Starvation and Cavitation Effects in Finite Grooved Journal Bearing

Lubricants and Lubrication / D. Dowson et al. (Editors) 1995 Elsevier Science B.V.

455

Starvation and Cavitation Effects in Finite Grooved Journal Bearing B. Vincent, P. Maspeyrot, J. F r b e Laboratoire de M M q u e des Solides - URA CNRS Universite de POITIERS 40, Avenue du Recteur Pineau, 86022 POITIERS Cedex

In this paper the effect of cavitation and starvation is analysed for a finite grooved journal bearing. The numerical procedure incorporates a cavitation algorithm based on the work of Elrod. Film rupture and reformation are predicted considering conservation of mass flow through the entire bearing. The effects of some degrees of starvation and lubricant supply pressure are studied in steady state conditions. A solution is proposed for dynamic loading conditions using the mobility method of Booker.

1. INTRODUCTION

Analytical and numerical investigation have used many algorithms to predict cavitation. Dowson and Taylor [ 11 and Booker [2] presented a review of the recent developments. A large part of these studies d a t e s cavitation with negative pressures. Indeed, cavitation areas can be determine coupling pressure in the cavitation region and fluid density rate [3] or setting negative pressures to some fixed cavitation pressure [4]. Except for particular cases [5], the bearing orbit and the positive film pressure region are known with enough accuracy. Although, oil flow can not be determined as mass is not conserved within the cavitation area. Some mass conservative algorithms have been developed with finite difference or finite element formulation. Jones [6] used internodal flow to determine the degree of filling of the clearance space.However, the change in the boundaries of the cavitation regions is not explicitly determined. Elrod and Adams [7]and Elrod [8] developed, with finite difference, a generalised form of Reynolds

differential equation. With this equation, the cavitation is taken into account without references to rupture or reformation of the oil film. This new equation can be applied in both cavitated and full film region (introducing a switch function) and for steady state or transient cases. D o w n , Taylor and Miranda [9,10] used the Elrod algorithm to predict the steady-state performance of finite groove bearings and compared their results to experimental data. Brewe [ll] applied this algorithm to model vapor cavitation in dynamically loaded journal bearings. Woods and Brewe [12] added multigrid techniques to the algorithm in order to improve solution speed. Vijayaraghavan and Keith [13,14] proposed a modification to the Elrod algorithm, specifically in the shear flow term, by introducing a type differencing procedure. This procedure automatically switches the form of differencing (central or upwind) of the shear flow terms in the full film and cavitated regions as required by the physics of the problems. Some authors [15,16] analysed noncircular journal bearing using this algorithm. The effects of groove

456 inlet conditions coupled with misalignment of the shaft has been developed by Vijayaraghavan and Keith [17].The misalignment considered varies in magnitude and direction with reference to the boundaries of the bearing. The authors [18]studied the effects of groove type, groove number and groove location on the performance of a finite length journal bearing. Both axial and circumferentialgrooves have been considered. The first algorithm using finite element cavitation has been developed by Kumar and Booker [19].This one Seems very simple to use and cavitation boundaries ( and subsequently regions ) are determined only with nodal pressure and densities. However, some problems occur,when the mesh or the integration method changed ( Goenka and Paranjpe [20]). Recently, Yu and Keith [21] presented a new numerical algorithm to predict gaseous cavitation in various fluid film bearings. Elrod's universal differential equation is transformed into a generalised bolindary integral equation, which is valid throughout both the fill film and cavitated regions. In this paper, the treatment of film rupture and reformation for the circular bearing is analysed. The effects of circumferential grooves of finite dimensions is considered. The grooves are positioned in such a way that they are symmetrical about the centerline of the bearing. More over, some degrees of starvation and lubricant supply pressure are treated in steady state conditions. The applied load is normal to the journal axis, the predicted parameters are presented with reference to the known vertical load line. For dynamically loaded journal bearing, a connecting-rod bearing with and without groove is studied. A mass conservation algorithm [13] is used and is coupled with the motion of a journal center that include the mass acceleration effects. The equation is solved using the Gauss-Seidel scheme and the mobility method of Booker [22]. Journal trajectories, minimum film thickness and pressure profile are presented and compared to the calculation without Elrod's algorithm. 1.1 Notations

c hh

-

Radialclearance Filmthickness Nondimensional film thickness [WC]

m-

Bearing length to diameter ratio Mobility vector Fluid pressure Switch function - Lubricant supply pressure Cavitation pressure - Bearing radius - Time u - Net d a c e velocity w - Applied load Coordinate axis in circumferential direction x z Coordinate axis in axial direction a - Mobility direction P - Bulk modulus Angular coordinate Y E Eccentricity ratio 4 - Attitude angle 0 - Fractional film content P - Fluid viscosity P - Fluid density P c - Fluid density at cavitation pressure 0 Angular velocity Angular velocity of the journal 0, Angular velocity of the bearing 0, i - Angular velocity of the load

