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Elasto-hydrodynamic lubrication analysis of journal bearings with combined use of boundary elements and finite elements
Analym with Boundary ~
13 (1994) 273-281
© 1994l~ev~ ScimceIJnfited Printedin GreatBritain.Allrights E~EVIER
Elasto-hydrodynamic lubrication analysis of journal bearings with combined use of boundary elements and finite elements K. Kohno Hiroshinm Research and Development Center, Mitsubishi Heavy Industries Co. Lid, Itozaki-cho, Mihara-shi 729-03, Japan
S. Takahashi & K. Said Nagasaki Research and Development Center, Mitsubishi Heavy In~stries Co. Ltd., Akunoura-cho, Nagasaki-shi 850-91, Japan
(Received 20 January 1994; acx:epted6 April 1994) In this paper, the boundary-element method is applied to analyze the elastohydrodynamic lubrication charactefistica of journal bearings with combined use of the boundary-element method and finite-element method. The boundaryelement method is used to calculate the elastic deformation of the bearing housing and the journal, and the finite-element method is applied to the solution of the Reynolds equation. The results of the iterative-solution method proposed are presented to illustrate its applicability to the actual journal bearings. Key words: Boundary-element method, finite-element method, journal bearing, elasto-hydrodynamic lubrication, elastic deformation, Reynolds equation.
INTRODUCTION
treatments have not been found as far as the authors are aware. From this viewpoint, the two-dimensional numerical technique, which accommodates the above two factors, was developed with the combined usage of the boundary-element method and the finite-difference method to treat the dynamic-pressure-type journal beating. 7 The boundary-element method8,9 is more suitable for analyzing the elastic deformation of the bearing than the domain-type method because the elastic problem involved requires only linear analysis as well as the deformation on the loading surface of the bearing of irregular shape, whereas the finite-difference method is easily applicable to the non-linear problem of the lubricant behavior, that is, the solution of the Reynolds equation. The results demonstrate the accuracy and the availability of the combined method. This paper proposes the extended-solution procedure to meet the much more severe requirements of the higher capacity and the lower speed of static-pressure-type journal bearings. The static-pressure-type bearing is needed to endure the extremely low-speed operation while keeping a sufficient film thickness. Considering that the elastic deformation of the bearing structure
In recent years, journal bearings used in iron- and steel-manufacturing plants have been required to bear severe loading conditions. It is fundamentally important in designing journal bearings to predict their lubric performance with high accuracy. The so-caUed hydrodynamic-lubrication theory has so far been applied to estimate the bearing performance. However, this theory cannot always give satisfactory results to meet the recent demands of the severe loading conditions of high load and low speed. Under the high-load and low-speed conditions, the elastic deformation of the bearing housing and the journal has sitmificant effects on the lubricant-film pressure and thickness. The heat flow between the bearing and the lubricant cannot be neglected when the rotating speed is not extremely low. On the other hand, these effects are not regarded in the classical lubrication theory, which assumes a rigid bearing and an adiabatic film-bearing interface. Although there have been several notable studies on the elasto-hydrodynamicl~ or thermo-hydrodynamic solution3-s by the formulation of the finite-element method or the finite-difference method, satisfactory 273
274
K. Kohno, S. Takahashi, K. Saki
has more significant effects on the lubricant performance, the three-dimensional boundary-element method is used for the elastic-deformation problem. The finiteelement method is used to solve the Reynolds equation of high non-linearity. First, the outline of the elastohydrodynamic-lubrication theory is presented to solve the Reynolds equation by the finite-element method and the elastic equation by the boundary-element method. An iteratlve-solution scheme is proposed to obtain the lubricant-film pressure and thickness. Finally, numerical results are presented to reveal its usefulness as well as its applicabiilty to the design of an actual system of journal bearings.
PRELIMINARY CONSIDERATIONS The performance analysis of staric-pressure-type bearings, which involves the thrce-dimensional elastic deformation of the bearing and the journal, has not been widely studied, although Oh and Goenka l°'n conducted numerical analyses considering the threedimensional-rod deformation of dynamic-pressure-type bearings. The finite-element method is currently widely used for solving a large range of engineering problems. In the area of bearing-performance analysis, however, it is less preferable to treat the three-dimensionality than the boundary-element method on the basis that the deformarion on the surface of the bearing is required only to calculate the lubricant film pressure and thickness. In order to solve the Reynolds equation, there have been several investigations using the finite-difference method,12 the finite-element method, 13-15 or the boundary-element method. 16 Among them, the finiteelement method is considered to be suitable for the solution of the Reynolds equation, because the Reynolds equation involves non-line~srity and the finite-element method can easily treat the arbitrary shape of static-pressure pockets, which supply the lubricant oil. ELASTO-HYDitODYNAMIC LUBRICATION
THEORY Reym~ eqmtlea Figure I shows a schematic view of the bearing structure and the notation used. Auumin~ that the relative velocity between two walls is neglected under the lowspeed condition, the Reynolds equation for the incompremible Newtonian lubricant in the steady state can be written as follows:
Lubricantfilm
Bearinghousing ~ L Fill. 1. Schematicview of journal bearing. where V is a differential operator O/Oxi, h is the lubricant-film thickness, and/~ is the oil viscosity. With the lubricant-film thickness and the lubricant-oil viscosity dependent on the oil-film pressure p, it requires nonlinear treatment to solve the Reynolds equation, eqn (1). Equation (1) is solved to obtain the pressure distribution under the pressure-boundary conditions, the configuration of the bearing housing, the journal, and the staricpressure pocket being taken into account.
