7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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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

983089 Department of Mathematics G H Patel College of Engineering and echnology Vallabh Vidyanagar 983091983096983096983089983090983088 India983090 Mechanical Engineering Department B V Mahavidyalaya Vallabh Vidyanagar 983091983096983096983089983090983088 India

Correspondence should be addressed to R C Sanghvi rajeshsanghvigcetacin

Received 983091983089 May 983090983088983089983092 Revised 983090983089 October 983090983088983089983092 Accepted 983092 November 983090983088983089983092 Published 983091983088 November 983090983088983089983092

Academic Editor Liwei Zhang

Copyright copy 983090983088983089983092 R C Sanghvi et al Tis 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 powersatisactorily but also can do so with uniorm motion Te designo gears requires aniterative approach to optimize the design parameters that take care o kinematics aspects as well as strength aspects Moreover thechoice o materials available or gears is limited Owing to the complex combinations o the above acts manual design o gears iscomplicated and time consuming In this paperthe volume and load carrying capacity areoptimized Treedifferent methodologies(i) MALAB optimization toolbox (ii) genetic algorithm (GA) and (iii) multiobjective optimization (NSGA-II) techniqueare usedto solve the problem In the 1047297rst two methods volume is minimized in the 1047297rst step and then the load carrying capacities o bothshafs are calculated In the third method the problem is treated as a multiobjective problem For the optimization purpose acewidth module and number o teeth are taken as design variables Constraints are imposed on bending strength surace atiguestrength and intererence It is apparent rom the comparison o results that the result obtained by NSGA-II is more superior thanthe results obtained by other methods in terms o both objectives

1 Introduction

Designing a new product consists o several parameters andphases which differ according to the depth o design inputdata design strategy procedures and results Mechanicaldesign includes an optimization process in which designersalways consider certain objectives such as strength de1047298ec-

tion weight wear and corrosion depending on the require-ments However designoptimization or a complete mechan-ical assembly leads to a complicated objective unction witha large number o design variables So it is a better practiceto apply optimization techniques or individual componentsor intermediate assemblies than a complete assembly Forexample in an automobile power transmission system opti-mization o gearbox is computationally and mathematically simpler than the optimization o complete system Tepreliminary design optimization o two-stage helical geartrain has been a subject o considerable interest since many high-perormance power transmission applications requirehigh-perormance gear train

A traditional gear design involves computations basedon tooth bending strength tooth surace durability toothsurace atigue intererence efficiency and so orth Geardesign involves empirical ormulas different graphs andtables which lead to a complicated design Manual designis very difficult considering the above acts and there is aneed or the computer aided design o gears With the aid o

computer design can be carried out iteratively and the design variables which satisy the given conditions can be deter-mined Te design so obtained may not be the optimum onebecause in the above process the design variables so obtainedsatisy only one condition at a time or example i moduleis calculated based on bending strength the same module issubstituted to calculate the surace durability It is accepted i it is withinthe strength limit o surace durability otherwise itis changed accordingly So optimization methodsare requiredto determine design variables which simultaneously satisy the given conditions As the optimization problem involvesthe objective unction and constraints that are not stated asexplicit unctions o the design variables it is hard to solve

Hindawi Publishing CorporationJournal of OptimizationVolume 2014 Article ID 670297 8 pageshttpdxdoiorg1011552014670297

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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983090 Journal o Optimization

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

Huang et al [983089] developed interactive physical program-ming approach o the optimization model o three-stagespur gear reduction unit with minimum volume maximum

surace atigue lie and maximum load-carrying capacity asdesign objectives and core hardness module ace width o gear tooth numbers o pinion tooth numbers o gear anddiameter o shaf as design variables In this modeling toothbending atigue ailure shaf torsional stress ace widthintererence and tooth number are considered as constraintsTe MALAB constrained optimization package is usedto solve this nonlinear programming problem Jhalani andChaudhary [983090] discussed the various parameters which canaffect the design o the gearbox or knee mounted energy harvester device and later it rames the optimization problemo mass unction based on the dimensions o gearbox or theproblem Te problem is solved using multistart approach o MALAB global optimization toolbox and value o globaloptimum unction is obtained considering all the localoptimum solutions o problem ong and Walton [983091] alsoselected center distance and volume as objectives or theinternal gears Numbers o teeth o gear and pinion andmodules are considered as variables or the optimizationand ldquobelt zone searchrdquo and ldquohal section algorithmrdquo areapplied as optimization methods Savsani et al [983092] presentedthe application o two advanced optimization algorithmsknown as particle swarm optimization (PSO) and simulatedannealing (SA) to 1047297nd the optimal combination o designparameters or minimum weight o a spur gear train Weiet al [983093] developed a mathematical model o optimizationconsidering the basic design parameters mainly tooth num-ber modulus ace width and helix angle o gearbox asdesign variables and reduction o weight or volume as anobjective Te model is illustrated by an example o thegearbox o medium-sized motor truck Optimization toolbox o MALAB and sequential quadratic programming(SQP) method were used to optimize the gearbox Te designcriterion and perormance conditions o gearbox are treatedas constraints

Mendi et al [983094] studied the dimensional optimizationo motion and orce transmitting components o a gearboxby GA It is aimed at obtaining the optimal dimensionsor gearbox shaf gear and the optimal rolling bearingto minimize the volume which can carry the system load

using GA Te results obtained by GA optimization arecompared to those obtained by analytical methods Mogaland Wakchaure [983095] used GA as evolutionary techniques oroptimization o worm and worm wheel Te main objectiveor optimization is minimizing the volume here other objec-tives are considered as constraints Gear ratio ace widthand pitch circle diameter o worm and worm wheel arethe design variables or objectives Constraints are centerdistance de1047298ection o worm andbeam strength o worm gearYokota et al [983096] ormulated an optimalweight design problemo a gear or a constrained bending strength o gear torsionalstrength o shafs and each gear dimension as a nonlinearinteger programming (NIP) problem and solved it directly

by using an improved GA Te efficiency o the proposedmethod is con1047297rmed by showing the improvement in weighto gears and space area Buiga and Popa [983097] presented anoptimal design mass minimization problem o a single-stagehelical gear unit complete with the sizing o shafs gearingand housing using GAs Mohan and Seshaiah [983089983088] discussed

the optimization o spur gear set orits centerdistance weightand tooth de1047298ections with module ace width and numbero teeth on pinion as decision variables subject to constraintson bending stress and contact stress Tree materials namelyCast Iron C-983092983093 and Alloy Steel (983089983093Ni983090 Cr983089) are consideredTe gear parameters obtained rom GA are compared withthe conventional results

Tompson et al [983089983089] presented a generalized optimaldesign ormulation with multiple objectives which is in prin-ciple applicable to a gear train o arbitrary complexity Temethodology is applied to the design o two-stage and three-stage spur gear reduction units subject to identical loadingconditions and other design criteria Te approach servesto extend traditional design procedures by demonstratingthe tradeoff between surace atigue lie and minimum

volume using a basic multiobjective optimization procedurePadmanabhan et al [983089983090] investigated that in many real-lieproblems objectives under consideration con1047298ict with eachother and optimizing a particular solution with respect toa single objective can result in unacceptable results withrespect to the other objectives Multiobjective ormulationsare realistic models or many complex engineering opti-mization problems Ant Colony Optimization was developedspeci1047297cally or a worm gear drive problem with multipleobjectives Deb and Jain [983089983091] demonstrated the use o a mul-tiobjective evolutionary algorithm namely Nondominated

Sorting Genetic Algorithm (NSGA-II) which is capable o solving the original problem involving mixed discrete andreal-valued parameters and more than one objective

In this paper two stages o helical gear train are consid-ered Tere are several actors which affect the assembly aswell as working condition Tey are not generally consid-ered in literature Te optimization model ormulated hereincludes these actors in constraints A GUI is developedwhich acilitates the input o various combinations o inputdata Moreover a code o GA is also developed Te optimiza-tion is carried out using optimization toolbox o MALABand GA and the results obtained by both o the methodsare compared Tese methods are applied to minimize the

volume only Te resulting values o the parameters areapplied to 1047297nd 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

Te optimization model o two-stage helical gear reductionunit is ormulated in this section with minimum volume andmaximum load carrying capacity as design objectives Teschematic illustration o two-stage helical gear reduction unitisshownin Figure 983089 Asit isa caseo two-stage gearreduction

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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Journal o Optimization 983091

A

B

D

C

Shaf (Ls ds)

F983145983143983157983154983141 983089 Schematic illustration o two-stage helical gear train

the gear ratios between 1047297rst pair and second pair are chosenin such a way that their values are easible and their productremains the same as that o required

983090983089 Design Variables Te mainly affected parameters o gearrom the volume point o view are ace width moduleand number o teeth o gear Tese parameters directly orindirectly affect the objectives widely So the design vector is

= 9831631038389 10383891038389 11039251103925 110392511039251038389 907317 907317907317 9073171038389 907317983165 (983089)

where

907317

907317907317

9073171038389and

907317 are thenumbero teeth o gears

and respectively 1038389 and 10383891038389 are the ace widths o gears and respectively 11039251103925 and 110392511039251038389 are the normal moduleso gears and respectively Here it is assumed that all gearsare o the same material (say with the same Brinell hardnessnumber) and are o the same helix angle

983090983090 Objective Functions For the optimization 1047297rst the vol-ume o the two-stage helical gear train is minimized Aferachieving the optimal value o design variables or minimum

volume those values o variables are applied to maximize theload carrying capacity o both o the stages From both o these stages the minimum load carrying capacity out o thetwo is chosen as the maximum capacity or the gear train

