearing is its oil feed arrangement. A normal journal bearing uses an oil feed hole, the location of which is a problem in dynamically loaded situations. In severe cases of dynamic loading, like in internal combustion engines, this problem is overcome by the use of a circumferential groove. In a submerged journal bearing, the oil flows out of the edges of the bearing in the pressure regions and flows into the bearing gap, in the cavitation regions, thus maintaining continuity of flow. Subambient pressures are therefore a necessary feature of successful submerged bearing operation. The aim of this study is to predict the stability of a submerged oil journal bearing under a unidirectional constant load considering thermal effects. Heat transfer is assumed to be three-dimensional in the fluid and the bush and two-dimensional in the shaft. Cavitation and oil film history effects are taken into account. The generalized Reynolds equation is solved and the fluid velocity vector is determined at all points. Energy and heat transfer equations are then solved simultaneously in the fluid and the solids to obtain a new temperature field. The procedure is repeated until the required convergence is satisfied. The force components together with the mass parameter and steady state load are used to solve the equations of motion by a fourth-order Runge-Kutta method.
2. Theory
The generalized Reynolds equation in non-dimensional Newtonian fluid is (see Appendix A and Figs. 1 and 2) afae(ii3F2a~lae)+(RIL)2alaz(PF2ap/a.f)=(1
-d+ld~)ahIae-a/ae(hF,
where
Fig. 1. Schematic
diagram
of a journal
form for an incompressible
bearing.
lFO)+akKb
(1)
21
*I
CCAV
= 3CAV/L
theorcticol
*
Fig. 2. Theoretical model of the cavitation aone (from ref. 15).
The non-dimensional &=1+acos
film thickness
is given by
(2)
8
and + the non-dimensional
viscosity by
+j= exp[ - y(T- TJ]
(3)
where y is the thermoviscosity coefficient and T the film temperature. The non-dimensional velocity com~nents are obtained from ii=Afi2ajV39+B1Fo w=A(RfL)&2i+~fZ
(4) (5)
where I A=
s
A dr\&-
(FJFo)B
0
i
B=
dhllj s 0
The fluid velocity 8 in the y direction, that is across the film, is obtained by integrating the continuity equation using finite differences. The continuity equation is written as
22
In the full film zone, the energy equation Pe(tiaT/M
+dlh(aTlay)
= ,+hz((aaiay)2
+ (RIL)*(a?yaf)
in non-dimensional
form is written
as
+ aTjar)
+ (awlay)2) + 1 /h2(a2?yap)
(7)
where Pe is the Peclet number and (Y is the dissipation number. The operator a/aB* is introduced to change the shape of the film into a rectangular field a/as* =a/ae-y/L(&/de)(a/ay) The energy equation
(8) in the zone where the pressure
Pe(ria~/ae* + g&(aT/a)i) + aT/ar) = ~+jR(ati/a)i)2
gradient
is nil is written
+ i/R(a’T/aF)
as (9)
because convective heat transfer in the axial direction will not be present as the fluid velocity vector in that direction is zero. Therefore the temperature is uniform in the axial direction. Heat transfer equations in the bush and shaft are as follows for the bush a2TlaP + l/f(aT/aq
+ llf2(a’T/ae’)
+ (R/L)2a2T/a3
= P%(aT/ar)
(IO)
and for the shaft a2fiai2 + l/~(a?ya~ It is assumed
3. Boundary
+ (Rn)2a2i-lal
that the temperature
*l/2)=0
o)=o
is independent
of B in the shaft.
at the edges of the bearing
is assumed
to be zero (atmospheric)
for OGeG27r
(2) The pressure aptaye,
(11)
conditions
(1) The pressure p(e,
= PG(aplar)
distribution
is symmetrical
about the bearing
mid-plane
for 09eG27
The Jakobsson-Floberg-Olsson boundary condition is used to determine the cavitation zone boundaries. With viscosity of the lubricant varying over the film thickness, this boundary condition takes the form [12] (~~-ae~~/a~)(i
- p)= ((apbe),,2-
(R/L)2(a~/a~))ln(aela~))ln)Rsl
(12)
where c&l+j(F, IF,) -
The suffixes l/2 indicate that the quantities and reformation boundaries.
determined
concern both the cavitation
23
The boundary conditions for temperature are as follows. surface, the free convection and radiation hypothesis gives (ap/a+_Rz
For the bush
outer
= - Bib(F+Rz - FaIh)
(13)
and (af’/a&_.,,=
-Bi&*rR--?‘~)
at the bush-fluid
interface,
the temperature
is given by the heat flux continuity
condition
1= - [k(ep(aT7ay~_,]!kbkb ci
(aF/a+
(14)
k(8) is the thermal conductivity of the fluid which is constant the active zone and is variable in the inactive zone of the film.
