A thermohydrodynamic method of bearing analysis AI. Nica*

The coupled Reynolds and energy equations are considered for both thrust and journal bearings. The one-dimensional solutions are provided with correction coefficients depending on the actual bearing width, resulting in ready-for-use relations. Diagrams obtained by numerical treatment are presented for finite bearings, besides the analytical relations, so that actual design requirements can be easily coped with. Comparisons with experimental results prove to be satisfactory, both for thrust and journal bearings; a numerical example of calculation is also included.

The mathematical difficulties inherent in the simultaneous treatment of the Reynolds and energy equations has led to simplified solutions, either by considering a constant value for the viscosity, or by admitting a very simple law for its variation (Norton ~ , for instance, considered a linear variation).

More recently, numerical methods have been developed for coupled Reynolds, energy and heat conduction equations for one-dimensional journal bearings 2 . In the same conditions the problem of Reynolds, elasticity and energy equations for the pressure, temperature and film thickness between two heavily loaded rolling and sliding cylinders was treated considering a mean viscosity across the film 3. Journal bearings of various arcs have also been considered on the assumption of constant lubricant viscosity, to obtain the hydrodynamic characteristics and a mean temperature rise in the film, by employing computer techniques 4. Valuable results were obtained in the same way for plane and sector thrust bearings s'a.

The analytical treatment was limited to the one- dimensional case, both for slider 9 and journal bearings (adiabatic flow, no side leakage) to determine the maximum temperature profile 1' ~ ~ ; the adiabatic temperature distribution was also obtained for both infinately long and short bearings 12.

The author's aim has been to establish an analytical solution for the coupled Reynolds and energy equations for sliding bearings, the only way of providing a general view on the problem. A solution for finite journal bearings was obtained by using the results regarding the pressure distribution in the film, and by integrating the energy

* Visiting Scientist, Abteilung Reibungsforsehung, Max-Planck- Institut f'fir Str&nungsforschung, GSttingen, West Germany

equation I a,l 4. After integration, the energy equation for the general case, i.e. considering the heat to be dissipated both by the lubricant flow and by the surfaces in contact, isla:

J P Vh ~ '~x l ! axl 121/ax3 .axa ]

hk - - - hk +Kh(T -T h) ax, ax3

(1)

where: P T

is the pressure at any point in the lubricating film is the temperature at any point in the lubricating film

h is the thickness of the oil film at any point To is the temperature of the solid surface for x2 = 0

(Fig 1) T h is the temperature of the solid surface for x= = h

(Fig 1) V is the relative velocity of the surfaces J is the mechanical equivalent of heat k is the coefficient of thermal conductivity of the

lubricant Ko, h is the coefficient of thermal conductivity between

the lubricant and the solid surfaces xl is the co-ordinate in the direction of motion

218 TRIBOLOGY November 1971

x~ is the co-ordinate perpendicular to the direction of motion in the direction of the oil film thickness (Fig 1)

xa is the co-ordinate in the direction of the bearing width (Fig 1)

p is the density of the lubricant 7/ is the absolute viscosity of the lubricant Cv is the specific heat of the lubricant for constant

volume

Since Equation 1 is generally valid, the temperatures can be calculated for any point of the lubricating film with help of the pressure distribution obtained for finite journal bearings from the complete Reynolds equation 14.

To simplify the calculations, the heat can be considered to be dissipated only through the lubricant, an assumption acceptable for pressure-fed bearings 1 s as well as for slider bearings ~ 6

Inf initely long bearings

Slider bearings For slider bearings (Fig 2) Equation 1 takes the form:

(~ h a dp) dT V z ha (dpt2 Jp Vh Cv - - = r/-- + - -

12n dxl dx, h 12n \~-~12 ] X 0 ~- L[ ]

while the pressure equation is 14:

dpoo [" 1 ho dxl = 6r/V[ (3) [( hi-h2 )2( hi-h2 ) 3

- - x , h~ - - x~ hi 1 1

ho being the thickness of the lubricating film where dp -- ~-0, dxl

and 1 the bearing length (Fig 2). For values of q (the parameter characterizing the viscosity variation in the film) equal to 0 and 1 :

h~ h2 ho =2- -

hi + h2 hi

hi In h2 ho- - -

hi - - - 1 h2

for q = 0

for q = 1 (4)

Equation 3 takes the form:

~2

I

X!

