research article multi-objective optimization of two-stage...

Research ArticleMulti-Objective Optimization of Two-Stage HelicalGear Train Using NSGA-II

R C Sanghvi1 A S Vashi2 H P Patolia2 and R G Jivani2

1 Department of Mathematics G H Patel College of Engineering and Technology Vallabh Vidyanagar 388120 India2Mechanical Engineering Department B V Mahavidyalaya Vallabh Vidyanagar 388120 India

Correspondence should be addressed to R C Sanghvi rajeshsanghvigcetacin

Received 31 May 2014 Revised 21 October 2014 Accepted 4 November 2014 Published 30 November 2014

Academic Editor Liwei Zhang

Copyright copy 2014 R C Sanghvi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Gears not only transmit themotion and power satisfactorily but also can do sowith uniformmotionThe design of gears requires aniterative approach to optimize the design parameters that take care of kinematics aspects as well as strength aspects Moreover thechoice of materials available for gears is limited Owing to the complex combinations of the above facts manual design of gears iscomplicated and time consuming In this paper the volume and load carrying capacity are optimizedThree differentmethodologies(i)MATLAB optimization toolbox (ii) genetic algorithm (GA) and (iii) multiobjective optimization (NSGA-II) technique are usedto solve the problem In the first two methods volume is minimized in the first step and then the load carrying capacities of bothshafts are calculated In the third method the problem is treated as a multiobjective problem For the optimization purpose facewidth module and number of teeth are taken as design variables Constraints are imposed on bending strength surface fatiguestrength and interference It is apparent from the comparison of results that the result obtained by NSGA-II is more superior thanthe results obtained by other methods in terms of both objectives

1 Introduction

Designing a new product consists of several parameters andphases which differ according to the depth of design inputdata design strategy procedures and results Mechanicaldesign includes an optimization process in which designersalways consider certain objectives such as strength deflec-tion weight wear and corrosion depending on the require-ments However design optimization for a completemechan-ical assembly leads to a complicated objective function witha large number of design variables So it is a better practiceto apply optimization techniques for individual componentsor intermediate assemblies than a complete assembly Forexample in an automobile power transmission system opti-mization of gearbox is computationally and mathematicallysimpler than the optimization of complete system Thepreliminary design optimization of two-stage helical geartrain has been a subject of considerable interest since manyhigh-performance power transmission applications requirehigh-performance gear train

A traditional gear design involves computations basedon tooth bending strength tooth surface durability toothsurface fatigue interference efficiency and so forth Geardesign involves empirical formulas different graphs andtables which lead to a complicated design Manual designis very difficult considering the above facts and there is aneed for the computer aided design of gears With the aid ofcomputer design can be carried out iteratively and the designvariables which satisfy the given conditions can be deter-minedThe design so obtained may not be the optimum onebecause in the above process the design variables so obtainedsatisfy only one condition at a time for example if moduleis calculated based on bending strength the same module issubstituted to calculate the surface durability It is accepted ifit is within the strength limit of surface durability otherwise itis changed accordingly So optimizationmethods are requiredto determine design variables which simultaneously satisfythe given conditions As the optimization problem involvesthe objective function and constraints that are not stated asexplicit functions of the design variables it is hard to solve

Hindawi Publishing CorporationJournal of OptimizationVolume 2014 Article ID 670297 8 pageshttpdxdoiorg1011552014670297

2 Journal of Optimization

it by classical optimization methods Moreover increasingdemand for compact efficient and reliable gears forces thedesigner to use optimal design methodology

Huang et al [1] developed interactive physical program-ming approach of the optimization model of three-stagespur gear reduction unit with minimum volume maximumsurface fatigue life and maximum load-carrying capacity asdesign objectives and core hardness module face width ofgear tooth numbers of pinion tooth numbers of gear anddiameter of shaft as design variables In this modeling toothbending fatigue failure shaft torsional stress face widthinterference and tooth number are considered as constraintsThe MATLAB constrained optimization package is usedto solve this nonlinear programming problem Jhalani andChaudhary [2] discussed the various parameters which canaffect the design of the gearbox for knee mounted energyharvester device and later it frames the optimization problemof mass function based on the dimensions of gearbox for theproblem The problem is solved using multistart approach ofMATLAB global optimization toolbox and value of globaloptimum function is obtained considering all the localoptimum solutions of problem Tong and Walton [3] alsoselected center distance and volume as objectives for theinternal gears Numbers of teeth of gear and pinion andmodules are considered as variables for the optimizationand ldquobelt zone searchrdquo and ldquohalf section algorithmrdquo areapplied as optimization methods Savsani et al [4] presentedthe application of two advanced optimization algorithmsknown as particle swarm optimization (PSO) and simulatedannealing (SA) to find the optimal combination of designparameters for minimum weight of a spur gear train Weiet al [5] developed a mathematical model of optimizationconsidering the basic design parameters mainly tooth num-ber modulus face width and helix angle of gearbox asdesign variables and reduction of weight or volume as anobjective The model is illustrated by an example of thegearbox of medium-sized motor truck Optimization toolbox of MATLAB and sequential quadratic programming(SQP)method were used to optimize the gearboxThe designcriterion and performance conditions of gearbox are treatedas constraints

