Roller Bearing Design

8/13/2019 Roller Bearing Design

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Engineering Optimization,Vol. 35, No. 6, December 2003, 649659

ROLLING ELEMENT BEARING DESIGN THROUGHGENETIC ALGORITHMS

INDRANEEL CHAKRABORTY , VINAY KUMAR a , ,SHIVASHANKAR B. NAIR a and RAJIV TIWARI b,

a Department of Computer Science & Engineering; b Department of Mechanical Engineering, Indian Institute of Technology Guwahati, North Guwahati, Guwahati 781039, Assam, India

(Received 17 August 2002; Revised 11 March 2003; In nal form 12 August 2003)

The design of rolling element bearings has been a challenging task in the eld of mechanical engineering. While mostof the real aspects of the design are never disclosed by bearing manufacturers, the common engineer is left with noother alternative than to refer to standard tables and charts containing the bearing performance characteristics. Thispaper presents a more viable method to solve this problem using genetic algorithms (GAs). Since the algorithm isbasically a guided random search, it weakens the chances of getting trapped in local maxima or minima. The methodused has yielded improved performance parameters than those catalogued in standard tables.

Keywords : Rolling element bearings; Optimum mechanical design; Genetic algorithms

1 INTRODUCTION

In general, a bearing is a machine element that plays a vital role in transferring loads betweentwo moving machine parts. The term rolling element bearing is used to describe that class of bearings wherein the main load is transferred through elements in rolling contact rather than insliding contact. Rolling element bearings are very popular because of their low starting torqueand kinetic friction. Rolling element bearings have wide applications from home appliancesto industrial machines such as gear boxes, electric motors, instruments and meters, internalcombustion engines, agricultural industries, textile industries, aircraft gas turbines, etc.

The design of rolling element bearings has a great impact on the performance, life andreliability of bearings. Consequently, it also affects the operating quality and economy of machines on which the bearings are used. The responsibility of selecting an optimum schemefrom all possible alternative designs, or more simply stated to make the best choice, rests solelyon the bearing designer.

The widespread use of rolling element bearings [1] has prompted many researchers tolook into simple and nave methods of computing optimal values of their associated design

Indraneel Chakraborty is currently with Massachusetts Institute of Technology and can be reached [email protected]

Vinay Kumar is currently with Mindtree Consulting, India and can be reached at [email protected] Corresponding author. E-mail: [email protected]

Engineering OptimizationISSN 0305-215X print; ISSN 1029-0273 online c 2003 Taylor & Francis Ltd

http: // www tandf co uk / journals


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650 I. CHAKRABORTY et al .

parameters. Conventionalmethods [2] are deterministic in nature anduse only a few geometricdesign variables because of their complexity and convergence problems.

Renements in the design can be made only by taking into account an adequate numberof geometric design variables and their associated constraints dened by the boundary dimen-sions, the mobility and the strength of the bearing parts. It is this increase in design variables

that tends to make the problem complex and computationally time consuming. Worse, there isno way of judging whether the solutions derived are optimal ones or those that were led astrayinto local optima. In other cases the problem may not converge to a solution.

The best design scheme has optimum technical and economic indices. For this purpose,various methods are adopted. A method of intuitional optimization means selecting the bestfrom several design schemes while a method of testing optimization means modifyinga designby sample tests gradually in order to seek the best one. Also a method of evolving optimizationrepresents the renewal of a design for improvement upon comments of customers or users.Mathematical optimization methods have an edge over all the above techniques in that theyare less time consuming and have better economic characteristics. Developments in com-puter technology have proved to be a great boon to the world of mathematical optimization.Computer-aided design (CAD), a dynamic eld of engineering science, is in fact based onmathematical optimization.

1.1 Motivation Behind Using Genetic Algorithms

Among the stochastic optimization techniques, genetic algorithms (GAs) [310] are popularand promising ones in that they overcome the problem of local minima (or maxima). Oneof the advantages of GAs is that they explore different areas of the search space in paralleland have no presumptions with regard to the search space. Better still, they provide severalalternative solutions to the problem at hand and can be easily combined with other meth-ods.

