The Effect of Compliance On Peak Oil Film Pressure in Connecting Rod Bearings

Thin Films in Tribology I D. Dowson et al. (Editors) Q 1993 Elsevier Science Publishers B.V. All rights reserved.

149

The Effect of Compliance On Peak Oil Film Pressure in Connecting Rod Bearings D.N.Fenner J.D.C.McIvor J.M.Conway-Jones H.Xu

Dept of Mechanical Engineering, King's College, University of London. Structural Mechanics & Dynamics, Sydney, Australia. Glacier Vandervell Ltd, London. T&N Technology Ltd, Rugby.

An elastohydrodynamic study of a highly-loaded bearing fatigue test rig shows that elastic deformation significantly increases the extent of the oil film and leads to a marked reduction in the maximum oil film pressure. Predictions for the ratio of peak oil film pressure to maximum specific load are reduced from about 10 when the bearing is assumed rigid, to approximately 2 for an elastic housing. At maximum load there is a marked reduction on the oil film thickness towards the bearing edges and a flattening of the pressure variation in the axial direction. These effects should be taken into account in less rigid automotive big-end bearings when interpreting bearing fatigue test data.

1. INTRODUCTION One of the criteria for the acceptance of a lubricated bearing material for a given duty is its fatigue strength under dynamic load. In the case of connecting-rod bearings maximum specific loads have increased over the years from 12-15 MPa for babitted white metal bearings to 35-50 MPa for current gasoline engines and 50-70 MPa for turbocharged truck diesel engines. In order to obtain fatigue data in less than the normal life expectancy of a modern engine, bearing fatigue test rigs may be run at even higher specific loads. When both bearing and shaft are assumed rigid the peak oil film pressures in these test rigs are typically predicted to be in the range 6-12 times the maximum specific load. However, it is well known that at such high pressures elastic deformation of the housing becomes significant and an elastohydrodynamic (EHD) analysis of the bearing is required. In this work the effects of elasticity on the predicted performance of a highly-loaded bearing fatigue test rig are assessed using the finite element (FE) based method for EHD analysis developed by the first two named authors [l-31. 2. EHDANALYSIS

A number of numerical methods, employing either a finite difference or finite element approach,

have been developed for solving dynamically loaded EHD bearing problems. These methods are based either on the short bearing theory [4-81 or entail a two-dimensional solution of Reynolds equation [2,3,9-111. The non-linearity of the EHD problem, resulting from the coupling of the elastic deflection of the bearing housing with the pressure derived from Reynolds equation, means that an iterative scheme is required for the solution of oil film pressures and thicknesses at each time step in the loading cycle of the bearing. The Newton-Raphson (N-R) method has been successfully applied to the problem [lo] and forms the basis of a robust and efficient solution algorithm. Even so, a prohibitive amount of computer CPU time may still be required for a converged solution. It has been shown recently [2,3], however, that use of higher-order rather than the more usually employed linear finite elements to represent the oil film, together with a fast N-R method can considerably reduce the CPU time required. The result is that EHD analysis, once felt to be an expensive luxury, can now be adopted as a practical tool for bearing design.

2.1 Governing equations For an incompressible, isoviscous, Newtonian fluid the two-dimensional distribution of pressure in the oil film satisfies Reynolds equation in a (e ,z) co-ordinate system (see Fig. 1) given by

150

In the FE formulation of the problem the film thickness at node i with a circumferential coordinate 8 , , is written as h,

=

c - (exsine,+eycos8,)+6 ,

(5)

where 6 , is the radial displacement at node i . With the aid of a structural FE model of the bearing housing the radial components { F } of the forces at nodes on the bearing surface are related to the displacements

16 }

at these nodes through

{ F } = [ K ] { 8 } where [K]is the stiffness matrix. The nodal forces can be expressed in terms of the

pressures { p } at nodes on the bearing surface In this equation p = p ( e , z , t ) and h = h ( 8 , z , t ) are the oil film pressure and thickness at time t, respectively, p is the viscosity, D is the bearing diameter and U = U ( t ) is the relative velocity of the bearing surfaces. If the problem is symmetric about the bearing centre-line z = 0 then the solution domain can be restricted to 0 5 z 5 L 1 2 , where L is the bearing width, and the pressure is required to satisfy the boundary conditions

p

-_

=

at

0 at the bearing edge, z

=

- 0 at bearing centre-line,

L =

where [C] = [K]-' [ M ] is the pressure compliance matrix for nodes on the bearing surface, from which it follows that

(2)

2

z

through I F } = [ M ] { p } where the matrix [MI depends upon the shape functions of the elements used [2]. Hence the nodal displacements can be written as

0

(3) where {

The oil film thickness, h, is given by

h = c - ( e x s h e + e y c o s 8 ) +6

1 is the i 'th row of the matrix [ C] .