Mendi et al [6] studied the dimensional optimizationof motion and force transmitting components of a gearboxby GA It is aimed at obtaining the optimal dimensionsfor gearbox shaft gear and the optimal rolling bearingto minimize the volume which can carry the system loadusing GA The results obtained by GA optimization arecompared to those obtained by analytical methods Mogaland Wakchaure [7] used GA as evolutionary techniques foroptimization of worm and worm wheel The main objectivefor optimization is minimizing the volume here other objec-tives are considered as constraints Gear ratio face widthand pitch circle diameter of worm and worm wheel arethe design variables for objectives Constraints are centerdistance deflection ofwormand beam strength ofwormgearYokota et al [8] formulated an optimalweight design problemof a gear for a constrained bending strength of gear torsionalstrength of shafts and each gear dimension as a nonlinearinteger programming (NIP) problem and solved it directly

by using an improved GA The efficiency of the proposedmethod is confirmed by showing the improvement in weightof gears and space area Buiga and Popa [9] presented anoptimal design mass minimization problem of a single-stagehelical gear unit complete with the sizing of shafts gearingand housing using GAs Mohan and Seshaiah [10] discussedthe optimization of spur gear set for its center distance weightand tooth deflections with module face width and numberof teeth on pinion as decision variables subject to constraintson bending stress and contact stressThreematerials namelyCast Iron C-45 and Alloy Steel (15Ni2 Cr1) are consideredThe gear parameters obtained from GA are compared withthe conventional results

Thompson et al [11] presented a generalized optimaldesign formulation withmultiple objectives which is in prin-ciple applicable to a gear train of arbitrary complexity Themethodology is applied to the design of two-stage and three-stage spur gear reduction units subject to identical loadingconditions and other design criteria The approach servesto extend traditional design procedures by demonstratingthe tradeoff between surface fatigue life and minimumvolume using a basic multiobjective optimization procedurePadmanabhan et al [12] investigated that in many real-lifeproblems objectives under consideration conflict with eachother and optimizing a particular solution with respect toa single objective can result in unacceptable results withrespect to the other objectives Multiobjective formulationsare realistic models for many complex engineering opti-mization problems Ant Colony Optimization was developedspecifically for a worm gear drive problem with multipleobjectives Deb and Jain [13] demonstrated the use of a mul-tiobjective evolutionary algorithm namely NondominatedSorting Genetic Algorithm (NSGA-II) which is capable ofsolving the original problem involving mixed discrete andreal-valued parameters and more than one objective

In this paper two stages of helical gear train are consid-ered There are several factors which affect the assembly aswell as working condition They are not generally consid-ered in literature The optimization model formulated hereincludes these factors in constraints A GUI is developedwhich facilitates the input of various combinations of inputdataMoreover a code of GA is also developedThe optimiza-tion is carried out using optimization toolbox of MATLABand GA and the results obtained by both of the methodsare compared These methods are applied to minimize thevolume only The resulting values of the parameters areapplied to find the maximum load carrying capacity In truesense the problem is solved as two single objective problemsone at a time Moreover NSGA-II is applied to the problemto solve it as a multiobjective problem

2 Formulation of Problem

The optimization model of two-stage helical gear reductionunit is formulated in this section withminimum volume andmaximum load carrying capacity as design objectives Theschematic illustration of two-stage helical gear reduction unitis shown in Figure 1 As it is a case of two-stage gear reduction

Journal of Optimization 3

A

B

D

C

Shaft (Ls ds)

Figure 1 Schematic illustration of two-stage helical gear train

the gear ratios between first pair and second pair are chosenin such a way that their values are feasible and their productremains the same as that of required

21 Design Variables Themainly affected parameters of gearfrom the volume point of view are face width moduleand number of teeth of gear These parameters directly orindirectly affect the objectives widely So the design vector119883 is

119883 = 119887119860 119887119862 119898119899119860

119898119899119862 119911119860 119911119861 119911119862 119911119863 (1)

where 119911119860 119911119861 119911119862 and 119911

119863are the number of teeth of gears119860119861

119862 and 119863 respectively 119887119860and 119887119862are the face widths of gears

119860 and 119862 respectively119898119899119860

and119898119899119862

are the normal modulesof gears119860 and119862 respectively Here it is assumed that all gearsare of the same material (say with the same Brinell hardnessnumber) and are of the same helix angle

22 Objective Functions For the optimization first the vol-ume of the two-stage helical gear train is minimized Afterachieving the optimal value of design variables for minimumvolume those values of variables are applied to maximize theload carrying capacity of both of the stages From both ofthese stages the minimum load carrying capacity out of thetwo is chosen as the maximum capacity for the gear train

The optimizationmodel of two-stage helical gear trains isderived as follows

Considering the dimensions of the three shafts constantthe volume of the gear train is

119881 =120587

4[(119889119860

2+ 119889119861

2) 119887119860+ (119889119862

2+ 119889119863

2) 119887119862

+ 1198891

21198711+ 1198892

21198712+ 1198893

21198713]

(2)

and the load carrying capacity 119875 is given as [14]

119875eff = 119862119904119875119905+ 119875119889 (3)

Referring to 119875eff of the two stages as 1198751 and 1198752further can

be written as

1198751=

21198621198781198721199051

119889119860

+

211205871198991(1198621198901119887119860119889119860

15cos2120595 + 2119889119860

051198721199051) cos120595

21120587119889119860

151198991+ 60 000radic119862119890

1119887119860119889119860cos2120595 + 2119872

1199051

1198752=

21198621198781198721199052

119889119862

+

211205871198992(1198621198902119887119862119889119862

15cos2120595 + 2119889119862

051198721199052) cos120595

21120587119889119862

151198992+ 60 000radic119862119890

2119887119862119889119862cos2120595 + 2119872

1199052

(4)

where 1198891 1198892 1198893and 119871

1 1198712 1198713represent the diameters of

shaft and lengths of shaft 1 2 3 respectively The factors119862119878and 119862 denote service factor and deformation factor

respectively 119872119905119894is the transmitted torque and 119890

119860and 119890119862are

sum of error between firstmeshing teeth and secondmeshingteeth respectively

Thus the objectives can be written for minimum volumeand maximum load carrying capacity as