Fig 1 Axes of co-ordinates

(2)

h2

t x,

Fig 2 Dimensions for an infinitely long plane bearing

The viscosity was considered to vary with h as shown by:

This is justified both theoretically and experimentally 1 a ,1 4 ; q was found to vary between 0 and 1, for most cases. The viscosity and the film thickness are denoted by rh and h~ at the beginning of the load carrying film, and by rh and h2 at its end.

By combining Relations 5 and 2 and by integrating with respect to x=, the temperature distribution in the lubricating film along the plane slider is obtained I 7:

2 rhV[ 4h __~1 h 1 6hox~ 3ho2x~(h~+h)] T=TI + In +

JPcvho h2 h hi h 2h~h 2 for q = 0 (7)

and

dp - 6rhV

dxl

dp

dxl

[(. ') ( hi - h2 2 hi - h2 - - x (h i +h2) hi

1 1

hi In hI

6~/1V I 1 _ ( h~ hi h~ - h2 hi - h2 hi - h2

- - x l - - hi hi 1 h2 1

2hi h2

__x,) ] for q=O

for q = 1 (5)

TRIBOLOGY November 1971 219

x3 x2

65

E

Fig 3 Dimensions for a finite plane bearing 55

T=TI 4 2~1V ( 61ho h 3h~xl]

JPCvhohi 4xl + In-- + for q = 1 hi - h2 hi hi h (8)

where ho is given by Relation 4, while:

hi - h2 h=hl - - - xl (9)

1

From numerical computations of Raimondi, Boyd and Kaufman 16, for values of the slenderness ratio ~ = b/1 (see Fig 3) > 1 (most usual cases) the temperature distribution does not differ very much from that corresponding to h = 0% and this is reflected by Equations 7 and 8. Moreover, a correction coefficient can be established for these relations, resulting in a simple and efficient method of calculating the temperature distribution in the lubricating film of slider bearings.

The errors are always on the safe side since, as will be seen when deducing the correction coefficient, the heat dissipation conditions correspond to ~. = 0% but in reality they improve with diminishing h (larger oil flow rates, etc).

The correction coefficient can be established by observing that the temperature in the bearing is a consequence of the frictional heat developed in the lubricating film, so that the thermal level is proportional to the friction. Thus, if the dimensionless friction coefficient, kf, in the bearing is considered ~ 4,1 s.

hi ln - -

r/m Vlb Fh2 hi h2 l (h l ) k f - - + - 1 ? (1o) 1 - - -

h2

where: F r/m V

is the friction force of the moving surface is the mean viscosity is a dimensionless load-carrying coefficient

Experimenta I~ Ku hn - Cooper

Theorctica I, T ipei - N ica - Biner

I % 0'.2 0'.4 or6 ,.o

Fig 4 Temperature distribution in a pivoted-pad thrust bearing

The correction coefficient, 3, will result as the ratio of kf for any real slenderness ratio of the slider to the value of kf corresponding to ~ = co:

kf ()k = real) - (12)

kf (X = oo)

The temperatures obtained by Equations 7 and 8 will be consequently affected by 3 (which is always < l) so that:

Treal = fir (13)

The added value of q results from Equation 14:

7"/2 ln - -

q = - - (14) h i

In-- h2

and the corresponding temperature is obtained from an interpolation formula:

Tq = Tq=0 +(Tq=l -Tq=0) q (15)

A numerical application of these relations 7, 8 and 12 to 15 to the geometrical and operating data of a large pivoted- pad thrust bearing investigated experimentally by Kuhn and Cooper 19, showed good agreement with the measured temperatures (Fig 4). The difference is mainly the pattern

h, - - - 1

~_ 6 ht h2 - In - - -2

(hi 1) 2 h2 hi - - -+1

~ h~

3.7 +

0.1 +~

li +16( h2 /

kh2 /

1-o.ls~ s/4 _ th ~- -~

(11)

220 TR IBOLOGY November 1971

kf k=2/kf ~.=oo of the temperature distribution, but the maximum temperature in the film only had a small error on the safe side 0.9

