Hindawi Publishing Corporation Journal of Optimization Volume 2014, Article ID 670297, 8 pages http://dx.doi.org/10.1155/2014/670297
Research Article Multi-Objective Optimization of Two-Stage Helical Gear Train Using NSGA-II R. C. Sanghvi, 1 A. S. Vashi, 2 H. P. Patolia, 2 and R. G. Jivani 2 �
Department of Mathematics, G. H. Patel College of Engineering and echnology, Vallabh Vidyanagar ������, India Mechanical Mecha nical Engineering Department, Department, B. V. Mahavidyalaya, Mahavidyalaya, Vallabh Vallabh Vidyanagar Vidyanagar ������, India
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Correspondence should be addressed to R . C. Sanghvi;
[email protected] [email protected] Received �� May ����; Revised �� October ����; Accepted � November ����; Published �� November ���� Academic Editor: Liwei Zhang Copyright © ���� R. C. Sanghvi et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction reproduction in any medium, provided the original work is properly cited. Gears no Gears nott onl onlyy tra transm nsmit it themotio themotion n an and d po powersatis wersatisac acto toril rilyy bu butt als also o ca can n do so wit with h uni unio orm rm mot motio ion. n. Te des designo igno gea gears rs req requir uires es an iterative approach approach to optimize the design parameters that take care o kinematics aspects as well as strength aspects. Moreover, Moreover, the choice o materials available or gears is limited. Owing to the complex combinations o the above acts, manual design o gears is complica com plicated ted and time con consumin suming. g. In this pape paper,the r,the volu volume me and load carryin carryingg cap capacit acityy areoptimized areoptimized.. Treedifferen Treedifferentt method methodolog ologies ies (i) MA MALAB LAB opt optimiza imization tion tool toolbox, box, (ii) gene genetic tic algo algorithm rithm (GA), and (iii) mult multiobje iobjective ctive opt optimiza imization tion (NSGA (NSGA-II) -II) tech techniqueare niqueare used to solve the problem. In the �rst two methods, volume is minimized in the �rst step and then the load carrying capacities o both shafs are calculated. In the third method, the problem is treated as a multiobjective problem. For the optimization purpose, ace width, module, and number o teeth are taken as design variables. Constraints are imposed on bending strength, surace atigue strength, and intererence. It is apparent rom the compariso comparison n o results that the result obtained by NSGA-II is more superior than the results obtained by other methods in terms o both objectives.
1. Introduction Designing a new product consists o several parameters and phases, which differ according to the depth o design, input data,, design stra data strategy tegy,, proce procedur dures, es, and resu results. lts. Mec Mechanic hanical al design includes an optimization process in which designers always consider certain objectives such as strength, de�ection, weight, wear, and corrosion depending on the requirements. ment s. How However ever,, design designoptimi optimizati zation on or a comp complete lete mecha mechannical assembly leads to a complicated objective unction with a large number o design variables. So it is a better practice to apply optimization techniques or individual components or intermediate assemblies than a complete assembly. For example, in an automobile power transmission system, optimization o gearbox is computationally and mathematically simple sim plerr tha than n the op optimi timiza zatio tion n o com comple plete te sys system tem.. Te preliminary design optimization o two-stage helical gear train has been a subject o considerable interest, since many high-perormance power transmission applications require high-perormance gear train.