Te optimization model o two-stage helical gear trains isderived as ollows

Considering the dimensions o the three shafs constantthe volume o the gear train is

= 4 98313110486162 + 907317

21048617 1038389 + 104861610383892 +

21048617 10383891038389

+ 121 + 2

22 + 323983133

(983090)

and the load carrying capacity is given as [983089983092]

eff = + (983091)

Reerring to eff o the two stages as 1 and 2 urther canbe written as

1 = 21

+ 211

10486161

1038389

15cos2

+ 2

05

1

1048617cos

21151 + 60000radic11038389cos2 + 21

2 = 221038389

+ 212 1048616210383891038389103838915cos2 + 21038389

0521048617 cos211038389

152 + 60 000radic2103838910383891038389cos2 + 22

(983092)

where 1 2 3 and 1 2 3 represent the diameters o shaf and lengths o shaf 983089 983090 983091 respectively Te actors and denote service actor and deormation actorrespectively

is the transmitted torque and

and

1038389

aresum o error between 1047297rst meshing teeth andsecond meshingteeth respectively

Tus the objectives can be written or minimum volumeand maximum load carrying capacity as

min

max

= 9831631 2983165 (983093)

983090983091 Constraints When the gear tooth is considered as acantilever beam the bending strength in working conditionshould not exceed standard endurance limit 1103925 From Lewisequation the constraint on bending strength is

1038389 le 1103925 (983094)

where = ( times 103)V V = (60 times 103) is diametralpitch 1038389 is ace width and is Lewis actor

However in this work the actors affecting bendingstrength during the production and assembly such as velocity actor overload actor and mounting actor to name a eware not taken into consideration So afer adding the effectso these actors the new constraints on bending strength orboth o the gear pairs can be expressed [983089983093] as

1038389

V 983080093983081 minus

1103925ms le 0

10383891038389103838910383891038389V 1038389 9830800931038389983081 minus

11039251038389ms le 0 (983095)

where is geometry actor which includes the Lewis ormactor and a stress concentration actor

V and denote velocity or dynamic actor overload actor and

mounting actor respectively 1103925 is standard R R Moore

endurance limit and denote load actor gradientactor and service actor respectively and ms denotetemperature actor reliability actor and mean stress actorrespectively

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

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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983092 Journal o Optimization

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

Afer including all the parameters the surace atigueconstraint ormula can be written [983089983093] as

991770 1038389

times cos095CR times

V 983080093983081minus Li le 0

991770 10383891038389103838910383891038389

times cos095CR 1038389times

V 1038389 9830800931038389983081minus Li le 0

(983096)

where

Li and

denote elastic coefficient actor

lie actor and reliability actor respectively and 1038389 aredimensionless constants and CR and CR 1038389 are contact ratios represents surace atigue strength

While designing the gear intererence is the main actorto consider Intererence usually takes place in the gear Soormulation o the optimization problem must take careo intererence o remove intererence the ollowing con-straints should be satis1047297ed (see [983089983093 983089983094])

minus radic2 +

2sin2 le 01038389 minus radic1038389

2 + 10383892sin2 le 0

2sin2 minus 907317 le 02

sin2 minus 907317907317 le 02

sin2 minus 9073171038389 le 02

sin2 minus 907317 le 0

(983097)

3 Methods of Solution

Since there are many input parameters such as dimensionso shafs gear train parameters material properties workingcondition o gear train and actor affecting production andassembly a GUI is prepared as shown in Figures 983090 983091 983092 and983093 Te problem is solved by ollowing three ways

(i) using optimization toolbox o MALAB

(ii) using code developed or GA

(iii) using multiobjective optimization (NSGA-II) tech-nique

Te ranges o the problem variables are taken as reerencerom manuacturerrsquos catalog [983089983095] and these ranges or

1038389

1103925

F983145983143983157983154983141 983090 Input data through ldquoData Shafrdquo

F983145983143983157983154983141 983091 Input data through ldquoData Geartrainrdquo

F983145983143983157983154983141 983092 Input data through ldquoData Factorrdquo

F983145983143983157983154983141 983093 Input data through ldquoData Factor983090rdquo

907317 907317907317 10383891038389 11039251038389 9073171038389 and 907317 are taken as 983094983088ndash983096983088 983092ndash983089983090 983089983092ndash983090983088983092983092ndash983094983093 983096983093ndash983089983088983093 983091ndash983089983088 983089983092ndash983090983088 and 983095983095ndash983089983089983088 respectively

983091983089 Using the Optimization oolbox of MALAB In thismethod 1047297rst the volume o the gear train is minimized Teresulting values o the parameters are used to determine theload carrying capacities o both o the shafs Te minimumo them is considered as the maximum load carrying capacityIn this way a multiobjective problem is reduced to a single

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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Journal o Optimization 983093

objective problem Te ldquooptimtoolrdquo eature o MALABis useul or different kinds o optimization problem Inthe problem discussed here constraints are nonlinear Soldquominconrdquo unction o MALAB applicable or nonlinearconstraint minimization is used or the optimization Tereare different algorithms and methods available under this

option in the optimization toolbox Interior-point algorithmis chosen among them as it handles large sparse problemsas well as small dense problems Moreover the algorithmsatis1047297es bounds at all iterations and can recover rom NaNor In results It is a large-scale algorithm widely used or thistype o problems

Tis unction requires a point to start with the choiceo which is arbitrary Te results obtained or ace width o gear module o gear (and ) number o teeth o gear number o teeth o gear ace width o gear moduleo gear (and ) number o teeth o gear and numbero teeth o gear are 983094983088 983092 983089983095983088983097983095 983093983091983095983091983095 983096983093 983091 983089983095983088983097983095 and983097983092983088983091983093 respectively Te corresponding volume is 1948 times10

7

mm

3

Te result remains invariant i other starting pointsare chosen For the value o load carrying capacity the values

or 1047297rst and second stages are 983091983091983091983093983090 times 9830899830884 N and 983091983091983097983088983097 times9830899830884 N So rom these values the load carrying capacity o the

gear train is selected as 983091983091983091983093983090 times 9830899830884 N

983091983090 Optimization Using Genetic Algorithm Te same strategy used in the 1047297rst method is also applied here to deal with amultiobjective problem First the volume is minimized andthen minimum o the resulting two load carrying capacitiesis chosen as the maximum load carrying capacity Te only difference is that to minimize the volume GA is used Asdiscussed in introduction many designs are characterized by

mixed continuous-discrete variables and discontinuous andnonconvex design spaces Standard nonlinear programmingtechniques are not capable o solving these types o problemsTey usually 1047297nd relative optimum that is closest to thestarting point GA is well suited or solving such problemsand in most cases they can 1047297nd the global optimum solutionwith high probability Actually the idea o evolutionary computing was introduced in the 983089983097983094983088s by I Rechenberg inhis work ldquoEvolution strategiesrdquo which was then developedby others GAs were invented and developed by Holland[983089983096] Te basic ideas o analysis and design based on theconcepts o biological evolution can be ound in the work o Rechenberg [983089983097] Philosophically GAs are based on Darwinrsquos

theory o survival o the 1047297ttest and also are based on theprinciples o natural genetics and natural selection Te basicelements o natural genetics-reproduction cross-over andmutation are used in the genetic search procedures

GA is a search algorithm based on the conjecture o nat-ural selection and genetics Te eatures o GA are differentrom the other search techniques in several aspects as ollows

(i) the algorithm is a multipath that searches many peaksin parallel hence reducing the possibility o localminimum trapping

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

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

(iv) GAs use objective unction inormation not deriva-tions or other auxiliary knowledge to determine the1047297tness o the solution

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

983091983090983089 Outline of Basic Genetic Algorithm Te basic procedureo GA as outlined in [983090983088] is as ollows

(983089) [Start] Generate random population o chromo-somes (suitable solution or problem)

(983090) [Fitness] Evaluate the 1047297tness () o each chromo-some in the population

(983091) [New population] Create a new population by repeat-ing ollowing steps until the new population is com-plete

(i) [Selection] Select two parent chromosomesrom a population according to their 1047297tness (thebetter 1047297tness the bigger chance to be selected)

(ii) [Crossover] With a crossover probability 1038389

crossover the two parents to rom two new off-spring I no crossover was perormed offspringis the exact copy o parents

(iii) [Mutation] With a mutation probability

mutate new offspring at each locus (position inchromosome)

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

(983092) [Replace] Use new generated population or a urtherrun o the algorithm

(983093) [est] I the end condition is satis1047297ed stop and returnthe best solution in current population

(983094) [Loop] Go to Step (983090)

983091983090983090 Implementation of Genetic Algorithm Extensive exper-iments are carried out or different combinations o popu-lation size and number o generations It is observed thatthe results remain consistent when the population size is 983097983088and number o generations is 983097983088 So eleven good results

with this population size and number o generations areshown in able 983089 in which the 983089983088th solution is the bestCorresponding load carrying capacities o the 1047297rst and thesecond pair are 983091983090983096983094 kN and 983091983092983089983094 kN respectively So theload carrying capacity o gear train is selected as 983091983090983096983094 kN or

which optimum volume is 983090983088983091983097983094 times 9830899830887 mm3

983091983091 Optimization Using NSGA-II In this case the problem isconsidered as a multiobjective problem So both objectivesare treated together In general in case o multiobjectiveoptimization the objectives are con1047298icting So a singlesolution cannot be accepted as the best solution Insteada set o solutions is obtained which are better than the