From Equations 7 to 15 a practical scheme for thermal calculations of slider bearings results: a final maximum ~ O 8 temperature, T2, is selected so that with the help of the inlet temperature, T~, the values of the viscosity are known O-7 at the beginning of the load-carrying film (rh corresponding g to T1) and at the end of the film (rh corresponding to T2 ). r, 6 q is deducted with the help of Equation 14 and/3, from Fig 5, so that the temperature distribution can be calculated from ~ 0-5 Equations 7, 8, 14 and 15. The calculations are pursued until the difference between two successive values T~ is 0.4 sufficiently small, e.g. less than 5%. The dimensions of the bearing (width b and length 1) as well as the film thickness (hi and h2) are known from previous hydrodynamic considerations (Equations 10, 11 and 14). If unacceptable temperature results, i.e. giving viscosity values less than ~2,

hi new values -- and bearing dimensions must be considered.

h2

Journal bearings The analyses of journal bearings, for the thermohydrodynamic situation have been performed for both infinitely long and finite bearings. In the first case relatively simple relations are available for pressure distribution, useful for bearings with light loads and moderate speeds; if a correction coefficient is established, as for slider bearings, realistic results are obtained for the temperature distribution.

The energy equation is yielded by Equation 2, while after integration the Reynolds equation becomes 14:

dp= 6~lVr , [ 1 ho ]

dO (1 +e) 2 c 2 (1 +ecosO) 2-q - c(1 +ecosO) 3"q (16)

ho =(1 - e 2) c (17)

where

and 0 is the angular co-ordinate of the lubricating film c is the radial clearance e is the bearing eccentricity (Fig 6) e = e/c is the eccentricity ratio

If the values q = 0 and q = 1 are considered successively in dp

Equation 16 and the resulting values of -- are introduced dO

into Equation 2 put into the form

JPcv 12r~rl dO dx h 12r~r~\dx]

after integration with respect to the position angle O, the temperature distribution for the convergent zone is obtained:

T=TI + 2rh Vr 1

Jpc v c 2 (1 +e)

2r/, Vr, I 2+ 3e___2_ 2 Jpc vc 2 ( l -e2) [ lx/i-L-~-e 2 arc tan

tan O 8 2

arc tan x /c ( I - e ( I - e )

# --x/l - e 2 tan-

2 -t

l+e

and:

T=T i +

I.O

I I I I I I I I

2 3 ,4 5 6 7 8 9 I0 h, /h 2

Fig 5 Correction coefficient,/3, for journal bearings

Fig 6 Characteristic angles and parameters of a journal bearing

In these relations TI is the temperature at the beginning of the pressure zone (0 = 0, h = hi ). For practical purposes it is possible to consider T~ = T i that is, the temperature at the bearing inlet.

It was deduced theoretically and confirmed experimentallyl 3,2 o,21 that in the divergent (unloaded) zone of the bearing, the temperature distribution is symmetrical to that established in the convergent (loaded) zone, with respect to the line of centres. In this case the

0 ] (19) tan -

i~_e2(c 2 ) 2 3 esin0 forq= 1 + - arc tan~2) c 1 +ecos0

3 esin0 3 (1 - e2)e sinO] (20)

2 ( l+ecos0) 2 ( l+ecos0) 2 fo rq=0

TRIBOLOGY November 1971 221

temperature distribution can be readily deduced, since:

T d (- 0) = T c (0) (21)

if T c is the temperature in the convergent zone and T d is temperature in the divergent zone.

Obviously, the direct method for calculating the tempera-

dp ture distribution in the divergent zone is to consider - - = 0

dx~

in Equation 18 and to integrate it by using the boundary conditions characteristic to this zone.

It is natural to find some difference between the results obtained in the above mentioned way and those given by Equations 19 to 21, since the integrating hypotheses differ from one zone to the other.

The maximum value of T2 corresponds to 0 = 180 e (h = h2 ) as found by my experimental investigations 22 and by other workers in the field 2 z and can be calculated from Equations 19 and 20. In this way it is possible to verify the maximum temperature rise in the bearing, so that a tendency to overheat can be eliminated. At the same time T2 is useful in calculating the modifications of the bearing clearance during operation 23

To obtain a realistic temperature distribution, a correction coefficient must be applied by considering the influence of the actual slenderness ratio of the bearing. This can be performed in a similar manner to that discussed for slider bearings. Thus, if the dimensionless friction coefficient Cmt for the bearing journal is considered 14:

-1 { oo Cm~ (l+e) q 2-e+n=ll~ [q l+( l -e) (q- l -2n)]

(q - 2) (q - 3__))_... (q - 2n) + 3__ee Cn (22) 2n! ' 7r

where C n is the normal component of the load-carrying coefficient:

~2 (1 + e) ~ W C~ = (23)

6rh Vb

and ~ = c /q , the clearance ratio. The correction coefficient /3 will result from:

Crn I (;k = real) /3 = (24) Cml (X = =)

Its values as a function of e are presented in Fig 6; it was assumed that the divergent zone is completely filled with lubricant, giving the highest friction values.