M P g Pal Po R t

-

-

-

d,-

Angular velocity of the attitude angle

2. ANALYTICAL FORMULATION

Elrod et Adams developed a single equation, called the universal equation, which is applicable for both full film and cavitated regions of the bearing. The derivation of this equation is based on the continuity of mass flow through the entire bearing. The authors [8] have discussed the development of this equation and for a two dimensional problem, a steady state load, a laminar flow,the Reynolds equation is written as:

457

where 8, g and p are defined as:

e={

3.1.1 Shear flow A finite difference version of Eq.(l) can be written as:

p/p, in full film regions V,/V, in cavitated regions

1 when 0 2 1 g = { 0 when 8 < 1

In the full film (g, = 1) ,this yield which upon integration gives P = P, +gPlnO V, is the total clearance volume and V, is the volume occupied by the fluid. In the full film region (g=l), Eq.(l) reduces to compressible form of the Reynolds equation. In the cavitated region (g=O), the pressure remains constant at the cavitation pressure and the flow is driven by shear. The boundary conditions for film rupture and reformation are implicitly applied by enforcing conservation of mass flow across these boundaries.

(3)

which does not take account of compressibility effects. (gi = 0), Eq.(l)

In the cavitation region produces

ax a (U --.he 2 ) i =-

[

(he)i -(he)i-l] Ax

(4)

3.1.2 Pressure induced flow 3. NUMERICAL FORMULATION

The pressure induced flow term in the x direction is differenced as:

3.1 Static case In the full film, Eq.(l) is a F a l differential equation of an elliptic type that is best treated using central differencing. On the other hand, in the cavitated region, the governing equation is hyperbolic that is best handled via upwind differencing. The development of the algorithm is discussed in Vijayaraghavan and Keith [ 13,141. In the following, the shear flow and the pressure induced flow components will be treated separately.

h3

;(-i&$

=-

12p.A??

[

v + l / 2 &+I(@,+,

- 1)

- (lf+IP + h?-I/Z) g,(e, - 1) + V+I/Z

-41

gl-I(~,-l

(5)

The expression in the z direction will be similar to the above and may be written by replacing x and i with z and j, respectively.

458 3.2 Dynamic case

3.2.2 Pressure induced flow

The partial differential equation governing hydrodynamic lubrication may be written to include journal rotation, journal center motion and Elrod's algorithm as follow

The pressure induced flow term in the x and z directions are the same as those defined in the static case.

4. SOLUTION PROCEDURE 4.1 Static case

where

- =-0

2

+

When the variables in Eq.(6) are nondimensionalized and included the mobility method, the expression can be written as

(7)

where

In this study, cavitation journal bearing is examined. The finite length journal bearing is considered. A real implicit finite difference scheme is used to numerically determine the distribution of 8 . By this method, the finite difference version of Eq.(l) which incorporates Eq.(2) and (9,can be written at each node in a general form. The system of equation for every node is solved using the Gauss-Seidel scheme. Initially all nodes are assumed to be in the full film, and at all non b o u n d a ~nodes ~ (e=1 and e l ) except for the nodes located within the finite groove. For these nodes the value of 8 is known, corresponding to the lubricant supply and its value is maintained throughout the calculation. The switch function distribution is updated after every time step. The time march process is continued until the change in 8 between two time intervals was less than lo-' . At this point, the converged steady state had been reached. 4.2 Dynamic case

3.2.1 Convection flow

The development of the algorithm is discussed in [13,14]. A finite difference version of Eq.(7) can be written in compact and efficient form as