A traditional gear design involves computations based on tooth bending strength, tooth surace durability, tooth surace atigue, intererence, efficiency, and so orth. Gear design desig n invo involves lves empirical ormulas, different different grap graphs hs and tables, which lead to a complicated design. Manual design is very difficult considering the above acts and there is a need or the computer aided design o gears. With the aid o compu com puter ter,, des design ign can be car carried ried out ite itera rativ tively ely and the des design ign variables which satisy the given conditions can be determined. Te design so obtained may not be the optimum one, because beca use in the abo above ve pr proces ocesss the des design ign va varia riable bless so obt obtain ained ed satisy only one condition at a time; or example, i module is calculated based on bending strength, the same module is substituted to calculate the surace durability durability.. It is accepted i it is wit withinthe hinthe str streng ength th limi limitt o su surac racee du durab rabili ility; ty; oth otherwi erwise se it is chan changed ged acco accordin rdingly gly.. So optim optimizat ization ion methodsare requ required ired to determine design variables which simultaneously satisy the given conditions. As the optimization problem involves the objective unction and constraints that are not stated as explicit unctions o the design variables, it is hard to solve
� it by classical optimization methods. Moreover, increasing demand or compact, efficient, and reliable gears orces the designer to use optimal design methodology. Huang et al. [�] developed interactive physical programming approach o the optimization model o three-stage spur gear reduction unit with minimum volume, maximum surace atigue lie, and maximum load-carrying capacity as design objectives and core hardness, module, ace width o gear, tooth numbers o pinion, tooth numbers o gear, and diameter o shaf as design variables. In this modeling, tooth bending atigue ailure, shaf torsional stress, ace width, intererence, and tooth number are considered as constraints. Te MALAB constrained optimization package is used to solve this nonlinear programming problem. Jhalani and Chaudhary [�] discussed the various parameters which can affect the design o the gearbox or knee mounted energy harvester device and later it rames the optimization problem o mass unction based on the dimensions o gearbox or the problem. Te problem is solved using multistart approach o MALAB global optimization toolbox and value o global optimum unction is obtained considering all the local optimum solutions o problem. ong and Walton [�] also selected center distance and volume as objectives or the internal gears. Numbers o teeth o gear and pinion and modules are considered as variables or the optimization and “belt zone search” and “hal section algorithm” are applied as optimization methods. Savsani et al. [ �] presented the application o two advanced optimization algorithms known as particle swarm optimization (PSO) and simulated annealing (SA) to �nd the optimal combination o design parameters or minimum weight o a spur gear train. Wei et al. [�] developed a mathematical model o optimization considering the basic design parameters, mainly tooth number, modulus, ace width, and helix angle o gearbox as design variables and reduction o weight or volume as an objective. Te model is illustrated by an example o the gearbox o medium-sized motor truck. Optimization tool box o MALAB and sequential quadratic programming (SQP) method were used to optimize the gearbox. Te design criterion and perormance conditions o gearbox are treated as constraints. Mendi et al. [�] studied the dimensional optimization o motion and orce transmitting components o a gearbox by GA. It is aimed at obtaining the optimal dimensions or gearbox shaf, gear, and the optimal rolling bearing to minimize the volume which can carry the system load using GA. Te results obtained by GA optimization are compared to those obtained by analytical methods. Mogal and Wakchaure [�] used GA as evolutionary techniques or optimization o worm and worm wheel. Te main objective or optimization is minimizing the volume; here other objectives are considered as constraints. Gear ratio, ace width, and pitch circle diameter o worm and worm wheel are the design variables or objectives. Constraints are center distance, de�ection o worm andbeam strength o worm gear. Yokota et al. [�] ormulated an optimalweight design problem o a gear or a constrained bending strength o gear, torsional strength o shafs, and each gear dimension as a nonlinear integer programming (NIP) problem and solved it directly
Journal o Optimization by using an improved GA. Te efficiency o the proposed method is con�rmed by showing the improvement in weight o gears and space area. Buiga and Popa [ �] presented an optimal design mass minimization problem o a single-stage helical gear unit, complete with the sizing o shafs, gearing, and housing using GAs. Mohan and Seshaiah [ ��] discussed the optimization o spur gear set orits centerdistance, weight and tooth de�ections with module, ace width, and number o teeth on pinion as decision variables subject to constraints on bending stress and contact stress. Tree materials, namely, Cast Iron, C-��, and Alloy Steel (��Ni� Cr�), are considered. Te gear parameters obtained rom GA are compared with the conventional results. Tompson et al. [��] presented a generalized optimal design ormulation with multiple objectives which is, in principle, applicable to a gear train o arbitrary complexity. Te methodology is applied to the design o two-stage and threestage spur gear reduction units, subject to identical loading conditions and other design criteria. Te approach serves to extend traditional design procedures by demonstrating the tradeoff between surace atigue lie and minimum volume using a basic multiobjective optimization procedure. Padmanabhan et al. [��] investigated that in many real-lie problems, objectives under consideration con�ict with each other, and optimizing a particular solution with respect to a single objective can result in unacceptable results with respect to the other objectives. Multiobjective ormulations are realistic models or many complex engineering optimization problems. Ant Colony Optimization was developed speci�cally or a worm gear drive problem with multiple objectives. Deb and Jain [��] demonstrated the use o a multiobjective evolutionary algorithm, namely, Nondominated Sorting Genetic Algorithm (NSGA-II), which is capable o solving the original problem involving mixed discrete and real-valued parameters and more than one objective. In this paper, two stages o helical gear train are considered. Tere are several actors, which affect the assembly as well as working condition. Tey are not generally considered in literature. Te optimization model ormulated here includes these actors in constraints. A GUI is developed which acilitates the input o various combinations o input data. Moreover, a code o GA is also developed. Te optimization is carried out using optimization toolbox o MALAB and GA and the results obtained by both o the methods are compared. Tese methods are applied to minimize the volume only. Te resulting values o the parameters are applied to �nd the maximum load carrying capacity. In true sense, the problem is solved as two single objective problems, one at a time. Moreover, NSGA-II is applied to the problem to solve it as a multiobjective problem.
