Optimum computerized design of hydrodynamic journal bearings

Int. I. mech. Sci., Vol. 18, pp. 215-222. Pergamon Press 1976. Printed in Great Britain

OPTIMUM COMPUTERIZED DESIGN OF HYDRODYNAMIC JOURNAL BEARINGS D. DowsoN Department of Mechanical Engineering, University of Leeds, Leeds

and J. N. ASHTON Department of Mechanical Engineering, University of Manchester Institute of Science and Technology, Sackville Street, Manchester M60 10D, England (Received 18 July 1975, and in revised form 2 April 1976)

Summary--Solutionsof the Reynolds equation have been computed for a bearing configuration which, previously, has received little attention theoretically. Operating characteristics evaluated from the computed solutions are presented graphically, and also form the basis for the development of a computerized optimum design technique. The optimum design objective is stated explicitly in terms of the operating characteristics and is minimised within both design and operative constraints. Optimum designs are given for a range of runningconditions for minimumpower loss. A comparison is made of the optimum designs obtained for three differing design objectives. Results from the computerised design technique are also compared with the solution given by a long-hand design procedure.

NOTATION c

radial clearance dimensionless operating characteristic F~ dimensionalizing term h film thickness H dimensionless film thickness l length/diameter ratio m M Ms P P, Q~ Q, R T, U W IV, x y /3 p E "O

design objective rotor mass weighting factor for load equality constraint pressure operating characteristic lubricant inlet flow rate lubricant side leakage flow rate bearing radius torque acting on shaft journal speed bearing load weighting factors applied to the operating characteristics circumferential co-ordinate axial co-ordinate attitude angle lubricant density eccentricity ratio lubricant viscosity

1. INTRODUCTION THE ART of journal bearing design has progressed considerably since the advent of high speed digital

computers. Analytical solutions of the Reynolds equation for infinitely long and very narrow bearings have been replaced by the computerised numerical solutions for the finite length journal bearing. In this paper an account is given of the way in which digital computer techniques have been applied to the actual design process, thus replacing the conventional long-hand-graphical procedure. It is a rather lengthy exercise to obtain a suitable design of bearing using hand techniques and to achieve something approaching an optimum design takes considerably longer. Raimondi and Boyd, 1 Rippel, 2 I. Mech. E. Data Sciences Item No. 660233 and others have presented methods by which a systematic search is made for an improved design. However, it will be shown in this paper that by using the high speed of a digital computer a systematic procedure can be used to advantage to produce an optimum, or near optimum, design very rapidly. More attention is being given to the whole concept of optimum design. Recently, Moes and Bosma, ~e Seireg and Ezzat, 6 Rowe et al. 7 and Unklesbay et al., s have presented work aimed at optimizing the design of various types of bearing. A computer program has been written which applies a powerful gradient search technique, 215

216

D, DOWSON and J. N. ASHTON

d e v e l o p e d by F l e t c h e r , 9'~° to the design of a finite length journal bearing within b o t h linear and non-linear constraints. The data n e c e s s a r y f o r the design optimisation p r o g r a m has b e e n evaluated f o r a bearing configuration which, although quite c o m m o n in practice, a p p e a r s to h a v e r e c e i v e d little a t t e n t i o n theoretically. Solutions of the R e y n o l d s e q u a t i o n for this bearing configuration h a v e b e e n c o m p u t e d and the required operating c h a r a c t e r i s t i c s h a v e b e e n s t o r e d as d i s c r e t e values on magnetic tape f o r s u b s e q u e n t use by the design optimisation program. 2. SOLUTIONS OF THE REYNOLDS EQUATION

w

(o)

~-~N

Numerical solutions for the Reynolds equation have been obtained to yield the pressure distribution for a finite length single inlet groove journal bearing. Unlike most previous solutions, the position of the inlet groove is fixed for all values of eccentricity ratio. It is situated on the load line in the unloaded half of the bearing. The arrangement is shown diagrammatically in Fig. l(a). Using the computed pressure distributions, the following dimensionless operating characteristics have been evaluated:

~

~gitrn thickness(h)

~lh~o~

1

¢x

OiL inlet groove Piretss~)r~i° n (P/ | [

Ih

~

~Fitrn thickness(h)

(c)

Load carrying capacity (flY) Torque acting on shaft (T,) Side leakage oil flow rate (t~) Inlet oil flow rate ((~,) Adiabatic temperature rise (Ai-) Attitude angle (/3).