The hydrodynamic "universal equation", Eq.(7), is discretized using the cavitation algorithm and

solved using the Gauss-Seidel scheme and the mobility method. The axial edge of the bearing is maintained at the ambient pressure. For the bearing with groove, the circumferential groove is extended through the entire circumference of bearing and is maintained at the lubricant supply pressure. The cavitation pressure is taken to be absolute zero. Initially the bearing is assumed to be filled with lubricant at the ambient pressure and released at an arbitrary location. The initially components of eccentricity, mobility direction and attitude angle are given. Quation (7) is numerically solved to determine the current 8 distribution. At every time

459 steps, the switch function distribution is updated. The force components are computed from the pressure profile. The attitude angle is computed. If the attitude angle is not equal with the initially attitude angle, the procedure is repeated. Otherwise the mobility vector is changed and a new attitude angle is computed. The procedure is repeated for every load increment. The trajectory of the journal and the minimum film thickness are recorded at specified time interval. The code developed is run on the DEC ALPHA A X P 3000-800 super computer.

while the Figure 4 presents the fractional film content profile. ing data

R

m

U

3.15 x 10'

m/S

m

C

E

Pas Nlm'

P P Pll -

0.0035

N/m2

P

5. NUMERICAL EXAMPLE

5.1 Static case

Some types of journal bearing lubricant supply, submerged, grooved and starved were modelled in this study. The geometric arrangement, as well as the direction of the angular coordinate measurement, are shown in Fig. 1. The physical and operating parameters are given in Table 1. The load is assumed to be applied along a vertical axis.

c,

E

"1

3 E

3 0.8 E

d

1E:

0

60 120 180 240 300 Angular coordinate (deg)

360

Fig. 2: Angular distribution of fractional film content

Fig. 1: Bearing Geometry The bearings were assumed to be at full film initially. At the axial end of the bearing, the inlet was assumed to be flooded at the atmospheric pressure. For the bearing without groove, Figure 2 is the 8 distribution at the bearing edge and the bearing middle section. At the edge of the bearing, one can note that the formation of the film arises after the one in the middle section, due to the axial leakage. Both the graph are almost similar. Figure 3 show the classical profile of pressure for this bearing,

Fig. 3: Pressure profile

460

-ff

Bearing middle scotion

Fig. 4: Profile of fractional film content In the case of grooved journal bearing, two part~alcircumferential grooves are considered. In the grooves the lubricant inlet is assumed to be at the pressure supply of 0.2 ma. The grooves are positioned in such a way that they are symmetrical about the centerline of the bearing. The bearing data are those defined in Table 1 and the groove parameters are given in Table 2. Figure 5 gives the 8 distribution for two sections of the bearing. In the middle one, the 1 1 1 film is almost complete along the circumference as it is the line where the grooves are located. On the other hand, at the edge of the bearing the minimum value of the fractional film content is near 0.3. Figure 6 permits to know the variation of the fractional film content in all the bearing. For this case,the groove increases the side leakage. Hence, at the edge of the bearing the fractional film content is lower than the case without groove. In the middle section, one can note that the groove increases the volume of fluid (0 z 0.8). Figure 7 presents the pressure profile deduced from these distribution. On this plot, the film reformation and rupture boundaries are given and the film reformation occurs along the trailing edge of the grooves.

Fig. 5: Angular distribution of fractional film content with two grooves

Fig. 6: Profile of fractional film content with two grooves

ITable 2: Circumferential groove arrangements I

I groove I start angle / I circumferential I axial I number I vertical axis I extent I extent 1'' I 45' I 90' I 0.2L

Fig. 7: Pressure profile of two grooves

46 1 values, when the inlet condition are starved I), except for the side leakage. (0.6< However, for the inlet condition is low (W 0.6), the performance parameters are in a drastic reduction.

0(0,z)=1 .oo1

5.2 Dynamic case

The specific case chosen is the connecting-rod bearing of the Ruston and Hornsby VEB diesel engine. This particular connecting-rod bearing is probably the most analysed bearing in the literature. The bearing data is given below in Table 3.

L

0 -

uo

0"

Fig. 8: Pressure distribution for various e(0,z)

Figure 8 indicates the location of reformation and rupture front for various degrees of starvation. The area of the full film reduces when the inlet condition is less than unity. In fact the formation of the complete film occurs later when there is not enough fluid in the bearing. The film rupture is quite identical.