011 vl elty The viscosity of the lubricant oil is generally dependent on the pressure. The relationship between the oil viscosity and the pressure is assumed in the following expression: =
(2)
Next, the press function • usually used in the elastohydrodynamic-lubrication analysis for roller bearings 1~ is introduced. This gives:
1 ( 1 - e °p) • = I~l;dp --~--~
(3)
l~m..eltmmt fermalatlm The non-linear relationship between the lubricant-oil viscosity and the lubricant pressure in the Reynolds equation, eqn (1), can be reduced to a linear relationship by using the press function ~ in the fonowingmanner. The press function • defined in eqn (3) satisfies the following dilferenrial relationship: ~V~ = Vp
(4)
Substituting eqn (4) into the Reynolds equation, eqn (1), gives: =
o
(5)
where ~/* = h3/12. The weighted-residual form of eqn (5) is expressed as
follows: ]Q VOT*V~)~dfl = 0
(6)
Elasto-hydrodynamics lubrication analysis of journal bearingz where w is a weighted function. Transforming eqn (6) and applying Gauss' divergent theorem result in: - Ia r/*Vo:V¢ d r / = - Ia V(r/*V¢o:) dr2 = Iv o:#~dF
275
calculated by using the inverted form of eqn (3) in the following expression: p = __1in (1 Ot
(14)
(7) where f/and F are the domain and the boundary of an objective element, respectively, q represents the oil-flow vector, which equals -~/*V~, and ~ is the outward normal vector. By using the shape function N as the weighted function ~o, the Galerkin method can be applied to eqn (7) to result in the basic equation for the finite-element formulation as follows:
- Jn~*VNrVN df~{~2p} f lrNrFI~dI"
(8)
which is rewritten on the assumption that ~* is constant in an element, as follows: [k]{~p} = {%}
(9)
DEFORMATION ANALYSIS BY BEM Lubricant-film tiddmess The lubricant-film thickness varies with the high lubricant-film pressure appl/ed to the active zones of the inner surface of the bearing and the outer surface of the journal. On taking account of the effects of the elastic deformation on the film pressure in the solution of the Reynolds equation, eqn (1), the lubricant-film thickness can be defined as a sum of the gap for an undeformed geometry and the elastic deformation of the bearing housing and the journal as shown in Fig. 2. This gives: h(0, Z) = C{1 -
where
[k] = -¢]* In VNTVN {e¢} = Ir
df~
NrqndF
Superposing eqn (9) for each element of the whole domain, the Reynolds equation, eqn (1), is represented in matrix form as follows: = {E}
(10)
where {W} and {E} are nodal vectors of the press function and oil flows, respectively. Separating the variables into two parts of one in the domain and the other on the boundary, namely, the inner and outer surfaces of the lubricant film, eqn (10) can be written as follows: KDD KDB] ][BD KBsJ{Ws } = { E : }
(11)
where subscripts D and B represent nodal variables in the domain and those on the boundary, respectively. Nodal oil flows in the domain, ED, prove to be null owing to the contribution of the surrounding element in the assembling process of eqn (9), while the nodal pressures on the boundary, tPB, are described by the boundary conditions. On condensing eqn (11), the following expression for the unknown variables, the nodal pressure in the domain ~PD and the nodal oil flow Es on the boundary, is obtained: {tPD} = [kDD] -! ({EB} -- [KDgtJ{tI/B})
(12)
{EB} = [KBD]{tI/D} "~"[]~BB]{tI/B}
(13)
Once nodal values of the press function ~I'v are obtained, those of the lubricant-oil pressure can be
cos ( 0 -
+ hE(0, Z)
(15)
where the first term on the right-hand side is a gap of the undeformed geometry and the second is an additional term owing to the total elastic deformation of active zones of the bearing housing and the journal. The parameters C, ~, and ~ are the radial clearance, eccentricity ratio, and rigid-rotational angle, respectively. Boeadary-demem method The elastic deformation of the bearing structure is formulated by the three-dimensional boundary-element method. Figure 3 illustrates the configurations and the boundary conditions for (a) the bearing housing and (b) the journal. The bearing housing being supported on the upper surface by the hydraulic cylinder, the center of the upper surface is constrained in the vertical direction to avoid the rigid movement with the surface
m/ z housing Fill. 2. Notation and co-ordinate system.
276
K. Kohno, S. Takahashi, K. SaM
.
.
.
.
on the supported surface:
.
Pz = -P~o on the loading zone of the bearing inner surface sin ( 0
Px = Pl
-
7r) '[
py = pl cos (0 - ~r)
¢.x
Z
where Pl is the lubricant-film pressure on the inner surface of the bearing. On taking into account the fact that the relationships between the deformation of the loading zone and the lubricant pressure are required only in the iterative numerical calculation, the condensation of eqn (16) leads t o :
{~s} = [~B]{p~} + {BB}
(a) Bearinghousing
(18)
Assuming the journal deformation is s3nnmetric at its center in the longitudinal direction, a half of the journal is discretized as shown in Fig. 3(b) with the loading zone subject to the lubricant-film pressure distribution and with the barrel assumed to be subjected to the uniform line load equivalent of the load capacity. By following the same procedures as in the case of the bearing housing mentioned above, the boundary-element expression for the journal is described as follows:
Bm~l ~
(17)
f
fflm
[ HJ
HiJo ] {
Hoi HL
J
[/iJ}
= [ GIJi GiJ°] { pJ }
(19)
aJi aL
where u J and pJ represent the displacements and the tractions of the journal, respectively. The subscripts i and o represent the loading zone subject to the lubricant-film pressure and the remaining part of the whole journal structure, respectively. Considering the following boundary conditions: at the barrel center:
Milling reason fo¢~ W (b) j o u r ~ l
Fig. 3. Configurations and boundary conditions.