FIO. 1. (a) Journal bearing configuration. (b) Actual inlet boundary conditions. (c) Assumed inlet boundary conditions.

2.2 Basic assumptions The solutions for a finite length journal bearing are subject to the following assumptions: (a) The bearing surfaces are rigid. (b) The lubricant is isoviscous and incompressible. (c) The lubricant behaviour is Newtonian. (d) Inertia and gravitational effects on the lubricant film are neglected. (e) The variation of pressure across the lubricant film is neglected. (f) The film thickness is small compared with the radius of curvature of the bearing surfaces. (g) Steady state conditions apply. (h) Bearing surfaces are parallel in the axial direction. 2.3 Reynolds equation For an isoviscous incompressible lubricant Reynolds equation may be written as, _

OiL

hmin

hinlctl P'l

2.1 Bearing configuration

(a) (b) (c) (d) (e) (f)

intet groove

hrnax

the

ah

c~x The equation has been written in finite difference form and solved iteratively using the Gauss-Seidel overrelaxation technique. A constant size step grid has been used in the axial (y) direction and a variable step grid in the circumferential (x) direction. In reaching a solution, the equation has been written in dimensionless terms and the substitution given by Hakannson" and Cameron ~: of 4~ = H3/2/~ has been adopted.

2.4 Boundary conditions In most previous numerical solutions of the Reynolds equation it has been assumed that the lubricant inlet groove lies at the point of maximum film thickness. The boundary condition is usually that p = 0 along the inlet groove. However, in practice it is usual to have a finite supply pressure (Ps). If, as has been assumed in this particular bearing configuration, the lubricant inlet groove lies some way before the point of maximum film thickness ( h ~ ) , the pressure will fall to a local minimum somewhere between the two, as shown in Fig. l(b). At this point the condition (ap/ax)=O will exist. As the supply pressure (Ps) is reduced, this condition will tend towards the limiting case of p =(ap/ax)=0, as shown in Fig. l(c). The effect of the pressure distribution between the inlet groove and the point where p = (ap/Ox)=0 has been ignored in the analysis since the accuracy with which the true physical situation can be incorporated is doubtful. Any contribution to the load components resulting from the relatively small hydrodynamic pressures is negligible. In evaluating the operating characteristics it has been assumed that the clearance space is completely filled with lubricant at atmospheric pressure in this region (i.e. p = 0). The usual assumption that p=(ap/ax)=O at the cavitation boundary has been included. At the edges of the bearing the pressures are said to be equal to atmospheric pressure (i.e. p = 9). 2.5 Computed results Solutions of the Reynolds equation over a range 0.05~
Optimum computerised design of hydrodynamic journal bearirtgs puted. For each value of ~ and (l/d) the dimensionless operating characteristics listed in section 2.1 have been evaluated. The results are shown plotted against ~ for differing values of (l/d) in Figs. 2 and 3. Since these values are required by the optimisation program the results are also stored as discrete values on magnetic tape.

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217

any operating characteristic may be written

P, = F~(W, l/d, c,R, U, ~)f,(~, l/d). Since the values of W, U and R are usually predefined by other considerations of the component of which the

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FIG. 2. Dimensionless operating characteristics: (a) load carrying capacity vs eccentricity ratio; (b) torque acting on the shaft vs eccentricity ratio; (c) side leakage oil flow rate vs eccentricity ratio; (d) inlet oil flow rate vs eccentricity ratio.

