Linear deformation of a journal bearing and its relationship to hydrodynamic pressure

Wear, 115 (1987)

47

41 - 52

LINEAR DEFORMATION OF A JOURNAL BEARING RELATIONSHIP TO HYDRODYNAMIC PRESSURE

YATAO

AND ITS

ZHANG

Division of Machine

Elements,

Chalmers

University

of Technology,

Gb;teborg (Sweden)

Summary

This study reconsiders the assumption of a linear relationship between oil film pressure and bearing liner deformation. The study begins by presenting a method for determining the liner deformation by means of a general solution of the stress function in the theory of elasticity. The functions of the oil pressure and deformation are expanded into a Fourier series after which numerical methods are applied. By calculating the pressure distribution and the displacement, the relationship between them is investigated. All the results fit a linear relationship but the slope and intercept of the linear equation vary with the parameters cy, /3, E and Y in a complicated way. For a Poisson ratio of v = 0.3 two approximate equations are given based on the computed results.

1. Introduction

In heavily loaded journal bearings the deformation of the bearing liner becomes a significant factor which cannot be ignored in the evaluation of bearing performance. Some previous studies have shown that there are great variations in the data, such as the load capacity, friction coefficient, maximum pressure etc., on undeformed journal bearings and deformed journal bearings. For example, with the same shaft eccentricity ratio the maximum pressure decreases by about 30% if the deformation is taken into account. In the field of full journal bearings Higginson [l] was the first to investigate the deformed liner. He used a linear force-displacement relationship for the bearing liner deformation. Conway [Z] developed Higginson’s model further and considered variable viscosity in the calculation of bearing characteristics. With the development of numerical techniques, Jain and Sinhasan [ 3, 41 and Oh and Huebner [ 51 adopted the finite element method to determine both the liner deformation and the oil film pressure distribution. O’Donoghue [6] also employed the elastic theory and expanded the *Paper Technology,

presented at the Nordic Symposium Luleg, Sweden, June 15 - 18, 1986.

0043-1648/87/$3.50

@ Elsevier

on Tribology,

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42

oil pressure into a Fourier series but their solution seems quite different from that presented here even if p reaches zero (or the thickness of the liner is infinite). Of the three methods discussed above, the linear model is, of course, the simplest method to apply. Its advantages are not only the simplicity in computation but also the easy treatment in the analysis of bearing performance. The finite element method takes a lot of computer time, The application of the three-dimensional theory of elasticity seems to be complicated in mathematical treatment. Although the first use of the linear model appeared almost 20 years ago, there has been no study to show whether the model is realistic or not. The following questions arise from further consideration of the linear model. (1) Is the deformation calculated using the linear model coincident with the real bearing deformation? (2) What conditions in a journal bearing operation introduce the quantitative errors between the linear model and the actual deformation? (3) What are the slope and intercept of a linear equation of the displacement us. the oil pressure if there is approximate linearity? The present study employs a general solution for the two-dimensional polar stress function in finding the displacement of a bearing liner. The purpose of the study is to find the actual relationship between the oil film pressure and the liner displacement and to calculate some characteristic values of the bearing operations. Some assumptions are made in the study. The housing outside the bearing liner is considered to be rigid. This assumption was also made by Conway, Jain and Sinhasan [2 - 41 and seems very reasonable because the rigidity of most liner elements is close to une order of magnitude lower than that of the housing. Another assumption made in the study is that the shear forces at the inner surface and the outer surface of the bearing liner are zero. Considering the shear forces, the only shear force exerted on the liner is the viscous shear force. In a normal situation of a fully lubricated bearing this kind of force is only about 0.5% of the total load. It is regarded small enough not to cause an error beyond the precision we require. For the same reason the shear force at the outer surface is also neglected. The shaft in the present analysis is considered to be rigid. The analysis is limited to one-dimensional fluid flow. A plane stress relationship is adopted for the behaviour of the liner. Poisson’s ratio is 0.3.