120-

LOAD (1 0' N)

80I

PMAX (1' 0 Pa)

f / =

=

0

ATTITUDE

AXIAL FLOW(lod m' 1s)

0 0.2

0.4

0.6

0.8

1.0

Degree of Starvation

1.2

Fig. 9: Performance of journal bearing Figure 9 shows that the performance parameters do not appreciably change, compared to the flooded

LID R C

P

P O Pd 0

P

m m Pa.s N/m2 N/m2 rads N/m2

0.626 0.1015 82.55~10~ 0.0 15 0 4.1~10' 62.84 (600 rpm) 1.8~10~

For the bearing without groove, the trajectories of the journal center of the connecting-rod axis are presented in Figure 10. The journal trajectory calculated with the Elrod's algorithm resembles to the trajectory predicted classically. The minimum film thickness is presented in Figure 11. The results are quite identical. The predicted minimum film thickness is 8.74 microns for the method considering cavitation and 8.46 microns without Elrod's algorithm. Figure 12 is a presentation of the journal center orbit for the connecting-rod bearing with a full circumferential groove. The groove is maintained to a pressure supply equal to 0.41 MPa. The trajectory calculated with Elrod's algorithm is more flat that the one plotted not taking account of cavitation. In this case, the minimum film thickness variations predicted by the two methods are presented in Figure 13. The gap between the two methods is very low, the minimum film thickness is 3.11 microns without cavitation algorithm as it is 3.22 microns taking account of cavitation.

462 These values of minimum film thickness are list in Table 4 and compared with some results obtained in the literature considering or not cavitation effects. The agreement between all results is very good. Furthermore, the minimum film thickness are sliehtlv ereater when the calculations are done with the cavitation algorithm.

an inversion of the cavitated area at this time and there is not any pressure against the shaft moving.

-& without Elrod's algo.

Table 4: Minimum film thickness (10"m) NO GROOVE no cavitation This study no cavitation This study cavitation FULL CIRCUMFERENTIALGROOVE no cavitation 3.36 Goenka no cavitation 3.11 This study 3.60 Vijayaraghavan cavitation cavitation 3.22 This study Figures 14 and 15 give the pressure profiles for the two method of calculation. Their are obtained for a crank angle equal to 68' and for a 1 1 1 circumferential groove. Over these figures, the black line represents the positive pressure boundaries.

Fig. 10: Trajectory of journal center

6. CONCLUSION

In the bearing, the study of gaseous cavitation area is essential. In the static case, the effect of a finite grooved bearing for starved bearing has been analysed. The Elrod's algorithm method is enough simple to use and automatically implements cavitation boundary conditions at film rupture and reformation. Nevertheless, this study can not be directly introduced for the problems of journal bearing under dynamic loaded. Because, in the dynamic case, the area of cavitation is not fixed and moved inside the bearing after every time step. The procedure uses the mobility method to solve the problem. This method is not always stable considering cavitation. Particularly, for low supply pressure, the velocity of the shaft becomes too important in the end of the load cycle. This is due to

+without Elmd's algo.

-+-

0

120

with Elrod's algo.

240 360 480 600 Crank Angle (deg)

Fig. 11: Minimum film thickness

720

463

+ without Elrod's algo.

--+-

with Elrod's algo

Fig. 14: Pressure profile with Elrod's algorithm

Fig. 12: Trajectory ofjournal center with groove

- 401 fi

W

35

+without Elrod's algo. +withElrod'salgo.

I Fig. 15: Pressure profile without Elrod's algorithm

REFERENCES

0

120

240

360

480

Crank Angle (deg)

600

720

Fig. 13: Minimum film thickness with groove

1. Dowson, D., and Taylor, C.M., "Cavitation in Bearings", Annual Reviews of Fluid Mechanics, Annual Reviews, Palo Alto, pp.3566,1979. 2. Booker, J.F., "Classic Cavitation Models for Finite Element Analysis", Current Research in Cavitating Fluid Films, Eds. D.E. Brewe, J.H.Bal1, and M.M. Khonsari, STLE Special Publication SP-28, pp.39-40, 1990.