3. DESIGN OPTIMISATION 3.1 Design variables The dimensionless operating characteristics can be expressed as a function of ~ and (l/d) (i.e. f(~,(l/d))). Using interpolation techniques the values of any of the operating characteristics, which are stored as discrete values on magnetic tape, may be evaluated at any point within the range 0.05~<~<0.95, 0.125~(1]d)<~2. In dimensionalizing the operating characteristics, the dimensionless expression is multiplied by some function of W, (lid), c, R, U and 71 (i.e. F~(W, (l/d) c, R, U, 77). Hence

bearing forms a part, the general term for any operating characteristic may be written as

P~ = F'~(l/d, c, ~l)f~(e, l/d). There are, therefore, four variables in the design of journal bearing, ~, l / d , c and 77.

a

3.2 Constraints More often than not the choice of bearing design is limited by a range of constraints which may be divided

218

D. DOWSON and J. N. ASHTON 4000

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Check constraints

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search I

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I I

Check for saddle point and uniqueIness using Iocal l L_ search J

(c) FIG. 3. Dimensionless operating characteristics: (a) adiabatic temperature rise vs eccentricity ratio; (b) attitude angle vs eccentricity ratio; (c) flow chart for optimisation program. I/d=~, = 0.125 lld=~ = 2.0 c=a, = 2.5 x 10-3 mm cm~ = 2.5 mm 7/mi. = 0 . 7 cp T/m,~ = 700 cp.

broadly into t w o categories: (1) design constraints (2) operative constraints. Design constraints are primarily linear constraints on the design variables and usually take the form of upper and lower bounds. Operative constraints are usually non-linear constraints on the operating characteristics. They often take the form of upper and/or lower bounds. However, other types of operative constraints are frequently involved. Below are given a number of constraints which have been applied in the present work. Design constraints: Emin = 0 . 0 5

~=.x = 0.95

Operative constraints: (a) Constant load capacity R3U l .

Define c~ =

[

RUI

W-n---;IL

E,

l

O p t i m u m c o m p u t e r i s e d design of h y d r o d y n a m i c journal bearings (b) R e y n o l d s n u m b e r R, = p U~< 1000 ~7 (c) Taylor n u m b e r T, = R (Re)2 ~< 1708. (d) Rotor stability, the rotor stability criterion given by L u n d and Saibel," as interpreted by Seireg and Ezzat, 6 h a s been incorporated. 2"oUR 3 [ l ~3

For tr ~<0-28 cMU 2 R2W

3 tr o-55

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(e) Guide to m i n i m u m clearance 3 c >~ hm~n 12-5/(11 - 1/ (l/d)) w h e r e for R = 1 0 0 m m h m ~ . = 6 . 3 5 x 10-3mm for n ~<800 r p m and hmin = (13.907 x 10-'n + 5"240) x 10-3 m m for n > 800 rpm, n = journal speed rpm. (f) Safe design domain; the safe design domain given in Fig. 6 of I. Mech. E. Data Sciences Item No. 660233 h a s been included in the p r o g r a m in the f o r m of non-linear constraints on • a n d lid. It is not suggested that the list of constraints given a b o v e are all that can be applied to a bearing design. T h e list r e p r e s e n t s a sample of the type of constraints likely to be involved and serve to give s o m e indication of the complexity of the task facing the would-be bearing designer.

3.3 Design objective T h e design objective m u s t be stated explicitly in t e r m s of the operating characteristics it is required to minimise or maximise. It m a y be that m o r e than one operating characteristic is to be minimised, with differing importances attached to each. T h e facility to incorporate this situation h a s been included in the design objective function. If m is the design objective expression then, m =~,(Wt,.P,)

where Wt are the weighting factors which reflect the relative importance of each operating characteristic in the design objective. It is t h e n n e c e s s a r y to add to this expression the load equality constraint s u c h that m = ~ (Wt,. P , ) + M g a .