2. Basic theories 2.1. Theory of elastic deformation The deformation of the bearing liner in this study is limited to two dimensions. The general solution to the problem of a stress function in polar coordinates found in ref. 7 is

43

a,rO sin 8

$ = a, In r + b,r2 + cOr 2 In r + d,r 28 + ahO +

+ (b,r3

+ a’,r-’ + bir In r) cos 6 -

2

c,rd cos 8 .___-2

+ (d,r3 + c’,r-l + dir In r) sin 0 + 2

(a,?

+ b,rn+2

+ a:r-”

n>2

+ bLr-“+2)

cos n0 + C (c,r”

+ d,r” c2 + cLr_” + dLr-“+2)sin

n0

(I)

n>2

To evaluate this equation fully, we need four independent boundary conditions because the coefficients before every sine and cosine term comprise four constants a,, b:, a,!, and b; or c,, d,, CA and d; (n = 0, 1,2, 3, . ..). Assuming zero shear stress at the inner surface and the outer surface of the bearing liner and zero displacement at the outer surface, gives the following three boundary conditions: 7,,9(a,e) = 6

(2)

r,e(b,o)

(3)

u(b,8)

=6 = 0

The stresses and be derived from inated. After the be written in the

(4) the displacement are related to the stress function and can eqn. (1). Therefore three of the four constants are elimelimination the radial stress at the liner’s inner surface can following form:

a,(a,B)=A,a,+Alalcos8+B,c,sinC3-

c

A,a,cosn8-

n>2

C

B,c,sinn~3

n>2

(5) In eqn. (5) A,, A,, B,, . . . A, and B, only depend on Poisson’s ratio V, the inner radius a and the outer radius b. In the same way we can write the radial displacement u(a,O) = F,aO + Fia, cos 0 + G,c, sin 8 + C F,,a,, cos ni3 + C n>2 n>2

G,c,

sin no (6)

F, and G, have the same meaning as A,, A 1, B,, . . . A, and F,, G,, .-. Detailed expressions for eqns. (5) and (6) are given in ref. 8. In these two equations only two sets of the constants are unknown. They are a,,, a,, Cl, . . . a, and c, and will be found with the fourth boundary condition F,,

B,.

44

rigid

housing

bearing

liner shaft

Fig. 1. Journal bearing geometry.

a&e) = -P(e)

(7)

We expand the oil pressure into a Fourier series and compare the two sides in eqn. (7). Then the coefficients ao, a,, cl, . . , a,, and c, can be found. 2.2. Lubrication theory of journal bearings Figure 1 is a diagram showing the bearing, shaft and angular coordinates. When we consider a deformable bearing the film thickness is expressed by h = C(1 + E cos 0) + u(a,B)

(8)

Reynolds’ equation for a one-dimensional problem with an isoviscous and incompressible fluid in polar coordinates is (the assumptions for the derivation of this equation are stated in ref. 9).

(9) where the constant h,, is determined by dp 0 z =

when h = h,

It is assumed that the bearing is supplied with enough oil from a groove which is located at 8 = 0 and the pressure of the oil supply and the pressure in the cavitation zone are zero since these two pressures are small compared with the oil pressure supporting the total load. The boundary conditions for Reynolds’ equation become P(0) = 0 and

ate=0

(10)

45

d&e,)

P(el)=

--$-

According

o

at0=8,

=

to the boundary

(11)

condition,

the angle 8i will be determined

by

‘1 h-ho

s 0

h”-

(12)

de=o

By non-dimensionalizing obtain

the film thickness

and the oil film pressure,

where h = h/C and h, = ho/C. The oil pressure can be written in the form of a Fourier coefficients are computed using numerical methods p(O) = PO + C (p,, n> 1 Finally,

cos n8 + Is,, sin no)

the non-dimensional

displacement

we

series whose (14)

of the inner surface is given

by u(a,8) = oti(p,e)

