Juma\ Kejuruteraan 5(1993) 47-58

Misalignment Effects on Isotropic Roughness Hydrodynamic Finite Journal Bearings

Che Hassan Che Haron

ABSTRACf

Hydrodynamically lubricated isotropic roughness bearing with the misalign­ment of the jour/Ull is analysed. A modified Reynolds' equation which includes surface roughness and misalignment effects is solved numerically using finite difference method with the aid of successive·over-relaxation method. The average flow model is employed in evaluating hydrodynamic pressure due to isotropic roughness and misalignment effects. The bearings' characteristics' values is solved numerically using Simpson's rule. The results of the numerical analysis explore the effects of both sUrface roughness and misalignment on the design parameler of journal bearing. The degree of misalignment and roughness structure greatly effects the performance of the hydrody/Ulmic bearings. Controlling the roughness and degree of misalign­ment is an important design factor particularly for bearings operating at minimum film thickness and high eccentricity ratio.

ABSTRAK

Galas hidrodinamik berpermukaan kasar jenis isotropik yang tidak sejajar dianalisiskan. Persamaan Reynold yang terubah-suai yang merangkumi kesan kekasaran permukaan dan tidak sejajaran diselesaikan secara berangka dengan menggunakan kaedah perbezaan lerhad dan kaedah 'successive­over-relaxation' . Model aUran purata digunakan bagi mengjra nilai tekanan hidrodinamik yang disebabkan oleh kekasaran permukaan dan salah jajaran. Nitai bagi parameter galas diselesaikan secara berangka dengan menggunakan hukum Simpson. Hasil kepufusan analisis berangka menerangkan kesan kekasaran permukaan dan salah jajaran keatas parameter~parameter reka bentuk galas jurnal. Salah jajaran dan kekasaran permukaan memberi kesan yang menonjol terhorklp kecekapall galas hidradinamik. Mengawal kekasaran permukaan dan salah jajaran adalah faktor reka ben/uk yang penling bagi galas yang beroperasi parkl keadaan lapisan filem yang minimum dan nisbah kepusatan yang tinggi.

INTRODUCTION

There has been a great deal of interest in predicting the perfonnance characteristics of the hydrodynamically lubricated rough surface finite jour­nal bearings. Different mathematical models have been suggested which have resulted in different Reynolds'-type equations. Generally, the peaks of the roughness asperities are of the same order as the nominal film thickness or slightly higher. These irregularities greatly influenced various bearing's

48

perfonnance parameters. Many researchers have reported lbeir prediction on the surface roughness effects in hydrodynamically lubricated bearings.

Tzeng and Saibel (1967) have introduced stochastic concept and applied it to slider bearing with one-dimensional transverse roughness. Christensen and Tonder (1973) developed the stochastic Reynolds equation for one­directional roughness, namely transverse and longitudinal, and applied this equation to analyse the hydrodynamic lubrication of slider and journal bearings. Chow and Cheng (1976) extended Christensen's stochastic theory to Elastohydrodyamic lubrication between two rollers. The analysis showed that the surface roughness can have a noticeable effect on the level of the mean film thiclrness between elastohydrodynamic contacts. Rhow and Elrod (1974) published the study on the effects of two sided-striated roughness on bearing load carrying capacity. Patir and Cheng (1978) introduced the "average flow model" and obtained an average Reynolds' equation in tenns of the pressure and shear flow factors. Majumdar and Hamrock (1981) have used the average flow model in their study on the surface roughness effect on fmite journal bearings. They found that surface roughness influenced the perfonnance of the journal bearings. Che Hassan c.H. and Nil< Abdullah N.S. (1991) have used the average flow model by evaluating temperature effects on isotropic roughness finite journal bearings.

Misalignment in hydrodynamic bearings has been recognised by several investigators. Non-parallelism between the axis of the bearing and that of the journal is caused by off-centre loads, assembly errors, an inaccuracy in manufacturing of the surfaces, elastic deflection of the shaft and thennal distortion of the bearing and shaft. This misalignment has a non-negligible effect on the bearing behaviour and parameter.