Both temperature, T, and/3 are deducted by interpolation for the real value of q from:

in nl r/2

q = ~ (25) l+e

ln~ l -e

while e can be determined from the graph ~ = ~ (e) in Fig 8; the load-carrying coefficient ~" is given by the geometric and operating characteristics of the bearing:

qj2W ~" = - - (26)

2r/l Vrl

The real temperature corresponding to a definite slenderness ratio is obtained from:

Treal =/3T (27)

1.0

0 .9

~_ 0-8

g "~0-7

~ 0.6 k)

0.5

I - X-CO

X- l .5~

-~ q-1

040 0'.2 0'.4 o'.6 o'.e ;.o Eccentricity n3tio,

Fig 7 Correction coefficient,/3, for plane bearings

-

U

u

Gn t -

(J g I

"O

O d

0 0"2 0"4 06 O.B Eccentr ic ity rat io,

Fig 8 Load parameter, ~, against eccentricity ratio, e

I.O

222 TR IBOLOGY November 1971

In conclusion, the method of thermal calculation for 50 infinitely long journal bearings consists of determining ~" from Equation 26, with the help of the actual geometric 48 and operating characteristics of the bearing already known 46 from hydrodynamic considerations. Then e and q are deduced from Fig 8 and Equation 25 by admitting an ~ 44

tJ arbitary value of T2 which allows the determination of o 42 r12. Next, the temperature distribution is obtained from Equations 19 to 21 and/3 from Fig 7. Equation 27 finally ~ 40 gives the real temperatures for the actual value of ~, = b/2q o for the bearing. When the difference between two calculated E 38 values of T2 is sufficiently small, the calculation is finished ~- 36 and a check on the viscosity variation and on other operating characteristics is easily performed. 34

Fig 9 presents a comparison between some actual 32 temperature measurements and the calculated values by the method above (Equations 19 to 21 and 24 to 27), for a bearing having the following characteristics: rl = 18 mm; b = 18 mm; ~, = 0.5; ff = 1.5%; V = 1.8 m/s; c v = 0.5 kcal/kg deg C; oil viscosity 120 cSt, SAE 20W; supply pressure, Pi = 9.81N/era'; e = 0.8; The differences between the theoretical and the experimental values are small (under 4 deg C) and the error for the maximum temperature is on the safe side.

Finite bearings

Slider bearings As already mentioned, the results obtained for the infinitely- long slider bearings, when corrected for real geometric parameters, prove to be reliable for practical calculations.

The general expression of the integrated energy equation (Equation 1) yields directly the temperature distribution for the finite bearings by help of the pressure distrubition. For finite slider bearings, the pressure distribution in the median plane of the sector was, necessary to calculate the temperature distribution, found as~ 4 :

120

Experimentol ~ .B Theoretical

(con~cted two-dimensional)

, f ' \

I ! I

180 240 300 ~60 O+e i

Fig 9 Temperature distribution in journal bearings

while 131 and/32 assume the values 14 1/8 and 1/2 respectively (see also Fig 3).

The profile of the temperature distribution allows the determination of the real viscosity variation in the film, and the calculations are repreated until the differences between two successive values of T2 are sufficiently small.

Journal bearings The temperature distribution in the lubricating film of f'mite bearings has been analysed 1 a by integrating the energy equation and by using the results obtained for the pressure distribution ~ 4. To provide data for design purposes, dimensionless diagrams of the temperature distribution are presented, by a numerical treatment of the analytical relations.