Ux = Uy = uz = O
subject to the uniform load equivalent to the load capacity of the bearing as shown in Fig. 3(a). Hence, distinguishing the pressurized loading zone and the rest of the surface, the elastic equations for the bearing housing are derived by following the boundary-element formulation as follows:
Hro
OBoi OBoo][fp~ vo j
(16)
where ue and pn represent the displacements and the tractions, respectively. The subscript i represents the loading zone of the bearing inner surface subject to the lubricant-film pressure, while the subscript o refers to the remaining part of the whole beating housing. The boundary conditions to eqn (16) for the bearing
housing result as follows: at the center of the upper surface: Ux = Uy f
u: = O
on the loading line of the barrel: Pz = PJo on the loading zone of the journal outer surface: Px = -Pl sin (0 - 7r) / py = - p l cos (e - ~r) J
(20)
and condensing eqn (19) results in the following:
{Ul} = [GJl{PiJ} + {B J}
(21)
The fundamental relationships of the lubricant-film pressure and the deformation for the bearing inner surface and the journal outer surface, both being at the interface of the lubricant film, are expressed in eqns (18) and (21). Hence the additional lubricant-film thickness hE due to the effects of the deformation of both the bearing and the journal is obtained as follows: he = (u~e - UJx)sin (0 - ~r) + (uyn - u J) cos (0 - ~r) (22)
Elasto-hydrodynamics lubrication analysis of journal bearings where the negative sign before the displacement u J means that the film thickness decreases with the positive UJ.
Evaluating the elastic deformation by using eqns (18) and (21) and summing them up by using eqn (22), the results are substituted into the Reynolds equation, eqn (10), to calculate the refined lubricant-film pressure. These iterative processes should be repeated until the convergence of the lubricant-film pressure and other variables is reached within the allowable tolerance. In this iterative process, the time-consuming calculation of the coefficient matrices such as G B, B e, G J, and B J need to be evaluated only once before the iterative computations.
NUMERICAL-ANALYSIS PROCEDURE The complete treatment of the elasto-hydrodynamiclubrication analysis formulated above requires the simultaneous solution of the two sets of governing equations: the Reynolds equations, equs (12) and (13) for the lubricant-film pressure and the elastic equations, eqns (18), (21), and (22) for the film thickness. The unknown variables here are the eccentricity
st|
!
ratio, the rigid-rotation angle ~, the lubricant-film pressure, and the elastic deformation of the bearing housing and the journal. In order to solve the problem presented, it is necessary to adopt the numerical iterative scheme because of the high degree of non-linearity involved. On the basis of the previous considerations, an effective computer program has been developed in this study, of which the main flow is shown in Fig. 4. The iterative scheme is described as follows: (1) Assuming the initial values of the eccentricity ratio and the rigid-rotation angle. (2) Calculate the lubricant-film thickness and judge the contact area to rectify the eccentricity. (3) Solve the Reynolds equation by taking account of the elastic deformation. (4) Calculate the loading capacity by integrating the lubricant-film pressure. (5) Rectify the eccentricity to obtain the given loading capacity. (6) Calculate the elastic deformation of the bearing housing and the journal to correct the lubricantfilm thickness for the next iteration. (7) Return to step (2) when convergence is not reached. In developing the program, special attention should be paid to avoiding the divergence of the temporary solution in the iterative process because the Reynolds equation is highly non-linear in the range of high loads, i.e. a high eccentricity ratio. Modification of the eccentricity ratio and the rigid-rotation angle are derived from the comparison of the vertical load and the moment subjected to the beating housing with the given loads.
I
I MoC'y P, I
277
APPLICATIONS
w I
I So,ve Z,,ynok~~. I FEM
[ ' c ,ou.ow I
Analysis model and coadlflomm Table 1 describes the calculating conditions of the journal bearing used in the application. Rotational speed is assumed to be zero for the sake of analytical simplicity, while that of the existing bearing is from three to four revolutions per minute. The bearing and the journal are discretized with the second-order boundary elements as shown in Fig. 5. The total number of nodes for the beating housing and the journal are 522 and 504, respectively. The sur-
Modify
[ Solveclinic ~
I BEM
4. Flow chart of elamto-hydrodynamio-lubrkationanalysis.
Table 1. Cakalatlall eea~iem Journal radius, Rs 723"9mm Maximum loading capacity, W 4350ton Rotational speed, Ns 0 r/rain (real speed 3--4r/rain) Other condifiom are corporate confidential.
K. Kolmo, S. Takahashi, K. Saki
278
Two slash-marked areas in the figure are the atatic pressure pockets, which supply the lubricant oil. Some examples are given to show the applications of the proposed coupling analysis in the following.
'I lr
E x m O e 1: eimte Jeamal This example shows the application to the elastic journal and the rigid bearing problem. Figure 7 shows the bearing performance of (a) the lubricant-film pressure, (b) the elastic deformation of the journal, and (c) the film thickness for 120% of the loading capacity. It can be seen that the deformation of the journal surface around the pressure pocket zone is swelling as is similar to the film thickness distribution due to the elastic effect of the journal. The film pressure has gentle slopes toward the pocket area in contrast to the case of the rigid journal shown in Fig. 8.
(a)
P
z
(a)
x
(b)
Fig. 5. Boundary-element models for the bearing housing and journal. (a) Bearing housing; (b) journal. faces subjected to the lubricant pressure have morerefined meshes than the other parts. The lubricant surface is discretized with 432 four-node isoparametric finite elements and 475 nodes as shown in Fig. 6. The allowable tolerance in the iteration is set within 1% for the lubricant-film pressure and the load.
,#z
(b)
0
b
g
o~ pocket
oilpocket
interofk),di~ m
oedeto f ~ ~
zone
l)rem.a'epo~__et
Fig. 6. Finite-element model for the lubricant film.
(c)
Fig. 7. Bearing performance with the rigid bearing and elastic journal. (a) Lubricant Rim pressure; (b) elastic deformation of the bearing surface; (c) lubricant a m ~ .