Safar et al. (1985) presented an analysis of a journal bearing running under adiabatic conditions for maximum allowable misalignment. The analy­sis showed that the journal misalignment is allowed to vary up to the axial plane of the load vector. Abdel-Latif et al. (1988) has modified Christensen­Reynolds-type equation by including the misalignment tenn and applied it to the hydrodynamically lubricated journal bearings with rough surfaces. It has been shown that misalignment has a significant effect on the design param­eter of the bearings. D. Vijayaraghavan and T.G. Keith (1990) have done the analysis for a finite grooved misaligned journal bearing considering cavita­tion and starvation effects. The study has show that for a small of misalign­ment, the bearing performance parameters close to the aligned bearings. However as the degree of misalignment increases, the perfonnance param­eters' changes considerably.

In the present work, the influence of both criteria namely, surface roughness and journal misalignment are combined. The "average flow model" developed by Patir eL al. (1978) is utilised. The average flow model is based on defining average Reynolds' equation in tenns of pressure and shear flow factors, which are functions of surface roughness characteristic. These flow factors can be derived independently from the mean flow quantities obtained from the numerical solution of a model bearing with randomly generated or measured surface roughness. The surface roughness is randomly generated by assuming the roughness height distribution is Gaussian and the roughness have different correlation lengths in x and y directions. The average flow

49

model can do the . analysis for three dimensional roughness structure by incorporating tile pressure and flow faclOrs in an average Reynolds' equation. One of the main advantage of this model is its capability in solving any type of roughness structure and in analysing the surface roughness effects in thin­film or partial lubrication regime. The Reynolds equation are modified by incorporating the pressure flow factor, shear flow factors and the misalign­ment term. The misalignment of the shaft is in the horizontal and vertical directions. The isotropic roughness pattern which is defined as width-to­length ratio or correlation length in x and y direction is equal to 1.0. The flow for the lubricant in journal bearings is assumed to be laminar. Since the temperature effects is not considered in the present study therfore the viscosity of the lubricant is assurned to be constant. The computer pro­gramme is written for the purpose of solving the misaligned rough surface finite journal bearings. The modified Reynolds' equation is solved numeri­cally by using a finite difference method. In order to speed-up the conver­gence of the iteration process the successive over relaxation (SOR) method is implemented.

GOVERNING EQUATIONS

REYNOLDS EQUATIONS

A modified Reynolds-type differential equation which gives the two-dimen­sional pressure distribution in the case of bearings with surface roughness and for a stationery bearings and moving journal is given as follows:

1lte following non.<Jimensional variables are used 10 convert the Reynolds equation from dimensional into non-dimensional fonn:

It - Y h C - c2

9 =- Y=-H=-;\.=_ P=p­R' L' C· .." ~R

oj> = (2 V . - I) 4i S rJ S

The non-dimensional average Reynolds equation is given by:

3/ii ( R) 2 3 a~ 2a H iJp 3 afi O¢I x <P,H -, + L <pyH -:-;: +3<P,H 1ii~ +H aeaii""

ae ay

01 ( l.H )~ aH 6 ~ • =11 + erf 7i~ae+rT(2V~ -I) (2.0)

where 9 and Y are non-dimensional co-ordinates, H is the dimensionless film thickness, q, and q, are the pressure flow factors and q, is the shear flow . , .

50

factor. These flow factors are introduced by Patir et al (1978). The Reynolds or Swift-Steiber boundary conditions are applied in solving the Reynolds­type equation. The cavitation effect is taken into account by imposing the cavitation boundary conditions to the problem. The cavitation occurred whenever the lubricant is separated and the negative pressure at this cavita­tion region is assumed to be zero. The negative pressure is assumed to be zero in order to simplified the numerical routine and also to reduce the computing time. Although this assumption will produce an error but it will be very small. The cavitation or the Swift-Steiber boundary conditions are:

- .~ P",, =ae-=Oat 8=9,=1t+8 2

where 9 is the angle measured from H to the point of film rapture (separate~l) and P is the non-dimensional ~~vitation pressure. The inlet and

m periodic boundary conditions used in solving Reynolds' equation are:

p= 0 at

~ =0 at dY

9=0

)"=0

FILM THICKNESS DISTRIBUTION

The approach employed by Mohtar et aI (1988) is used to represent the fluid thickness. The film thickness varies in both circumferential and axial direc­tions. Referring to Figure I, the non-dimensional fIlm thickness is repre­sented by

c

L/2

l'