P=Po + 1 - ch - - i

Alo +kA20 +(Al l +kA21)x 1 +(A12 +kA22) x 2

1 hi - h2 2 lh~ xl)

(28)

where Po is the atmospheric pressure and:

a =

Alo=-a l ; A2o=al

At1 =0.833a ; A2t =0.167a a a

Al2 =0.04175- ; A22 =1.04175- 1 1

h~ 1 - - -1 1

6~lV h2 ch ~ ~, ; k -

h~ hi 1 - -+1 1 h2 ch X

(29)

TRIBOLOGY November 1971 223

For the convergent, load-carrying zone (0 ~< 0 < n) the temperature distribution has the form 13.

0 2.71Vr I ! 1

Jpc v c 2 (1 +e) q (1 +e cos0) 2-q T=TI +

1+-

/ COS 0 + 2 aq cos 20 +

K 2 (q, X) \ 12

2e sin 0 (sin 0 aq sin 20) \ :

) 1 +e cosO (1 +e COS0) 2q

cos 0 + 2 aq cos 20 + K (q, X)

6 (1 + e cos O) q

2e sin 0 (sin 0 + aq sin 20)

1 +e cos0

=T1 + 2.71Vrl

Jpc vc 2 (1 +6) q I (0 'q 'e '~)

dO

(30)

where

k (q, h) - 6e

(1 - e2) vq " - - - - - 1 -

l+~e2(1 q) 1

A1 2q 7. a, = . - A l lq I

1

1

ch 2X 2 q _

lq

ch X ~X/~q

' [ t tl 1-2(1 - q) 1 +e 2 1 - l - -e 2 ( l -q ) -

2

m12 q = 0.05386q - 0.03816 + (0.6524 - 0.0254q) e

A l l q 3 lq = 0.83292 + 0.09828q + (0.7412 - 0.4992q) e

/

(31)

while the divergent zone Or ~< 0 ~< 2n) the temperature distribution takes the form:

0 2.71 Vrl t" dO

T=T1 + c2 JP c v (1 + e) q (l + e cos 0) 2"q

(32)

In these relations (Equations 30 and 32) the temperature distribution is considered in the median plane of the bearing since the axial variation of the temperature (along x3) is negligible and the error is on the safe side. At the same time, the heat is considered as dissipated only through the lubricant and the viscosity as varying according to Equation 3. Comparisons of the temperature distribution obtained from Equations 30 and 32, have shown good agreement 13 with actual temperature measurements.

To avoid the laborious calculations inherent in the direct utilization of Equations 30 to 32, the temperature rise AT = T - T1 is put in a dimensionless form as:

Jp c v c 2 1 - - AT= I (0 , q ,e , ~) (33) '71Vrl (1 + e) q

With the usual values, J = 427 kgfm/kcal, c v = 0.5 kcal/kg deg C, and P = 93.78 kg/m 3 , Equation 33 takes the form:

104 c 2 1 - - AT=104 KTAT=~I (0 , q ,e, )Q (34) rh Vrl (1 +e) q

By intermediate of a Fortran 1V language and an IBM 360/30 Digital Computer, the temperature distribution (in the form of the product K T AT) was obtained for various values of ), and e, for the characteristic cases q = 0 and q = 1. The results are presented in Figs 10 to 13. When a value K T AT = y is read on one of these diagrams for some actual bearing operating parameters, in a given point, O, of the load- carrying film, the temperature rise AT for this point will result as:

y rhVq AT = - - = y (35)

K T c 2

Where .71, V, rl and c are the operating parameters of the bearing.

Thus, the thermal calculation of finite journal bearings can be performed in the following way. From the geometrical (r l , b, and c) and operating (V and W) characteristics of the bearing and the lubricant properties (*7, p and Cv) the load- carrying capacity, ~', can be calculated from Equation 26, Then e and q are deducted from Fig 8 and Equation 25, by assuming a probable value for T2 to give *72- These data are sufficient to obtain from one of the Figs 10 to 13 (corresponding to the actual value of ~ for the bearing) the temperature rise, by interpolation for the calculated value q:

ATreaJ = ATq = 0 + (ATq = 1 - ATq = 0) q (36)

224 TRIBOLOGY November 1971

I--

t - 0

"E

u Q. E

10-3

io-4

iO-S

i d 6

$ / /

0"2

I i I I I

30 60 90 120 150 180 0 [o]

Fig 10 K t AT against 8 for k = 0.5

iO-3

I -

t - O

..O

E

ICY 4

iO-S

X-I ~'*/ / I ., q=l / / / " "

. . . . q = 0 0 '8 s////i,, /

o-2./,-~ 0'8 s////i,, /

d.6

i(~ ' , , , , 30 60 90 120 150 e [o]

180

Fig 11 K t AT against 8 for X = 1

iO-2

iO-~

I-

X c O

.IO

e~

E

~ io-S

),-I-5 q-l

q=O 9 ~ e-O.