Elasto-hydrodynamics lubrication analysis of journal bearings P
279
z
•
s
ej
e
°
.'4
l
f
characteristics. Figure 9 shows the bearing performance of the pressure, the elasticdeformation, and the film thickness for 50% of the loading capacity and Fig. 10 illustratesthe
(a)
(a)
(b) Fig. lO. Elastic deformation of the bearing and journal. (a) Bearing housing; (b) journal.
(b)
h
(c) Fill. 9. Bearingperfor~,m,~ewith the elastic beating and elastic journal (a) Lubricant tilm palmalre;(b) elastic deformation of the bearing mrfaze; (c) lubricant film thickness.
elastic deformations of the bearing and the journal. The film thickness has a tendency to increase slightly toward the edges in the circnmfereDtial direction 0 in comparison with the case of the elastic journal and rigid bearing shown in Fig. 7(c). This fact indicates that the bearing deformation has a sitmificant infltlence on the beating performances. Next, on calculating the performances under the more severe loading conditions of 80%, 100%, and 120% load, all cases result in the bearing inner surface making contact with the journal, which leads to serious bearing troubles. Further investigations indicate that the unexpectedly low capacity is attributed to the bending deformation of the bearing housing to bring about the film-thickness decrease as shown in Fig. 11, i.e. the oval deformation of the beating inner surface makes the film thicknem widen from the pressure pockets around the center to the oil pockets at the frinl~, which significantly reduces the oil-flow remtanm in the circumferential direction to lower the lubricant pressure in the pressure pockets and result in the low capacity. Figure 12 illustrates the effects of the elaslic deformation on the minimal aim thiekne~ the ___*~een_ tricity ratio, and the pocket pressure under 50% loading capacity, where the vertical ~ is the ratio of the value for each condition to that of the rigid-bearing and rigid-journal
280
K. Kohno, S. Takahashi, K. S a M
i""
This also means that the elastic deformation of the bearing structure reduces the loading capacity of the journal bearing with the same pump pressure. From a practical point of view in d m i g n i ~ the actual staticpressure type of journal bearing, it proves to be an essential factor to take account of the elastic effects of the bearing structure.
.~...... ~ '.,
_i
i:
"'~
F i ,"
,,,'~'~. _~"'-. . . . . .
•
,'
,:7 "'"J ' """X._
h"
q (a)
(b)
CONCLUSIONS This paper has proposed a new numerical procedure for coupled used of the boundary elements and the finite elements for solving the elasto-hydrodynamic-lubrication problem of the static-type journal bearing. The boundary-element method is applied to evaluate the three-dimensional elastic deformation of the bearing housing and the journal, and the finite-element method is used to solve the Reynolds equation. The coupling method proposed in this paper provides an effective numerical technique for treating high non-linearity. In addition, some applications to ~ problems demonstrate that the elastic deformation of the bearing structure involved here plays an essential role in investigating the bearing performance and that the proposed method is applicable to actual systems of journal bearings.
Fig. 11. Lubricant-film thickness under 80% load. (a) Elastic deformation of the bearing housing; (b) lubricant film thickness.
ACKNOWLEDGEMENTS
condition. This figure shows that including the elastic effects of the bearing and the journal reduces the minimal film thidfJ3es8 and the pocket pressure required and has a sianificant influence on the eccentricity ratio.
The authors wish to express cordial thanks to Dr Masataka Tanaka, Professor of Shiag~ University, for his helpful guidance and ~ t .
I minimalfiknthickness [] eccenlricity ratio
[]
REFERENCES
Axset ge..ure
2 1.8 1.6 .~1.4
~ 1.2 1 0.8
[H E!;
0.4 0.2 0
Ri~0djonrr~ Analyticalcondition
LF.~ic journal
Fig. 12. Effects of the elastic deformation of the bearing and journal.
1. Oh, K. P. & Huebner, K. H., Solution of the elastohydrodynamic finite journal bearing problem, Trans. ASME J. Lubr. Technol., 1973, 95, 342-52. 2. Taylor, C. & O'Callaghun, J. F., A numerical solution of the elastohydrodynamic lubrication problem using finite elements, J. Mech. Engng ScL, 1972, 14, 229-37. 3. ]k~itsui, J. & Yamada, J., A s ~ of the lubrication film characteristics of journal ~ ~ I), Trans. Japan Soc. Mech. Engrs, 1979, 45, Set. C, 235-47. 4. Mitsui, J., Hod, Y. & Taatk& M., Thermohydrodynamic Analysis of Cooling Elect of Supply Oil in Circular Journal Bearing, Trans. A S M E J. I.a~. TechnoL, 1983, 105, 414-21. 5. Suganami, T. & Smri, A. Z., Thermohydrodynami¢ an~y~ of j ~ ~ , Tra,~s. A S M E I. Lubr Technol., 1979, ~ * 21"7. 6. Khonsari, M. M. & Wang, S. H., On the fluid-solid interaction in reference to thermoelastohydrodynamic analysis of journal bearings, Trans. ASME I. Tribol., 1991, 113, 398--404. 7. Kohno, K., Takahui, S. & Sudoh, K,, Thenno.elutohydrodynamic lubrication analyds of journal bearings
Elasto-hydrodynamics lubrication analysis of journal bearings with combined use of boundary elements and finite c£flferenoes, Boundary Elements IX, Vol. 2, ed. C. A. Brebbia, W. L. Wendland & G. Kuhn~ Springer-Verlag, Berlin, 1987, pp. 599-612. 8. Brebbia, C. A., Telles, J. C. F. & Wrobel, L. C., Boundary Element Technique - - Theory and Application in Engineering, Springer-Verlag, Berlin. 9. Tanaka, M., Matsumoto, T. & Nakamura, M., Boundary Element Method, Balhukan Publishing Co., 1992. 10. Oh, K. P. & Goenka, P. K., The elastohydrodyvamic solution of journal bearings under dyDami¢loading, Trans. ASME J. Tribol., 1985, 107, 389-95. 11. Goenka, P. K. & Oh, K. P., An optimum short bearing theory for the elastohydrodynamic solution of journal bearings, Trans. ASME J. Tribol., 1986, 108, 294-9. 12. Castelli, V. & Pirvics, J., Review of numerical methods in
281
gas bearing film analysis, Trans. ASME J. Lubr. Technol., 1968, 90, 777-92. 13. Reddi, M. M., Finite-element solution of the incompressible lubrication problem, Trans. ASME J. Lubr. Technol., 1969, 91, 524-33. 14. Nguyen, S. H., p-Version incompressible lubrication finite element analysis of large width bearings, Trans. ASME J. Lubr. Technol,, 1991, 113, 116-19. 15. Booker, J. F. & Hucbner, K. H., Application of finite element methods to lubrication: an engineering approach, Trans. ASME J. Lubr. Technol., 1972, 94, 313-23. 16. Khader, M. S., A generalized integral numerical solution method for lubrication problems, Trans. ASME J. Tribol., 1984, 106, 255-9. 17. Dowson, D. & Higginson, G. R., Elasto-Hydrodynamic Lubrication, Pergamon, Oxford, UK, 1977.