and equal to k. in
k(e) = k, - P(k - b)
(15)
For f = f l/2, that is at the ends of the shaft, a free convection assumed. This gives
hypothesis
(a~/a~~,.,,=Bi,(T;-_.,,-huh)
is
(16)
Assuming the shaft temperatures to be independent fluid-shaft interface is given by the heat flux continuity
of 0, the temperature condition as below
at the
(17)
4. Solution
method
An initial temperature distribution is assumed in the fluid and the viscosity is found at every point using eqn. (3). The generalized Reynolds equation is solved using finite differences and the Gauss-Seidel method with over-relaxation scheme, satisfying the boundary conditions. The fluid velocity vector is calculated at all points of the fluid. The energy equation is solved using an implicit finite difference method and the Ritchmeyer technique [17] is used. The heat conduction equations in the bush and shaft are solved using finite differences and the Gauss-Seidel method with SOR, satisfying the boundary conditions. With the new temperature field thus obtained, the viscosity is evaluated at all points in the fluid and the above procedure is repeated until the relative difference between two successive steps, at each point on the boundary between the film and the bush, is less than 0.1%. The force components are then evaluated from 1R BI
wr=2
~COS
ss 0.92 l/2
bv,=2
ededz-
(18)
81
ss 082
p sin 8 dtI &
These are used in the solution motion are
(19 of the equations
of motion.
The equations
of
24
(21) Equations
(20) and (21) are non-dimensionalized
Mi-Me@-I@JIV~-cos
as
C$=o
Me~$+2h?<~--W~/W~+sin
(22)
c#J=O
(23)
where
and WOE -
w,c2
rpR9L Initially & and 4 are set equal to zero to obtain the steady state load capacity and force components. These are used in the above eqn. (20) which are second-order differential equations in E and 4. They are solved by a fourth-order Runge-Kutta method to obtain E and 4 values for the next time step. The entire procedure is repeated to obtain E and 4 values for successive time steps. These are then used to obtain a trajectory of the journal centre, using computer graphics. 5. Results and discussion
Figure 3 shows the variation in load capacity with eccentricity for both THD and isothermal conditions. Figure 4 gives a comparison of the maximum bush surface and Frene [9]. temperature vs. eccentric& with-that obtained by Boncompain present r--
_
--- -- t----
andys,s
-----I--- --
'........BoyonPaln et,al ref.9
.._. .._ I_._.~.~~ -__-__-___c :T , ...___i_.__.. ___)_.-. .~
g1.00 s
.-.__!__ ..,. ;. ~__.._ ,, ,,. I ... ~:.._~-_.____~.._._._ ~
0.50
i:/
0.00 0.0
0.2
0.4
_J
0.6
0.8
0.0
0.2 0.4 ECCENTRICITY RATIO
ECCENTRICITY RATIO
Fig. 3. Variation
in load capacity with eccentricity
Fig. 4. Maximum bush surface temperature
ratio.
vs. eccentricity
ratio.
0.6
0.8
25
LEfWlNCE IRCLE
LEWWE IRCLE
CLEMNCE CTRCI F
EARAKE RCLE
LERONCE IRCLE
LEARANCE IRCLE
LEFIRANCE IRCLE
(0
LEAMINCE IRCLE
Fig. 5. Trajectory for journal centre with unidirectional constant load. (a) Isothermal solution, ~,,=0.4,~=180”,L/D=1.00,~=3.0 , m= - 0.1. (b) Thermohydrodynamic solution, l,, = 0.4,4 = 180”, L/D=l.OO, h?=3.0, T= -0.1. (c) Isothermal solution, l,,=0.4, 4=180”, L/D=l.OO, A?=5.29, TT= -0.1. (d) Thermohydrodynamic solution, l=0.4, d= 180”, L/D= 1.00, &=5.29, rr= -0.1. (e) Isothermal solution, E,,= 0.4, += 180”,L/D= 1.00,&f= 10.96, W= - 0.1. (f) Thermohydrodynamic solution, e0=0.4, +=lSOO, L/D=l.OO, A?= 10.96, z-= -0.1. (g) Isothermal solution, ~=0.6, 4=270”, L/D=l.OO, h?=3.0, T= -0.1. (h) Thermohydrodynamic solution, l0=O.6, +=270”, Ll D=1.00,1@=3.O,?r= -0.l.(i)Isothe~alsolution,q=0.6,~=2700,L/D=1.00,~=5.29,~=-0.1. (j) Thermohydrodynamic solution, G = 0.6,4 = 270”, L/D = 1.00, it? = 5.29, T= - 0.1. (k) Isothermal solution, l,,=0.6, +=270”, L/D= 1.00, u= 10.96, r= -0.1. (1) Thermohydrodynamic solution, l,,=O.6, +=270”, L/D=l.OO, ni=10.96, vi-= -0.1.