I I / ,, ,"

0"8. / ' / ,,"

. . .~ .6 o'.4 j , -

IC)6 30 ' i , , 60 90 120 150 e [o]

Fig 12 K t AT against 8 for X = 1.5

10-21 q=l)"2

I . . . . q=O

I--

E .O

.O

e (~.

E

iO-4

ICr s

f ~ jS

# t

'Z 0.8 / / / "

~-0"2

180 i d 6

0.4 0.6

30 60 90 120 150 180 e [o]

Fig 13 K t AT against 8 for ;k = 2

so that the temperature of any point of the load-carrying film (0

as unacceptable, the whole hydrodynamic calculation must be reconsidered.

Example An electric motor bearing with the following characteristics:

Shaft radius, r~ = 0.04 m Bearing width, b = 0.08 m Slenderness ratio, k = 1 Radial clearance, c = 0.06 X 10 -3 (~k = 1.5%) Velocity, V = 6.7 m/s Load, W = 650 kgf Temperature, T] = 40C Oil viscosity SAE 10W; rh = 20 cSt Assumed value T2 = 70Cgiving

7/2 = 9.5 cSt

Equation 26 yields ~" = 1.23, so from Fig 8, e = 0.64 for q = 0 and e = 0.8 for q = 1. For these values of e, Equation 25 gives q = 0.588 and q = 0.405 respectively. Fig 11 will lead to a value K T AT = 1.3 X l0 -4 for the mean values e = 0.72 and q = 0.5. Since c v = 0.5 kcal/kg deg C and p = 93.78 kg/m 3 , the temperature rise is easily deduced from Equation 35: AT = 7hVrl 1.3 X 10-4/c 2 , that is AT = 21.5vC and T2 = T] + AT = 40 + 21.5 = 61.5C. For this new T2, Equation 25 yields q = 0.322 for the mean value e = 0.72. From Fig 11 and the new value e = 0.69(deductedfrom Fig 7 for ~" = 1.23 and the new value q = 0.322) AT = 20.5C, so that T2 = 60;5C. Since the difference between these last two values T2 is sufficiently small it can be considered that T2 = 61C. Actual measurements gave T2 ~ 58C.

Conclusions The importance of the calculation of thermal levels in bearings is obvious. It is no use determining a 'mean' temperature value since the extreme values can lead to unacceptable operating situations.

An analysis is given for finite bearings. However, especially for slider bearings, the infinitely long assumption leads to realistic results when conveniently corrected, giving the advantage of simplicity.

Analytical relations have been numerically treated for finite journal bearings, so that every important geometrical configuration is provided with ready-for-use diagrams and calculation methods. The various simplifying hypotheses regarding the boundary conditions and heat dissipation have been checked by experimental investigations.

Determination of the temperature distribution verifies all geometrical and operating bearing parameters and, at the same time, allows the real clearance of the running bearing to be calculated as a result of the thermal regime and load 2 a

Further work is necessary in order to provide ready-to- handle relations for all possible transitory situations of heat dissipation, from starved bearings where the heat is dissipated only through the solid surfaces in contact, to pressure-fed bearings where the heat can be considered to be dissipated through the lubricant flow I s ,l 6,24.

References 1 Norton, A. E. 'Lubrication', McGraw-Hill Book Co, New

York, (1942) 2 Dowson, D., and March, C. N. 'A thermohydrodynamic

analysis of journal bearings', Proceedings of the Institution o[MechanicalEngineers, Vol 181, Part 30, (1966-1967)

3 Cheng, H. S. and Sternlicht, B. 'A numerical solution for the pressure, temperature and film thickness between two infinitely long, lubricated rolling and sliding cylinders, under heavy loads', Journal of Basic Engineering, Transactions of the American Society of Mechanical Engineers, Series D, Vol 87, No 3 (1965) p 695