2. Formulation of Problem Te optimization model o two-stage helical gear reduction unit is ormulated in this section, with minimum volume and maximum load carrying capacity as design objectives. Te schematic illustration o two-stage helical gear reduction unit isshownin Figure �. Asit isa caseo two-stage gearreduction,
Journal o Optimization
�
Reerring to be written as
D
Shaf (L s , ds )
eff o the two stages as
1 and
= 2 � � + 2 + 21 21 + 60,000√ � = 2 � � + 2 + 21 21 + 60,000√ �
1
1.5
1
2
C
1
1.5
�
�
the gear ratios between �rst pair and second pair are chosen in such a way that their values are easible and their product remains the same as that o required. �.�. Design Variables. Te mainly affected parameters o gear rom the volume point o view are ace width, module, and number o teeth o gear. Tese parameters directly or indirectly affect the objectives widely. So, the design vector is
= �� ,� ,� ,� ,� ,� ,� ,� �, � � � � � �� �
�
�
��
�
�
(�)
where , � , �,and are thenumbero teeth o gears , , , and , respectively; and � are the ace widths o gears and , respectively; � and �� are the normal modules o gears and , respectively. Here it is assumed that all gears are o the same material (say with the same Brinell hardness number) and are o the same helix angle. �.�. Objective Functions. For the optimization, �rst the volume o the two-stage helical gear train is minimized. Afer achieving the optimal value o design variables or minimum volume, those values o variables are applied to maximize the load carrying capacity o both o the stages. From both o these stages, the minimum load carrying capacity out o the two is chosen as the maximum capacity or the gear train. Te optimization model o two-stage helical gear trains is derived as ollows. Considering the dimensions o the three shafs constant, the volume o the gear train is
= 4 �� + �� + � + �� + + + � = + . 2
2
�
2
2
1
2
1
2
�
(�)
2
2
3
3
and the load carrying capacity is given as [��] eff
� , + 2 1
cos
1
2 � �
1.5
cos2
�
0.5
(�)
(�)
� , + 2 2
2 � �cos
2
cos
2
2
where 1 , 2 , 3 and 1 , 2 , 3 represent the diameters o shaf and lengths o shaf �, �, �, respectively. Te actors denote service actor and deormation actor, and respectively. is the transmitted torque and and � are sum o error between �rst meshing teeth andsecond meshing teeth, respectively. Tus the objectives can be written or minimum volume and maximum load carrying capacity as
, =� , �. min
1
max
(�)
2
�.�. Constraints. When the gear tooth is considered as a cantilever beam, the bending strength in working condition should not exceed standard endurance limit, � . From Lewis equation, the constraint on bending strength is
� ≤ , � = ( × 10 )/ = /(60 × 10 )
(�)
�
3 3 where , is diametral V , V pitch, is ace width, and is Lewis actor. However, in this work, the actors affecting bending strength during the production and assembly, such as velocity actor, overload actor, and mounting actor to name a ew, are not taken into consideration. So, afer adding the effects o these actors, the new constraints on bending strength or both o the gear pairs can be expressed [ ��] as
� �0.93 � − ≤ 0, � �0.93 � − ≤ 0,
V
� �
�
V
� �
�
�
�
ms
(�)
� ms
where is geometry actor which includes the Lewis orm actor and a stress concentration actor. , , and denote velocity or dynamic actor, overload actor, and mounting actor, respectively. � is standard R. R. Moore endurance limit. , , and denote load actor, gradient actor, and service actor, respectively. , , and ms denote temperature actor, reliability actor, and mean stress actor, respectively. Gear teeth are vulnerable to various types o surace damage. As was the case with rolling-element bearings, V
2
�
0.5
2 1 cos
1
1.5
F����� �: Schematic illustration o two-stage helical gear train.