for both isothermal and Journal centre trajectories, Figs. S(a)-S(1) were obtained THD conditions for similar input data. It was observed that the bearing system which was stable at lower mass for isothermal conditions became marginally stable under THD conditions. This indicates that an isothermal analysis is not adequate because it does not give an accurate picture of the actual state of the system. A stability diagram was constructed as shown in Fig. 6. A consideration of thermal effects in the stability analysis will reduce the range of the stable region. Figure 7 gives a comparison of the maximum bush surface temperature VS. load capacity with that obtained by McCallion et al. [6]. The higher temperatures attained
85
150 146
u
130 120
u
.McCal
Lionet al ‘ref.6
_
110 100 90 g 2
80 70
ij 2
60
--;-i’-L I /
‘il
60
2
55
g M
40 50
2
50. _................ I
/
g
30 20
p
45
1 /
, I
/
~.._._...__._...... i..................1..
1
/I
40
10 0
35 , 0.0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 ECCENTRICITY RATIO
0.8
0.9
1.0
2Ga3
/
-
I I /
I 6Gm
I
Km 1400 LORD CAPACITY (N)
1603
2206
Fig. 6. Stability diagram - mass parameter VS. eccentricity ratio. Fig. 7. Maximum bush surface temperature
vs. load capacity.
here may be due to the boundary conditions (Jakobsson-Floberg-Olsson) because recirculating flow and reverse flow were not considered.
used and
6. Conclusions The boundary conditions used account for both cavitation and reformation which are important in enclosed cavities like those of submerged bearings. This, together with a consideration of oil film history effects, is hoped to give a reliable description of the dynamic state of the bearing. It was observed that an isothermal solution gave a stable system while a THD solution showed the bearing system to be marginally stable. The range of the stable region is reduced with the inclusion of thermal effects. This indicates that THD analysis is essential for an accurate description of the state of the system.
Acknowledgment
The authors thank the referees for sparing the time to go through their manuscript and make useful suggestions. Their suggestions have been incorporated in the revised manuscript.
References 1 D. Dowson, J. Hudson, B. Hunter and C. March, An experimental investigation of the thermal equilibrium of steadily loaded journal bearings, Proc. Inst. Me& Eng., I81 (3B) (1966-1967) 70-80. 2 R. S. Gregory, Performance of thrust bearings at high operating speeds, Trans. ASME, J. Lube. Techad, 96 (1974) 7-14. 3 W. F. Cope, The hydrodynamical theory of film lubrication, Proc. R Sot. Lmdon, 197 (1949) 201-217.
28 4 D. Dowson, A generalized Reynolds equation for fluid film lubrication, Znt. .Z Mech. Sci., 4 (1962) 159-170. 5 D. Dowson and C. March, A thermohydrodynamic analysis of journal bearings, Proc. Inst. Mech. Eng., 181 (3B) (1966) 117-126. 6 H. McCallion, F. Yousif and T. Lloyd, The analysis of thermal effects in a full journal bearing, Trans. ASME, .I. Lubr. Technol., 92 (1970) 578-587. 7 B. C. Majumdar, The thermohydrodynamic solution of oil journal bearings, Wear, 31 (1975) 287-294. 8 H. Ezzat and S. Rohde, Thermal transients in finite slider bearings, Trans. ASME, J. Lubr. Technol., 96 (1974) 315-320. 9 R. Boncompain and J. Frene, Thermohydrodynamic analysis of a finite journal bearing static and dynamic characteristics, Proc. 6th Leeds-Lyons Symp. on Tribology, 1980, pp. 33-41. 10 J. Ferron, J. Frene and R. Boncompain, A study of the thermohydrodynamic performance of a plain journal bearing - comparison between theory and experiments, Trans. ASME, J. Lubr. Technol., 105 (1983) 422-428. 11 R. Boncompain, M. Fillon and J. Frene, Analysis of thermal effects in hydrodynamic bearings, Tram ASME, J. Tribal., 108 (1986) 219-224. 12 H. H. Ott and G. Paradissiadis, Thermohydrodynamic analysis of journal bearings considering cavitation and reverse flow, Trans. ASME, .I. Tribal., 110 (1988) 439-447. 13 B. Jakobsson and L. Floberg, The finite journal bearing, considering vaporization, Trans. Chalmers Univ. Technol., Gothenburg 1957. 14 K. 0. Olsson, Cavitation in dynamically loaded bearings, Trans. Chalmers Univ. TechnoZ., Gothenburg 1965. 15 J. Kicinski, Influence of the flow pre-history in the cavitation zone on the dynamic characteristics of slide bearings, Wear, 111 (1987) 289-311. 16 R. Pai and B. C. Majumdar, Stability of submerged oil journal bearings under dynamic loads, Wear, 146 (1991) 125-135. 17 R. D. Ritchmeyer, Difference Methods for Initial value Problems, Interscience, New York, 1957, p. 101, and references cited therein.