(15)

where ii(fl,e) = ii, + uic cos8+ii,,sin8+

2 (ti,,cOSne+ii,,sinnO) 12> 2

(16)

and 2(/3,0) = u(a,O)/C, LY= ~TJ/E(u/C)~ and 0 = u/b. quantities In eqn. (16) UO, iii,, Uls, . . . ii,, and ii,, are non-dimensional and only dependent on v, ,!?and p. Detailed expressions are given in ref. 8. After the pressure distribution is found, the non-dimensional load capacity and the friction coefficient are easily calculated. f’ 1

i& =

_I-p(e)

cos

e de

0

and w, =

i

01 p(0) sin 8 d8

(17)

0

jg

=

( Wt”

+

Wn2)1/2

(16)

and

(19)

46

TABLE 1 Comparison of results (data: (II= 0.075; Number

of terms

Angle of pressure zone cp Attitude angle y Total load W Maximum pressure 3 Minimum film thickgess hmh Friction coefficient f

3. Numerical computation

fl= 0.8; E = 0.8) 31

41

51

219.124 38.534 11.056 8.662 0.3130 0.7919

219.212 38.560 11.055 8.658 0.3130 0.7923

219.122 38.539 11.057 8.660 0.3135 0.7920

and discussion

3.1. Computational considerations It is quite clear that the displacement ii and the pressure fi have a mutually functional relationship. In the computation a direct iterative method is used as shown in the following: Pressure -

Displacement

---+ Pressure -

Displacement

This procedure is continued until the oil film pressure terion for the convergence applied here is

----+ Pressure converges.

The cri-

jE@ikAOi) _- E:(iiii”+ ‘A6i)l < 0 005 .__(20) ’ * W% k+lA&) In the selection of the Fourier series terms for both the pressure and the displacement we prefer to adopt as few terms as possible without losing the accuracy of the results. 31 terms are used for the pressure and the displacement because, when the number of the terms is greater than 31, the values of the displacement series coefficients are actually less than 10m4. Table 1 shows the comparison of some values which are computed with 31, 41 and 51 terms. The maximum deviation is about 0.2%. The parameter o is evaluated according to the following data: E = 5 X 10” N m-* (Cu-Pb); N = 1000 rev min-’ (w = 105 s-l); a/C = 1000; v = 0.04 Pa s; (x = q0/E(a/C)3 = 0.084. 3.2. Linear relationship Figures 2 - 4 show the relationship between the pressure and the liner displacement at the same location 8 in the bearing where the pressure and the displacement are in equilibrium. All the curves show a good linear relationship. The figures also prove that the linear force-displacement assumption is correct and can be adopted in the analysis of the bearing distortion caused by the hydrodynamic pressure. The following important points can be deduced from these figures. Firstly, the slopes of the linear equations vary with the radial ratio /3 and

47

u

ii 0.25

0.25 /

a=0.1000

6=0.75

I/ 0.20

0.20

-0.0875 =0.0750 =0.0625

cl.13

0.15

0.10

0,lO

11.05

0.05

0.0 0

I

,

5

10

I

I

15

~

I

20

25

0

5

F

Fig. 2. Displacement

us. pressure

(CY= 0.075, E = 0.9 and Y = 0.3).

Fig. 3. Displacement

us. pressure

(@= 0.8, E = 0.9 and v = 0.3).

us. pressure

(a = 0.075, p = 0.8 and v = 0.3).

/

I

10

/=0.0500

I

15

20 F

ii 0.15

0.10

0.05

Fig. 4. Displacement

the compliant coefficient ~11.The increase in LYand the decrease in fl will increase the slope. Secondly, the slope is almost independent of the eccentricity ratio according to Fig. 3. Thirdly, the displacement is not zero at the zero pressure point. This effect is attributed to Poisson’s ratio.