FIGURE 1. Journal bearing with the misalignment

51

H(e, y) = 1+ £ cos (e - 'V) (3.0)

The eccentricity ratio, E, and the attitude angle, '" are functions of y as shown in Fig. I, and are given by

(4.0)

and

2 " ( ') eo + 21.e ocos ($ - 'V 0) + I. (5.0) e( y) =

where £." is the eccentricity ratio at Ihe mid-plane. The misalignment ratio is defined as

, '- (R) I.=±~Y c (6.0)

The degree of misalignment for the bearing is defend by a parameter, Om' where

A e As o =-=

rn l J 2 m I - ( I! sin ( .,, - ,¥ )) _ £ COS (t - 'I' )

0 00 0

(7.0)

Z. S. Safar (1 984) found that the effect of the directional angle on the bearing performance characteristics is very small and therefore can be neglected (<1>=0). Therefore, for a given £0' LID ratio, Om and <1> , the distribu­tions of the eccentricity ratio and the attitude angle along the axial direction are computed and consequently the film thickness distribution is determined. By integrating the pressure distribution in the fluid film and calculating the surface roughness parameter, the load carrying capacity and side leakage are calculated.

BEARING CHARACTERISTIC V ALVES

HYDRODYNAMtC LOAD CALCULATION

The hydrodynamic load in the x and y direction are calculated by employing these equations.

and

w =­, J L/'J 0, o 0

pR cos BdBdy

JLI2J" W = 2 ' pR sin BdBd y

y 0 0

(8.0)

(9.0)

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or in the nondimensional form

JLI2J ", Wx = - Pcos ededy (10.0)

o 0

and

w =-y LI2J" J 'Psin ededy o 0

(11.0)

where 9, is the angle at. which the film rapture occurred. The total nondimensional hydrodynamic load is

(12.0)

and the attitude angel is (13.0)

SIDE LEAKAGE CALCULATION

The side flow of the oil film due to rotating effect is calculated by using this formula.

(14.0)

or in the nondimensional fonn

J" 3 Q = 2<1> H 2!.d8 o y ay (15.0)

NUMERICAL SOLUTION TECHNIQUE

The finite difference method is used in solving the modified-Reynolds c:quation. The Reynolds equation is discretized by implementing the central difference scheme. The Reynolds equation is solved numerically by including the surface roughness parameters, misalignment terms and total film thick­ness. Initially, the attitude angle is calculated by using the short bearing theorem. Having known the attitude angle, the calculation for mesh, film thickness, surface roughness is calculated. The pressure distribution is solved by implementing the successive over relaxation method for a rapid conver­gence. The negative values for the pressure at the cavitation region is set to zero. The Reynolds equation is converged when the difference between two iterations for the pressure distribution is less than 0.001. The solution from the converge pressure distribution is used to calculate hydrodynamic load, attitude angle and side leakage, When the difference between two iterations

53

for the attitude angel is less than 0.001, the solution for the whole problem is calculated. The integration's of the characteristic values of the bearing are calculated numerically using Simpson's rule. The computer progranune will repeat the same procedure for different degree of misalignment, mid-plane eccentricity ratios and roughness parameters.

RESULT AND DISCUSS[ON

The result obtained from computer code generated is presented. The isotropic roughness will obstruct the oil flow both axially and circumferentially. Therefore the isotropic roughness will increased fluid film pressure distribu­tion. Thus, the bearings performance parameter will be affected.

Graphs from Figure 2 have shown that the hydrodynamic load for higher eccentricity ratio (E = 0.6) is greater than hydrodynamic load for lower eccentricity ratio (e = 0.2). Hydrodynamic load for all values degree of misalignment is higher in the very rough region and getting smaller as the surface become smoother. Figure 2 also have shown that hydrodynamic load getting higher with he increase in degree of misalignment. A small changes in degree of misalignment at the bigger value in degree of misalignment (Dm = 0.7) or 1.0) will affect hydrodynamic load significantly. Figure 3 shows the differences in values between the misaligned bearings and the aligned bearings (from Majumdar et al. (1981». The misaligned bearings will give a higher load carrying capacity compared with the aligned bearings. [n