Appendix
A: Nomenclature
-4 B Bit,
integration functions Biot number for the bush,
Bi,
Biot number for the shaft, Bi,=h,L/k, radial clearance (m) specific heat of the lubricant (J kg-’ Y-‘) specific heat of solids (J kg-’ T-r) integration functions of viscosity film thickness (dimensionless) convection heat transfer coefficient for bush and shaft (W mm2 ‘C-l) thermal conductivity of the lubricant (W m-r T-‘) thermal conductivity of air (W m-l OC-‘) thermal conductivity of the bush (W m-l “C-l) thermal conductivity of the shaft (W m-l “C-r) bearing width (m) mass per bearing Jkg) mass parameter, M=MCo21Wo film pressure, j =pC2/qowR2, p is the film pressure (dimensionless) vapour pressure Peclet number, Pe = poC~uC2/k~ Peclet number for the bush, Peb=psC,d2/kb
C
CO CS fo, FI, h h b, s
ko k, kb k, L M
A? B PV
Pe Peb
F2
Bib= hbRt/kb
29
Pe, R & P t T
W W K WO w, % r, 8, 2 x3 Y, 2 6 7, f
Greek symbols a P Y E 170 75 6, A 7r PO
PS ; 0
f4
Peclet number for the shaft, Pe,=psCsuR21k, journal radius (m) dimensionless bush outside radius, R2 = RJR dimensionless radius, P= r/R time temperature (“C) ambient temperature (“C) dimensionless temperature, ?‘= T/T, dimensionless bush temperature, Eb = T,/T, dimensionless shaft temperature, T, = TJT, journal peripheral speed, lJ=(lrR (m s-l) components of fluid velocity in the x, y and z directions respectively, zi =u/U, 0 = vRICU, I+= w/U (dimensionless) load capacity (N) dimensionless load cap_acity, m= WC2JqouR3L hydrodynamic forces, W, = WC2f~o&3L, pa = WC?qodZ3L steady state load capacity (N) dimensionless steady state load capacity, q. = WC2/~o&3L coordinates coordinates dimensionless coordinates, 0 =x/R, jj =yfh, t =zfL
dissipation number, (Y= q. U2/koTa oil gap filling coefficient thermoviscosity coefficient eccentricity ratio fluid viscosity at ambient temperature (Pa s) dimensionless viscosity, f = q/q0 angular coordinates at which the film cavitates and reforms respectively integration variable dimensionless vapour pressure, a=p,C2/q00R2 lubricant density (kg m-‘) density of solids (kg mp3) dimensionless time, T- wf attitude angle angular velocity of journal (rad s-l)
157 (1992)
19
19-29
Stability of submerged thermal effects
oil journal bearings, considering
R. Pai and B. C. Majumdar Deparhnent (Received
of Mechanical
Engineering,
Indian
July 24, 1991; revised and accepted
Institute January
of Technology,
Kharagpur
(India)
14, 1992)
Abstract
The stability of submerged oil journal bearings was investigated considering thermal effects. The generalized Reynolds equation was integrated using the cavitation model proposed by Kicinski along the lines of Jakobsson-Floberg-Olsson theory. Heat transfer between the film and both the bush and the shaft is takdn into account. The viscosity is assumed to vary with temperature according to an exponential law. The results are compared with those obtained when temperature effects were neglected.