13 (1994) 273-281
© 1994l~ev~ ScimceIJnfited Printedin GreatBritain.Allrights E~EVIER
Elasto-hydrodynamic lubrication analysis of journal bearings with combined use of boundary elements and finite elements K. Kohno Hiroshinm Research and Development Center, Mitsubishi Heavy Industries Co. Lid, Itozaki-cho, Mihara-shi 729-03, Japan
S. Takahashi & K. Said Nagasaki Research and Development Center, Mitsubishi Heavy In~stries Co. Ltd., Akunoura-cho, Nagasaki-shi 850-91, Japan
(Received 20 January 1994; acx:epted6 April 1994) In this paper, the boundary-element method is applied to analyze the elastohydrodynamic lubrication charactefistica of journal bearings with combined use of the boundary-element method and finite-element method. The boundaryelement method is used to calculate the elastic deformation of the bearing housing and the journal, and the finite-element method is applied to the solution of the Reynolds equation. The results of the iterative-solution method proposed are presented to illustrate its applicability to the actual journal bearings. Key words: Boundary-element method, finite-element method, journal bearing, elasto-hydrodynamic lubrication, elastic deformation, Reynolds equation.
INTRODUCTION
treatments have not been found as far as the authors are aware. From this viewpoint, the two-dimensional numerical technique, which accommodates the above two factors, was developed with the combined usage of the boundary-element method and the finite-difference method to treat the dynamic-pressure-type journal beating. 7 The boundary-element method8,9 is more suitable for analyzing the elastic deformation of the bearing than the domain-type method because the elastic problem involved requires only linear analysis as well as the deformation on the loading surface of the bearing of irregular shape, whereas the finite-difference method is easily applicable to the non-linear problem of the lubricant behavior, that is, the solution of the Reynolds equation. The results demonstrate the accuracy and the availability of the combined method. This paper proposes the extended-solution procedure to meet the much more severe requirements of the higher capacity and the lower speed of static-pressure-type journal bearings. The static-pressure-type bearing is needed to endure the extremely low-speed operation while keeping a sufficient film thickness. Considering that the elastic deformation of the bearing structure
In recent years, journal bearings used in iron- and steel-manufacturing plants have been required to bear severe loading conditions. It is fundamentally important in designing journal bearings to predict their lubric performance with high accuracy. The so-caUed hydrodynamic-lubrication theory has so far been applied to estimate the bearing performance. However, this theory cannot always give satisfactory results to meet the recent demands of the severe loading conditions of high load and low speed. Under the high-load and low-speed conditions, the elastic deformation of the bearing housing and the journal has sitmificant effects on the lubricant-film pressure and thickness. The heat flow between the bearing and the lubricant cannot be neglected when the rotating speed is not extremely low. On the other hand, these effects are not regarded in the classical lubrication theory, which assumes a rigid bearing and an adiabatic film-bearing interface. Although there have been several notable studies on the elasto-hydrodynamicl~ or thermo-hydrodynamic solution3-s by the formulation of the finite-element method or the finite-difference method, satisfactory 273
274
K. Kohno, S. Takahashi, K. Saki
has more significant effects on the lubricant performance, the three-dimensional boundary-element method is used for the elastic-deformation problem. The finiteelement method is used to solve the Reynolds equation of high non-linearity. First, the outline of the elastohydrodynamic-lubrication theory is presented to solve the Reynolds equation by the finite-element method and the elastic equation by the boundary-element method. An iteratlve-solution scheme is proposed to obtain the lubricant-film pressure and thickness. Finally, numerical results are presented to reveal its usefulness as well as its applicabiilty to the design of an actual system of journal bearings.