219

w h e r e Mg is a weighting factor introduced to e n s u r e that the value of the load equality function does not b e c o m e insignificant in comparison with the other terms. Initially, during the optimisation p r o c e s s Mg is set equal to zero until a constrained m i n i m u m value of E (Wti. P~) has been located. Mg is then set equal to 10-5 and the search is continued. W h e n a n e w m i n i m u m has been reached Mg is multiplied by a factor of 10 and the search is re-started. This is repeated until two s u c c e s s i v e minima are equal. At this point it is a s s u m e d that the constrained o p t i m u m design h a s been located and the search is terminated. 3.4 Search techniques T h e basic optimisation technique is of the gradient search type as developed by Fletcher 9'1° for optimisation within linear constraints. In order to take account of the non-linear constraints a local univariate search in each of the design variables has been included. In the optimisation program, the gradient search technique optimises the design objective m. At each n e w point in the search a check is m a d e for violations of both the design and operative constraints. If a non-linear constraint has been violated the gradient search is halted and a local univariate search is carried out to bring the search back into the feasible design domain. The univariate search is also u s e d to e n s u r e that the initial estimate of the design is within the design d o m a i n and also to e n s u r e that the final solution is not a saddle point and is unique. A flow chart for the design optimisation program is s h o w n in Fig. 3(c). Further details and a program listing are available in ref. (15). In order to keep the c o m p u t i n g time reasonable, the step size in the univariate search h a s been 10% over a m a x i m u m range of -+ 90%. This r e p r e s e n t s a possible m a x i m u m error in the c o m p u t e d values for the design variables of -+10%. 3.5 Optimum design results O p t i m u m designs for a design objective of m i n i m u m power loss have been c o m p u t e d for two different loads over a range of running speeds. T h e results are for a 1 0 2 m m dia. bearing with all the design and operative constraints given in section 3.2 applied. Figs. 4(a), (b) and (c) show the o p t i m u m values for ~, lid and c, respectively, and Fig. 5(d) gives the o p t i m u m values of the design objective m. In this case, of course, m is the power loss. For each solution the value of viscosity ~ selected h a s been that of r/m~n, the lower b o u n d design constraint on viscosity. To further illustrate the technique and to d e m o n s t r a t e h o w entirely different designs are achieved for different design objectives, o p t i m u m designs have been c o m p u t e d for the following three design objectives: (1) m i n i m u m power loss; (2) m i n i m u m adiabatic temperature rise; (3) m i n i m u m lubricant flow rate. Again the designs are for a 102 m m dia. bearing with the constraints described in section 3.2 applied. The o p t i m u m values for each of the design variables and the value of the design objective m are given in Table 1. A s a further example to illustrate the difference b e t w e e n the results of a hand technique and a computerised o p t i m u m design technique, a worked example f r o m the I. Mech. E. Engineering Data Sciences Item No. 66023 ~ has been re-analysed. T h e o p t i m u m design solution w a s reached in three m i n u t e s computing time. It is

220

D. DOWSON and J. N. ASHTON

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FIG. 4. (a) Optimum eccentricity ratio vs speed; (b) optimum length-to-diameter ratio vs speed; (c) optimum radial clearance vs speed; (d) design objective vs speed. TABLE 1

Design objective Minimum power loss Minimum adiabatic temp. rise Minimum oil flow rate

c

1/

~

l/d

(mm)

(cp)

0.841

0.317

1.45 × 10 -2

0.85

1.48

8.9 x 10 _2

21.9

0.98

0.081

0.2

1.45 x 10 -2

323.0

0.051

considered that this is much less than the time necessary to reach a solution, using the hand technique. The example considered is a 10in. (254mm) dia. bearing, supporting a steady load of 8500 lb (37.8kN) running at a speed of 400 rpm. A further design constraint of l [d ~< 1.2 has been applied in addition to the constraints given in section 3.2. It has, however, not been possible to reproduce all the constraints applied in the hand technique, particularly those relating to the viscosity of the lubricant and this is reflected in the results. The design

0.69

m

0.03

objective has been to minimise power loss, as this was thought appropriate to the application, i.e. a bearing in an electric motor. A comparison of the two solutions is given in Table 2. 4. D I S C U S S I O N 4.1 Solutions o[ the Reynolds equation Figs. 2 and 3 show the dimensionless operating characteristics plotted against eccentricity ratio for several length to diameter ratios. The results obtained for