1. Introduction
Most studies on bearing performance routinely assume the oil film to be isothermal. This greatly simplifies the problem at the cost of accuracy. For a more accurate prediction of bearing performance, it is important to carry out a thermodynamic (THD) analysis. Experimental studies [l, 21 indicate that thermal effects should be considered in lubrication because of the strong dependence of viscosity on temperature. Although thermal effects were recognized by early researchers, Cope [3] was probably the first to derive the governing equations systematically for viscous flow in bearings. Assuming that oil carried away the entire frictional heat, he derived a simplified energy equation in which the variation of viscosity across the film thickness was neglected. Dowson [4] modified the classical Reynolds equation in order to take into account variations in lubricant viscosity and density both along and across the film. He obtained a new equation called the generalized Reynolds equation, which when associated with the energy equation describes properly the thermal phenomenon in hydrodynamic lubrication. Dowson and March [5] numerically simulated an infinitely long journal bearing. Comparing results with their experiments, they found a similar trend for the oil temperature distribution. McCallion ef al. [6] reported a solution technique which has received much attention owing to its simplicity. Majumdar [7] solved the energy and Reynolds equations numerically. He showed that the extent and load capacity of the film decrease if one includes the variation in density and viscosity in the lubricant as a function of temperature. Ezzat and Rohde [S] examined the thermal transient problem in finite slider bearings. Using an explicit scheme, they solved the energy equation by marching forward in time. Boncompain and coworkers [g-11] performed a rigorous THD analysis.
0043-1648/92/$5.00
0 1992 - Elsevier
Sequoia.
All rights reserved
20
Ott and Paradissiadis [12] developed a method of integrating the energy equation to take care of cavitation areas and the reverse flow at the film inlet. Submerged bearings were first investigated by Jakobsson and Floberg [13]. They proposed a cavitation model for stationary cavitation and evaluated bearing performance characteristics. Olsson 1141 and Kicinski [15] extended this cavitation model to nonstationary cavitation and to account for film history effects. More recently, we [16] have performed a stability analysis of submerged bearings under dynamic load without including thermal effects. The novelty of the submerged journal bearing is its oil feed arrangement. A normal journal bearing uses an oil feed hole, the location of which is a problem in dynamically loaded situations. In severe cases of dynamic loading, like in internal combustion engines, this problem is overcome by the use of a circumferential groove. In a submerged journal bearing, the oil flows out of the edges of the bearing in the pressure regions and flows into the bearing gap, in the cavitation regions, thus maintaining continuity of flow. Subambient pressures are therefore a necessary feature of successful submerged bearing operation. The aim of this study is to predict the stability of a submerged oil journal bearing under a unidirectional constant load considering thermal effects. Heat transfer is assumed to be three-dimensional in the fluid and the bush and two-dimensional in the shaft. Cavitation and oil film history effects are taken into account. The generalized Reynolds equation is solved and the fluid velocity vector is determined at all points. Energy and heat transfer equations are then solved simultaneously in the fluid and the solids to obtain a new temperature field. The procedure is repeated until the required convergence is satisfied. The force components together with the mass parameter and steady state load are used to solve the equations of motion by a fourth-order Runge-Kutta method.
2. Theory
The generalized Reynolds equation in non-dimensional Newtonian fluid is (see Appendix A and Figs. 1 and 2) afae(ii3F2a~lae)+(RIL)2alaz(PF2ap/a.f)=(1
-d+ld~)ahIae-a/ae(hF,
where
Fig. 1. Schematic
diagram
of a journal
form for an incompressible
bearing.
lFO)+akKb
(1)
21
*I
CCAV
= 3CAV/L
theorcticol
*
Fig. 2. Theoretical model of the cavitation aone (from ref. 15).
The non-dimensional &=1+acos
film thickness
is given by
(2)
8
and + the non-dimensional
viscosity by
+j= exp[ - y(T- TJ]
(3)
where y is the thermoviscosity coefficient and T the film temperature. The non-dimensional velocity com~nents are obtained from ii=Afi2ajV39+B1Fo w=A(RfL)&2i+~fZ
(4) (5)
where I A=
s
A dr\&-
(FJFo)B
0
i
B=
dhllj s 0
The fluid velocity 8 in the y direction, that is across the film, is obtained by integrating the continuity equation using finite differences. The continuity equation is written as
22
In the full film zone, the energy equation Pe(tiaT/M
+dlh(aTlay)
= ,+hz((aaiay)2
+ (RIL)*(a?yaf)
in non-dimensional
form is written
as
+ aTjar)
+ (awlay)2) + 1 /h2(a2?yap)
(7)
where Pe is the Peclet number and (Y is the dissipation number. The operator a/aB* is introduced to change the shape of the film into a rectangular field a/as* =a/ae-y/L(&/de)(a/ay) The energy equation
(8) in the zone where the pressure
Pe(ria~/ae* + g&(aT/a)i) + aT/ar) = ~+jR(ati/a)i)2
gradient
is nil is written
+ i/R(a’T/aF)
as (9)
because convective heat transfer in the axial direction will not be present as the fluid velocity vector in that direction is zero. Therefore the temperature is uniform in the axial direction. Heat transfer equations in the bush and shaft are as follows for the bush a2TlaP + l/f(aT/aq
+ llf2(a’T/ae’)
+ (R/L)2a2T/a3
= P%(aT/ar)
(IO)
and for the shaft a2fiai2 + l/~(a?ya~ It is assumed
3. Boundary
+ (Rn)2a2i-lal
that the temperature
*l/2)=0
o)=o
is independent
of B in the shaft.