119881min 119875max = 1198751 1198752 (5)

23 Constraints When the gear tooth is considered as acantilever beam the bending strength in working conditionshould not exceed standard endurance limit 119878

119899 From Lewis

equation the constraint on bending strength is

119865119905119875

119887119884le 119878119899 (6)

where 119865119905= (119896119882 times 10

3)V V = 120587119889119899(60 times 10

3) 119875 is diametral

pitch 119887 is face width and 119884 is Lewis factorHowever in this work the factors affecting bending

strength during the production and assembly such as velocityfactor overload factor and mounting factor to name a feware not taken into consideration So after adding the effectsof these factors the new constraints on bending strength forboth of the gear pairs can be expressed [15] as

119865119905119860119875119860

119887119860119869119860

119870V119860119870119900 (093119870119898119860) minus 1198781015840

119899119862119871119862119866119860

119862119878119896119903119896119905119896ms le 0

119865119905119862119875119862

119887119862119869119862

119870V119862119870119900 (093119870119898119862) minus 1198781015840

119899119862119871119862119866119862

119862119878119896119903119896119905119896ms le 0

(7)

where 119869 is geometry factor which includes the Lewis formfactor 119884 and a stress concentration factor 119870V 119870

119900 and

119870119898denote velocity or dynamic factor overload factor and

mounting factor respectively 1198781015840119899is standard R R Moore

endurance limit 119862119871 119862119866 and 119862

119878denote load factor gradient

factor and service factor respectively 119896119905 119896119903 and 119896ms denote

temperature factor reliability factor and mean stress factorrespectively

Gear teeth are vulnerable to various types of surfacedamage As was the case with rolling-element bearings


gear teeth are subjected to Hertz contact stresses and thelubrication is often elastohydrodynamic Excessive loadingand lubrication breakdown can cause various combinationsof abrasion pitting and scoring It will become evident thatgear-tooth surface durability is a more complex matter thanthe capacity to withstand gear-tooth-bending fatigue

After including all the parameters the surface fatigueconstraint formula can be written [15] as

119862119901radic

119865119905119860

119887119860119889119860119868119860

timescos120595

095CR119860

times 119870V119860119870119900 (093119870119898119860)

minus 119878119891119888119862Li119862119877 le 0

119889119862119901radic

119865119905119862

119887119862119889119862119868119862

timescos120595

095CR119862

times 119870V119862119870119900 (093119870119898119862)

minus 119878119891119888119862Li119862119877 le 0

(8)

where 119862119901 119862Li and 119862

119877denote elastic coefficient factor

life factor and reliability factor respectively 119868119860and 119868119862are

dimensionless constants andCR119860andCR

119862are contact ratios

119878119891119888represents surface fatigue strengthWhile designing the gear interference is the main factor

to consider Interference usually takes place in the gear Soformulation of the optimization problem must take careof interference To remove interference the following con-straints should be satisfied (see [15 16])

119903119886119860

minus radic1199031198871198602 + 1198881198602sin2120601 le 0

119903119886119862


2

sin2120601minus 119911119860le 0

2

sin2120601minus 119911119861le 0

2

sin2120601minus 119911119862le 0

2

sin2120601minus 119911119863le 0

(9)

3 Methods of Solution

Since there are many input parameters such as dimensionsof shafts gear train parameters material properties workingcondition of gear train and factor affecting production andassembly a GUI is prepared as shown in Figures 2 3 4 and5 The problem is solved by following three ways

(i) using optimization toolbox of MATLAB(ii) using code developed for GA(iii) using multiobjective optimization (NSGA-II) tech-

nique

The ranges of the problem variables are taken as referencefrommanufacturerrsquos catalog [17] and these ranges for 119887

119860119898119860

Figure 2 Input data through ldquoData Shaftrdquo

Figure 3 Input data through ldquoData Geartrainrdquo

Figure 4 Input data through ldquoData Factorrdquo

Figure 5 Input data through ldquoData Factor2rdquo

119911119860 119911119861 119887119862 119898119862 119911119862 and 119911

119863are taken as 60ndash80 4ndash12 14ndash20

44ndash65 85ndash105 3ndash10 14ndash20 and 77ndash110 respectively

31 Using the Optimization Toolbox of MATLAB In thismethod first the volume of the gear train is minimized Theresulting values of the parameters are used to determine theload carrying capacities of both of the shafts The minimumof them is considered as themaximum load carrying capacityIn this way a multiobjective problem is reduced to a single


objective problem The ldquooptimtoolrdquo feature of MATLABis useful for different kinds of optimization problem Inthe problem discussed here constraints are nonlinear Soldquofminconrdquo function of MATLAB applicable for nonlinearconstraint minimization is used for the optimization Thereare different algorithms and methods available under thisoption in the optimization toolbox Interior-point algorithmis chosen among them as it handles large sparse problemsas well as small dense problems Moreover the algorithmsatisfies bounds at all iterations and can recover from NaNor Inf results It is a large-scale algorithm widely used for thistype of problems

This function requires a point to start with the choiceof which is arbitrary The results obtained for face width ofgear 119860 module of gear 119860 (and 119861) number of teeth of gear119860 number of teeth of gear 119861 face width of gear 119862 moduleof gear 119862 (and 119863) number of teeth of gear 119862 and numberof teeth of gear 119863 are 60 4 17097 53737 85 3 17097 and94035 respectively The corresponding volume is 1948 times