O p t i m u m c o m p u t e r i s e d design of h y d r o d y n a m i c journal bearings TABLE 2

Eccentricity ratio L e n g t h / d i a m e t e r ratio Radial clearance in. (ram) Viscosity Reynolds (cp) P o w e r loss H.P. (W) Oil flow rate gal/min (cm3/sec) Adiabatic temp. rise °F (°C)

I.Mech.E. data sciences item No. 66023 results

Computed optimum results

0.82 0.35 0.005 (0.127) 4.8 × 10-6 (33.0) 0.965 (720) 0.323 (24.5) 17 (9.5)

0.8 1.13 0.0025 (0.0635) 1.0 × 10-7 (0.69) 0.181 (135) 1.15 (87.0) 1.2 (0.66)

a length to diameter ratio of infinity with the inlet groove on the load line s h o w a g r e e m e n t with the results predicted and m e a s u r e d by Floberg) 3 A c o m p a r i s o n with the results for a bearing with the inlet groove at the point of m a x i m u m film thickness can be made. The load carrying capacity s h o w s close a g r e e m e n t at high and low values of eccentricity ratio. At intermediate values the load carrying capacity is greater for the bearing with the inlet groove on the load line. This difference increases with increasing length to diameter ratio. T h e d i m e n s i o n l e s s shaft and b u s h torques are also greater for a bearing with the inlet groove on the load line. This is due to two factors: Firstly, the pressure region extends over a greater arc and secondly, the region of cavitating flow is reduced by the clearance space being completely filled with lubricant in the region b e t w e e n the inlet groove and the point of m a x i m u m film thickness. As the eccentricity ratio t e n d s to zero the two solutions approach the s a m e value. T h e inlet oil flow rate for a bearing with the inlet groove on the load line is in general higher than for a bearing with the inlet groove at the point of m a x i m u m film thickness. At high and low values of eccentricity ratio the two solutions tend to the s a m e value. T h e difference in the two t y p e s of solution increases with increasing length to diameter ratio. T h e p r e s s u r e gradient at the point of m a x i m u m film thickness will decrease as the start of the pressure region m o v e s towards the inlet groove, thus increasing the inlet oil flow rate. A s the eccentricity ratio t e n d s to zero and unity the solutions tend to the same value since as • -o 0, h~ -~ hm~ --->c and as ~ -~ 1, hmax--> h, T h e side leakage oil flow rate is higher for the bearing with the inlet groove on the load line, due to the increased circumferential extent of the h y d r o d y n a m i c pressure region. T h e friction coefficient ratio is defined as Frictional Torque Load. Since t h e s e two characteristics are affected in a similar mann~.r by having the inlet groove on the load line, the variation in values for the two types of solution is not so pronounced. The difference is m o s t p r o n o u n c e d at low values of eccentricity ratio. T h e Adiabatic T e m p e r a t u r e Rise is similarly affected since it too is a ratio of two operating characteristics, the shaft torque and the lubricant flow rate. Both are greater for a bearing with the inlet groove on the load line, the