cos2
2
2
A
can
1
B
2 urther
�
Journal o Optimization
gear teeth are subjected to Hertz contact stresses, and the lubrication is ofen elastohydrodynamic. Excessive loading and lubrication breakdown can cause various combinations o abrasion, pitting, and scoring. It will become evident that gear-tooth surace durability is a more complex matter than the capacity to withstand gear-tooth-bending atigue. Afer including all the parameters the surace atigue constraint ormula can be written [��] as
� � × 0.95 × �0.93 � − ≤ 0, � � × 0.95 × �0.93 � − ≤ 0, cos CR
Li
�
� � �
Li
V
cos CR �
�
V
F����� �: Input data through “Data Shaf.”
(�)
�
where , Li , and denote elastic coefficient actor, lie actor, and reliability actor, respectively. and � are dimensionless constants and CR and CR � are contact ratios. represents surace atigue strength. While designing the gear, intererence is the main actor to consider. Intererence usually takes place in the gear. So ormulation o the optimization problem must take care o intererence. o remove intererence, the ollowing constraints should be satis�ed (see [ ��, ��]):
− √ + ≤ 0, − √ + ≤ 0, 2 − � ≤ 0, 2 − � ≤ 0, 2 − � ≤ 0, 2 − � ≤ 0.
�
�
2
2
2 sin2
�
2 sin2
sin2
sin2
�
sin2
�
sin2
F����� �: Input data through “Data Geartrain.”
(�)
F����� �: Input data through “Data Factor.”
3. Methods of Solution Since there are many input parameters such as dimensions o shafs, gear train parameters, material properties, working condition o gear train, and actor affecting production and assembly, a GUI is prepared as shown in Figures �, �, �, and �. Te problem is solved by ollowing three ways:
F����� �: Input data through “Data Factor�.”
����� � ,
� , �,
�,
�,
and are taken as ��–��, �–��, ��–��, ��–��, ��–���, �–��, ��–��, and ��–���, respectively.
(i) using optimization toolbox o MALAB, (ii) using code developed or GA, (iii) using multiobjective optimization (NSGA-II) technique. Te ranges o the problem variables are taken as reerence rom manuacturer’s catalog [��] and these ranges or , ,
��
�.�. Using the Optimization oolbox of MALAB. In this method, �rst the volume o the gear train is minimized. Te resulting values o the parameters are used to determine the load carrying capacities o both o the shafs. Te minimum o them is considered as the maximum load carrying capacity. In this way, a multiobjective problem is reduced to a single
Journal o Optimization
�
objective problem. Te “optimtool” eature o MALAB is useul or different kinds o optimization problem. In the problem discussed here, constraints are nonlinear. So “mincon” unction o MALAB applicable or nonlinear constraint minimization is used or the optimization. Tere are different algorithms and methods available under this option in the optimization toolbox. Interior-point algorithm is chosen among them as it handles large, sparse problems, as well as small dense problems. Moreover, the algorithm satis�es bounds at all iterations and can recover rom NaN or In results. It is a large-scale algorithm widely used or this type o problems. Tis unction requires a point to start with, the choice o 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 , module o gear (and ), number o teeth o gear , and number o teeth o gear are ��, �, ��.���, ��.���, ��, �, ��.���, and ��.���, respectively. Te corresponding volume is 7 mm3 . Te result remains invariant i other starting points are chosen. For the value o load carrying capacity, the values or �rst and second stages are �.���� �� 4 N and �.���� ��4 N. So, rom these values, the load carrying capacity o the gear train is selected as �.���� ��4 N.