107mm3The result remains invariant if other starting points

are chosen For the value of load carrying capacity the valuesfor first and second stages are 33352 times 104N and 33909 times

104N So from these values the load carrying capacity of thegear train is selected as 33352 times 104N

32 OptimizationUsingGenetic Algorithm Thesame strategyused in the first method is also applied here to deal with amultiobjective problem First the volume is minimized andthen minimum of the resulting two load carrying capacitiesis chosen as the maximum load carrying capacity The onlydifference is that to minimize the volume GA is used Asdiscussed in introduction many designs are characterized bymixed continuous-discrete variables and discontinuous andnonconvex design spaces Standard nonlinear programmingtechniques are not capable of solving these types of problemsThey usually find relative optimum that is closest to thestarting point GA is well suited for solving such problemsand in most cases they can find the global optimum solutionwith high probability Actually the idea of evolutionarycomputing was introduced in the 1960s by I Rechenberg inhis work ldquoEvolution strategiesrdquo which was then developedby others GAs were invented and developed by Holland[18] The basic ideas of analysis and design based on theconcepts of biological evolution can be found in the work ofRechenberg [19] Philosophically GAs are based on Darwinrsquostheory of survival of the fittest and also are based on theprinciples of natural genetics and natural selectionThe basicelements of natural genetics-reproduction cross-over andmutation are used in the genetic search procedures

GA is a search algorithm based on the conjecture of nat-ural selection and genetics The features of GA are differentfrom the other search techniques in several aspects as follows

(i) the algorithm is amultipath that searches many peaksin parallel hence reducing the possibility of localminimum trapping

(ii) GAs work with coding of the parameter set not theparameters themselves

(iii) GAs evaluate a population of points not a singlepoint

(iv) GAs use objective function information not deriva-tions or other auxiliary knowledge to determine thefitness of the solution

(v) GAs use probabilistic transition rules not determin-istic rules in the generation of the new population

321 Outline of Basic Genetic Algorithm Thebasic procedureof GA as outlined in [20] is as follows

(1) [Start] Generate random population of 119899 chromo-somes (suitable solution for problem)

(2) [Fitness] Evaluate the fitness 119891(119909) of each chromo-some 119909 in the population

(3) [New population] Create a new population by repeat-ing following steps until the new population is com-plete

(i) [Selection] Select two parent chromosomesfrom a population according to their fitness (thebetter fitness the bigger chance to be selected)

(ii) [Crossover] With a crossover probability 119875119862

crossover the two parents to from two new off-spring If no crossover was performed offspringis the exact copy of parents

(iii) [Mutation] With a mutation probability 119875119898

mutate new offspring at each locus (position inchromosome)

(iv) [Accepting] Place new offspring in the newpopulation

(4) [Replace] Use new generated population for a furtherrun of the algorithm

(5) [Test] If the end condition is satisfied stop and returnthe best solution in current population

(6) [Loop] Go to Step (2)

322 Implementation of Genetic Algorithm Extensive exper-iments are carried out for different combinations of popu-lation size and number of generations It is observed thatthe results remain consistent when the population size is 90and number of generations is 90 So eleven good resultswith this population size and number of generations areshown in Table 1 in which the 10th solution is the bestCorresponding load carrying capacities of the first and thesecond pair are 3286 kN and 3416 kN respectively So theload carrying capacity of gear train is selected as 3286 kN forwhich optimum volume is 20396 times 107mm3

33 Optimization Using NSGA-II In this case the problem isconsidered as a multiobjective problem So both objectivesare treated together In general in case of multiobjectiveoptimization the objectives are conflicting So a singlesolution cannot be accepted as the best solution Insteada set of solutions is obtained which are better than the


Table 1 Results of GA for population size of 90 and 90 generations

Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863

Volume (107 timesmm3)

1 6706 4 18 57 9018 3 18 99 21662 6407 4 19 60 8529 3 18 99 20883 6207 4 18 57 867 3 18 99 20584 751 4 18 57 8591 3 18 99 21115 6607 4 20 63 8523 3 18 99 21326 637 4 19 60 8774 3 18 99 21047 62 4 18 57 8682 3 18 99 20598 6071 4 18 57 8604 3 18 99 20479 6177 4 18 57 8647 3 18 99 205510 6001 4 18 57 8531 3 18 99 203911 6285 4 19 60 8531 3 19 99 2159

Table 2 Results of NSGA-II for population size of 500 and 500 generations

Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863

Volume(107 timesmm3) Load carrying capacity (kN)

1 8000 4 18 57 8762 3 18 99 2048 353062 6039 4 18 57 8500 3 18 99 1951 337183 7904 4 18 57 9083 3 18 99 2066 358374 8000 4 18 57 9098 3 18 99 2070 359415 7148 4 18 57 8762 3 18 99 2013 349966 7477 4 18 57 8789 3 18 99 2027 353587 6680 4 18 57 8633 3 18 99 1993 343838 6345 4 18 57 8500 3 18 99 1963 34076

other solutions in terms of both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice for solving thiskind of problem In NSGA-II the iterative procedure startsfrom an arbitrary population of solutions and gradually thealgorithm converges to a population of solutions lying onthe Pareto optimal front with higher diversity The operatorsapplied are the same as those of GA namely selectioncrossover and mutation The tournament selection operatoris applied which also takes care of constraints Howeverin case of multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing function approach is used Crossover and mutationoperators are applied as usual A detailed discussion of thisalgorithm is found in [21]The standard code available at [22]is modified according to authorsrsquo need