48

The figures clearly demonstrate that the relationship pressure and the displacement can be expressed well by ii = tan(K

+ iii

between

the (21)

The question is, what are tan K and iii for the different values of cy, /3 and v? In Higginson and Conway’s studies the following formulae were used respectively for the slope (see refs. 1 and 2). tEUlK=

-

(22)

and tan

K =

(23)

where t is the thickness of the liner. These two formulae will give different values of the slope. Theoretically analysing the bearing lubrication characteristics, it is not necessary to know an exact expression for the slope but for practical applications it seems important to have a more accurate expression. 3ased on the results obtained from the present study and statistical methods, an approximate expression was formulated for a Poisson’s ratio of 0.3. tan

K =

1.313(0.97

- /3)(~

(24)

This coefficient is related to Poisson’s ratio. In the present paper v = 0.3. This is a very limited case and further study is needed. In the study 60 cases were checked with this formula. The checked slopes were calculated with the elastic theory as described in Section 2 and it was found that the maximum error was less than 1.5% (0.4 < E < 0.85, 0.6 < /3 < 0.9 and 0.05 G (Y < 0.10). A more complicated relationship exists between the three parameters (;Y,/I, E and the intercept than for the slope Ui = 5.556(1.0212

- p)(O.2877 - OI)(~- 0.0445)a

It should be noted that if we neglect it will yield about 5% error.

the intercept

(25) in the linear equation,

3.3. Some characteristics of the performances If any parameter, for example CY,p, E etc., has a tendency to increase the displacement of the bearing liner, the total load will decrease and the friction coefficient will increase. Figures 5 and 6 shows these points clearly. From Figs. 5 and 6 we see that a small eccentricity ratio causes a small variation in the load and the friction coefficient for different values of 0 and a large eccentricity ratio yields large variations. This also demonstrates that for heavy loads the deformation can no longer be ignored.

49

ii

w

f

20

20 -

1.25

1.25

1.00

15

1.00

15

0.75

0.75 10

10

0.50

0.50

5

5 0.25

0.25

I

0 0.0

0.2

0.4

0

0.0

I

0.6

0.8

1.0

0.0

0.0 0.2

0.4

0.6

E

Fig. 5. Load (-) Fig. 6. Load 0.3).

coefficient

and friction

(- - -)

coefficient

us. eccentricity

(- - --) us. eccentricity

!;j 20

15

10

5

0 0.0

I

I

I

1

I

0.1

0.2

0.3

0.4

0.5

hOy6 mJ.n

Fig. 7. Load us. minimum

l.C E

and friction

(-)

0.8

film thickness

(0 = 0.8 and v = 0.3).

(0 = 0.8 and v = 0.3). (CU= 0.075

and v =

50 TABLE

2

Comparison

of performance

P

cr

0.9

0.08242 0.03663 0.02137

0.8 0.7

aFrom

(E = 0.75,

Co = 0.05)

w

7

01 (deg)

1.846 1.856 1.867 1.895a

4.753 4.745 4.739 4.715a

213.567 213.109 212.523 212.3a

-..

PI1Ul.k

- _.-__1.4924 1.5082 I .5285 1.51P

ref. 2.

TABLE

3

Comparison

of performance

P

a

0.9 0.8 0.7

0.16484 0.07326 0.04274

8From

characteristics

characteristics

(e = 0.75,

C,, = 0.1)

w

a

01(deg)

&ax

1.649 1.656 1.684 1.720a

4.951 4.939 4.915 4.870a

219.911 219.243 219.438 219.78

1.2530 1.2717 1.2973 1.287a

ref. 2.