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

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983094 Journal o Optimization

983137983138983148983141 983089 Results o GA or population size o 983097983088 and 983097983088 generations

Sr number 1038389

(mm)1103925

(mm) 907317 907317907317

10383891038389

(mm)11039251038389

(mm) 9073171038389 907317 Volume (983089983088983095 times mm983091)

983089 983094983095983088983094 983092 983089983096 983093983095 983097983088983089983096 983091 983089983096 983097983097 983090983089983094983094

983090 983094983092983088983095 983092 983089983097 983094983088 983096983093983090983097 983091 983089983096 983097983097 983090983088983096983096

983091 983094983090983088983095 983092

983089983096 983093983095 983096983094983095 983091

983089983096 983097983097 983090983088983093983096

983092 983095983093983089 983092 983089983096 983093983095 983096983093983097983089 983091 983089983096 983097983097 983090983089983089983089

983093 983094983094983088983095 983092 983090983088 983094983091 983096983093983090983091 983091 983089983096 983097983097 983090983089983091983090

983094 983094983091983095 983092 983089983097 983094983088 983096983095983095983092 983091 983089983096 983097983097 983090983089983088983092

983095 983094983090 983092 983089983096 983093983095 983096983094983096983090 983091 983089983096 983097983097 983090983088983093983097

983096 983094983088983095983089 983092 983089983096 983093983095 983096983094983088983092 983091 983089983096 983097983097 983090983088983092983095

983097 983094983089983095983095 983092 983089983096 983093983095 983096983094983092983095 983091 983089983096 983097983097 983090983088983093983093

983089983088 983094983088983088983089 983092 983089983096 983093983095 983096983093983091983089 983091 983089983096 983097983097 983090983088983091983097

983089983089 983094983090983096983093 983092 983089983097 983094983088 983096983093983091983089 983091 983089983097 983097983097 983090983089983093983097

983137983138983148983141 983090 Results o NSGA-II or population size o 983093983088983088 and 983093983088983088 generations

Sr number 1038389

(mm)1103925

(mm)

907317

907317907317

10383891038389

(mm)11039251038389

(mm)

9073171038389

907317

Volume(983089983088983095

times mm983091)

Load carrying capacity (kN)

983089 983096983088983088983088 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983092983096 983091983093983091983088983094

983090 983094983088983091983097 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983093983089 983091983091983095983089983096

983091 983095983097983088983092 983092 983089983096 983093983095 983097983088983096983091 983091 983089983096 983097983097 983090983088983094983094 983091983093983096983091983095

983092 983096983088983088983088 983092 983089983096 983093983095 983097983088983097983096 983091 983089983096 983097983097 983090983088983095983088 983091983093983097983092983089

983093 983095983089983092983096 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983089983091 983091983092983097983097983094

983094 983095983092983095983095 983092 983089983096 983093983095 983096983095983096983097 983091 983089983096 983097983097 983090983088983090983095 983091983093983091983093983096

983095 983094983094983096983088 983092 983089983096 983093983095 983096983094983091983091 983091 983089983096 983097983097 983089983097983097983091 983091983092983091983096983091

983096 983094983091983092983093 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983094983091 983091983092983088983095983094

other solutions in terms o both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice or solving thiskind o problem In NSGA-II the iterative procedure startsrom an arbitrary population o solutions and gradually thealgorithm converges to a population o solutions lying onthe Pareto optimal ront with higher diversity Te operatorsapplied are the same as those o GA namely selectioncrossover and mutation Te tournament selection operatoris applied which also takes care o constraints Howeverin case o multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing unction approach is used Crossover and mutationoperators are applied as usual A detailed discussion o thisalgorithm is ound in [983090983089] Te standard code available at [983090983090]

is modi1047297ed according to authorsrsquo needAs a result o NSGA-II out o the population size o

983093983088 and number o generations o 983093983088983088 eight better resultsare selected and shown in able 983090 It has been observedthat ul1047297lling both o the objectives together the second lastsolution is the compromised one Corresponding optimum

volume and load carrying capacity o the train are 983089983097983097983091 times9830899830887 mm3 and 983091983092983091983096 kN respectively

4 Results and Discussion

Tere are several comments in order Te number o teetho gear

and gear

in the manuacturerrsquos design is 983089983092 It

creates intererence in working condition o eliminate itmanuacturer produces stub tooth instead o normal toothwhich is not advisable Te introduction o the constraintson intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de1047297nitely exceed 983089983095 Te major problem with the inbuiltldquominconrdquo unction o MALAB is that it considers all the

variables real As a result one has to round the optimum valueo integer variable to the nearest integer So the optimum

value o number o teeth o gear and gear is roundedoff to 983089983096 o maintain the gear ratios the numbers o teetho gear and gear have to be selected as 983093983094 and 983089983088983088respectively which are quite ar rom their actual optimum

values obtained using toolbox

However GA can deal with both types o variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers o teeth o gear andgear have to be changed to 983093983096 and 983089983088983088 respectively becauseo manuacturing inconveniences NSGA-II selects 983093983095 and 983097983097as the numbers o teeth o gear and gear which is betterthan both o the above results Te results are presented inable 983091

5 Conclusion and Future Scope

Result comparison table shows that in the 1047297rst two casesminimization o volume took place while load carrying

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 39

Journal o Optimization 983091

A

B

D

C

Shaf (Ls ds)

F983145983143983157983154983141 983089 Schematic illustration o two-stage helical gear train

the gear ratios between 1047297rst pair and second pair are chosenin such a way that their values are easible and their productremains the same as that o required

983090983089 Design Variables Te mainly affected parameters o gearrom the volume point o view are ace width moduleand number o teeth o gear Tese parameters directly orindirectly affect the objectives widely So the design vector is

= 9831631038389 10383891038389 11039251103925 110392511039251038389 907317 907317907317 9073171038389 907317983165 (983089)

where

907317

907317907317

9073171038389and

907317 are thenumbero teeth o gears

and respectively 1038389 and 10383891038389 are the ace widths o gears and respectively 11039251103925 and 110392511039251038389 are the normal moduleso gears and respectively Here it is assumed that all gearsare o the same material (say with the same Brinell hardnessnumber) and are o the same helix angle

983090983090 Objective Functions For the optimization 1047297rst the vol-ume o the two-stage helical gear train is minimized Aferachieving the optimal value o design variables or minimum

volume those values o variables are applied to maximize theload carrying capacity o both o the stages From both o these stages the minimum load carrying capacity out o thetwo is chosen as the maximum capacity or the gear train

Te optimization model o two-stage helical gear trains isderived as ollows

Considering the dimensions o the three shafs constantthe volume o the gear train is

= 4 98313110486162 + 907317

21048617 1038389 + 104861610383892 +

21048617 10383891038389

+ 121 + 2

22 + 323983133

(983090)

and the load carrying capacity is given as [983089983092]

eff = + (983091)

Reerring to eff o the two stages as 1 and 2 urther canbe written as

1 = 21

+ 211

10486161

1038389

15cos2

+ 2

05

1

1048617cos

21151 + 60000radic11038389cos2 + 21

2 = 221038389

+ 212 1048616210383891038389103838915cos2 + 21038389

0521048617 cos211038389

152 + 60 000radic2103838910383891038389cos2 + 22

(983092)

where 1 2 3 and 1 2 3 represent the diameters o shaf and lengths o shaf 983089 983090 983091 respectively Te actors and denote service actor and deormation actorrespectively

is the transmitted torque and

and

1038389

aresum o error between 1047297rst meshing teeth andsecond meshingteeth respectively

Tus the objectives can be written or minimum volumeand maximum load carrying capacity as

min

max

= 9831631 2983165 (983093)

983090983091 Constraints When the gear tooth is considered as acantilever beam the bending strength in working conditionshould not exceed standard endurance limit 1103925 From Lewisequation the constraint on bending strength is

1038389 le 1103925 (983094)

where = ( times 103)V V = (60 times 103) is diametralpitch 1038389 is ace width and is Lewis actor

However in this work the actors affecting bendingstrength during the production and assembly such as velocity actor overload actor and mounting actor to name a eware not taken into consideration So afer adding the effectso these actors the new constraints on bending strength orboth o the gear pairs can be expressed [983089983093] as

1038389

V 983080093983081 minus

1103925ms le 0

10383891038389103838910383891038389V 1038389 9830800931038389983081 minus

11039251038389ms le 0 (983095)

where is geometry actor which includes the Lewis ormactor and a stress concentration actor

V and denote velocity or dynamic actor overload actor and

mounting actor respectively 1103925 is standard R R Moore

endurance limit and denote load actor gradientactor and service actor respectively and ms denotetemperature actor reliability actor and mean stress actorrespectively

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

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 49

983092 Journal o Optimization

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

Afer including all the parameters the surace atigueconstraint ormula can be written [983089983093] as

991770 1038389

times cos095CR times

V 983080093983081minus Li le 0

991770 10383891038389103838910383891038389

times cos095CR 1038389times

V 1038389 9830800931038389983081minus Li le 0

(983096)

where

Li and

denote elastic coefficient actor

lie actor and reliability actor respectively and 1038389 aredimensionless constants and CR and CR 1038389 are contact ratios represents surace atigue strength

While designing the gear intererence is the main actorto consider Intererence usually takes place in the gear Soormulation o the optimization problem must take careo intererence o remove intererence the ollowing con-straints should be satis1047297ed (see [983089983093 983089983094])

minus radic2 +

2sin2 le 01038389 minus radic1038389

2 + 10383892sin2 le 0

2sin2 minus 907317 le 02

sin2 minus 907317907317 le 02

sin2 minus 9073171038389 le 02

sin2 minus 907317 le 0

(983097)