The components Wx and W, of the external (4)

where the first term, c, is the nominal radial clearance. The second term is due to the motion of the journal in the clearance space, where e, and ey are the x and y components of the eccentricity vector. The third term, 6 , is the radial elastic displacement of the bearing housing resulting from the oil film pressure on the bearing surface.

load, W , on the journal at any instant in time must be exactly balanced by the forces found by

integrating the pressure forces acting over the bearing surface area A . Thus

1pcose rdedz 1psino rdedz

=

A

A

=

W,

151

2.2 Newton-Raphson method If p"), ez('), ey(') denotes the r'th trial solution for the oil film pressure and journal eccentricity the N-R method can be used to obtain the ( r + 1) 'th trial solution given by

(9)

where ~ p ', A~ez(r) ) , A ey(r)are the corrections. The Galerkin method of weighted residuals is used to obtain the finite element formulation of the problem. For a mesh containing n nodes the correction vector

[ A ] { XI

=

I RI

is used to decompose the solution domain into : i) Lubricated region where Reynolds equation (1) is satisfied and : f = 0 and p > 0 (12) ii) Cavitated region where : fsOandp=O (13) Using the Murty algorithm [13] a trial decomposition of the solution domain is set up by assigning an initial status to each node in the mesh, and the EHD problem is then solved for p and f . The status of any node where the complementarity conditions (12) and (13) are not satisfied is then altered and further iterations are performed until convergence is achieved. 2.4 Finite element meshes A mesh of eight-node isoparametric elements (see Fig.2) is used to model the oil film, these having been shown [ 11 to be computationally more

(10)

where the matrices [ A ] and { R } are given in [2]. If the elements of the correction matrix { X} tend to zero as the sequence of iterations proceeds then the method is convergent. The solution is advanced to the next time step using implicit time-stepping. One of the most time consuming operations in the N-R method is the reformulation of the matrix [ A ] after each iteration. In an attempt to minimise computing time the first two named authors have developed a fast N-R method [2,3] in which a complete reformulation only takes place when the rate of convergence becomes unacceptably slow.

2.3 Cavitation

The boundary separating the lubricated and cavitated regions of the oil film is determined using the complementarity formulation [ 121 in which the net flow rate, f , at any pojnt in the fluid, given by f

=

v ( h 3 v p ) -1 2 p

Fig 2. F.E.mesh for oil film efficient than the linear elements normally used. The pressure compliance matrix [C] is obtained using 20-node elastic brick elements for the bearing housing.

3. PREDICTIONS FOR BEARING FATIGUE TEST RIG The Sapphire bearing fatigue test rig, details of which are given in the Appendix, is used for comparing the fatigue strength of bearing materials employed in reciprocating engines. The load case listed has a maximum specific load,

152

MPa. At maximum load the elastic bearing housing conforms with the journal over an arc of approximately 100 degree extent (Fig.5). This is comparable with the Hertzian contact condition modified by the classic calculation for EHD lubrication [ 141. The extent of the load carrying 90 80 70

p

-50

440 5

-€ 3 0 .L 2 0

10 0 -180 -135 -90 4 5

Fig 3 Half-width model of bearing housing

P = Wmax I (DL)of 109MPa and will cause fatigue cracking of aluminium bimetal bearings after about 25x106 load cycles. The finite element mesh used to obtain the compliance matrix for the Sapphire bearing housing is shown in Fig 3 and the location of the constraints is shown in Fig 14. On the bearing surface there are 60 nodes in the circumferential direction and 11 across the half-width. The journal is assumed to be rigid.

0

45

90

1 3 5 180

Bearing angle, deg.

Fig.5 Oil film thickness variation along bearing centre-line at maximum load. region of the oil film is approximately 5 times greater than that predicted for the rigid bearing (Fig.6). When the deformation of the journal is included it is expected that the reduction in p,, due to elasticity will be even more pronounced. 1400

1400,

I

I

I

1200

I

g loo0 800

g

In600 g400 200 0 -180-135-90 -45 0

0

90

180

270

360

Crank angle, deg.

45 90 135 180

Bearing angle, deg.

Fig. 4 Maximum oil film pressure variation

Fig 6 Oil film pressure variation along bearing centre-line at maximum load.