at the edges of the bearing
is assumed
to be zero (atmospheric)
for OGeG27r
(2) The pressure aptaye,
(11)
conditions
(1) The pressure p(e,
= PG(aplar)
distribution
is symmetrical
about the bearing
mid-plane
for 09eG27
The Jakobsson-Floberg-Olsson boundary condition is used to determine the cavitation zone boundaries. With viscosity of the lubricant varying over the film thickness, this boundary condition takes the form [12] (~~-ae~~/a~)(i
- p)= ((apbe),,2-
(R/L)2(a~/a~))ln(aela~))ln)Rsl
(12)
where c&l+j(F, IF,) -
The suffixes l/2 indicate that the quantities and reformation boundaries.
determined
concern both the cavitation
23
The boundary conditions for temperature are as follows. surface, the free convection and radiation hypothesis gives (ap/a+_Rz
For the bush
outer
= - Bib(F+Rz - FaIh)
(13)
and (af’/a&_.,,=
-Bi&*rR--?‘~)
at the bush-fluid
interface,
the temperature
is given by the heat flux continuity
condition
1= - [k(ep(aT7ay~_,]!kbkb ci
(aF/a+
(14)
k(8) is the thermal conductivity of the fluid which is constant the active zone and is variable in the inactive zone of the film.
and equal to k. in
k(e) = k, - P(k - b)
(15)
For f = f l/2, that is at the ends of the shaft, a free convection assumed. This gives
hypothesis
(a~/a~~,.,,=Bi,(T;-_.,,-huh)
is
(16)
Assuming the shaft temperatures to be independent fluid-shaft interface is given by the heat flux continuity
of 0, the temperature condition as below
at the
(17)
4. Solution
method
An initial temperature distribution is assumed in the fluid and the viscosity is found at every point using eqn. (3). The generalized Reynolds equation is solved using finite differences and the Gauss-Seidel method with over-relaxation scheme, satisfying the boundary conditions. The fluid velocity vector is calculated at all points of the fluid. The energy equation is solved using an implicit finite difference method and the Ritchmeyer technique [17] is used. The heat conduction equations in the bush and shaft are solved using finite differences and the Gauss-Seidel method with SOR, satisfying the boundary conditions. With the new temperature field thus obtained, the viscosity is evaluated at all points in the fluid and the above procedure is repeated until the relative difference between two successive steps, at each point on the boundary between the film and the bush, is less than 0.1%. The force components are then evaluated from 1R BI
wr=2
~COS
ss 0.92 l/2
bv,=2
ededz-
(18)
81
ss 082
p sin 8 dtI &
These are used in the solution motion are
(19 of the equations
of motion.
The equations
of
24
(21) Equations
(20) and (21) are non-dimensionalized
Mi-Me@-I@JIV~-cos
as
C$=o
Me~$+2h?<~--W~/W~+sin
(22)
c#J=O
(23)
where
and WOE -
w,c2
rpR9L Initially & and 4 are set equal to zero to obtain the steady state load capacity and force components. These are used in the above eqn. (20) which are second-order differential equations in E and 4. They are solved by a fourth-order Runge-Kutta method to obtain E and 4 values for the next time step. The entire procedure is repeated to obtain E and 4 values for successive time steps. These are then used to obtain a trajectory of the journal centre, using computer graphics. 5. Results and discussion
Figure 3 shows the variation in load capacity with eccentricity for both THD and isothermal conditions. Figure 4 gives a comparison of the maximum bush surface and Frene [9]. temperature vs. eccentric& with-that obtained by Boncompain present r--
_
--- -- t----
andys,s
-----I--- --
'........BoyonPaln et,al ref.9
.._. .._ I_._.~.~~ -__-__-___c :T , ...___i_.__.. ___)_.-. .~
g1.00 s
.-.__!__ ..,. ;. ~__.._ ,, ,,. I ... ~:.._~-_.____~.._._._ ~
0.50
i:/
0.00 0.0
0.2
0.4
_J
0.6
0.8
0.0
0.2 0.4 ECCENTRICITY RATIO
ECCENTRICITY RATIO
Fig. 3. Variation
in load capacity with eccentricity
Fig. 4. Maximum bush surface temperature
ratio.
vs. eccentricity
ratio.