3 Methods of Solution

Since there are many input parameters such as dimensionso shafs gear train parameters material properties workingcondition o gear train and actor affecting production andassembly a GUI is prepared as shown in Figures 983090 983091 983092 and983093 Te problem is solved by ollowing three ways

(i) using optimization toolbox o MALAB

(ii) using code developed or GA

(iii) using multiobjective optimization (NSGA-II) tech-nique

Te ranges o the problem variables are taken as reerencerom manuacturerrsquos catalog [983089983095] and these ranges or

1038389

1103925

F983145983143983157983154983141 983090 Input data through ldquoData Shafrdquo

F983145983143983157983154983141 983091 Input data through ldquoData Geartrainrdquo

F983145983143983157983154983141 983092 Input data through ldquoData Factorrdquo

F983145983143983157983154983141 983093 Input data through ldquoData Factor983090rdquo

907317 907317907317 10383891038389 11039251038389 9073171038389 and 907317 are taken as 983094983088ndash983096983088 983092ndash983089983090 983089983092ndash983090983088983092983092ndash983094983093 983096983093ndash983089983088983093 983091ndash983089983088 983089983092ndash983090983088 and 983095983095ndash983089983089983088 respectively

983091983089 Using the Optimization oolbox of MALAB In thismethod 1047297rst the volume o the gear train is minimized Teresulting values o the parameters are used to determine theload carrying capacities o both o the shafs Te minimumo them is considered as the maximum load carrying capacityIn this way a multiobjective problem is reduced to a single

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 59

Journal o Optimization 983093

objective problem Te ldquooptimtoolrdquo eature o MALABis useul or different kinds o optimization problem Inthe problem discussed here constraints are nonlinear Soldquominconrdquo unction o MALAB applicable or nonlinearconstraint minimization is used or the optimization Tereare different algorithms and methods available under this

option in the optimization toolbox Interior-point algorithmis chosen among them as it handles large sparse problemsas well as small dense problems Moreover the algorithmsatis1047297es bounds at all iterations and can recover rom NaNor In results It is a large-scale algorithm widely used or thistype o problems

Tis unction requires a point to start with the choiceo which is arbitrary Te results obtained or ace width o gear module o gear (and ) number o teeth o gear number o teeth o gear ace width o gear moduleo gear (and ) number o teeth o gear and numbero teeth o gear are 983094983088 983092 983089983095983088983097983095 983093983091983095983091983095 983096983093 983091 983089983095983088983097983095 and983097983092983088983091983093 respectively Te corresponding volume is 1948 times10

7

mm

3

Te result remains invariant i other starting pointsare chosen For the value o load carrying capacity the values

or 1047297rst and second stages are 983091983091983091983093983090 times 9830899830884 N and 983091983091983097983088983097 times9830899830884 N So rom these values the load carrying capacity o the

gear train is selected as 983091983091983091983093983090 times 9830899830884 N

983091983090 Optimization Using Genetic Algorithm Te same strategy used in the 1047297rst method is also applied here to deal with amultiobjective problem First the volume is minimized andthen minimum o the resulting two load carrying capacitiesis chosen as the maximum load carrying capacity Te only difference is that to minimize the volume GA is used Asdiscussed in introduction many designs are characterized by

mixed continuous-discrete variables and discontinuous andnonconvex design spaces Standard nonlinear programmingtechniques are not capable o solving these types o problemsTey usually 1047297nd relative optimum that is closest to thestarting point GA is well suited or solving such problemsand in most cases they can 1047297nd the global optimum solutionwith high probability Actually the idea o evolutionary computing was introduced in the 983089983097983094983088s by I Rechenberg inhis work ldquoEvolution strategiesrdquo which was then developedby others GAs were invented and developed by Holland[983089983096] Te basic ideas o analysis and design based on theconcepts o biological evolution can be ound in the work o Rechenberg [983089983097] Philosophically GAs are based on Darwinrsquos

theory o survival o the 1047297ttest and also are based on theprinciples o natural genetics and natural selection Te basicelements o natural genetics-reproduction cross-over andmutation are used in the genetic search procedures

GA is a search algorithm based on the conjecture o nat-ural selection and genetics Te eatures o GA are differentrom the other search techniques in several aspects as ollows

(i) the algorithm is a multipath that searches many peaksin parallel hence reducing the possibility o localminimum trapping

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

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

(iv) GAs use objective unction inormation not deriva-tions or other auxiliary knowledge to determine the1047297tness o the solution

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

983091983090983089 Outline of Basic Genetic Algorithm Te basic procedureo GA as outlined in [983090983088] is as ollows

(983089) [Start] Generate random population o chromo-somes (suitable solution or problem)

(983090) [Fitness] Evaluate the 1047297tness () o each chromo-some in the population

(983091) [New population] Create a new population by repeat-ing ollowing steps until the new population is com-plete

(i) [Selection] Select two parent chromosomesrom a population according to their 1047297tness (thebetter 1047297tness the bigger chance to be selected)

(ii) [Crossover] With a crossover probability 1038389

crossover the two parents to rom two new off-spring I no crossover was perormed offspringis the exact copy o parents

(iii) [Mutation] With a mutation probability

mutate new offspring at each locus (position inchromosome)

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

(983092) [Replace] Use new generated population or a urtherrun o the algorithm

(983093) [est] I the end condition is satis1047297ed stop and returnthe best solution in current population

(983094) [Loop] Go to Step (983090)

983091983090983090 Implementation of Genetic Algorithm Extensive exper-iments are carried out or different combinations o popu-lation size and number o generations It is observed thatthe results remain consistent when the population size is 983097983088and number o generations is 983097983088 So eleven good results

with this population size and number o generations areshown in able 983089 in which the 983089983088th solution is the bestCorresponding load carrying capacities o the 1047297rst and thesecond pair are 983091983090983096983094 kN and 983091983092983089983094 kN respectively So theload carrying capacity o gear train is selected as 983091983090983096983094 kN or

which optimum volume is 983090983088983091983097983094 times 9830899830887 mm3

983091983091 Optimization Using NSGA-II In this case the problem isconsidered as a multiobjective problem So both objectivesare treated together In general in case o multiobjectiveoptimization the objectives are con1047298icting So a singlesolution cannot be accepted as the best solution Insteada set o solutions is obtained which are better than the

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 69

983094 Journal o Optimization

983137983138983148983141 983089 Results o GA or population size o 983097983088 and 983097983088 generations

Sr number 1038389

(mm)1103925

(mm) 907317 907317907317

10383891038389

(mm)11039251038389

(mm) 9073171038389 907317 Volume (983089983088983095 times mm983091)

983089 983094983095983088983094 983092 983089983096 983093983095 983097983088983089983096 983091 983089983096 983097983097 983090983089983094983094

983090 983094983092983088983095 983092 983089983097 983094983088 983096983093983090983097 983091 983089983096 983097983097 983090983088983096983096

983091 983094983090983088983095 983092

983089983096 983093983095 983096983094983095 983091

983089983096 983097983097 983090983088983093983096

983092 983095983093983089 983092 983089983096 983093983095 983096983093983097983089 983091 983089983096 983097983097 983090983089983089983089

983093 983094983094983088983095 983092 983090983088 983094983091 983096983093983090983091 983091 983089983096 983097983097 983090983089983091983090

983094 983094983091983095 983092 983089983097 983094983088 983096983095983095983092 983091 983089983096 983097983097 983090983089983088983092

983095 983094983090 983092 983089983096 983093983095 983096983094983096983090 983091 983089983096 983097983097 983090983088983093983097

983096 983094983088983095983089 983092 983089983096 983093983095 983096983094983088983092 983091 983089983096 983097983097 983090983088983092983095

983097 983094983089983095983095 983092 983089983096 983093983095 983096983094983092983095 983091 983089983096 983097983097 983090983088983093983093

983089983088 983094983088983088983089 983092 983089983096 983093983095 983096983093983091983089 983091 983089983096 983097983097 983090983088983091983097

983089983089 983094983090983096983093 983092 983089983097 983094983088 983096983093983091983089 983091 983089983097 983097983097 983090983089983093983097

983137983138983148983141 983090 Results o NSGA-II or population size o 983093983088983088 and 983093983088983088 generations

Sr number 1038389

(mm)1103925

(mm)

907317

907317907317

10383891038389

(mm)11039251038389

(mm)

9073171038389

907317

Volume(983089983088983095

times mm983091)

Load carrying capacity (kN)

983089 983096983088983088983088 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983092983096 983091983093983091983088983094

983090 983094983088983091983097 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983093983089 983091983091983095983089983096

983091 983095983097983088983092 983092 983089983096 983093983095 983097983088983096983091 983091 983089983096 983097983097 983090983088983094983094 983091983093983096983091983095

983092 983096983088983088983088 983092 983089983096 983093983095 983097983088983097983096 983091 983089983096 983097983097 983090983088983095983088 983091983093983097983092983089

983093 983095983089983092983096 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983089983091 983091983092983097983097983094

983094 983095983092983095983095 983092 983089983096 983093983095 983096983095983096983097 983091 983089983096 983097983097 983090983088983090983095 983091983093983091983093983096

983095 983094983094983096983088 983092 983089983096 983093983095 983096983094983091983091 983091 983089983096 983097983097 983089983097983097983091 983091983092983091983096983091