4 Raimondi, A. A. and Boyd, J. 'A solution for the finite journal bearing and its application to analysis and design: Part I, II, III,', Transactions of the American Society of Lubrication Engineers Vol 1, No 1 (1958) p 113

5 Sternlicht, B. 'Energy and Reynolds considerations in thrust-bearing analysis', The Institution of Mechanical Engineers, Conference on Lubrication and Wear, Paper 21, London (1957)

6 Guilinger, W. H. and Saibel, E. 'Numerical methods of solution of adiabatic slider bearing without slide leakage', Proceedings of the Fifth Midwestern Conference on Fluid Mechanics (1957) p 285

7 Cameron, A. and Wood, W. L. 'Parallel surface thrust bearings', Transactions of the American Society of Lubrication Engineers, Vol 1, No 2 (1958)

8 Pinkus, O. and Sternlicht, B. 'Theory of hydrodynamic lubrication', McGraw-Hill Book Co, New York (1961)

9 Osterle, F., Charnes, A. and Saibel, E 'On the solution of Reynolds equation for sliders beating lubrication. IX The stepped slider with adiabatic lubricant flow', Transactions of the American Society of Mechanical Engineers, Vol 77, No 2 (1955)

10 Pinkus, O. and Sternlicht, B. 'The maximum temperature profile in journal bearings', Transactions of the A merican Society of Mechanical Engineers, Vo179, No 2 (1957) p 315

11 Hughes, W. F. and Osterle, F. 'Temperature effects m journal bearing lubrication', Transactions of the American Society of Lubrication Engineers, Vol 1, No 1 (1958) p 210

12 Purvis, M. B., Mayer, W. E. and Benton, T. C. 'Temperature distribution in the journal bearing lubricant film', Transactions of the American Society of Mechanical Engineers, Vo179, No 2 (1957) p 327

13 Tipei, N. and Nica, AL 'On the field of temperature in lubricating films', Transactions of the American Society of Mechanical Engineers, Journal of Lubrication Technology, Series F, Vol 89, No 4 (1967) p 483

14 Tipei, N. 'Theory of lubrication', Stanford University Press, Stanford, California (1962)

15 Niea, AL 'Note on the heat dissipation in journal bearings', Revue Roumaine des Sciences Techniques - S&ie de M~canique Appliqu~e, Vol 15, No 1 (1970) p 219

16 O'Connor, J. J. and Boyd, J. 'Standard handbook of lubrication engineering', Chapter 5, Section 1, McGraw-Hill Book Co, New York (1968)

17 Tipei, N., Nica, A1, and Beiner, L. 'A thermohydrodynamic analysis of sliding bearings', Rumanian Journal of Technical Sciences - Applied Mechanics, Vol 15, No 1 (1970) p 193

18 Tipei, N. et al 'Lagare cu alunecare (Sliding beatings)', Eclitura Academiei RPR, Bucharest (1961)

19 Kuhn, E. C. and Cooper, G. D. 'Some factors affecting performance of large pivoted-pad thrust bearings', Lubrication Engineering, Vol 15, No 6 (1959) p 264

20 Nica, AL and Matin, Gh. 'Cercetari experimentale asupra distributiei de temperaturi in lagarele circulare cu alunecare (Experimental investigations on the temperature distrubution in journal bearings)', Studii si Cercetari de Mecanica Aplicata, Bucharest, Vo130, No 5 (1969) p 1053

21 Woolacott, R. G. and Cooke, W. L. 'Thermal aspects of hydrodynamic journal bearing performance at high speeds', Proceedings of the Instutution of Mechanical Engineers, Vol 181, Part 30 (1966-1967)

22 Tipei, N. and Nica, AL 'Investigations on the operating conditions of journal beatings. I-The influence of viscosity variation', Revue de M$canique Appliqube, Bucharest, Vol 4, No 4, (1959) p 609

23 Nica, AL 'Contributions to the determination of the real clearance in sliding bearings', Transactions of the American Society o[ Mechanical Engineers, Journal of Basic Engineering, Vol 87, No 3 (1965) p 781

24 Nica, AI. 'Thermal behaviour and friction in journal bearings', Transactions of the American Society of Mechanical Engineers, Journal of Lubrication Technology, Vo192, Series F, No 3 (Jul 1970) p a73

226 TRIBOLOGY November 1971