The predictions for the variations of the maximum oil film pressure, p,,,, obtained using rigid and EHD analyses of the Sapphire bearing are shown in Fig.4. Elasticity reduces the predicted maximum oil film pressure from 1262MPa to 223

The variation of the ratio p,, I P with P is shown in Fig.7 where it is clear that, as well as the ratio having a much lower value, the trend is reversed for the elastic housing. Predictions for a rigid housing show an increase in the ratio with P

153

The variation of the oil film thickness in the axial direction is found to be very sensitive to changes in the housing geometry. In some test procedures it is common practice to increase the maximum specific load by reducing the width of the bearing shell, whilst maintaining the original width of the bearing housing. This has the effect of concentrating the load around the centre-line of the housing and leads to an increase in the relative curvature of the axial profile since the outer part of

100

3 a

10

a

1

10

100

P. MP.

250

Fig 7 Variation of maximum oil film pressure with maximum specific load whilst predictions for an elastic housing show a decrease with P as deformation spreads the load carrying film over a greater arc. At maximum load the oil film thickness is relatively uniform (Fig.5) along the centre-line of the bearing within the deformed arc, and is more than 5 times that predicted for a rigid bearing. There is, however, a marked reduction in oil film thickness towards the edge of the bearing (see Fig 8) where the minimum is similar to that for the rigid case. The reduced film thickness at the bearing edge restricts oil flow and leads to a flattening of the pressure variation in the axial direction (Fig 9). This is in marked contrast to the parabolic pressure variation assumed in short bearing theory [4-81. 2.5

s

,

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

2

200 0

a

I

!?2

z2

150

100

a

50

0

0

0.1

0.2

0.3

0.4

0.5

z/ L

Fig. 9 Axial variation of pressure at 8 =O deg. for maximum load. the housing is not loaded but can exert an exaggerated restraint on the effective edges of the bearing (Figlob). An EHD analysis of one such bearing predicts the occurrence of contact at the edges during the period of maximum load. This is in accordance with observations made during bearing tests which show clear evidence of metalto-metal-contact at the edges.

m-

g 1.5 c

Y

. v_

1

.?

0.5

5 t

01 0

2 IL

Fig. 8 Axial variation of oil film thickness at 8 =0 deg. for maximum load.

Fig. 10 Bearing housing geometry: (a) full-width bearing shell, (b) reduced-width bearing shell, (c) automotive big-end.

154

The empirical development of automotive connecting rods has resulted in relief of the back of the bearing housing on the rod side (see Fig.10~). This allows the edges of the bearing to deflect away from the journal thereby counteracting the tendency for the oil film thickness to diminish towards the edges. Excessive relief may, however, lead to unacceptably high stress concentrations in the central region.

4. PIEZO-VISCOUS EFFECTS

piezo-viscous effects into the solution procedure is currently in progress. 250

H

200

zln

150

s

100

$

.-5

s

50

0 0

It is well known that at high pressures the viscosity of lubricating oil may increase by a factor of 10 or more. The relationship commonly used is the Barus equation

180

90

270

360

Crank angle, deg.

Fig. 11 Maximum oil film pressure variation

7 .

where

p

is the viscosity at pressure p ,

po

is the

viscosity at atmospheric pressure p = 0 and a is the piezo-viscosity coefficient. Data for typical engine oils [15] up to pressures of 200 MPa indicate that a reduces with oil film pressure, p (MPa), and with temperature, T ("C), in a manner which can be represented by

0

0.1

0.3

0.2

0.4

0.5

ZIL

Integration over the pressure contours predicted for the elastic Sapphire bearing at the instant of maximum load indicates an instantaneous mean viscosity of 41cP over the conforming area of the bearing and journal. Results obtained using this increased viscosity for the whole load cycle indicate that there is very little change in the extent of the deformed arc and the peak oil film pressure is only increased by about 3 % as shown in Fig.11. However, Fig 12 shows that at maximum load the oil film thickness is more than doubled at the centre-line of the bearing and is more than 10 times thicker at the edge. Since the pressure is high for only about 30% of each cycle the effect is likely to be less than that indicated here. Incorporation of

Fig. 12 Axial variation of oil film thickness at 8 =O deg. for maximum load.

5. CONCLUSIONS The actual mechanism of cracking of the surface layer of lubricated bearings is not completely understood. It is becoming clear, however, that it arises from a combination of stresses which are primarily related to the condition of contact stress. The results presented in this work show the importance of using EHD analysis to predict the performance of bearings in a relatively rigid, highly-loaded test rig. It will be important to take into account the effect of flexibility in less rigid automotive type big-end bearings when interpreting results from fatigue test rigs and this matter is currently in hand.

155

Predictions for the peak oil film pressure, obtained assuming a rigid bearing, will not only greatly exaggerate the severity of conditions but will also give the wrong trends. Allowance for the flexibility of the shaft is likely to further reduce the severity of the peak oil film pressure and the stresses in the bearing. Results obtained for the axial variations of pressure and film thickness clearly indicate the inadequacies of the assumptions in short bearing theory. The omission of an allowance for piezoviscous effects has a relatively small effect on the computed peak oil film pressure and consequently on the stresses likely to cause fatigue cracking. The effect on the predicted minimum oil film thickness is, however, significant.