PRELIMINARY CONSIDERATIONS The performance analysis of staric-pressure-type bearings, which involves the thrce-dimensional elastic deformation of the bearing and the journal, has not been widely studied, although Oh and Goenka l°'n conducted numerical analyses considering the threedimensional-rod deformation of dynamic-pressure-type bearings. The finite-element method is currently widely used for solving a large range of engineering problems. In the area of bearing-performance analysis, however, it is less preferable to treat the three-dimensionality than the boundary-element method on the basis that the deformarion on the surface of the bearing is required only to calculate the lubricant film pressure and thickness. In order to solve the Reynolds equation, there have been several investigations using the finite-difference method,12 the finite-element method, 13-15 or the boundary-element method. 16 Among them, the finiteelement method is considered to be suitable for the solution of the Reynolds equation, because the Reynolds equation involves non-line~srity and the finite-element method can easily treat the arbitrary shape of static-pressure pockets, which supply the lubricant oil. ELASTO-HYDitODYNAMIC LUBRICATION
THEORY Reym~ eqmtlea Figure I shows a schematic view of the bearing structure and the notation used. Auumin~ that the relative velocity between two walls is neglected under the lowspeed condition, the Reynolds equation for the incompremible Newtonian lubricant in the steady state can be written as follows:
Lubricantfilm
Bearinghousing ~ L Fill. 1. Schematicview of journal bearing. where V is a differential operator O/Oxi, h is the lubricant-film thickness, and/~ is the oil viscosity. With the lubricant-film thickness and the lubricant-oil viscosity dependent on the oil-film pressure p, it requires nonlinear treatment to solve the Reynolds equation, eqn (1). Equation (1) is solved to obtain the pressure distribution under the pressure-boundary conditions, the configuration of the bearing housing, the journal, and the staricpressure pocket being taken into account.
011 vl elty The viscosity of the lubricant oil is generally dependent on the pressure. The relationship between the oil viscosity and the pressure is assumed in the following expression: =
(2)
Next, the press function • usually used in the elastohydrodynamic-lubrication analysis for roller bearings 1~ is introduced. This gives:
1 ( 1 - e °p) • = I~l;dp --~--~
(3)
l~m..eltmmt fermalatlm The non-linear relationship between the lubricant-oil viscosity and the lubricant pressure in the Reynolds equation, eqn (1), can be reduced to a linear relationship by using the press function ~ in the fonowingmanner. The press function • defined in eqn (3) satisfies the following dilferenrial relationship: ~V~ = Vp
(4)
Substituting eqn (4) into the Reynolds equation, eqn (1), gives: =
o
(5)
where ~/* = h3/12. The weighted-residual form of eqn (5) is expressed as
follows: ]Q VOT*V~)~dfl = 0
(6)
Elasto-hydrodynamics lubrication analysis of journal bearingz where w is a weighted function. Transforming eqn (6) and applying Gauss' divergent theorem result in: - Ia r/*Vo:V¢ d r / = - Ia V(r/*V¢o:) dr2 = Iv o:#~dF
275
calculated by using the inverted form of eqn (3) in the following expression: p = __1in (1 Ot
(14)
(7) where f/and F are the domain and the boundary of an objective element, respectively, q represents the oil-flow vector, which equals -~/*V~, and ~ is the outward normal vector. By using the shape function N as the weighted function ~o, the Galerkin method can be applied to eqn (7) to result in the basic equation for the finite-element formulation as follows:
- Jn~*VNrVN df~{~2p} f lrNrFI~dI"
(8)
which is rewritten on the assumption that ~* is constant in an element, as follows: [k]{~p} = {%}
(9)
DEFORMATION ANALYSIS BY BEM Lubricant-film tiddmess The lubricant-film thickness varies with the high lubricant-film pressure appl/ed to the active zones of the inner surface of the bearing and the outer surface of the journal. On taking account of the effects of the elastic deformation on the film pressure in the solution of the Reynolds equation, eqn (1), the lubricant-film thickness can be defined as a sum of the gap for an undeformed geometry and the elastic deformation of the bearing housing and the journal as shown in Fig. 2. This gives: h(0, Z) = C{1 -
where
[k] = -¢]* In VNTVN {e¢} = Ir
df~
NrqndF
Superposing eqn (9) for each element of the whole domain, the Reynolds equation, eqn (1), is represented in matrix form as follows: = {E}
(10)
where {W} and {E} are nodal vectors of the press function and oil flows, respectively. Separating the variables into two parts of one in the domain and the other on the boundary, namely, the inner and outer surfaces of the lubricant film, eqn (10) can be written as follows: KDD KDB] ][BD KBsJ{Ws } = { E : }
(11)
where subscripts D and B represent nodal variables in the domain and those on the boundary, respectively. Nodal oil flows in the domain, ED, prove to be null owing to the contribution of the surrounding element in the assembling process of eqn (9), while the nodal pressures on the boundary, tPB, are described by the boundary conditions. On condensing eqn (11), the following expression for the unknown variables, the nodal pressure in the domain ~PD and the nodal oil flow Es on the boundary, is obtained: {tPD} = [kDD] -! ({EB} -- [KDgtJ{tI/B})
(12)
{EB} = [KBD]{tI/D} "~"[]~BB]{tI/B}
(13)
Once nodal values of the press function ~I'v are obtained, those of the lubricant-oil pressure can be
cos ( 0 -
+ hE(0, Z)
(15)
where the first term on the right-hand side is a gap of the undeformed geometry and the second is an additional term owing to the total elastic deformation of active zones of the bearing housing and the journal. The parameters C, ~, and ~ are the radial clearance, eccentricity ratio, and rigid-rotational angle, respectively. Boeadary-demem method The elastic deformation of the bearing structure is formulated by the three-dimensional boundary-element method. Figure 3 illustrates the configurations and the boundary conditions for (a) the bearing housing and (b) the journal. The bearing housing being supported on the upper surface by the hydraulic cylinder, the center of the upper surface is constrained in the vertical direction to avoid the rigid movement with the surface
m/ z housing Fill. 2. Notation and co-ordinate system.
276
K. Kohno, S. Takahashi, K. SaM
.
.
.
.
on the supported surface:
.