As a result of NSGA-II out of the population size of50 and number of generations of 500 eight better resultsare selected and shown in Table 2 It has been observedthat fulfilling both of the objectives together the second lastsolution is the compromised one Corresponding optimumvolume and load carrying capacity of the train are 1993 times

107mm3 and 3438 kN respectively

4 Results and Discussion

There are several comments in order The number of teethof gear 119860 and gear 119862 in the manufacturerrsquos design is 14 It

creates interference in working condition To eliminate itmanufacturer produces stub tooth instead of normal toothwhich is not advisable The introduction of the constraintson interference in the proposed formulation takes care ofthis problem as the number of teeth of gear 119860 and gear 119862

will definitely exceed 17 The major problem with the inbuiltldquofminconrdquo function of MATLAB is that it considers all thevariables real As a result one has to round the optimumvalueof integer variable to the nearest integer So the optimumvalue of number of teeth of gear 119860 and gear 119862 is roundedoff to 18 To maintain the gear ratios the numbers of teethof gear 119861 and gear 119863 have to be selected as 56 and 100respectively which are quite far from their actual optimumvalues obtained using toolbox

However GA can deal with both types of variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers of teeth of gear 119861 andgear119863 have to be changed to 58 and 100 respectively becauseof manufacturing inconveniences NSGA-II selects 57 and 99as the numbers of teeth of gear 119861 and gear 119863 which is betterthan both of the above results The results are presented inTable 3

5 Conclusion and Future Scope

Result comparison table shows that in the first two casesminimization of volume took place while load carrying


Table 3 Comparison of results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

GA (round off) NSGA-II (round off)

Face width of gear 119860 (mm) 70 60 60 67Module of gear 119860 (mm) 7 4 4 4Number of teeth of gear 119860 14 18 18 18Number of teeth of gear 119861 44 56 58 57Face width of gear 119862 (mm) 95 85 85 86Module of gear 119862 (mm) 35 3 3 3Number of teeth of gear 119862 14 18 18 18Number of teeth of gear119863 77 100 100 99Volume (mm3) 2293 times 107 20408 times 107 20602 times 107 1993 times 107

Load carrying capacity (N) 3405 times 104 32861 times 104 32874 times 104 34383 times 104

capacity is reducedmarginally lowWhile using optimizationtoolbox volume is reduced by 1504 but when their nearerinteger value of variable is selected because of inconveniencesin manufacturing volume is reduced by 109 For the GAthe volume is reduced by 1105 but when their nearer integervalue of variable is selected volume is reduced by 1015Though these results show that optimization tool box givesbetter result than GA it is better to use GA for globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGAfinds themore convenientsolution with high probability of manufacturing HoweverNSGA-II gives the best result compared to both of the abovemethods as it is superior in terms of both of the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 1308 and loadcarrying capacity is increased by 1

The problem can be extended to more than two stagesOther recently developed evolutionary algorithms such asPSO and cuckoo search can also be tried to solve thisproblem Similar approach can be followed in case of otherapplications such as minimization of weight of spring andminimization of weight of pulley system

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Z Huang Z-G Tian and M J Zuo ldquoMultiobjectiveoptimization of three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and Technology vol 19 no 5 pp 1080ndash1086 2005

[2] D Jhalani and H Chaudhary ldquoOptimal design of gearbox forapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scientific amp Engineering Research vol 3no 10 pp 1071ndash1075 2012

[3] B S Tong and D Walton ldquoThe optimisation of internal gearsrdquoInternational Journal of Machine Tools andManufacture vol 27no 4 pp 491ndash504 1987

[4] V Savsani RV Rao andD PVakharia ldquoOptimalweight designof a gear train using particle swarm optimization and simulatedannealing algorithmsrdquoMechanism andMachineTheory vol 45no 3 pp 531ndash541 2010

[5] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoThe structural optimization of gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 2ndInternational Conference on Intelligent Computing Technologyand Automation (ICICTA rsquo09) pp 356ndash359 Hunan ChinaOctober 2009

[6] F Mendi T Baskal K Boran and F E Boran ldquoOptimizationof module shaft diameter and rolling bearing for spur gearthrough genetic algorithmrdquo Expert Systems with Applicationsvol 37 no 12 pp 8058ndash8064 2010

[7] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-mization approach for design of worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of ManMachine Interface vol 3 pp 8ndash12 2013

[8] T Yokota T Taguchi and M Gen ldquoA solution method foroptimal weight design problem of the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 35 no 3-4 pp 523ndash526 1998

[9] O Buiga andC-O Popa ldquoOptimalmass design of a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics TechnicalSciences Information Science vol 13 no 3 pp 243ndash250 2012

[10] Y Mohan and T Seshaiah ldquoSpur gear optimization byusing genetic algorithmrdquo International Journal of EngineeringResearch and Applications vol 2 pp 311ndash318 2012

[11] D F Thompson S Gupta and A Shukla ldquoTradeoff analysisin minimum volume design of multi-stage spur gear reductionunitsrdquoMechanism and Machine Theory vol 35 no 5 pp 609ndash627 2000

[12] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization of worm Gear driverdquo International Jour-nal ofMiningMetallurgy andMechanical Engineering vol 1 pp57ndash61 2013

[13] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of MechanicalDesign Transactions of the ASME vol 125 no 3 pp 609ndash6192003

[14] V B BhandariDesign of Machine Elements Tata McGraw-Hill2010


[15] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 2011

[16] G Maitra Handbook of Gear Design Tata McGraw-Hill 2ndedition 2003

[17] Design Catalog of Hi-Tech Drive Pvt Ltd Plot No 443A GIDCV U Nagar Gujarat India