Modal analysis of elastohydrodynamic lubrication: a connecting rod application, ASME J.Tribology,

REFERENCES

112, 524-534, 1990. 12. Oh, K.P., The

1. McIvor, J.D.C., & Fenner, D.N., An evaluation of eight-node quadrilateral finite elements for the analysis of a dynamically loaded journal bearing, Proc.I.Mech.E., Part C, 202(C2), 95-101, 1988. 2. McIvor, J.D.C., The analysis of dynamically loaded flexible journal bearings using higher-order finite elements. PhD thesis, University of h n d o n , 1988. 3. McIvor, J.D.C., & Fenner, D.N., Finite

element analysis of dynamically loaded flexible journal bearings: a fast Newton-Raphson method, ASME J. Tribology, 111,597-604, 1989. 4. Fantino, B.& Frene, J. Comparison of dynamic behaviour of elastic connecting rod bearings in both petrol and diesel engines, ASME J. Tribology,

Partl:theory, Proc.I.Mech E., Part C, 205(C2), 99-106, 1991. 8. Xu, H. & Smith, E.H, A new approach to the solution of elastohydrodynamic lubrication of crankshaft bearings, Proc.1.Mech.E. ,204, 187-197, 1990. 9. LaBouff, G.A., & Booker, J.F., Dynamically

loaded journal bearings: a finite element treatment for rigid and elastic surfaces, ASME J.Tribology, 107, 505-515, 1985. 10. Oh, K.P., &

Goenka, P.K., The elastohydrodynamic solution of journal bearings under dynamic loading, ASME J.Tribology, 107, 389-395, 1985. 11. Kumar, A., Goenka, P.K. & Booker, J.F.,

numerical solution of dynamically loaded elastohydrodynamic contact as a nonlinear complementarity problem, ASME J.Tribology, 106, 88-95, 1984. 13. Murty, K.G., Note on a Bard-type scheme for solving the complementarity problem, Opsearch, 11,123-130, 1974. 14. Dowson, D. & Higginson, G.R., Theory of

roller-bearing lubrication and deformation, Proceedings of 1963 Lubrication and Wear Group Convention, 216-227, I.Mech.E., London, 1964. 15. Hutton, J.F., Jones, B. & Bates, T.W. Effects of isotropic pressure on the high temperature shear rate viscosity of motor oils, SAE Paper 830030, 1983.

APPENDIX

107, 87-91, 1985

5. van der Tempel, L.,Moes, H. & Bosma, R., Numerical simulation of dynamically loaded flexible short journal bearings, ASME J. Tribology, 107, 396-401, 1985. 6. Goenka, P.K. & Oh, K.P., An optimal short bearing theory for the elastohydrodynamic solution of journal bearings, ASME J.Tribology, 108, 294299, 1986. 7. Aitken,

M.B. & McCallion, H., Elastohydrodynamic lubrication of big-end bearings

In the Sapphire hydrodynamic bearing fatigue test rig (see Fig 13) dynamic loading is applied to a connecting-rod bearing. The test bearing engages with a journal which rotates with a small (0.381mm) eccentricity at 3000 rev/min. The connecting rod is attached to a double-acting piston operating in an oil-filled cylinder with relief valves to control the pressure. Adjustment of the valves enables the foward and reverse loads on the bearing to be controlled, as indicated by a strain

156

m Reverse load

52.9mm 28.7mm 0.07mm 3000 rev/min 4.6cP 109 MPa 0.381mm 259 mm

Diameter Length Diametral clearance SPd Effective viscosity Maximum specific load Crank length Con-rod length

I 1

I

Forward load

:

6

Fig. 13 Sapphire bearing fatigue test rig gauge on the connecting-rod. The load is increased stepwise after inspections at 20 hour (3.6~10~ load cycles) intervals until fatigue cracking is observed. Oil is supplied through a 6mm diameter hole in the bearing cap half at 45' from the split line. The oil used is Shell Rotella T 1OW with no viscosity index improvers. The dimensions of the bearing housing are shown in Fig. 14 and the bearing data and loads are given in Table 1.

.

-.

crank angle deg. 0

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340

wx N

wy

N

165322 158721 133605 8996 41045 -2816 -16294 -19455 -14043 0 -3920 -45 -903 -84 455 -114 7730 -129 9130 -129 27772 -114 70052 -85 112368 -45 149430 0

45 85 114 129 129 114 84 45

crank wx angle N deg

.

10 30 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350

wy

N

23 162676 66 149474 101 117751 124 6616 131 1461 123 -7229 101 -18201 66 -16741 23 -8707 -23 -2336 -66 -946 -101 1842 -123 7637 -131 17256 -124 47581 -101 99168 -66 133546 -23 158694

-~

I

w 31.0

128.0

M 66.0

area of constraint

Fig 14 Sappire bearing housing (all dimensions in mm)

*

Table 1 Bearing data and loads