Pz = -P~o on the loading zone of the bearing inner surface sin ( 0
Px = Pl
-
7r) '[
py = pl cos (0 - ~r)
¢.x
Z
where Pl is the lubricant-film pressure on the inner surface of the bearing. On taking into account the fact that the relationships between the deformation of the loading zone and the lubricant pressure are required only in the iterative numerical calculation, the condensation of eqn (16) leads t o :
{~s} = [~B]{p~} + {BB}
(a) Bearinghousing
(18)
Assuming the journal deformation is s3nnmetric at its center in the longitudinal direction, a half of the journal is discretized as shown in Fig. 3(b) with the loading zone subject to the lubricant-film pressure distribution and with the barrel assumed to be subjected to the uniform line load equivalent of the load capacity. By following the same procedures as in the case of the bearing housing mentioned above, the boundary-element expression for the journal is described as follows:
Bm~l ~
(17)
f
fflm
[ HJ
HiJo ] {
Hoi HL
J
[/iJ}
= [ GIJi GiJ°] { pJ }
(19)
aJi aL
where u J and pJ represent the displacements and the tractions of the journal, respectively. The subscripts i and o represent the loading zone subject to the lubricant-film pressure and the remaining part of the whole journal structure, respectively. Considering the following boundary conditions: at the barrel center:
Milling reason fo¢~ W (b) j o u r ~ l
Fig. 3. Configurations and boundary conditions.
Ux = Uy = uz = O
subject to the uniform load equivalent to the load capacity of the bearing as shown in Fig. 3(a). Hence, distinguishing the pressurized loading zone and the rest of the surface, the elastic equations for the bearing housing are derived by following the boundary-element formulation as follows:
Hro
OBoi OBoo][fp~ vo j
(16)
where ue and pn represent the displacements and the tractions, respectively. The subscript i represents the loading zone of the bearing inner surface subject to the lubricant-film pressure, while the subscript o refers to the remaining part of the whole beating housing. The boundary conditions to eqn (16) for the bearing
housing result as follows: at the center of the upper surface: Ux = Uy f
u: = O
on the loading line of the barrel: Pz = PJo on the loading zone of the journal outer surface: Px = -Pl sin (0 - 7r) / py = - p l cos (e - ~r) J
(20)
and condensing eqn (19) results in the following:
{Ul} = [GJl{PiJ} + {B J}
(21)
The fundamental relationships of the lubricant-film pressure and the deformation for the bearing inner surface and the journal outer surface, both being at the interface of the lubricant film, are expressed in eqns (18) and (21). Hence the additional lubricant-film thickness hE due to the effects of the deformation of both the bearing and the journal is obtained as follows: he = (u~e - UJx)sin (0 - ~r) + (uyn - u J) cos (0 - ~r) (22)
Elasto-hydrodynamics lubrication analysis of journal bearings where the negative sign before the displacement u J means that the film thickness decreases with the positive UJ.
Evaluating the elastic deformation by using eqns (18) and (21) and summing them up by using eqn (22), the results are substituted into the Reynolds equation, eqn (10), to calculate the refined lubricant-film pressure. These iterative processes should be repeated until the convergence of the lubricant-film pressure and other variables is reached within the allowable tolerance. In this iterative process, the time-consuming calculation of the coefficient matrices such as G B, B e, G J, and B J need to be evaluated only once before the iterative computations.
NUMERICAL-ANALYSIS PROCEDURE The complete treatment of the elasto-hydrodynamiclubrication analysis formulated above requires the simultaneous solution of the two sets of governing equations: the Reynolds equations, equs (12) and (13) for the lubricant-film pressure and the elastic equations, eqns (18), (21), and (22) for the film thickness. The unknown variables here are the eccentricity
st|
!
ratio, the rigid-rotation angle ~, the lubricant-film pressure, and the elastic deformation of the bearing housing and the journal. In order to solve the problem presented, it is necessary to adopt the numerical iterative scheme because of the high degree of non-linearity involved. On the basis of the previous considerations, an effective computer program has been developed in this study, of which the main flow is shown in Fig. 4. The iterative scheme is described as follows: (1) Assuming the initial values of the eccentricity ratio and the rigid-rotation angle. (2) Calculate the lubricant-film thickness and judge the contact area to rectify the eccentricity. (3) Solve the Reynolds equation by taking account of the elastic deformation. (4) Calculate the loading capacity by integrating the lubricant-film pressure. (5) Rectify the eccentricity to obtain the given loading capacity. (6) Calculate the elastic deformation of the bearing housing and the journal to correct the lubricantfilm thickness for the next iteration. (7) Return to step (2) when convergence is not reached. In developing the program, special attention should be paid to avoiding the divergence of the temporary solution in the iterative process because the Reynolds equation is highly non-linear in the range of high loads, i.e. a high eccentricity ratio. Modification of the eccentricity ratio and the rigid-rotation angle are derived from the comparison of the vertical load and the moment subjected to the beating housing with the given loads.
I
I MoC'y P, I
277
APPLICATIONS
w I
I So,ve Z,,ynok~~. I FEM
[ ' c ,ou.ow I
Analysis model and coadlflomm Table 1 describes the calculating conditions of the journal bearing used in the application. Rotational speed is assumed to be zero for the sake of analytical simplicity, while that of the existing bearing is from three to four revolutions per minute. The bearing and the journal are discretized with the second-order boundary elements as shown in Fig. 5. The total number of nodes for the beating housing and the journal are 522 and 504, respectively. The sur-
Modify
[ Solveclinic ~
I BEM
4. Flow chart of elamto-hydrodynamio-lubrkationanalysis.
Table 1. Cakalatlall eea~iem Journal radius, Rs 723"9mm Maximum loading capacity, W 4350ton Rotational speed, Ns 0 r/rain (real speed 3--4r/rain) Other condifiom are corporate confidential.
K. Kolmo, S. Takahashi, K. Saki
278
Two slash-marked areas in the figure are the atatic pressure pockets, which supply the lubricant oil. Some examples are given to show the applications of the proposed coupling analysis in the following.