1.2 Genetic Algorithms in Mechanical Engineering

A plethora of mechanical engineering applications have been known to exploit GAs [1114].Choi and Yoon [15] have used GAs to design an automotive wheel-bearing unit with discretedesign variables. The authors are not aware, to date, of any literature available on the use of GAs for designing rolling element bearings with continuous design variables.

Standard nonlinear programming techniques to solve the discrete optimization problem [2]are inefcient and computationally expensive. Also, in most cases, such techniques nd alocal optimum that is closest to the starting point. In contrast, GAs are well suited to solvesuch problems and in most cases can nd the global optimum with a high probability since apopulation of design points, rather than a single point, is used for starting the procedure.

1.3 Basic Overview of Genetic Algorithm

The algorithm is provided with a set of possible solutions (represented by chromosomes ) termeda population . Solutions from one population are taken and used to form a new population. Thisis motivated by a hope that the new population will perform better than its predecessors.Solutions chosen to form new solutions ( offsprings ) are selected based on their tness themore suitable they are, the better their chances of being reproduced. This process of selectionis repeated till some predetermined condition based on, for instance, the numberof populations


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ROLLING BEARING DESIGN 651

This paperdescribes an attempt to solve thedesign optimization problem for rolling elementbearingswith ve design parameters usingGAs based on the requirement of the longest fatiguelife. This is a singularlydifcult problem andhardlyany satisfactorysolution hasbeen reportedin the literature before this effort [1].

2 DESIGN OPTIMIZATION PROBLEM OF ROLLING ELEMENT BEARINGS

On the basis of operating requirements, different objective functions for rolling element bear-ings may be proposed, the most importantof these being the requirement of the longest fatiguelife. In normal operating conditions of rolling element bearings, the main mode of failureis contact fatigue . Without a special note, bearing life means contact fatigue life. The basicrequirement for conventional rolling element bearings is thus a long fatigue life. To solve thisproblem for a given size of the bearing outline or boundary dimensions (i.e. bearing bore, d ,and outside diameter, D ), the dynamic load rating C should be maximum. The dynamic load

rating C is dened as the constant radial load which a group of apparently identical bearingscan endure for a rating life of one million revolutions of the inner ring (stationary load and sta-tionary outer ring). The fatigue life [16], L, of the bearing (in millionsof revolutions) subjectedto any other applied load F is given by

L =C F

a

, ( 1)

where a = 3 in the present case (for ball bearings).

2.1 Design Parameters

Design parameters are basically internal structure dimensions (see Figs. 1 and 2) and othervariables, called main parameters, which need to be determined in the bearing design. The ve


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FIGURE 2 Cut sections of bearing races.

design parameters [1] for the given problem are represented as

X = [ Db , Z , Dm , f o , f i ]T , ( 2)

where

Db = diameter of the ball Z = number of balls

Dm = pitch diameter ( i.e. diameter of locus of center of the balls) f o = r o / Db = curvature radius coefcient of the outer raceway groove f i = r i / Db = curvature radius coefcient of the inner raceway groove

r o and r i are the outer and the inner raceway groove curvature radius, respectively.

2.2 Objective Function

Based on the dynamic load rating the objective function can be expressed as [1]

max[ f ( x )] = max[ f c Z 2/ 3 D1.8b ] Db 25 .4 mm ,

= max[3 .647 f c Z 2/ 3 D1.4b ] Db > 25 .4 mm ,(3)

where

f c = 37 .91 1 + 1.041 1 +

1.72 f i (2 f o 1) f o (2 f i 1)

0.41 10 / 3 0.3

0.3(1 ) 1.39

(1 + ) 1/ 3 2 f i

2 f i 1

0.41

.