983096 983094983091983092983093 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983094983091 983091983092983088983095983094

other solutions in terms o both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice or solving thiskind o problem In NSGA-II the iterative procedure startsrom an arbitrary population o solutions and gradually thealgorithm converges to a population o solutions lying onthe Pareto optimal ront with higher diversity Te operatorsapplied are the same as those o GA namely selectioncrossover and mutation Te tournament selection operatoris applied which also takes care o constraints Howeverin case o multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing unction approach is used Crossover and mutationoperators are applied as usual A detailed discussion o thisalgorithm is ound in [983090983089] Te standard code available at [983090983090]

is modi1047297ed according to authorsrsquo needAs a result o NSGA-II out o the population size o

983093983088 and number o generations o 983093983088983088 eight better resultsare selected and shown in able 983090 It has been observedthat ul1047297lling both o the objectives together the second lastsolution is the compromised one Corresponding optimum

volume and load carrying capacity o the train are 983089983097983097983091 times9830899830887 mm3 and 983091983092983091983096 kN respectively

4 Results and Discussion

Tere are several comments in order Te number o teetho gear

and gear

in the manuacturerrsquos design is 983089983092 It

creates intererence in working condition o eliminate itmanuacturer produces stub tooth instead o normal toothwhich is not advisable Te introduction o the constraintson intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de1047297nitely exceed 983089983095 Te major problem with the inbuiltldquominconrdquo unction o MALAB is that it considers all the

variables real As a result one has to round the optimum valueo integer variable to the nearest integer So the optimum

value o number o teeth o gear and gear is roundedoff to 983089983096 o maintain the gear ratios the numbers o teetho gear and gear have to be selected as 983093983094 and 983089983088983088respectively which are quite ar rom their actual optimum

values obtained using toolbox

However GA can deal with both types o variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers o teeth o gear andgear have to be changed to 983093983096 and 983089983088983088 respectively becauseo manuacturing inconveniences NSGA-II selects 983093983095 and 983097983097as the numbers o teeth o gear and gear which is betterthan both o the above results Te results are presented inable 983091

5 Conclusion and Future Scope

Result comparison table shows that in the 1047297rst two casesminimization o volume took place while load carrying

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 49

983092 Journal o Optimization

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

Afer including all the parameters the surace atigueconstraint ormula can be written [983089983093] as

991770 1038389

times cos095CR times

V 983080093983081minus Li le 0

991770 10383891038389103838910383891038389

times cos095CR 1038389times

V 1038389 9830800931038389983081minus Li le 0

(983096)

where

Li and

denote elastic coefficient actor

lie actor and reliability actor respectively and 1038389 aredimensionless constants and CR and CR 1038389 are contact ratios represents surace atigue strength

While designing the gear intererence is the main actorto consider Intererence usually takes place in the gear Soormulation o the optimization problem must take careo intererence o remove intererence the ollowing con-straints should be satis1047297ed (see [983089983093 983089983094])

minus radic2 +

2sin2 le 01038389 minus radic1038389

2 + 10383892sin2 le 0

2sin2 minus 907317 le 02

sin2 minus 907317907317 le 02

sin2 minus 9073171038389 le 02

sin2 minus 907317 le 0

(983097)

3 Methods of Solution

Since there are many input parameters such as dimensionso shafs gear train parameters material properties workingcondition o gear train and actor affecting production andassembly a GUI is prepared as shown in Figures 983090 983091 983092 and983093 Te problem is solved by ollowing three ways

(i) using optimization toolbox o MALAB

(ii) using code developed or GA

(iii) using multiobjective optimization (NSGA-II) tech-nique

Te ranges o the problem variables are taken as reerencerom manuacturerrsquos catalog [983089983095] and these ranges or

1038389

1103925

F983145983143983157983154983141 983090 Input data through ldquoData Shafrdquo

F983145983143983157983154983141 983091 Input data through ldquoData Geartrainrdquo

F983145983143983157983154983141 983092 Input data through ldquoData Factorrdquo

F983145983143983157983154983141 983093 Input data through ldquoData Factor983090rdquo

907317 907317907317 10383891038389 11039251038389 9073171038389 and 907317 are taken as 983094983088ndash983096983088 983092ndash983089983090 983089983092ndash983090983088983092983092ndash983094983093 983096983093ndash983089983088983093 983091ndash983089983088 983089983092ndash983090983088 and 983095983095ndash983089983089983088 respectively

983091983089 Using the Optimization oolbox of MALAB In thismethod 1047297rst the volume o the gear train is minimized Teresulting values o the parameters are used to determine theload carrying capacities o both o the shafs Te minimumo them is considered as the maximum load carrying capacityIn this way a multiobjective problem is reduced to a single

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 59

Journal o Optimization 983093

objective problem Te ldquooptimtoolrdquo eature o MALABis useul or different kinds o optimization problem Inthe problem discussed here constraints are nonlinear Soldquominconrdquo unction o MALAB applicable or nonlinearconstraint minimization is used or the optimization Tereare different algorithms and methods available under this

option in the optimization toolbox Interior-point algorithmis chosen among them as it handles large sparse problemsas well as small dense problems Moreover the algorithmsatis1047297es bounds at all iterations and can recover rom NaNor In results It is a large-scale algorithm widely used or thistype o problems

Tis unction requires a point to start with the choiceo which is arbitrary Te results obtained or ace width o gear module o gear (and ) number o teeth o gear number o teeth o gear ace width o gear moduleo gear (and ) number o teeth o gear and numbero teeth o gear are 983094983088 983092 983089983095983088983097983095 983093983091983095983091983095 983096983093 983091 983089983095983088983097983095 and983097983092983088983091983093 respectively Te corresponding volume is 1948 times10

7

mm

3

Te result remains invariant i other starting pointsare chosen For the value o load carrying capacity the values

or 1047297rst and second stages are 983091983091983091983093983090 times 9830899830884 N and 983091983091983097983088983097 times9830899830884 N So rom these values the load carrying capacity o the

gear train is selected as 983091983091983091983093983090 times 9830899830884 N

983091983090 Optimization Using Genetic Algorithm Te same strategy used in the 1047297rst method is also applied here to deal with amultiobjective problem First the volume is minimized andthen minimum o the resulting two load carrying capacitiesis chosen as the maximum load carrying capacity Te only difference is that to minimize the volume GA is used Asdiscussed in introduction many designs are characterized by

mixed continuous-discrete variables and discontinuous andnonconvex design spaces Standard nonlinear programmingtechniques are not capable o solving these types o problemsTey usually 1047297nd relative optimum that is closest to thestarting point GA is well suited or solving such problemsand in most cases they can 1047297nd the global optimum solutionwith high probability Actually the idea o evolutionary computing was introduced in the 983089983097983094983088s by I Rechenberg inhis work ldquoEvolution strategiesrdquo which was then developedby others GAs were invented and developed by Holland[983089983096] Te basic ideas o analysis and design based on theconcepts o biological evolution can be ound in the work o Rechenberg [983089983097] Philosophically GAs are based on Darwinrsquos

theory o survival o the 1047297ttest and also are based on theprinciples o natural genetics and natural selection Te basicelements o natural genetics-reproduction cross-over andmutation are used in the genetic search procedures

GA is a search algorithm based on the conjecture o nat-ural selection and genetics Te eatures o GA are differentrom the other search techniques in several aspects as ollows

(i) the algorithm is a multipath that searches many peaksin parallel hence reducing the possibility o localminimum trapping

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

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

(iv) GAs use objective unction inormation not deriva-tions or other auxiliary knowledge to determine the1047297tness o the solution

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

983091983090983089 Outline of Basic Genetic Algorithm Te basic procedureo GA as outlined in [983090983088] is as ollows

(983089) [Start] Generate random population o chromo-somes (suitable solution or problem)

(983090) [Fitness] Evaluate the 1047297tness () o each chromo-some in the population

(983091) [New population] Create a new population by repeat-ing ollowing steps until the new population is com-plete

(i) [Selection] Select two parent chromosomesrom a population according to their 1047297tness (thebetter 1047297tness the bigger chance to be selected)

(ii) [Crossover] With a crossover probability 1038389

crossover the two parents to rom two new off-spring I no crossover was perormed offspringis the exact copy o parents

(iii) [Mutation] With a mutation probability

mutate new offspring at each locus (position inchromosome)

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

(983092) [Replace] Use new generated population or a urtherrun o the algorithm

(983093) [est] I the end condition is satis1047297ed stop and returnthe best solution in current population

(983094) [Loop] Go to Step (983090)

983091983090983090 Implementation of Genetic Algorithm Extensive exper-iments are carried out or different combinations o popu-lation size and number o generations It is observed thatthe results remain consistent when the population size is 983097983088and number o generations is 983097983088 So eleven good results

with this population size and number o generations areshown in able 983089 in which the 983089983088th solution is the bestCorresponding load carrying capacities o the 1047297rst and thesecond pair are 983091983090983096983094 kN and 983091983092983089983094 kN respectively So theload carrying capacity o gear train is selected as 983091983090983096983094 kN or

which optimum volume is 983090983088983091983097983094 times 9830899830887 mm3

983091983091 Optimization Using NSGA-II In this case the problem isconsidered as a multiobjective problem So both objectivesare treated together In general in case o multiobjectiveoptimization the objectives are con1047298icting So a singlesolution cannot be accepted as the best solution Insteada set o solutions is obtained which are better than the

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 69

983094 Journal o Optimization

983137983138983148983141 983089 Results o GA or population size o 983097983088 and 983097983088 generations