'I lr
E x m O e 1: eimte Jeamal This example shows the application to the elastic journal and the rigid bearing problem. Figure 7 shows the bearing performance of (a) the lubricant-film pressure, (b) the elastic deformation of the journal, and (c) the film thickness for 120% of the loading capacity. It can be seen that the deformation of the journal surface around the pressure pocket zone is swelling as is similar to the film thickness distribution due to the elastic effect of the journal. The film pressure has gentle slopes toward the pocket area in contrast to the case of the rigid journal shown in Fig. 8.
(a)
P
z
(a)
x
(b)
Fig. 5. Boundary-element models for the bearing housing and journal. (a) Bearing housing; (b) journal. faces subjected to the lubricant pressure have morerefined meshes than the other parts. The lubricant surface is discretized with 432 four-node isoparametric finite elements and 475 nodes as shown in Fig. 6. The allowable tolerance in the iteration is set within 1% for the lubricant-film pressure and the load.
,#z
(b)
0
b
g
o~ pocket
oilpocket
interofk),di~ m
oedeto f ~ ~
zone
l)rem.a'epo~__et
Fig. 6. Finite-element model for the lubricant film.
(c)
Fig. 7. Bearing performance with the rigid bearing and elastic journal. (a) Lubricant Rim pressure; (b) elastic deformation of the bearing surface; (c) lubricant a m ~ .
Elasto-hydrodynamics lubrication analysis of journal bearings P
279
z
•
s
ej
e
°
.'4
l
f
characteristics. Figure 9 shows the bearing performance of the pressure, the elasticdeformation, and the film thickness for 50% of the loading capacity and Fig. 10 illustratesthe
(a)
(a)
(b) Fig. lO. Elastic deformation of the bearing and journal. (a) Bearing housing; (b) journal.
(b)
h
(c) Fill. 9. Bearingperfor~,m,~ewith the elastic beating and elastic journal (a) Lubricant tilm palmalre;(b) elastic deformation of the bearing mrfaze; (c) lubricant film thickness.
elastic deformations of the bearing and the journal. The film thickness has a tendency to increase slightly toward the edges in the circnmfereDtial direction 0 in comparison with the case of the elastic journal and rigid bearing shown in Fig. 7(c). This fact indicates that the bearing deformation has a sitmificant infltlence on the beating performances. Next, on calculating the performances under the more severe loading conditions of 80%, 100%, and 120% load, all cases result in the bearing inner surface making contact with the journal, which leads to serious bearing troubles. Further investigations indicate that the unexpectedly low capacity is attributed to the bending deformation of the bearing housing to bring about the film-thickness decrease as shown in Fig. 11, i.e. the oval deformation of the beating inner surface makes the film thicknem widen from the pressure pockets around the center to the oil pockets at the frinl~, which significantly reduces the oil-flow remtanm in the circumferential direction to lower the lubricant pressure in the pressure pockets and result in the low capacity. Figure 12 illustrates the effects of the elaslic deformation on the minimal aim thiekne~ the ___*~een_ tricity ratio, and the pocket pressure under 50% loading capacity, where the vertical ~ is the ratio of the value for each condition to that of the rigid-bearing and rigid-journal
280
K. Kohno, S. Takahashi, K. S a M
i""
This also means that the elastic deformation of the bearing structure reduces the loading capacity of the journal bearing with the same pump pressure. From a practical point of view in d m i g n i ~ the actual staticpressure type of journal bearing, it proves to be an essential factor to take account of the elastic effects of the bearing structure.
.~...... ~ '.,
_i
i:
"'~
F i ,"
,,,'~'~. _~"'-. . . . . .
•
,'
,:7 "'"J ' """X._
h"
q (a)
(b)
CONCLUSIONS This paper has proposed a new numerical procedure for coupled used of the boundary elements and the finite elements for solving the elasto-hydrodynamic-lubrication problem of the static-type journal bearing. The boundary-element method is applied to evaluate the three-dimensional elastic deformation of the bearing housing and the journal, and the finite-element method is used to solve the Reynolds equation. The coupling method proposed in this paper provides an effective numerical technique for treating high non-linearity. In addition, some applications to ~ problems demonstrate that the elastic deformation of the bearing structure involved here plays an essential role in investigating the bearing performance and that the proposed method is applicable to actual systems of journal bearings.
Fig. 11. Lubricant-film thickness under 80% load. (a) Elastic deformation of the bearing housing; (b) lubricant film thickness.
ACKNOWLEDGEMENTS
condition. This figure shows that including the elastic effects of the bearing and the journal reduces the minimal film thidfJ3es8 and the pocket pressure required and has a sianificant influence on the eccentricity ratio.
The authors wish to express cordial thanks to Dr Masataka Tanaka, Professor of Shiag~ University, for his helpful guidance and ~ t .
I minimalfiknthickness [] eccenlricity ratio
[]
REFERENCES
Axset ge..ure
2 1.8 1.6 .~1.4
~ 1.2 1 0.8
[H E!;
0.4 0.2 0
Ri~0djonrr~ Analyticalcondition
LF.~ic journal
Fig. 12. Effects of the elastic deformation of the bearing and journal.
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281
gas bearing film analysis, Trans. ASME J. Lubr. Technol., 1968, 90, 777-92. 13. Reddi, M. M., Finite-element solution of the incompressible lubrication problem, Trans. ASME J. Lubr. Technol., 1969, 91, 524-33. 14. Nguyen, S. H., p-Version incompressible lubrication finite element analysis of large width bearings, Trans. ASME J. Lubr. Technol,, 1991, 113, 116-19. 15. Booker, J. F. & Hucbner, K. H., Application of finite element methods to lubrication: an engineering approach, Trans. ASME J. Lubr. Technol., 1972, 94, 313-23. 16. Khader, M. S., A generalized integral numerical solution method for lubrication problems, Trans. ASME J. Tribol., 1984, 106, 255-9. 17. Dowson, D. & Higginson, G. R., Elasto-Hydrodynamic Lubrication, Pergamon, Oxford, UK, 1977.