10 ×
×
1.948 × ×
�.�.Optimization Using Genetic Algorithm. Te same strategy used in the �rst method is also applied here to deal with a multiobjective problem. First the volume is minimized and then minimum o the resulting two load carrying capacities is chosen as the maximum load carrying capacity. Te only difference is that to minimize the volume, GA is used. As discussed in introduction, many designs are characterized by mixed continuous-discrete variables and discontinuous and nonconvex design spaces. Standard nonlinear programming techniques are not capable o solving these types o problems. Tey usually �nd relative optimum that is closest to the starting point. GA is well suited or solving such problems, and in most cases, they can �nd the global optimum solution with high probability. Actually the idea o evolutionary computing was introduced in the ����s by I. Rechenberg in his work “Evolution strategies” which was then developed by others. GAs were invented and developed by Holland [��]. Te basic ideas o analysis and design based on the concepts o biological evolution can be ound in the work o Rechenberg [��]. Philosophically, GAs are based on Darwin’s theory o survival o the �ttest and also are based on the principles o natural genetics and natural selection. Te basic elements o natural genetics-reproduction, cross-over, and mutation are used in the genetic search procedures. GA is a search algorithm based on the conjecture o natural selection and genetics. Te eatures o GA are different rom the other search techniques in several aspects as ollows: (i) the algorithm is a multipath that searches many peaks in parallel, hence reducing the possibility o local minimum trapping; (ii) GAs work with coding o the parameter set, not the parameters themselves;
(iii) GAs evaluate a population o points, not a single point; (iv) GAs use objective unction inormation, not derivations or other auxiliary knowledge, to determine the �tness o the solution; (v) GAs use probabilistic transition rules, not deterministic rules in the generation o the new population. �.�.�. Outline of Basic Genetic Algorithm. Te basic procedure o GA as outlined in [��] is as ollows:
(�) [Start] Generate random population o chromosomes (suitable solution or problem) (�) [Fitness] Evaluate the �tness some in the population
()
o each chromo-
(�) [New population] Create a new population by repeating ollowing steps until the new population is complete: (i) [Selection] Select two parent chromosomes rom a population according to their �tness (the better �tness, the bigger chance to be selected.) (ii) [Crossover] With a crossover probability, � crossover the two parents to rom two new offspring. I no crossover was perormed, offspring is the exact copy o parents. (iii) [Mutation] With a mutation probability, mutate new offspring at each locus (position in chromosome.) (iv) [Accepting] Place new offspring in the new population.
(�) [Replace] Use new generated population or a urther run o the algorithm. (�) [est] I the end condition is satis�ed, stop and return the best solution in current population. (�) [Loop] Go to Step (�). �.�.�. Implementation of Genetic Algorithm. Extensive experiments are carried out or different combinations o population size and number o generations. It is observed that the results remain consistent when the population size is �� and number o generations is ��. So, eleven good results with this population size and number o generations are shown in able � in which the ��th solution is the best. Corresponding load carrying capacities o the �rst and the second pair are ��.�� kN and ��.�� kN, respectively. So, the load carrying capacity o gear train is selected as ��.�� kN or which optimum volume is �.���� ��7 mm3 .
×
�.�. Optimization Using NSGA-II. In this case, the problem is considered as a multiobjective problem. So, both objectives are treated together. In general, in case o multiobjective optimization, the objectives are con�icting. So, a single solution cannot be accepted as the best solution. Instead, a set o solutions is obtained which are better than the
�
Journal o Optimization ���� �: Results o GA or population size o �� and �� generations.
Sr. number � � � � � � � � � �� ��
�
�
(mm)
(mm)
��.�� ��.�� ��.�� ��.� ��.�� ��.� �� ��.�� ��.�� ��.�� ��.��
� � � � � � � � � � �
�
�
� �
�
(mm)
(mm)
�� �� �� �� �� �� �� �� �� �� ��
�� �� �� �� �� �� �� �� �� �� ��
��.�� ��.�� ��.� ��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.��
� � � � � � � � � � �
�
� �
�
�
�� �� �� �� �� �� �� �� �� �� ��
�� �� �� �� �� �� �� �� �� �� ��
×
Volume (��� mm� ) �.��� �.��� �.��� �.��� �.��� �.��� �.��� �.��� �.��� �.��� �.���
���� �: Results o NSGA-II or population size o ��� and ��� generations. Sr. number � � � � � � � �
�
�
(mm)
(mm)
��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.��
� � � � � � � �
�
�
� �
�
(mm)
(mm)
�� �� �� �� �� �� �� ��
�� �� �� �� �� �� �� ��
��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.�� ��.��
� � � � � � � �
�
�
� � �
Volume (��� mm� )
�� �� �� �� �� �� �� ��
�� �� �� �� �� �� �� ��
×
Load carrying capacity (kN)
�.��� �.��� �.��� �.��� �.��� �.��� �.��� �.���
��.��� ��.��� ��.��� ��.��� ��.��� ��.��� ��.��� ��.���
other solutions in terms o both objectives which are called Pareto optimal solutions. Since evolutionary algorithms are population based, they are the natural choice or solving this kind o problem. In NSGA-II, the iterative procedure starts rom an arbitrary population o solutions and gradually the algorithm converges to a population o solutions lying on the Pareto optimal ront with higher diversity. Te operators applied are the same as those o GA, namely, selection, crossover, and mutation. Te tournament selection operator is applied which also takes care o constraints. However, in case o multiobjective optimization, additional task is to obtain solutions which are as diverse as possible. For that, the sharing unction approach is used. Crossover and mutation operators are applied as usual. A detailed discussion o this algorithm is ound in [��]. Te standard code available at [��] is modi�ed according to authors’ need. As a result o NSGA-II, out o the population size o �� and number o generations o ���, eight better results are selected and shown in able �. It has been observed that ul�lling both o the objectives together, the second last solution is the compromised one. Corresponding optimum volume and load carrying capacity o the train are �.��� ��7 mm3 and ��.�� kN, respectively.