464 3. LaBouff, G.A., and Booker, J.F., "Dynamically Loaded Journal Bearings: A Finite Element Treatment for Rigid and Elastic Surfaces", ASME Journal of Tribology, Vol.107, pp.505515, 1985. 4. Oh, K.P., and Goenka, P.K., "The Elastohydrodymmc Solution of Journal Bearings under Dynamic Loading", ASME Journal of Tribology, Vol. 107, pp.389-395, 1985. 5 . Kumar, A., and Booker, J.F., " A Finite Element Cavitation Algorithm", ASME Journal of Tribology, Vol. 113, pp.276-284, 1991. 6. Jones, G.J., "Crankshaft Bearings: Oil Film History", Tribology of Reciprocating Engines, Buttenvorths, London, England, pp.83-88, 1983. 7. Elrod, Jr., H.G. and Adams, M.L., "A Computer Program for Cavitation and Starvation Problems", Cavitation and Related Phenomena in Lubrication, Mechanical Engineering Publications, New York, pp.37-41, 1974. 8. Elrod, Jr., H.G, "A Cavitation Algorithm", ASME, Journal of Lubrication Technology, Vol. 103, pp 350-354, 1981. 9. Dowson, D., Miranda, A.A.M., and Taylor, C.M., "Implementation of an Algorithm Enabling the Determinationof Film Rupture and Reformation Boundaries in a Film Bearing", Proceedings of 10th Leeds-Lyon Symposium of Tribology, Butterworths, U.K., Paper III(ii), 1984. lO.Dowson, D., Taylor, C.M., and Miranda, A.A.M., "The Prediction of Liquid Film Journal Bearing Performance with Consideration of Lubricant Film Reformation Part I: Theoretical Results, Part 11: Experimental Results", Proceedings of Institution of Mechanical Engineers, Vol.199, No. C2, Institute of Mechanical Engineers, pp.93-Ill, 1985. ll.Brewe, D.E., "Theoretical Modeling of the Vapor Cavitation in Dynamically Loaded Journal Bearings", ASME Journal of Tribology, Vol. 108, pp.628-637, 1986. 12. Woods,C.M., and Brewe, D.E., "The Solution of the Elrod Algorithm for a Dynamically Loaded Journal Bearing Using Multigrid

Techniques", A W E Journal of Tribology, Vol. 111, pp.302-308, 1989. 13.Vijayaraghavan, D. and Keith, Jr., T.G., "Development and Evaluation of a Cavitation Algorithm", STLE Tribology Transaction, V01.32, NO. 2, pp.225-233, 1989. 14.Vijayaraghavan, D. and Keith, Jr., T.G., "Grid Transformation and Adaptation Techniques Applied in the Analysis of Cavitated Journal Bearings", A N E Journal of Tribology, Vol. 112, pp.52-59, 1990. 15.Vaidyanathaq K. and Keith, Jr., T.G., "Numerical Prediction of Cavitation in Noncircular Journal Bearings", STLE Tribology Transaction, Vo1.32, No. 2, pp.215-224, 1989. 16. Vijayaraghavan, D., Brewe, D.E., and Keith, Jr., T.G., "Effect of Outsf-Roundness on the Performance of a Diesel Engine Connecting-Rod Bearing", A W E Journal of Tribology, Vol. 115, pp.538-543, 1993. 17,Vijayaraghavan, D. and Keith, Jr., T.G., "Analysis of a Finite Grooved Misaligned Journal Bearing Considering Cavitation and Starvation Effects", ASME Journal of Tribology, Vol. 112, pp.60-68, 1990. 18.Vijayaraghavan, D. and Keith, Jr., T.G., "Effects of Type and Location of Oil Groove on the Performance of Journal Bearings", STLE Tribology Transaction, Vo1.35, No. 1, pp.98-106, 1992. 19.Kumar, A. and Booker, J.F., "A Finite Element Cavitation Algorithm: ApplicationNalidation", ASME Journal of Tribology, Vol.113, pp.255260, 1991. 20. Goenka, P.K.,and Paranjpe, R.S., Discussion of the paper "A Finite Element Cavitation Algorithm" by A. Kumar and J.F. Booker, A W E Journal of Tribology, Vol. 113, pp.284285,1991. 21. Yu, Q. and Keith, Jr., T.G., 'I A Boundary Element Cavitation Algorithm", STLE Tribology Transaction, Vo1.37, N0.2, pp.217-226, 1994. 22. Booker, J.F., "Dynamically Loaded Journal Bearings: Numerical Application of the Mobility Method", ASME Journal of Tribology, series F, pp. 168-174, 1971