Because = Db cos / D

m (which appears in the objective function, (Eq. (3)) is not an

independent parameter, it does not gure in the list of design parameters. Here , the freecontact angle, depends upon the type of bearing. In the current discussion, the deep groove


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2.3 Constraint Conditions

For the convenience of the bearing assembly, the number and diameter of the balls shouldsatisfy the following requirement [1]

2( Z 1) sin 1 Db Dm

0 . (4)

Therefore, a constraint condition is

g1( X ) =0

2sin 1 ( Db / Dm ) Z + 1 0, ( 5)

where, 0 is the maximum tolerable assembly angle (see Fig. 3) which depends upon thebearing geometry.

The diameter of balls should be within a certain range given by

K D min D d

2 Db K D max

D d 2

, ( 6)

where K D min and K D max are the minimum and the maximum values of the ball diameterconstants respectively, that is connected with the diametric series of bearings and ball strength.

The corresponding constraint conditions are given by

g2( X ) = 2 Db K D min ( D d ) 0, ( 7)

g3( X ) = K D max ( D d ) 2 Db 0. (8)


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In order to guarantee the running mobility of bearings, the difference between the pitchdiameter and the average diameter in a bearing should be less than a certain given value.Therefore,

g4( X ) = Dm (0.5 e)( D + d ) 0, ( 9)

g5( X ) = (0.5 + e)( D + d ) Dm 0, ( 10 )

where, e is a constant and is obtained based on the mobility conditions of the balls.In the design optimization of bearings, the pitch diameter is usually larger than the corres-

ponding average diameter of a bearing. Hence, the thickness of a bearing ring at the outerraceway bottom should not be less than Db , where is a constant which is obtained from thesimple strength consideration of the outer ring. The constraint condition is then

g6( X ) = 0.5( D Dm Db ) Db 0. (11 )

The groove curvature radii of the inner and outer raceways of a bearing should not be less

than 0 .515 Db . No upper limit is specied for the groove curvature radius, but if it is larger than0.52 Db and 0 .53 Db , respectively for inner and outer raceways, the dynamic load rating of thebearing will decline. Therefore

g7( X ) = 0.52 f i 0.515 , ( 12 )

g8( X ) = 0.53 f o 0.515 . (13 )

In brief, a design optimization problem is summed up as the optimization of an objectivefunction (Eq. (3)) and eight inequality constraints expressed by Eqs. (5) and (713).

3 COMPUTATIONAL PROCEDURE

This section describes the procedure used for implementation of GAs in solving the givenproblem.

3.1 Handling Constraints

The method of handling constraints in optimization problems is a critical issue in the domainof GAs. Constraints are usually classied as equality or inequality relations. The problem in

hand has only inequality constraints. Genetic algorithms generate a sequence of parametersto be tested using the system model, the objective function and the constraints. After runningthe model the objective function is evaluated and a check is performed to verify whether anyconstraints are violated. If not, the parameter set is assigned the tness value corresponding tothe objective function evaluation. If constraints are violated, the solution is infeasible and thushas no tness. This solution works in the current optimization problem given that the domainof feasible solutions is large.

3.2 Optimization Procedure

The overall procedure for solving the discrete optimization problem mentioned using GAsis illustrated in Figure 4. The parameters of the GA are specied in the beginning and aninitial population is generated. Given pop design points, the values of tness and the design


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FIGURE 4 Flowchart for the solution of a problem using GA.

violating any of the constraints are given a tness value of zero. A new population is created byselection, crossover and mutation . The design variables viz. pitch diameter Dm , ball diameter Db , number of balls Z , f i and f o are varied at this stage. The wholeprocess is repeateduntil thenumber of generations reaches the maximum number specied. Since superior individuals inthe population have higher probabilities of survival to the next generation, the procedure con-verges to the best design point after maxgen generations. The design point normally representsthe global optimum, owing to the evolutionary nature of the algorithm.