[18] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[19] I Rechenberg Cybernetic Solution Path of an ExperimentalProblem Library Translation 1122 Royal Aircraft Establish-ment Farnborough Hampshire UK 1965

[20] P E Amiolemhen and A O A Ibhadode ldquoApplication ofgenetic algorithmsmdashdetermination of the optimal machiningparameters in the conversion of a cylindrical bar stock into acontinuous finished profilerdquo International Journal of MachineTools and Manufacture vol 44 no 12-13 pp 1403ndash1412 2004

[21] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 2009

[22] httpwwwmathworksinmatlabcentralfileexchange31166-ngpm-a-nsga-ii-program-in-matlab-v1-4

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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it by classical optimization methods Moreover increasingdemand for compact efficient and reliable gears forces thedesigner to use optimal design methodology

Huang et al [1] developed interactive physical program-ming approach of the optimization model of three-stagespur gear reduction unit with minimum volume maximumsurface fatigue life and maximum load-carrying capacity asdesign objectives and core hardness module face width ofgear tooth numbers of pinion tooth numbers of gear anddiameter of shaft as design variables In this modeling toothbending fatigue failure shaft torsional stress face widthinterference and tooth number are considered as constraintsThe MATLAB constrained optimization package is usedto solve this nonlinear programming problem Jhalani andChaudhary [2] discussed the various parameters which canaffect the design of the gearbox for knee mounted energyharvester device and later it frames the optimization problemof mass function based on the dimensions of gearbox for theproblem The problem is solved using multistart approach ofMATLAB global optimization toolbox and value of globaloptimum function is obtained considering all the localoptimum solutions of problem Tong and Walton [3] alsoselected center distance and volume as objectives for theinternal gears Numbers of teeth of gear and pinion andmodules are considered as variables for the optimizationand ldquobelt zone searchrdquo and ldquohalf section algorithmrdquo areapplied as optimization methods Savsani et al [4] presentedthe application of two advanced optimization algorithmsknown as particle swarm optimization (PSO) and simulatedannealing (SA) to find the optimal combination of designparameters for minimum weight of a spur gear train Weiet al [5] developed a mathematical model of optimizationconsidering the basic design parameters mainly tooth num-ber modulus face width and helix angle of gearbox asdesign variables and reduction of weight or volume as anobjective The model is illustrated by an example of thegearbox of medium-sized motor truck Optimization toolbox of MATLAB and sequential quadratic programming(SQP)method were used to optimize the gearboxThe designcriterion and performance conditions of gearbox are treatedas constraints

Mendi et al [6] studied the dimensional optimizationof motion and force transmitting components of a gearboxby GA It is aimed at obtaining the optimal dimensionsfor gearbox shaft gear and the optimal rolling bearingto minimize the volume which can carry the system loadusing GA The results obtained by GA optimization arecompared to those obtained by analytical methods Mogaland Wakchaure [7] used GA as evolutionary techniques foroptimization of worm and worm wheel The main objectivefor optimization is minimizing the volume here other objec-tives are considered as constraints Gear ratio face widthand pitch circle diameter of worm and worm wheel arethe design variables for objectives Constraints are centerdistance deflection ofwormand beam strength ofwormgearYokota et al [8] formulated an optimalweight design problemof a gear for a constrained bending strength of gear torsionalstrength of shafts and each gear dimension as a nonlinearinteger programming (NIP) problem and solved it directly

by using an improved GA The efficiency of the proposedmethod is confirmed by showing the improvement in weightof gears and space area Buiga and Popa [9] presented anoptimal design mass minimization problem of a single-stagehelical gear unit complete with the sizing of shafts gearingand housing using GAs Mohan and Seshaiah [10] discussedthe optimization of spur gear set for its center distance weightand tooth deflections with module face width and numberof teeth on pinion as decision variables subject to constraintson bending stress and contact stressThreematerials namelyCast Iron C-45 and Alloy Steel (15Ni2 Cr1) are consideredThe gear parameters obtained from GA are compared withthe conventional results

Thompson et al [11] presented a generalized optimaldesign formulation withmultiple objectives which is in prin-ciple applicable to a gear train of arbitrary complexity Themethodology is applied to the design of two-stage and three-stage spur gear reduction units subject to identical loadingconditions and other design criteria The approach servesto extend traditional design procedures by demonstratingthe tradeoff between surface fatigue life and minimumvolume using a basic multiobjective optimization procedurePadmanabhan et al [12] investigated that in many real-lifeproblems objectives under consideration conflict with eachother and optimizing a particular solution with respect toa single objective can result in unacceptable results withrespect to the other objectives Multiobjective formulationsare realistic models for many complex engineering opti-mization problems Ant Colony Optimization was developedspecifically for a worm gear drive problem with multipleobjectives Deb and Jain [13] demonstrated the use of a mul-tiobjective evolutionary algorithm namely NondominatedSorting Genetic Algorithm (NSGA-II) which is capable ofsolving the original problem involving mixed discrete andreal-valued parameters and more than one objective

In this paper two stages of helical gear train are consid-ered There are several factors which affect the assembly aswell as working condition They are not generally consid-ered in literature The optimization model formulated hereincludes these factors in constraints A GUI is developedwhich facilitates the input of various combinations of inputdataMoreover a code of GA is also developedThe optimiza-tion is carried out using optimization toolbox of MATLABand GA and the results obtained by both of the methodsare compared These methods are applied to minimize thevolume only The resulting values of the parameters areapplied to find the maximum load carrying capacity In truesense the problem is solved as two single objective problemsone at a time Moreover NSGA-II is applied to the problemto solve it as a multiobjective problem