221

ratio of which remains very nearly the same for the two t y p e s of solution. More heat is generated and an increased lubricant flow is p r e s e n t to carry the heat away. In general, it m a y be said that the operating characteristics for the beating with the inlet groove on the load line do not vary significantly f r o m t h o s e for a bearing with the oil groove at the point of m a x i m u m film thickness for length to diameter ratios of unity or less. A b o v e a length to diameter ratio of unity the difference b e c o m e s significant and should be taken into account. A more detailed comparison is given in ref. (15). 4.2 Optimum design results T h e results given in Fig. 4 for a 102 m m dia. bearing supporting two loads over a range of running speeds, follow the expected trends. Since the design objective has been to minimise power loss the design procedure would attempt to minimise ,, ~ and 1/d whilst trying to m a x i m i s e c. This it cannot do completely b e c a u s e of the constraints and particularly b e c a u s e of the load equality constraint a. T h e result h a s been that in all c a s e s the value of selected is that of ~=in and as a c o n s e q u e n c e high values of E are selected. Although the design search attempts to minimise l[d, at low speeds l/d m u s t be large to provide the n e c e s s a r y load carrying capacity. T h e radial clearance c is m a x i m i s e d as far as possible. However, at low speeds, because lid cannot be increased due to the safe design domain constraint, a small value of c is obtained. As the speed increases c increases and lid decreases. At higher speeds ~ = em,x and in order to ensure the n e c e s s a r y film thickness c is increased. Finally, E d e c r e a s e s and lid increases as the safe design domain constraint b e c o m e s inactive. T h e increase in lid is n e c e s s a r y to maintain the required load carrying capacity. Again the results given in Table 1 for three different design objectives show the expected trends. For m i n i m u m power loss the results agree with those given above. T h e results for m i n i m u m Adiabatic T e m p e r a t u r e Rise are as expected, with a high value of l/d, and c, to give a high lubricant flow rate and a low value of *7 to give a low power loss. For m i n i m u m lubricant flow rate the results are again to be expected with low values of ~, lid and c, the three parameters on which the lubricant flow rate depends. The results given in Table 2 clearly d e m o n s t r a t e the need for a complete mathematical representation of all the constraints. Since it was not possible to reproduce the constraints on viscosity applied in the worked example, a true comparison cannot be made. An a s s e s s m e n t can be made, however, on the basis of speed and clearly the computerised solution is by far the faster. 4.3 Conclusions T h e solutions of the Reynolds equation for the bearing configuration presented here vary little f r o m the more c o m m o n l y a s s u m e d configuration for lid ~< 1. A b o v e this value the differences are significant and should be taken into account. T h e o p t i m u m design technique presented here demonstrates the feasibility of this type of approach. It is clear f r o m the results that the design objective m u s t be explicitly stated in terms of the design variables within equally explicitly defined constraints. T h e s e constraints m u s t completely define the design domain. Failure to achieve this will permit the c o m p u t e r to select an unfeasible design. Unlike the bearing designer, the c o m p u t e r has no powers of 'discretion'. Nevertheless, it is

222

D. DOWSON and J. N. ASHTON

believed that the design optimisation technique has been proven as a valuable design tool.

REFERENCES 1. A. A. RAIMONDIand J. BOYD, A Solution [or the Finite Journal Bearing and its Application to Analysis and Design I, II & I I I , Trans. A.S.LE. (1958). 2. H. C. RIPPEL, Cast Bronze Bearing Design Manual, Cast Bronze Bearing Inst. Inc., Ohio (1967). 3. Calculation Methods for Steadily Loaded Pressure Fed Hydrodynamic Journal Bearings, I. Mech. Engineering Sciences Data Item No. 66023 (1966). 4. H. MOES and R. BOSMA, Trans. A.S.M.E. (1970). 5. H. MoEs and R. BOSMA, Trans. A.S.M.E. (1970). 6. A. SEREG and H. EZZAT, Trans. A.S.M.E. (1969). 7. W. B. ROWE, J. P. O'DONOGHUE and A. CAMERON, Tribology, August 1970.

8. K. UNKLESBAY, G. E. STAATS and D. L. CREIGHTON, Int. J. Engng. II, 973 1973. 9. R. FLETCHER, I.C.I. Report MSDM/68/130 (1968). 10. R. FLETCHER, I.C.I. Report MSDH/68/19 (1968). 11. B. HAKANSSON, The Journal Bearing considering variable viscosity, Report No. 25, Institute of Machine Elements, No. 298 Trans., Chalmers University of Technology (1964). 12. A. CAMERON, Principles of Lubrication, Longmans Green, London (1966). 13. L. FLOBERG, The Infinite Journal Bearing considering vaporisation, Report No. 2. Institute of Machine Elements, No. 189 Trans, Chalmers University of Technology, Gothenburg (1957). 14. J. W. LUND and E. SEIBEL, Oil Whip Whirl Orbits of a Rotor in Sleeve Bearings, J. Engng. Ind. pp. 813-823 Nov. (1967). 15. J. N. ASHTON, Ph.D. Thesis, University of Leeds (1973).