4 ENCODING THE PROBLEM

Representing the problem to suit the algorithm is a crucial issue. The following has to be


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In order to use GAs to solve problems, variables x i of the function f ( x ) are rst coded insome string structures. Generally binary-coded strings having 1s and 0s are used. The lengthof the string is usually determined depending on the desired accuracy of the solutions. Forinstance, if four bits are used to code each variable in a two-variable function optimizationproblem, the strings (00000000) and (11111111) would represent the extreme points respec-

tively, because the sub strings (0000) and(1111) have the minimum andthe maximum decodedvalues.In the problem at hand, decimal mapping has been used and the solution has the least

signicant bit of the order of 10 6 Newton, which is sufcient for the given problem. Sincethere are ve parameters there are ve sub-strings each of length ve digits. Each point x canbe denoted by x = ( x 1 , x 2, x 3 , x 4 , x 5) . The function value at the point x can be calculated bysubstituting x in the given objective function f ( x ) .

The operation of GAs begins with a population of random strings representing design ordecision variables. Thereafter, each string is evaluated to nd the tness value. The popula-tion is then operated by three main genetic operations reproduction, crossover , and muta-tion to create a new population of points. The new population is further evaluated andtested for termination. If the termination criterion (the maximum number of iterations speci-ed by the user) is not met, the population is iteratively operated by the above three operatorsand evaluated. This procedure is continued till the termination criterion is met. One cycle of these operations and the subsequent evaluation procedure is known as a generation in GAterminology.

5 RESULTS AND CONCLUSIONS

Classical optimization approaches usually consider only three design parameters, viz. pitchdiameter, Dm , ball diameter, Db and the number of balls, Z , for reasons of complexity [1, 2].In the present analysis we have used ve design parameters (viz. Dm , Db , Z , f i and f o ). Thebest solutions obtained from the present analysis of the given problem are provided in Table Ifor different sets of boundary dimensions i.e. the outer diameter, D and the bore, d . In theconstraints, the following values have been used for the constants: K D min = 0.5, K D max =0.8, e = 0.1, = 0.1 and 0 = 4.7124 radians, which are obtained from considerations of themobility of balls and the strength of balls and rings. The GA parameters used are: populationsize = 300, cross-over probability = 0.5 and mutation probability = 0.15.

In order to compare the increase in the fatigue life of the bearings designed using GAs asagainst those using available standard values, the following relation has been used.

= Lg Ls

=C gC s

3

. (14 )

This hasbeen derived from Eq. (1) using a constant applied load F to both thebearings designedusing GAs and those designed using available values. The subscripts g and s represent thevalues computed using GAs and those currently available, respectively. From Table I it can beobserved that there is a quantum jump in the fatigue life of the bearings designed using the GAapproach as compared to the ones designed using the standard values, the order of increase inthe life ranging from four to eight.

Table II (a)(c) throw more light on the convergence and the fact that the solution is mostlikely an optimal one and not one trapped in local maxima. Table II(a) shows that even withan increase in mutation probability, the reported values of the dynamic capacity have hardly


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T A B L E I

O p t i m u m

D e s

i g n

P a r a m e t e r s o f

R o

l l i n g

E l e m e n t B e a r i n g s

T h r o u g

h G A

.

B o u n d a r y d i m e n s i o n s

P e r f o r m a n c e c o m p a r i s o n s

( i n p u t )

D e s i g n v a r i a b l e s ( o u t p u t )

( c a l c u l a t i o n s a r e b a s e d o n t h e o u t p u t )

D ( m m )

d ( m m )

D b

( m m )

D m

( m m )

f i

f o

Z

C g ( N e w t o n )

C s

( N e w t o n ) [ 1 6 ]