Sr number 1038389

(mm)1103925

(mm) 907317 907317907317

10383891038389

(mm)11039251038389

(mm) 9073171038389 907317 Volume (983089983088983095 times mm983091)

983089 983094983095983088983094 983092 983089983096 983093983095 983097983088983089983096 983091 983089983096 983097983097 983090983089983094983094

983090 983094983092983088983095 983092 983089983097 983094983088 983096983093983090983097 983091 983089983096 983097983097 983090983088983096983096

983091 983094983090983088983095 983092

983089983096 983093983095 983096983094983095 983091

983089983096 983097983097 983090983088983093983096

983092 983095983093983089 983092 983089983096 983093983095 983096983093983097983089 983091 983089983096 983097983097 983090983089983089983089

983093 983094983094983088983095 983092 983090983088 983094983091 983096983093983090983091 983091 983089983096 983097983097 983090983089983091983090

983094 983094983091983095 983092 983089983097 983094983088 983096983095983095983092 983091 983089983096 983097983097 983090983089983088983092

983095 983094983090 983092 983089983096 983093983095 983096983094983096983090 983091 983089983096 983097983097 983090983088983093983097

983096 983094983088983095983089 983092 983089983096 983093983095 983096983094983088983092 983091 983089983096 983097983097 983090983088983092983095

983097 983094983089983095983095 983092 983089983096 983093983095 983096983094983092983095 983091 983089983096 983097983097 983090983088983093983093

983089983088 983094983088983088983089 983092 983089983096 983093983095 983096983093983091983089 983091 983089983096 983097983097 983090983088983091983097

983089983089 983094983090983096983093 983092 983089983097 983094983088 983096983093983091983089 983091 983089983097 983097983097 983090983089983093983097

983137983138983148983141 983090 Results o NSGA-II or population size o 983093983088983088 and 983093983088983088 generations

Sr number 1038389

(mm)1103925

(mm)

907317

907317907317

10383891038389

(mm)11039251038389

(mm)

9073171038389

907317

Volume(983089983088983095

times mm983091)

Load carrying capacity (kN)

983089 983096983088983088983088 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983092983096 983091983093983091983088983094

983090 983094983088983091983097 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983093983089 983091983091983095983089983096

983091 983095983097983088983092 983092 983089983096 983093983095 983097983088983096983091 983091 983089983096 983097983097 983090983088983094983094 983091983093983096983091983095

983092 983096983088983088983088 983092 983089983096 983093983095 983097983088983097983096 983091 983089983096 983097983097 983090983088983095983088 983091983093983097983092983089

983093 983095983089983092983096 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983089983091 983091983092983097983097983094

983094 983095983092983095983095 983092 983089983096 983093983095 983096983095983096983097 983091 983089983096 983097983097 983090983088983090983095 983091983093983091983093983096

983095 983094983094983096983088 983092 983089983096 983093983095 983096983094983091983091 983091 983089983096 983097983097 983089983097983097983091 983091983092983091983096983091

983096 983094983091983092983093 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983094983091 983091983092983088983095983094

other solutions in terms o both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice or solving thiskind o problem In NSGA-II the iterative procedure startsrom an arbitrary population o solutions and gradually thealgorithm converges to a population o solutions lying onthe Pareto optimal ront with higher diversity Te operatorsapplied are the same as those o GA namely selectioncrossover and mutation Te tournament selection operatoris applied which also takes care o constraints Howeverin case o multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing unction approach is used Crossover and mutationoperators are applied as usual A detailed discussion o thisalgorithm is ound in [983090983089] Te standard code available at [983090983090]

is modi1047297ed according to authorsrsquo needAs a result o NSGA-II out o the population size o

983093983088 and number o generations o 983093983088983088 eight better resultsare selected and shown in able 983090 It has been observedthat ul1047297lling both o the objectives together the second lastsolution is the compromised one Corresponding optimum

volume and load carrying capacity o the train are 983089983097983097983091 times9830899830887 mm3 and 983091983092983091983096 kN respectively

4 Results and Discussion

Tere are several comments in order Te number o teetho gear

and gear

in the manuacturerrsquos design is 983089983092 It

creates intererence in working condition o eliminate itmanuacturer produces stub tooth instead o normal toothwhich is not advisable Te introduction o the constraintson intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de1047297nitely exceed 983089983095 Te major problem with the inbuiltldquominconrdquo unction o MALAB is that it considers all the

variables real As a result one has to round the optimum valueo integer variable to the nearest integer So the optimum

value o number o teeth o gear and gear is roundedoff to 983089983096 o maintain the gear ratios the numbers o teetho gear and gear have to be selected as 983093983094 and 983089983088983088respectively which are quite ar rom their actual optimum

values obtained using toolbox

However GA can deal with both types o variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers o teeth o gear andgear have to be changed to 983093983096 and 983089983088983088 respectively becauseo manuacturing inconveniences NSGA-II selects 983093983095 and 983097983097as the numbers o teeth o gear and gear which is betterthan both o the above results Te results are presented inable 983091

5 Conclusion and Future Scope

Result comparison table shows that in the 1047297rst two casesminimization o volume took place while load carrying

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 59

Journal o Optimization 983093

objective problem Te ldquooptimtoolrdquo eature o MALABis useul or different kinds o optimization problem Inthe problem discussed here constraints are nonlinear Soldquominconrdquo unction o MALAB applicable or nonlinearconstraint minimization is used or the optimization Tereare different algorithms and methods available under this

option in the optimization toolbox Interior-point algorithmis chosen among them as it handles large sparse problemsas well as small dense problems Moreover the algorithmsatis1047297es bounds at all iterations and can recover rom NaNor In results It is a large-scale algorithm widely used or thistype o problems

Tis unction requires a point to start with the choiceo which is arbitrary Te results obtained or ace width o gear module o gear (and ) number o teeth o gear number o teeth o gear ace width o gear moduleo gear (and ) number o teeth o gear and numbero teeth o gear are 983094983088 983092 983089983095983088983097983095 983093983091983095983091983095 983096983093 983091 983089983095983088983097983095 and983097983092983088983091983093 respectively Te corresponding volume is 1948 times10

7

mm

3

Te result remains invariant i other starting pointsare chosen For the value o load carrying capacity the values

or 1047297rst and second stages are 983091983091983091983093983090 times 9830899830884 N and 983091983091983097983088983097 times9830899830884 N So rom these values the load carrying capacity o the

gear train is selected as 983091983091983091983093983090 times 9830899830884 N

983091983090 Optimization Using Genetic Algorithm Te same strategy used in the 1047297rst method is also applied here to deal with amultiobjective problem First the volume is minimized andthen minimum o the resulting two load carrying capacitiesis chosen as the maximum load carrying capacity Te only difference is that to minimize the volume GA is used Asdiscussed in introduction many designs are characterized by

mixed continuous-discrete variables and discontinuous andnonconvex design spaces Standard nonlinear programmingtechniques are not capable o solving these types o problemsTey usually 1047297nd relative optimum that is closest to thestarting point GA is well suited or solving such problemsand in most cases they can 1047297nd the global optimum solutionwith high probability Actually the idea o evolutionary computing was introduced in the 983089983097983094983088s by I Rechenberg inhis work ldquoEvolution strategiesrdquo which was then developedby others GAs were invented and developed by Holland[983089983096] Te basic ideas o analysis and design based on theconcepts o biological evolution can be ound in the work o Rechenberg [983089983097] Philosophically GAs are based on Darwinrsquos

theory o survival o the 1047297ttest and also are based on theprinciples o natural genetics and natural selection Te basicelements o natural genetics-reproduction cross-over andmutation are used in the genetic search procedures

GA is a search algorithm based on the conjecture o nat-ural selection and genetics Te eatures o GA are differentrom the other search techniques in several aspects as ollows

(i) the algorithm is a multipath that searches many peaksin parallel hence reducing the possibility o localminimum trapping

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

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

(iv) GAs use objective unction inormation not deriva-tions or other auxiliary knowledge to determine the1047297tness o the solution

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

983091983090983089 Outline of Basic Genetic Algorithm Te basic procedureo GA as outlined in [983090983088] is as ollows

(983089) [Start] Generate random population o chromo-somes (suitable solution or problem)

(983090) [Fitness] Evaluate the 1047297tness () o each chromo-some in the population

(983091) [New population] Create a new population by repeat-ing ollowing steps until the new population is com-plete

(i) [Selection] Select two parent chromosomesrom a population according to their 1047297tness (thebetter 1047297tness the bigger chance to be selected)

(ii) [Crossover] With a crossover probability 1038389

crossover the two parents to rom two new off-spring I no crossover was perormed offspringis the exact copy o parents

(iii) [Mutation] With a mutation probability

mutate new offspring at each locus (position inchromosome)

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

(983092) [Replace] Use new generated population or a urtherrun o the algorithm

(983093) [est] I the end condition is satis1047297ed stop and returnthe best solution in current population

(983094) [Loop] Go to Step (983090)

983091983090983090 Implementation of Genetic Algorithm Extensive exper-iments are carried out or different combinations o popu-lation size and number o generations It is observed thatthe results remain consistent when the population size is 983097983088and number o generations is 983097983088 So eleven good results

with this population size and number o generations areshown in able 983089 in which the 983089983088th solution is the bestCorresponding load carrying capacities o the 1047297rst and thesecond pair are 983091983090983096983094 kN and 983091983092983089983094 kN respectively So theload carrying capacity o gear train is selected as 983091983090983096983094 kN or

which optimum volume is 983090983088983091983097983094 times 9830899830887 mm3

983091983091 Optimization Using NSGA-II In this case the problem isconsidered as a multiobjective problem So both objectivesare treated together In general in case o multiobjectiveoptimization the objectives are con1047298icting So a singlesolution cannot be accepted as the best solution Insteada set o solutions is obtained which are better than the