0.6
0.8
25
LEfWlNCE IRCLE
LEWWE IRCLE
CLEMNCE CTRCI F
EARAKE RCLE
LERONCE IRCLE
LEARANCE IRCLE
LEFIRANCE IRCLE
(0
LEAMINCE IRCLE
Fig. 5. Trajectory for journal centre with unidirectional constant load. (a) Isothermal solution, ~,,=0.4,~=180”,L/D=1.00,~=3.0 , m= - 0.1. (b) Thermohydrodynamic solution, l,, = 0.4,4 = 180”, L/D=l.OO, h?=3.0, T= -0.1. (c) Isothermal solution, l,,=0.4, 4=180”, L/D=l.OO, A?=5.29, TT= -0.1. (d) Thermohydrodynamic solution, l=0.4, d= 180”, L/D= 1.00, &=5.29, rr= -0.1. (e) Isothermal solution, E,,= 0.4, += 180”,L/D= 1.00,&f= 10.96, W= - 0.1. (f) Thermohydrodynamic solution, e0=0.4, +=lSOO, L/D=l.OO, A?= 10.96, z-= -0.1. (g) Isothermal solution, ~=0.6, 4=270”, L/D=l.OO, h?=3.0, T= -0.1. (h) Thermohydrodynamic solution, l0=O.6, +=270”, Ll D=1.00,1@=3.O,?r= -0.l.(i)Isothe~alsolution,q=0.6,~=2700,L/D=1.00,~=5.29,~=-0.1. (j) Thermohydrodynamic solution, G = 0.6,4 = 270”, L/D = 1.00, it? = 5.29, T= - 0.1. (k) Isothermal solution, l,,=0.6, +=270”, L/D= 1.00, u= 10.96, r= -0.1. (1) Thermohydrodynamic solution, l,,=O.6, +=270”, L/D=l.OO, ni=10.96, vi-= -0.1.
for both isothermal and Journal centre trajectories, Figs. S(a)-S(1) were obtained THD conditions for similar input data. It was observed that the bearing system which was stable at lower mass for isothermal conditions became marginally stable under THD conditions. This indicates that an isothermal analysis is not adequate because it does not give an accurate picture of the actual state of the system. A stability diagram was constructed as shown in Fig. 6. A consideration of thermal effects in the stability analysis will reduce the range of the stable region. Figure 7 gives a comparison of the maximum bush surface temperature VS. load capacity with that obtained by McCallion et al. [6]. The higher temperatures attained
85
150 146
u
130 120
u
.McCal
Lionet al ‘ref.6
_
110 100 90 g 2
80 70
ij 2
60
--;-i’-L I /
‘il
60
2
55
g M
40 50
2
50. _................ I
/
g
30 20
p
45
1 /
, I
/
~.._._...__._...... i..................1..
1
/I
40
10 0
35 , 0.0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 ECCENTRICITY RATIO
0.8
0.9
1.0
2Ga3
/
-
I I /
I 6Gm
I
Km 1400 LORD CAPACITY (N)
1603
2206
Fig. 6. Stability diagram - mass parameter VS. eccentricity ratio. Fig. 7. Maximum bush surface temperature
vs. load capacity.
here may be due to the boundary conditions (Jakobsson-Floberg-Olsson) because recirculating flow and reverse flow were not considered.
used and
6. Conclusions The boundary conditions used account for both cavitation and reformation which are important in enclosed cavities like those of submerged bearings. This, together with a consideration of oil film history effects, is hoped to give a reliable description of the dynamic state of the bearing. It was observed that an isothermal solution gave a stable system while a THD solution showed the bearing system to be marginally stable. The range of the stable region is reduced with the inclusion of thermal effects. This indicates that THD analysis is essential for an accurate description of the state of the system.
Acknowledgment
The authors thank the referees for sparing the time to go through their manuscript and make useful suggestions. Their suggestions have been incorporated in the revised manuscript.
References 1 D. Dowson, J. Hudson, B. Hunter and C. March, An experimental investigation of the thermal equilibrium of steadily loaded journal bearings, Proc. Inst. Me& Eng., I81 (3B) (1966-1967) 70-80. 2 R. S. Gregory, Performance of thrust bearings at high operating speeds, Trans. ASME, J. Lube. Techad, 96 (1974) 7-14. 3 W. F. Cope, The hydrodynamical theory of film lubrication, Proc. R Sot. Lmdon, 197 (1949) 201-217.