3 0

1 0

7 . 4

7 0 5

1 9 . 5 3 2 1

0 . 5

1 5 0

0 . 5 1 5 0

7

7 3 0 6 . 6

3 5 8 0

8 . 5

0

3 5

1 5

7 . 9

0 1 9

2 3 . 9 3 5 5

0 . 5

1 5 0

0 . 5 1 5 0

8

9 5 5 3 . 7

5 8 7 0

4 . 3

1

4 7

2 0

1 0

. 5 9 6 8

3 2 . 1 6 2 6

0 . 5

1 5 0

0 . 5 1 5 2

8

1 6 2 1 3 . 4

9 4 3 0

5 . 0

8

6 2

3 0

1 2

. 7 9 4 2

4 4 . 0 7 8 5

0 . 5

1 5 0

0 . 5 1 5 0

9

2 5 7 8 5 . 0

1 4

, 9 0 0

5 . 1

8

8 0

4 0

1 5

. 9 9 7 1

5 7 . 5 1 7 9

0 . 5

1 5 0

0 . 5 1 5 7

9

3 8 9 7 9 . 1

2 2

, 5 0 0

5 . 2

0

9 0

5 0

1 5

. 8 2 1 4

6 7 . 8 0 4 0

0 . 5

1 5 0

0 . 5 1 5 0

1 1

4 5 1 6 1 . 4

2 6

, 9 0 0

4 . 7

3

1 1 0

6 0

1 9

. 3 3 3 1

8 2 . 9 1 2 4

0 . 5

1 5 2

0 . 5 1 5 2

1 1

6 4 5 4 2 . 3

4 0

, 3 0 0

4 . 1

1

1 2 5

7 0

2 1

. 9 8 8 7

9 4 . 1 9 2 4

0 . 5

1 5 0

0 . 5 1 5 0

1 1

8 1 7 0 1 . 2

4 7

, 6 0 0

5 . 0

6

1 4 0

8 0

2 3

. 9 9 5 2

1 0 5 . 8 9 2 6

0 . 5

1 5 0

0 . 5 1 5 0

1 1

9 5 9 1 5 . 9

5 5

, 6 0 0

5 . 1

3

1 6 0

9 0

2 7

. 9 9 5 5

1 2 0 . 7 3 1 0

0 . 5

1 5 0

0 . 5 1 5 1

1 1

1 2 1 4 0 1 . 9

7 3

, 9 0 0

4 . 4

3

N o t e :

P o p u l a t

i o n s i z e =

3 0 0 , c r o s s - o v e r p r o

b a b

i l i t y =

0 . 5 a n

d m u t a t i o n p r o

b a b

i l i t y =

0 . 1 5

.


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TABLE II Variation of Best Solution (for D = 30 mm, d = 10 mm) with Respect to GAParameters.

Mutation probability Dynamic capacity, C g

(a) Population size = 300, Crossover probability = 0.5, Number of generation = 10 , 000

0.01 7297.40.05 7305.90.10 7295.60.15 7286.90.20 7302.60.25 7307.50.30 7285.10.35 7296.7

Crossover probability Dynamic capacity, C g

(b) Population size = 300, Mutation probability = 0.15, Number of generation = 10 , 000

0.1 7305.30.2 7305.30.3 7308.00.4 7291.20.5 7286.90.6 7300.0

Number of generations Dynamic capacity, C g

(c) Population size = 300, Mutation probability = 0.15, Crossover probability = 0.5

100 7191.7200 7286.6300 7286.6400 7286.6500 7286.6700 7286.6

1000 7286.65000 7302.87000 7302.8

minima). Normally the value of mutation rate is kept low to ensure that the search does notbecome too random. Even at a high mutation rate of 0.35, the value of C does not differ muchfrom that reported when the optimum rate is 0.05.

Table II(b) shows solutions arrived at by varying values of crossover probability. Here too

the dynamic capacities have hardly varied. Hovering around the neighbourhood of 7300, thesevalues aremuch the same as those reported in Table II(a). Increasing the numberof generations(Tab. II(c)) seem to produce better values of dynamic capacity but for higher numbers theyseem to saturate to the values reported in Table II(a) and (b). Of special signicance are thosereported for population sizes of 5000 and 7000 (Tab. II(c)) wherein the results derived areidentical, thus clearly indicating high precision.