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 69

983094 Journal o Optimization

983137983138983148983141 983089 Results o GA or population size o 983097983088 and 983097983088 generations

Sr number 1038389

(mm)1103925

(mm) 907317 907317907317

10383891038389

(mm)11039251038389

(mm) 9073171038389 907317 Volume (983089983088983095 times mm983091)

983089 983094983095983088983094 983092 983089983096 983093983095 983097983088983089983096 983091 983089983096 983097983097 983090983089983094983094

983090 983094983092983088983095 983092 983089983097 983094983088 983096983093983090983097 983091 983089983096 983097983097 983090983088983096983096

983091 983094983090983088983095 983092

983089983096 983093983095 983096983094983095 983091

983089983096 983097983097 983090983088983093983096

983092 983095983093983089 983092 983089983096 983093983095 983096983093983097983089 983091 983089983096 983097983097 983090983089983089983089

983093 983094983094983088983095 983092 983090983088 983094983091 983096983093983090983091 983091 983089983096 983097983097 983090983089983091983090

983094 983094983091983095 983092 983089983097 983094983088 983096983095983095983092 983091 983089983096 983097983097 983090983089983088983092

983095 983094983090 983092 983089983096 983093983095 983096983094983096983090 983091 983089983096 983097983097 983090983088983093983097

983096 983094983088983095983089 983092 983089983096 983093983095 983096983094983088983092 983091 983089983096 983097983097 983090983088983092983095

983097 983094983089983095983095 983092 983089983096 983093983095 983096983094983092983095 983091 983089983096 983097983097 983090983088983093983093

983089983088 983094983088983088983089 983092 983089983096 983093983095 983096983093983091983089 983091 983089983096 983097983097 983090983088983091983097

983089983089 983094983090983096983093 983092 983089983097 983094983088 983096983093983091983089 983091 983089983097 983097983097 983090983089983093983097

983137983138983148983141 983090 Results o NSGA-II or population size o 983093983088983088 and 983093983088983088 generations

Sr number 1038389

(mm)1103925

(mm)

907317

907317907317

10383891038389

(mm)11039251038389

(mm)

9073171038389

907317

Volume(983089983088983095

times mm983091)

Load carrying capacity (kN)

983089 983096983088983088983088 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983092983096 983091983093983091983088983094

983090 983094983088983091983097 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983093983089 983091983091983095983089983096

983091 983095983097983088983092 983092 983089983096 983093983095 983097983088983096983091 983091 983089983096 983097983097 983090983088983094983094 983091983093983096983091983095

983092 983096983088983088983088 983092 983089983096 983093983095 983097983088983097983096 983091 983089983096 983097983097 983090983088983095983088 983091983093983097983092983089

983093 983095983089983092983096 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983089983091 983091983092983097983097983094

983094 983095983092983095983095 983092 983089983096 983093983095 983096983095983096983097 983091 983089983096 983097983097 983090983088983090983095 983091983093983091983093983096

983095 983094983094983096983088 983092 983089983096 983093983095 983096983094983091983091 983091 983089983096 983097983097 983089983097983097983091 983091983092983091983096983091

983096 983094983091983092983093 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983094983091 983091983092983088983095983094

other solutions in terms o both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice or solving thiskind o problem In NSGA-II the iterative procedure startsrom an arbitrary population o solutions and gradually thealgorithm converges to a population o solutions lying onthe Pareto optimal ront with higher diversity Te operatorsapplied are the same as those o GA namely selectioncrossover and mutation Te tournament selection operatoris applied which also takes care o constraints Howeverin case o multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing unction approach is used Crossover and mutationoperators are applied as usual A detailed discussion o thisalgorithm is ound in [983090983089] Te standard code available at [983090983090]

is modi1047297ed according to authorsrsquo needAs a result o NSGA-II out o the population size o

983093983088 and number o generations o 983093983088983088 eight better resultsare selected and shown in able 983090 It has been observedthat ul1047297lling both o the objectives together the second lastsolution is the compromised one Corresponding optimum

volume and load carrying capacity o the train are 983089983097983097983091 times9830899830887 mm3 and 983091983092983091983096 kN respectively

4 Results and Discussion

Tere are several comments in order Te number o teetho gear

and gear

in the manuacturerrsquos design is 983089983092 It

creates intererence in working condition o eliminate itmanuacturer produces stub tooth instead o normal toothwhich is not advisable Te introduction o the constraintson intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de1047297nitely exceed 983089983095 Te major problem with the inbuiltldquominconrdquo unction o MALAB is that it considers all the

variables real As a result one has to round the optimum valueo integer variable to the nearest integer So the optimum

value o number o teeth o gear and gear is roundedoff to 983089983096 o maintain the gear ratios the numbers o teetho gear and gear have to be selected as 983093983094 and 983089983088983088respectively which are quite ar rom their actual optimum

values obtained using toolbox

However GA can deal with both types o variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers o teeth o gear andgear have to be changed to 983093983096 and 983089983088983088 respectively becauseo manuacturing inconveniences NSGA-II selects 983093983095 and 983097983097as the numbers o teeth o gear and gear which is betterthan both o the above results Te results are presented inable 983091

5 Conclusion and Future Scope

Result comparison table shows that in the 1047297rst two casesminimization o volume took place while load carrying

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 69

983094 Journal o Optimization

983137983138983148983141 983089 Results o GA or population size o 983097983088 and 983097983088 generations

Sr number 1038389

(mm)1103925

(mm) 907317 907317907317

10383891038389

(mm)11039251038389

(mm) 9073171038389 907317 Volume (983089983088983095 times mm983091)

983089 983094983095983088983094 983092 983089983096 983093983095 983097983088983089983096 983091 983089983096 983097983097 983090983089983094983094

983090 983094983092983088983095 983092 983089983097 983094983088 983096983093983090983097 983091 983089983096 983097983097 983090983088983096983096

983091 983094983090983088983095 983092

983089983096 983093983095 983096983094983095 983091

983089983096 983097983097 983090983088983093983096

983092 983095983093983089 983092 983089983096 983093983095 983096983093983097983089 983091 983089983096 983097983097 983090983089983089983089

983093 983094983094983088983095 983092 983090983088 983094983091 983096983093983090983091 983091 983089983096 983097983097 983090983089983091983090

983094 983094983091983095 983092 983089983097 983094983088 983096983095983095983092 983091 983089983096 983097983097 983090983089983088983092

983095 983094983090 983092 983089983096 983093983095 983096983094983096983090 983091 983089983096 983097983097 983090983088983093983097

983096 983094983088983095983089 983092 983089983096 983093983095 983096983094983088983092 983091 983089983096 983097983097 983090983088983092983095

983097 983094983089983095983095 983092 983089983096 983093983095 983096983094983092983095 983091 983089983096 983097983097 983090983088983093983093

983089983088 983094983088983088983089 983092 983089983096 983093983095 983096983093983091983089 983091 983089983096 983097983097 983090983088983091983097

983089983089 983094983090983096983093 983092 983089983097 983094983088 983096983093983091983089 983091 983089983097 983097983097 983090983089983093983097

983137983138983148983141 983090 Results o NSGA-II or population size o 983093983088983088 and 983093983088983088 generations

Sr number 1038389

(mm)1103925

(mm)

907317

907317907317

10383891038389

(mm)11039251038389

(mm)

9073171038389

907317

Volume(983089983088983095

times mm983091)

Load carrying capacity (kN)

983089 983096983088983088983088 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983092983096 983091983093983091983088983094

983090 983094983088983091983097 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983093983089 983091983091983095983089983096

983091 983095983097983088983092 983092 983089983096 983093983095 983097983088983096983091 983091 983089983096 983097983097 983090983088983094983094 983091983093983096983091983095

983092 983096983088983088983088 983092 983089983096 983093983095 983097983088983097983096 983091 983089983096 983097983097 983090983088983095983088 983091983093983097983092983089

983093 983095983089983092983096 983092 983089983096 983093983095 983096983095983094983090 983091 983089983096 983097983097 983090983088983089983091 983091983092983097983097983094

983094 983095983092983095983095 983092 983089983096 983093983095 983096983095983096983097 983091 983089983096 983097983097 983090983088983090983095 983091983093983091983093983096

983095 983094983094983096983088 983092 983089983096 983093983095 983096983094983091983091 983091 983089983096 983097983097 983089983097983097983091 983091983092983091983096983091

983096 983094983091983092983093 983092 983089983096 983093983095 983096983093983088983088 983091 983089983096 983097983097 983089983097983094983091 983091983092983088983095983094

other solutions in terms o both objectives which are calledPareto optimal solutions Since evolutionary algorithms arepopulation based they are the natural choice or solving thiskind o problem In NSGA-II the iterative procedure startsrom an arbitrary population o solutions and gradually thealgorithm converges to a population o solutions lying onthe Pareto optimal ront with higher diversity Te operatorsapplied are the same as those o GA namely selectioncrossover and mutation Te tournament selection operatoris applied which also takes care o constraints Howeverin case o multiobjective optimization additional task is toobtain solutions which are as diverse as possible For that thesharing unction approach is used Crossover and mutationoperators are applied as usual A detailed discussion o thisalgorithm is ound in [983090983089] Te standard code available at [983090983090]

is modi1047297ed according to authorsrsquo needAs a result o NSGA-II out o the population size o