Another important feature of a deformed journal bearing is that the deformed bearing is able to increase the minimum film thickness with the same total load compared with a rigid journal bearing (see Fig. 7). For example, for a non-dimensional load over 15 the film thickness increases by approximately 10%. This brings some benefits to the journal bearing operation, e.g. a thicker film carries away more fluid and prevents contact between the surfaces of the shaft and bearing. 3.4. Comparison with a previous study A comparison of the results in this study with the results obtained by Conway [2] may allow us to see some differences or errors between the two models. Making an exact comparison, however, is quite difficult. Here we only consider the cases when the slopes are taken with the same values. The slope in Conway’s paper is c,=

6cqa2( 1 - v2)

(26)

C3E

It can be related

Co = 5.46~~ ; i

to cxand /3 for v = 0.3 as follows

WY

-- 1 1

Tables 2 and 3 illustrate different values of cy and /?.

the results obtained

for a fixed value of Co and

51

A maximum relative error of about 4.5% is found for the load. The errors are mainly caused by neglecting the intercept. It is clear that the oil pressure becomes slightly smaller without the intercept and the pressure increases. Therefore the total load is greater than that with the intercept. The other point is that the errors for C, = 0.1 are greater than that for C,, = 0.05. This is because the intercept increases with the slope and a comparatively large intercept is ignored.

4. Conclusions The relationship between the oil film pressure and the bearing liner deformation is studied. The study shows that a very good linear relationship exists between them and also demonstrates the significance of the liner deformation in the lubrication characteristics, especially at heavy loads. With the application of the linear model two coefficients, the slope and the intercept, should be carefully set. Ignoring the intercept will cause about 5% error in the total loads. Two formulae based on the data obtained for calculating the slope and the intercept are proposed for a Poisson’s ratio of v = 0.3. For a fixed Poisson’s value the slope shows a linear variation with the radial ratio /3 and the compliant constant cr and is independent of the eccentricity ratio.

Acknowledgments I wish to express my sincere thanks to Professor of the Division of Machine Elements, for his guidance tion. I also wish to thank my colleagues for their help.

Goran Gerbert, head and valuable instruc-

References G. R. Higginson, The theoretical effects of elastic deformation of the bearing liner on journal bearing performances, Proc. Symp. on Elastohydrodynamic Lubrication, Institute of Mechanical Engineers, London, 1965, pp. 31 - 38. H. D. Conway, The analysis of the lubrication of a flexible journal bearing, J. Lubr. Technol., 97(4) (1975) 599 - 604. S. C. Jain, Elastohydrodynamic analysis of a cylindrical journal bearing with a flexible bearing shell, Wear, 78 (1982) 325 - 335. S. C. Jain and R. Sinhasan, Performance of flexible shell journal bearings with variable viscosity lubricant, Tribal. Int., 16 (6) (1983) 331 - 339. K. P. Oh and K. H. Huebner, Solution of the elastohydrodynamic finite journal bearing problem, J. Lubr. Technol., 95(3) (1973) 342 - 352. J. O’Donoghue, The effect of elastic distortion on journal bearing performance, J. Lubr. Technol., 89(l) (1967) 409 - 417. S. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1953.

52 8 Y. Zhang, Deformation of a journal bearing and its lubrication behaviour, Inner Rep. 1985-l O-30, 1985 (Division of Machine Elements, Chalmers University of Technology, Sweden). 9 0. Pinkus and B. Sternlicht, Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.

Appendix A: Nomenclature a b c

E

f h

P U Ui

W

inner radius of bearing liner outer radius of bearing liner clearance between shaft and bearing modulus of elasticity friction coefficient film thickness oil pressure displacement of bearing liner in radial direction intercept load

Ci, di, ai, bj, C; and dj . . . Aip Biy o--p Fi, Gi, .+.ypic,pisy

ai, bi,

series

coefficients

wq/E(a/C)3 compliant coefficient 0 attitude angle eccentricity ratio viscosity coordinate angle angle of oil pressure zone slope angle Poisson’s ratio tensile stress shear stress stress function angular speed

non-dimensional quantity

of stress function coefficients

a.. and iiic, iii, .a.

of Fourier