practical world, bearings with higher degree of misalignment will have a

h Y d r 0 d Y n a m i c

0 a d

0 .7

0 .6

0 .5

G El

0.4

0 .3

0.2

0 . 1

a

Rp-3.e-O .2

-+- Smooth. e-O .2

-*- Rp-3,e '"O.6

~ Smoolh, e-O .6

o~--~--~--~----~--~--~ o 0.2 0.4 0 .6 0.8 1.2

Degree of misalignment

FIGURE 2. Hydrodynamic load vs. degree of misalignment for different eccentricity ration

54

h Y d

0 d Y n a m ; c

0 a d

n 0 n d

m

• n

• ; 0 n a I

• ; d

•

• a k a g

•

0.2,---------------,

0.1

Majumdar

0.05 -+- Om 1.0

-+- Om 0.7

..e- Om 0.3

~ Om 0.1

roughn ••• parameter

FIGURE 3. Nondimensional hydrodynamic load vs. roughness parameter for different degree of

misa1ignment

0 .35

0.3

0.25

t-----> V 0 .2 v-: ~

0 . 15

~

0 . 1

0 .05

o 1 2

: - m'lum~.r

I -+- dm 1.00

-+- dm 0 .7

, -8- dm 0 .3

l - .m 0 .1

3 • 5

roughness parameter

6

FIGURE 4. Slide leakage vs. roughness parameter for different degree of misalignment

bigger chance of rubbing each other and hence wear process will occurred. Therefore, after certain time limit the bearing will become useless or fail.

Since isotropic roughness blocked the oil flow tn the axial direction, therefore isotropic roughness will produce less side leakage if compared with smooth surface solution (Figure 4). Figure 5 shows that for a very rough

55

surface, the oil side leakage is very minimum. Side leakage will increase with the increase of roughness parameter (surface becoming smoother). Figure 5 also shows that when degree of misalignment increases side leakage will· decreases. Misaligned bearings will produced less side leakage when com­pared to aligned bearings (see Figure 5). Figure 6 has shown that the hydrodynamic load increases significantly as the eccentricity ratio increases.

h y ~ r 0 ~ y n a m I e

0 a ~

n o n d ; 0.6

m e n 5 i o 0.6 n a I

• i 0.4 d

• • 0.2 a k a 9 •

0 0

Rp-3,.·O .2

-+-- Smooth. e-0.2

"""'*'""" Rp-a,.·O.6

--B- Smooth, .-0.6

0.2 0.4 0.6 0.8 Degree of misalignment

FIGURE 5. Side leakage vs. degree of misalignment for different eccentricity ratios

2.5

legend

Om ' 0 .1 2 I~ Dm • 0.3

-+- 001 • 0.7

1.5 -& Om·l.O

0.5

° 0.2 0.4 0.6 center eccentriCitY ratios

1.2

0.8

FIGURE 6. Hydrodynamic load vs. eccentricty ration for roughness parameter 3.0

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The graph also shown that the journal with a larger misalignment will give a larger effeclS. The amount of side leakage also increases as the eccentricity ratio increases. These effects can be seen from the graph from Figure 7. The increment of the side leakage and hydrodynamic load is due to a high pressure area developed for .a high 'eccentricity ratio. Figure 8 shows an example of the effects of surface roughness and misalignment on the film thickness profile as compared with smooth surface solution.

1.2

legend

Dm·O.\

S -+- Dm • 0.3

! 0.8 ...;t<- Dm • 0.7 d

e -B- Dm • \.0

e 0.6

a k a 0.4 g e

0.2

0 0.2 0.4 0.6

center eccentricity fatiO

FIGURE 7. Side leakage vs eccentricity ratio for roughness parameter 3.0

en en

'" 12 ~ ... :'l r;: c "' " 'l z 0 1il Z

~ c I

Z

fi1

."

----~ - - -

" --------

OJ -- ---~ ~

d

= d

~

d

;J 0 d

~

" " d

OJ o

Rou&h Surface 1.-,1.0

RouSh Surface A:: 2.0

Rough Surface A- 3.0

Rough Surface 11.- 4.0 Rouah Surface A= 5.0

Rough Surface 11.- 6.0 /.