983093983088 and number o generations o 983093983088983088 eight better resultsare selected and shown in able 983090 It has been observedthat ul1047297lling both o the objectives together the second lastsolution is the compromised one Corresponding optimum

volume and load carrying capacity o the train are 983089983097983097983091 times9830899830887 mm3 and 983091983092983091983096 kN respectively

4 Results and Discussion

Tere are several comments in order Te number o teetho gear

and gear

in the manuacturerrsquos design is 983089983092 It

creates intererence in working condition o eliminate itmanuacturer produces stub tooth instead o normal toothwhich is not advisable Te introduction o the constraintson intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de1047297nitely exceed 983089983095 Te major problem with the inbuiltldquominconrdquo unction o MALAB is that it considers all the

variables real As a result one has to round the optimum valueo integer variable to the nearest integer So the optimum

value o number o teeth o gear and gear is roundedoff to 983089983096 o maintain the gear ratios the numbers o teetho gear and gear have to be selected as 983093983094 and 983089983088983088respectively which are quite ar rom their actual optimum

values obtained using toolbox

However GA can deal with both types o variablesinteger and real very easily by choosing appropriate stringlength But in this case also numbers o teeth o gear andgear have to be changed to 983093983096 and 983089983088983088 respectively becauseo manuacturing inconveniences NSGA-II selects 983093983095 and 983097983097as the numbers o teeth o gear and gear which is betterthan both o the above results Te results are presented inable 983091

5 Conclusion and Future Scope

Result comparison table shows that in the 1047297rst two casesminimization o volume took place while load carrying

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 79

Journal o Optimization 983095

983137983138983148983141 983091 Comparison o results

Variables and objectives Catalog valueOptimization toolbox

value(round off)

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

Face width o gear (mm) 983095983088 983094983088 983094983088 983094983095

Module o gear

(mm) 983095 983092 983092 983092

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983092983092 983093983094 983093983096 983093983095

Face width o gear (mm) 983097983093 983096983093 983096983093 983096983094

Module o gear (mm) 983091983093 983091 983091 983091

Number o teeth o gear 983089983092 983089983096 983089983096 983089983096

Number o teeth o gear 983095983095 983089983088983088 983089983088983088 983097983097

Volume (mm983091) 983090983090983097983091 times 983089983088983095983090983088983092983088983096 times 983089983088983095 983090983088983094983088983090 times 983089983088983095 983089983097983097983091 times 983089983088983095

Load carrying capacity (N) 983091983092983088983093 times 983089983088983092983091983090983096983094983089 times 983089983088983092 983091983090983096983095983092 times 983089983088983092 983091983092983091983096983091 times 983089983088983092

capacity is reduced marginally low While using optimization

toolbox volume is reduced by 983089983093983088983092 but when their nearerinteger value o variable is selected because o inconveniencesin manuacturing volume is reduced by 983089983088983097 For the GAthe volume is reduced by 983089983089983088983093 but when their nearerinteger

value o variable is selected volume is reduced by 983089983088983089983093Tough these results show that optimization tool box givesbetter result than GA it is better to use GA or globaloptimum value as optimization toolbox which gives resultsclosest to the starting point andGA 1047297nds the more convenientsolution with high probability o manuacturing HoweverNSGA-II gives the best result compared to both o the abovemethods as it is superior in terms o both o the objectivesminimum volume and maximum load carrying capacity Forthe NSGA-II the volume is reduced by 983089983091983088983096 and loadcarrying capacity is increased by 983089

Te 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 ollowed in case o otherapplications such as minimization o weight o spring andminimization o weight o pulley system

Conflict of Interests

Te authors declare that there is no con1047298ict o interestsregarding the publication o this paper

References

[983089] H-Z Huang Z-G ian and M J Zuo ldquoMultiobjectiveoptimization o three-stage spur gear reduction units usinginteractive physical programmingrdquo Journal of Mechanical Sci-ence and echnology vol 983089983097 no 983093 pp 983089983088983096983088ndash983089983088983096983094 983090983088983088983093

[983090] D Jhalani and H Chaudhary ldquoOptimal design o gearbox orapplication in knee mounted biomechanical energy harvesterrdquoInternational Journal of Scienti1047297c amp Engineering Research vol983091no 983089983088 pp 983089983088983095983089ndash983089983088983095983093 983090983088983089983090

[983091] B S ong and D Walton ldquoTe optimisation o internal gearsrdquoInternational Journal of Machine ools and Manufacture vol 983090983095no 983092 pp 983092983097983089ndash983093983088983092 983089983097983096983095

[983092] V Savsani R V Raoand DPVakharia ldquoOptimal weight design

o a gear train using particle swarm optimization and simulatedannealing algorithmsrdquo Mechanism and Machine Teory vol 983092983093no 983091 pp 983093983091983089ndash983093983092983089 983090983088983089983088

[983093] H Wei F Lingling L Xiohuai W Zongyian and Z LeishengldquoTe structural optimization o gearbox based on sequentialquadratic programming methodrdquo in Proceedings of the 983090nd International Conference on Intelligent Computing echnology and Automation (ICICA rsquo983088983097) pp 983091983093983094ndash983091983093983097 Hunan ChinaOctober 983090983088983088983097

[983094] F Mendi Baskal K Boran and F E Boran ldquoOptimizationo module shaf diameter and rolling bearing or spur gearthrough genetic algorithmrdquo Expert Systems with Applications vol 983091983095 no 983089983090 pp 983096983088983093983096ndash983096983088983094983092 983090983088983089983088

[983095] Y K Mogal and V D Wakchaure ldquoA multi-objective opti-

mization approach or design o worm and worm wheel basedon genetic algorithmrdquo Bonfring International Journal of Man Machine Interface vol 983091 pp 983096ndash983089983090 983090983088983089983091

[983096] Yokota aguchi and M Gen ldquoA solution method oroptimal weight design problem o the gear using geneticalgorithmsrdquo Computers amp Industrial Engineering vol 983091983093 no 983091-983092 pp 983093983090983091ndash983093983090983094 983089983097983097983096

[983097] O Buiga andC-O PopaldquoOptimal mass designo a single-stagehelical gear unit with genetic algorithmsrdquo Proceedings of theRomanian Academy Series AmdashMathematics Physics echnical Sciences Information Science vol 983089983091 no 983091 pp 983090983092983091ndash983090983093983088 983090983088983089983090

[983089983088] Y Mohan and Seshaiah ldquoSpur gear optimization by using genetic algorithmrdquo International Journal of Engineering Research and Applications vol 983090 pp 983091983089983089ndash983091983089983096 983090983088983089983090

[983089983089] D F Tompson S Gupta and A Shukla ldquoradeoff analysisin minimum volume design o multi-stage spur gear reductionunitsrdquo Mechanism and Machine Teory vol 983091983093 no 983093 pp 983094983088983097ndash983094983090983095 983090983088983088983088

[983089983090] S Padmanabhan M Chandrasekaran and V SrinivasaldquoDesign optimization o worm Gear driverdquo International Jour-nal of Mining Metallurgy and Mechanical Engineering vol983089pp983093983095ndash983094983089 983090983088983089983091

[983089983091] K Deb and S Jain ldquoMulti-speed gearbox design usingmulti-objective evolutionary algorithmsrdquo Journal of Mechanical Design ransactions of the ASME vol 983089983090983093 no 983091 pp 983094983088983097ndash983094983089983097983090983088983088983091

[983089983092] V B Bhandari Design of Machine Elements ata McGraw-Hill983090983088983089983088

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 89

983096 Journal o Optimization

[983089983093] R C Juvinall and K M Marshek Fundamentals of MachineComponent Design John Wiley amp Sons 983090983088983089983089

[983089983094] G Maitra Handbook of Gear Design ata McGraw-Hill 983090ndedition 983090983088983088983091

[983089983095] Design Catalog of Hi-ech Drive Pvt Ltd Plot No 983092983092983091A GIDCV U Nagar Gujarat India

[983089983096] J H Holland Adaptation in Natural and Arti1047297cial SystemsUniversity o Michigan Press Ann Arbor Mich USA 983089983097983095983093

[983089983097] I Rechenberg Cybernetic Solution Path of an Experimental Problem Library ranslation 983089983089983090983090 Royal Aircraf Establish-ment Farnborough Hampshire UK 983089983097983094983093

[983090983088] P E Amiolemhen and A O A Ibhadode ldquoApplication o genetic algorithmsmdashdetermination o the optimal machiningparameters in the conversion o a cylindrical bar stock into acontinuous 1047297nished pro1047297lerdquo International Journal of Machineools and Manufacture vol 983092983092 no 983089983090-983089983091 pp 983089983092983088983091ndash983089983092983089983090 983090983088983088983092

[983090983089] K Deb Multi-Objective Optimization Using Evolutionary Algo-rithms John Wiley amp Sons New York NY USA 983090983088983088983097

[983090983090] httpwwwmathworksinmatlabcentral1047297leexchange983091983089983089983094983094-ngpm-a-nsga-ii-program-in-matlab-v983089-983092

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom

7212019 Multi-objective Optimization of Two-Stage Helical Gear Train using NSGA-II

httpslidepdfcomreaderfullmulti-objective-optimization-of-two-stage-helical-gear-train-using-nsga-ii 99

Submit your manuscripts at

httpwwwhindawicom