2 Formulation of Problem

The optimization model of two-stage helical gear reductionunit is formulated in this section withminimum volume andmaximum load carrying capacity as design objectives Theschematic illustration of two-stage helical gear reduction unitis shown in Figure 1 As it is a case of two-stage gear reduction


A

B

D

C

Shaft (Ls ds)




119883 = 119887119860 119887119862 119898119899119860

119898119899119862 119911119860 119911119861 119911119862 119911119863 (1)

where 119911119860 119911119861 119911119862 and 119911




and119898119899119862





119881 =120587

4[(119889119860

2+ 119889119861

2) 119887119860+ (119889119862

2+ 119889119863

2) 119887119862

+ 1198891

21198711+ 1198892

21198712+ 1198893

21198713]

(2)


119875eff = 119862119904119875119905+ 119875119889 (3)


be written as

1198751=

21198621198781198721199051

119889119860

+

211205871198991(1198621198901119887119860119889119860

15cos2120595 + 2119889119860

051198721199051) cos120595

21120587119889119860

151198991+ 60 000radic119862119890

1119887119860119889119860cos2120595 + 2119872

1199051

1198752=

21198621198781198721199052

119889119862

+

211205871198992(1198621198902119887119862119889119862

15cos2120595 + 2119889119862

051198721199052) cos120595

21120587119889119862

151198992+ 60 000radic119862119890

2119887119862119889119862cos2120595 + 2119872

1199052

(4)

where 1198891 1198892 1198893and 119871




119860and 119890119862are



119881min 119875max = 1198751 1198752 (5)


119899 From Lewis


119865119905119875

119887119884le 119878119899 (6)

where 119865119905= (119896119882 times 10

3)V V = 120587119889119899(60 times 10




119865119905119860119875119860

119887119860119869119860

119870V119860119870119900 (093119870119898119860) minus 1198781015840

119899119862119871119862119866119860

119862119878119896119903119896119905119896ms le 0

119865119905119862119875119862

119887119862119869119862

119870V119862119870119900 (093119870119898119862) minus 1198781015840

119899119862119871119862119866119862

119862119878119896119903119896119905119896ms le 0

(7)


119900 and











119862119901radic

119865119905119860

119887119860119889119860119868119860

timescos120595

095CR119860

times 119870V119860119870119900 (093119870119898119860)

minus 119878119891119888119862Li119862119877 le 0

119889119862119901radic

119865119905119862

119887119862119889119862119868119862

timescos120595

095CR119862

times 119870V119862119870119900 (093119870119898119862)

minus 119878119891119888119862Li119862119877 le 0

(8)

where 119862119901 119862Li and 119862







119903119886119860


119903119886119862


2

sin2120601minus 119911119860le 0

2

sin2120601minus 119911119861le 0

2

sin2120601minus 119911119862le 0

2

sin2120601minus 119911119863le 0

(9)




nique


119860119898119860





119911119860 119911119861 119887119862 119898119862 119911119862 and 119911


































Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 6706 4 18 57 9018 3 18 99 21662 6407 4 19 60 8529 3 18 99 20883 6207 4 18 57 867 3 18 99 20584 751 4 18 57 8591 3 18 99 21115 6607 4 20 63 8523 3 18 99 21326 637 4 19 60 8774 3 18 99 21047 62 4 18 57 8682 3 18 99 20598 6071 4 18 57 8604 3 18 99 20479 6177 4 18 57 8647 3 18 99 205510 6001 4 18 57 8531 3 18 99 203911 6285 4 19 60 8531 3 19 99 2159


Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 8000 4 18 57 8762 3 18 99 2048 353062 6039 4 18 57 8500 3 18 99 1951 337183 7904 4 18 57 9083 3 18 99 2066 358374 8000 4 18 57 9098 3 18 99 2070 359415 7148 4 18 57 8762 3 18 99 2013 349966 7477 4 18 57 8789 3 18 99 2027 353587 6680 4 18 57 8633 3 18 99 1993 343838 6345 4 18 57 8500 3 18 99 1963 34076














value(round off)








References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

B

D

C

Shaft (Ls ds)




119883 = 119887119860 119887119862 119898119899119860

119898119899119862 119911119860 119911119861 119911119862 119911119863 (1)

where 119911119860 119911119861 119911119862 and 119911




and119898119899119862





119881 =120587

4[(119889119860

2+ 119889119861

2) 119887119860+ (119889119862

2+ 119889119863

2) 119887119862

+ 1198891

21198711+ 1198892

21198712+ 1198893

21198713]

(2)


119875eff = 119862119904119875119905+ 119875119889 (3)


be written as

1198751=

21198621198781198721199051

119889119860

+

211205871198991(1198621198901119887119860119889119860

15cos2120595 + 2119889119860

051198721199051) cos120595

21120587119889119860

151198991+ 60 000radic119862119890

1119887119860119889119860cos2120595 + 2119872

1199051

1198752=

21198621198781198721199052

119889119862

+

211205871198992(1198621198902119887119862119889119862

15cos2120595 + 2119889119862

051198721199052) cos120595

21120587119889119862

151198992+ 60 000radic119862119890

2119887119862119889119862cos2120595 + 2119872

1199052

(4)

where 1198891 1198892 1198893and 119871




119860and 119890119862are



119881min 119875max = 1198751 1198752 (5)