28 4 D. Dowson, A generalized Reynolds equation for fluid film lubrication, Znt. .Z Mech. Sci., 4 (1962) 159-170. 5 D. Dowson and C. March, A thermohydrodynamic analysis of journal bearings, Proc. Inst. Mech. Eng., 181 (3B) (1966) 117-126. 6 H. McCallion, F. Yousif and T. Lloyd, The analysis of thermal effects in a full journal bearing, Trans. ASME, .I. Lubr. Technol., 92 (1970) 578-587. 7 B. C. Majumdar, The thermohydrodynamic solution of oil journal bearings, Wear, 31 (1975) 287-294. 8 H. Ezzat and S. Rohde, Thermal transients in finite slider bearings, Trans. ASME, J. Lubr. Technol., 96 (1974) 315-320. 9 R. Boncompain and J. Frene, Thermohydrodynamic analysis of a finite journal bearing static and dynamic characteristics, Proc. 6th Leeds-Lyons Symp. on Tribology, 1980, pp. 33-41. 10 J. Ferron, J. Frene and R. Boncompain, A study of the thermohydrodynamic performance of a plain journal bearing - comparison between theory and experiments, Trans. ASME, J. Lubr. Technol., 105 (1983) 422-428. 11 R. Boncompain, M. Fillon and J. Frene, Analysis of thermal effects in hydrodynamic bearings, Tram ASME, J. Tribal., 108 (1986) 219-224. 12 H. H. Ott and G. Paradissiadis, Thermohydrodynamic analysis of journal bearings considering cavitation and reverse flow, Trans. ASME, .I. Tribal., 110 (1988) 439-447. 13 B. Jakobsson and L. Floberg, The finite journal bearing, considering vaporization, Trans. Chalmers Univ. Technol., Gothenburg 1957. 14 K. 0. Olsson, Cavitation in dynamically loaded bearings, Trans. Chalmers Univ. TechnoZ., Gothenburg 1965. 15 J. Kicinski, Influence of the flow pre-history in the cavitation zone on the dynamic characteristics of slide bearings, Wear, 111 (1987) 289-311. 16 R. Pai and B. C. Majumdar, Stability of submerged oil journal bearings under dynamic loads, Wear, 146 (1991) 125-135. 17 R. D. Ritchmeyer, Difference Methods for Initial value Problems, Interscience, New York, 1957, p. 101, and references cited therein.
Appendix
A: Nomenclature
-4 B Bit,
integration functions Biot number for the bush,
Bi,
Biot number for the shaft, Bi,=h,L/k, radial clearance (m) specific heat of the lubricant (J kg-’ Y-‘) specific heat of solids (J kg-’ T-r) integration functions of viscosity film thickness (dimensionless) convection heat transfer coefficient for bush and shaft (W mm2 ‘C-l) thermal conductivity of the lubricant (W m-r T-‘) thermal conductivity of air (W m-l OC-‘) thermal conductivity of the bush (W m-l “C-l) thermal conductivity of the shaft (W m-l “C-r) bearing width (m) mass per bearing Jkg) mass parameter, M=MCo21Wo film pressure, j =pC2/qowR2, p is the film pressure (dimensionless) vapour pressure Peclet number, Pe = poC~uC2/k~ Peclet number for the bush, Peb=psC,d2/kb
C
CO CS fo, FI, h h b, s
ko k, kb k, L M
A? B PV
Pe Peb
F2
Bib= hbRt/kb
29
Pe, R & P t T
W W K WO w, % r, 8, 2 x3 Y, 2 6 7, f
Greek symbols a P Y E 170 75 6, A 7r PO
PS ; 0
f4
Peclet number for the shaft, Pe,=psCsuR21k, journal radius (m) dimensionless bush outside radius, R2 = RJR dimensionless radius, P= r/R time temperature (“C) ambient temperature (“C) dimensionless temperature, ?‘= T/T, dimensionless bush temperature, Eb = T,/T, dimensionless shaft temperature, T, = TJT, journal peripheral speed, lJ=(lrR (m s-l) components of fluid velocity in the x, y and z directions respectively, zi =u/U, 0 = vRICU, I+= w/U (dimensionless) load capacity (N) dimensionless load cap_acity, m= WC2JqouR3L hydrodynamic forces, W, = WC2f~o&3L, pa = WC?qodZ3L steady state load capacity (N) dimensionless steady state load capacity, q. = WC2/~o&3L coordinates coordinates dimensionless coordinates, 0 =x/R, jj =yfh, t =zfL
dissipation number, (Y= q. U2/koTa oil gap filling coefficient thermoviscosity coefficient eccentricity ratio fluid viscosity at ambient temperature (Pa s) dimensionless viscosity, f = q/q0 angular coordinates at which the film cavitates and reforms respectively integration variable dimensionless vapour pressure, a=p,C2/q00R2 lubricant density (kg m-‘) density of solids (kg mp3) dimensionless time, T- wf attitude angle angular velocity of journal (rad s-l)