Smooth Surface / . / / I

/1

~+--'-'--'-'--'-'--.-.--.-.--r~ o 3Q 60 90 120 150 IBO 210 240 270 300 330 360

ANGULAR POSITION (DEGREE)

FIGURE 8. Example of film thickness profile for smooth and rough surfaces with £ :::; 0.2

0.8

57

CONCLUSION

The results obtained in this paper have shown that both journal roughness and misalignment give significant effect to the perfonnance of hydrodynamic finite journal bearings. The result have shown that the misaligned bearings will support higher load compared with the aligned bearings. However, the misalignment in the bearings will shorten bearing's life and therefore should be avoided. The result also have shown that a small changes in degnee of misalignment will not give a significant effects on the bearings performance parameters. As the degree of misalignment increases, hydrodynamic load increases and side leakage decreases. The result obtained in the pn:sent study have shown that the higher the degree of misalignment the greater the effect on the bearings perfonnance parameters will be. These effects will be maximum at the higher eccentricity rdtio where the oil film thickness is minimum. The combined effect of journal roughness and misalignment is more pronounced at higher eccentricity ratio.

Symbol

C D Dm h H Hmin

H, L P p

p ~, q Q R u,v,w U w, W.., x, y. z

13 • '0 A X'

f' <!> <!>. <!>.<!> tjI 6 6,

NOMENCLA TIJRE

Explanation

Clearance, m Bearing Diameter, m Degree of misalignment Film thickness, m Nondimensional film thickness. H=h/C Minimum film thickness. m Total film thickness. m Bearing length, m Pressure distribution, N/m2 Nondimensional Pressure distribution Cavitation pressure distribution side leakage, m3/s Nondimensional Side leakage Bearing radius, m oil velocity in X.y, z direction, mls Journal speed, mis Total hydrodynamic load, N Hydrodynamic load in x, y direction, N Cartesian co-ordinate misalignment angle eccentricity ratio. e = e/ centre eccentricity ratio Roughness parameter, Misalignment ratio, Oil viscosity. poise Angle for direction of misalignment Shear flow faclor Pressure flow factors attitude angle Circumferential co-ordinate Cavitation co-ordinate

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REFERENCES

Abdel-Latif, L.A. & Mokhtar, M.O.A. 1988. Misalignment effects on hydrodynamically lubricated journal bearings with TOugh surfaces. Wear 128, : 225-237.

Che Hassan C.H. & Nil< Abdullah N.S. 1991. The effect of isotropic roughness on the thennohydrodynamic solution of finite journal bearings. furnoi Kejuruteraan 3: 27-40.

Chow. L.S.H. & Cheng, H.S. J 976. TIle effect of surface roughness on average film thickness between lubricated roUers . Journal of Lubrication Technology 98: 117.

Christensen, H. & Tonder K. 1973 (April). The hydrodynamic lubrication of rough journa1 bearings. Journal of Lubrication Technology 98: 166.

Majumdar, B.S. & Hamrock, BJ. 1981. Surface roughness effect on fmite oil journal bearings. NASA Tech. Memo. No. 32639.

Patir, N. & Cheng, H,S. 1978 (Jan). An average flow model for detennining effects of three-dimensional roughness on partial hydrodynamic lubrication. Journal of Lubrication Technology 100: 12-1~ .

Rhow, S.K. & Elrod, H.G. 1974. The effects on bearing load canying capacity of two­sided striated roughness. Journal of Lubrication Technology 96: .554.

Safar, Z. 1984. Energy loss due to bearing misalignment. Tribology International 17(2): 107·109.

Safar, Z.o Mokhtar, M.O.A. & Peeken, H. 1985. Thennal characteristics of misaligned finite journal bearings. Tribology Int. 18(2): 13-16.

Tzeng, S.T. Saibel , E. 1%7. Surface roughness effect on slider bearing lubrication. ASLE Trans. 10: 334.

Vijayaraghavan, D. & Keith, T.G. 1990 (Jan.). Analysis of a finite grooved mis­aligned journal bearing considering cavitaliop aoo starvation effecls. Journal of Tribology 112: 62-67.

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