119899 From Lewis


119865119905119875

119887119884le 119878119899 (6)

where 119865119905= (119896119882 times 10

3)V V = 120587119889119899(60 times 10




119865119905119860119875119860

119887119860119869119860

119870V119860119870119900 (093119870119898119860) minus 1198781015840

119899119862119871119862119866119860

119862119878119896119903119896119905119896ms le 0

119865119905119862119875119862

119887119862119869119862

119870V119862119870119900 (093119870119898119862) minus 1198781015840

119899119862119871119862119866119862

119862119878119896119903119896119905119896ms le 0

(7)


119900 and











119862119901radic

119865119905119860

119887119860119889119860119868119860

timescos120595

095CR119860

times 119870V119860119870119900 (093119870119898119860)

minus 119878119891119888119862Li119862119877 le 0

119889119862119901radic

119865119905119862

119887119862119889119862119868119862

timescos120595

095CR119862

times 119870V119862119870119900 (093119870119898119862)

minus 119878119891119888119862Li119862119877 le 0

(8)

where 119862119901 119862Li and 119862







119903119886119860


119903119886119862


2

sin2120601minus 119911119860le 0

2

sin2120601minus 119911119861le 0

2

sin2120601minus 119911119862le 0

2

sin2120601minus 119911119863le 0

(9)




nique


119860119898119860





119911119860 119911119861 119887119862 119898119862 119911119862 and 119911


































Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 6706 4 18 57 9018 3 18 99 21662 6407 4 19 60 8529 3 18 99 20883 6207 4 18 57 867 3 18 99 20584 751 4 18 57 8591 3 18 99 21115 6607 4 20 63 8523 3 18 99 21326 637 4 19 60 8774 3 18 99 21047 62 4 18 57 8682 3 18 99 20598 6071 4 18 57 8604 3 18 99 20479 6177 4 18 57 8647 3 18 99 205510 6001 4 18 57 8531 3 18 99 203911 6285 4 19 60 8531 3 19 99 2159


Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 8000 4 18 57 8762 3 18 99 2048 353062 6039 4 18 57 8500 3 18 99 1951 337183 7904 4 18 57 9083 3 18 99 2066 358374 8000 4 18 57 9098 3 18 99 2070 359415 7148 4 18 57 8762 3 18 99 2013 349966 7477 4 18 57 8789 3 18 99 2027 353587 6680 4 18 57 8633 3 18 99 1993 343838 6345 4 18 57 8500 3 18 99 1963 34076














value(round off)








References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119862119901radic

119865119905119860

119887119860119889119860119868119860

timescos120595

095CR119860

times 119870V119860119870119900 (093119870119898119860)

minus 119878119891119888119862Li119862119877 le 0

119889119862119901radic

119865119905119862

119887119862119889119862119868119862

timescos120595

095CR119862

times 119870V119862119870119900 (093119870119898119862)

minus 119878119891119888119862Li119862119877 le 0

(8)

where 119862119901 119862Li and 119862







119903119886119860


119903119886119862


2

sin2120601minus 119911119860le 0

2

sin2120601minus 119911119861le 0

2

sin2120601minus 119911119862le 0

2

sin2120601minus 119911119863le 0

(9)




nique


119860119898119860





119911119860 119911119861 119887119862 119898119862 119911119862 and 119911


































Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 6706 4 18 57 9018 3 18 99 21662 6407 4 19 60 8529 3 18 99 20883 6207 4 18 57 867 3 18 99 20584 751 4 18 57 8591 3 18 99 21115 6607 4 20 63 8523 3 18 99 21326 637 4 19 60 8774 3 18 99 21047 62 4 18 57 8682 3 18 99 20598 6071 4 18 57 8604 3 18 99 20479 6177 4 18 57 8647 3 18 99 205510 6001 4 18 57 8531 3 18 99 203911 6285 4 19 60 8531 3 19 99 2159


Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 8000 4 18 57 8762 3 18 99 2048 353062 6039 4 18 57 8500 3 18 99 1951 337183 7904 4 18 57 9083 3 18 99 2066 358374 8000 4 18 57 9098 3 18 99 2070 359415 7148 4 18 57 8762 3 18 99 2013 349966 7477 4 18 57 8789 3 18 99 2027 353587 6680 4 18 57 8633 3 18 99 1993 343838 6345 4 18 57 8500 3 18 99 1963 34076














value(round off)








References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 6706 4 18 57 9018 3 18 99 21662 6407 4 19 60 8529 3 18 99 20883 6207 4 18 57 867 3 18 99 20584 751 4 18 57 8591 3 18 99 21115 6607 4 20 63 8523 3 18 99 21326 637 4 19 60 8774 3 18 99 21047 62 4 18 57 8682 3 18 99 20598 6071 4 18 57 8604 3 18 99 20479 6177 4 18 57 8647 3 18 99 205510 6001 4 18 57 8531 3 18 99 203911 6285 4 19 60 8531 3 19 99 2159


Sr number 119887119860

(mm)119898119860

(mm) 119911119860

119911119861

119887119862

(mm)119898119862

(mm) 119911119862

119911119863


1 8000 4 18 57 8762 3 18 99 2048 353062 6039 4 18 57 8500 3 18 99 1951 337183 7904 4 18 57 9083 3 18 99 2066 358374 8000 4 18 57 9098 3 18 99 2070 359415 7148 4 18 57 8762 3 18 99 2013 349966 7477 4 18 57 8789 3 18 99 2027 353587 6680 4 18 57 8633 3 18 99 1993 343838 6345 4 18 57 8500 3 18 99 1963 34076














value(round off)








References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












value(round off)








References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article multi-objective optimization of two-stage...

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