creates intererence in working condition. o eliminate it, manuacturer produces stub tooth instead o normal tooth which is not advisable. Te introduction o the constraints on intererence in the proposed ormulation takes care o this problem as the number o teeth o gear and gear will de�nitely exceed ��. Te major problem with the inbuilt “mincon” unction o MALAB is that it considers all the variables real. As a result, one has to round the optimum value o integer variable to the nearest integer. So, the optimum value o number o teeth o gear and gear is rounded off to ��. o maintain the gear ratios, the numbers o teeth o gear and gear have to be selected as �� and ���, respectively, which are quite ar rom their actual optimum values obtained using toolbox. However, GA can deal with both types o variables, integer and real, very easily by choosing appropriate string length. But in this case also, numbers o teeth o gear and gear have to be changed to �� and ���, respectively, because o manuacturing inconveniences. NSGA-II selects �� and �� as the numbers o teeth o gear and gear which is better than both o the above results. Te results are presented in able �.
4. Results and Discussion
5. Conclusion and Future Scope
Tere are several comments in order. Te number o teeth o gear and gear in the manuacturer’s design is ��. It
Result comparison table shows that in the �rst two cases minimization o volume took place while load carrying
×
Journal o Optimization
� ���� �: Comparison o results.
Variables and objectives Face width o gear Module o gear
(mm)
(mm)
Number o teeth o gear Number o teeth o gear Face width o gear Module o gear
(mm)
(mm)
Number o teeth o gear Number o teeth o gear �
Catalog value
Volume (mm ) Load carrying capacity (N)
Optimization toolbox value (round off)
GA (round off)
NSGA-II (round off)
��
��
�
�� �
�
�
��
��
��
��
��
�� ��
��
��
��
��
� ��
�
�
��
��
��
�� �.� �� ��
× ×
���
��� �
�.��� �� �.��� ���
× ×
�
�.���� �� �.���� ���
capacity is reduced marginally low. While using optimization toolbox, volume is reduced by ��.��% but when their nearer integer value o variable is selected because o inconveniences in manuacturing, volume is reduced by ��.�%. For the GA the volume is reduced by ��.��% but when their nearerinteger value o variable is selected, volume is reduced by ��.��%. Tough these results show that optimization tool box gives better result than GA, it is better to use GA or global optimum value as optimization toolbox which gives results closest to the starting point andGA �nds the more convenient solution with high probability o manuacturing. However, NSGA-II gives the best result compared to both o the above methods as it is superior in terms o both o the objectives, minimum volume and maximum load carrying capacity. For the NSGA-II, the volume is reduced by ��.��% and load carrying capacity is increased by �%. Te problem can be extended to more than two stages. Other recently developed evolutionary algorithms such as PSO and cuckoo search can also be tried to solve this problem. Similar approach can be ollowed in case o other applications, such as minimization o weight o spring and minimization o weight o pulley system.
Conflict of Interests Te authors declare that there is no con�ict o interests regarding the publication o this paper.
References [�] H.-Z. Huang, Z.-G. ian, and M. J. Zuo, “Multiobjective optimization o three-stage spur gear reduction units using interactive physical programming,” Journal of Mechanical Science and echnology , vol. ��, no. �, pp. ����–����, ����. [�] D. Jhalani and H. Chaudhary, “Optimal design o gearbox or application in knee mounted biomechanical energy harvester,” International Journal of Scienti�c & Engineering Research, vol.�, no. ��, pp. ����–����, ����. [�] B. S. ong and D. Walton, “Te optimisation o internal gears,” International Journal of Machine ools and Manufacture, vol. ��, no. �, pp. ���–���, ����.
× ×
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�.���� �� �.���� ���
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Journal o Optimization
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