The dynamic load rating, C , also referred to as the optimum technical index, is used tocompare the results of the existing design [16] with those of the present ones (see Tab. I).A marked improvement in the value of C has been observed. While the standard values of C for most of the sets are low, the ones found using GAs are about one and a half to twotimes higher. The marginally larger values of C in these cases may however be an indicationof the global optimality of the solutions. It can thus be clearly seen that the computed valuesof dynamic load rating C and the increase in the fatigue life are far better than those reported


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The software developed for the design of the rolling element bearings thus serves as a toolto generate the values of the dynamic load rating C and Db , Dm , f i , f o , Z , parameters thatdene the internal geometry, given various values of boundary dimensions ( D , d ). From themanufacturing perspective, knowledge of the internal geometry could be used in analyzing thebearing which gives insights into the stresses that the bearing can sustain, the deformations

that may occur, kinematical issues and other such performance characteristics even before theactual fabrication of the bearing.The currently available sources for bearing design provide only the boundary dimensions

and performance characteristics. Conventional optimization methods fail to incorporate somany design variables (ve used in the present work) and do not take care of the inherentlocal maxima (or minima) problems. Another advantage of using the GA approach is that thenumber of design variables can be easily increased, if required, without much computationaleffort.

With hardly any source that can provide these values, manufacturers as well as prospectivebearing designers will nd this tool to be efcient and highly productive. Furthermore thecorresponding values of C serve as a measure of the fatigue life of the bearings.

References

[1] Changsen, W. (1991) Analysis of Rolling Element Bearings , Mechanical Engineering Publications Ltd.[2] Rao, S. S. (1996) Optimisation Theory and Applications: Third Edition . John Wiley & Sons, NY.[3] Belew, R. K. andBooker, L. B. (Eds.) (1991) Proc. of the Fourth InternationalConference on Genetic Algorithms ,

Morgan Kaufmann, San Mateo, California.[4] Davis, L. (1991) Handbook of Genetic Algorithms . Van Nostrand Reinhold, New York.[5] Fogel, D. B. and Ghozeil, A. (2000) Schema processing, proportional selection, and the misallocation of trials

in genetic algorithms. Information Sciences , 122 (24), 93119.[6] Goldberg, D. (1989) Genetic Algorithms in Search, Optimization and Machine Learning , Addison-Wesley.[7] Holland, J. H. (1975) Adaptation in Natural and Articial systems . University of Michigan, Ann Arbor Press.[8] Michalewicz, Z. (1996) Genetic Algorithms + Data Structures = Evolution Programs . Springer-Verlag, Berlin.[9] Rawlins, G. J. E. (1991) Foundations of Genetic Algorithms . Morgan Kaufmann, San Mateo, California.

[10] Whitley, D. (Ed.) (1993) Foundations of Genetic Algorithms , Vol. II. Morgan Kaufmann, San Mateo, California.[11] Gen, M. and Cheng, R. (1997) Genetic Algorithms and Engineering Design . Wiley, New York.[12] Deb, K. (1995) Optimization for Engineering Design, Algorithms and Examples . Prentice Hall of India Pvt. Ltd.,

New Delhi.[13] Marcelin, J. L. (2001) Genetic optimisation of gears. International Journal of Advanced Manufacturing Tech-

nology , 19 , 910915.[14] Periaux, P. (2002) Genetic Algorithms in Aeronautics and Turbomachinery . John Wiley & Sons, NY.[15] Choi, D. H. andYoon, K. C. (2001) A design method of an automotive wheel bearing unit with discrete design

variables using genetic algorithms. Journal of Tribology, Transactions of the ASME , 123 (1), 181187.[16] Shigley, J. E. (1986) Mechanical Engineering Design . McGraw-Hill Book Company, NewYork.


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