Research Article Design Optimization of Mechanical Components … · 2019. 7. 31. · Design Optimization of Mechanical Components Using an ... to develop a new optimization technique

Research ArticleDesign Optimization of Mechanical Components Using anEnhanced Teaching-Learning Based Optimization Algorithmwith Differential Operator

B Thamaraikannan and V Thirunavukkarasu

Department of Mechanical Engineering United Institute of Technology Coimbatore 641020 India

Correspondence should be addressed to B Thamaraikannan thamaraikannanphd2011gmailcom

Received 19 March 2014 Revised 12 July 2014 Accepted 27 July 2014 Published 21 September 2014

Academic Editor Albert Victoire

Copyright copy 2014 B Thamaraikannan and V Thirunavukkarasu This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

This paper studies in detail the background and implementation of a teaching-learning based optimization (TLBO) algorithmwith differential operator for optimization task of a few mechanical components which are essential for most of the mechanicalengineering applications Like most of the other heuristic techniques TLBO is also a population-based method and uses apopulation of solutions to proceed to the global solution A differential operator is incorporated into the TLBO for effective searchof better solutions To validate the effectiveness of the proposed method three typical optimization problems are consideredin this research firstly to optimize the weight in a belt-pulley drive secondly to optimize the volume in a closed coil helicalspring and finally to optimize the weight in a hollow shaft have been demonstrated Simulation result on the optimization(mechanical components) problems reveals the ability of the proposed methodology to find better optimal solutions comparedto other optimization algorithms

1 Introduction

The problem of volume minimization of a closed coil helicalspring was solved using some traditional technique undersome constraints A graphical technique was used by Y V MReddy and B S Reddy to optimize weight of a hollow shaftafter satisfying a few constraints Moreover Reddy et al opti-mized weight of a belt-pulley drive under some constraintsusing geometric programming [1ndash3]

Majority of mechanical design includes an optimizationtask in which engineers always consider certain objectivessuch as weight wear strength deflection corrosion and vol-ume depending on the requirements However design opti-mization for a complete mechanical system leads to a cum-bersome objective function with a large number of designvariables and complex constraints [4ndash6] Hence it is a generalprocedure to apply optimization techniques for individualcomponents or intermediate assemblies rather than a com-plete assembly or system For example in a centrifugal pumpthe optimization of the impeller is computationally andmath-ematically simpler than the optimization of the complete

pump Analytical or numerical methods for calculating theextremes of a function have long been applied to engineeringcomputations Perhaps these traditional optimization proce-dures perform well in many practical cases they may fail toperform inmore complex design situations In real time opti-mization (design) problems the number of design variableswill be very large and their influence on the objective func-tion to be optimized can be very complicated (nonconvex)with a nonlinear character The objective function may havemany local optima whereas the designer is interested in theglobal optimum or a reasonable and acceptable optimum[7 8]

Optimization is a method of obtaining the best resultunder the given circumstances It plays a vital role inmachine design because themechanical components are to bedesigned in an optimal manner While designing machineelements optimization helps in a number of ways to reducematerial cost to ensure better service of components toincrease production rate and many such other parameters[9ndash12] Thus optimization techniques can effectively be used

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 309327 10 pageshttpdxdoiorg1011552014309327

2 Mathematical Problems in Engineering

to ensure optimal production rateThere are several methodsavailable in the literature of optimization Some of them aredirect search methods and others are gradient methods Indirect search method only the function value is necessarywhereas the gradient based methods require gradient infor-mation to determine the search direction However thereare some difficulties with most of the traditional methods ofoptimization and these are given below

A large body of literature is available on traditional meth-ods for solving the above problems Perhaps the traditionaltechniques have a variety of drawbacks paving the way foradvent of new and versatile methodologies to solve suchoptimization problems Such problems cannot be handledby classical methods (eg gradient methods) that onlycompute local optima So there remains a need for efficientand effective optimization methods for mechanical designproblems Continuous research is being conducted in thisfield and nature-inspired heuristic optimization methods areproving to be better than deterministic methods and thus arewidely used [13ndash16]

The most commonly used evolutionary optimizationtechnique is the genetic algorithm (GA) However GA pro-vides a near optimal solution for a complex problem havinglarge number of variables and constraints This is mainly dueto the difficulty in determining the optimum controllingparameters such as population size crossover rate andmuta-tion rate A change in the algorithm parameters changes theeffectiveness of the algorithmThe same is the case with PSOwhich uses inertia weight and social and cognitive param-eters Similarly ABC [17] requires optimum controllingparameters of number of bees (employed scout and onlook-ers) limit and so forth HS requires the harmony memoryconsideration rate the pitch adjusting rate and the numberof improvisations Therefore the efforts must be continuedto develop a new optimization technique which is free fromthe algorithm parameters that is no algorithm parametersare required for the working of the algorithm This aspect isconsidered in the present work

Recently a new optimization technique known asteaching-learning based optimization (TLBO) has beendeveloped byRao et al [5 18ndash20] It is one of the recent evolu-tionary algorithms and is based on the natural phenomenonof teaching and learning process It has already proved itssuperiority over other existing optimization techniques suchas GA ABC PSO harmony search (HS) DE and hybrid-PSOThis research also proposes a hybridmethod combiningthe teaching-learning based optimization (TLBO) and adifferential mechanism Here TLBO will be performed asa base level search procedure which makes a decision todirect the search towards the optimal region Later the exactmethod (SQP) will be used to fine-tune that region to get thefinal solution

2 Mathematical Formulation

In this section the detailed design considerations of closedcoil helical spring optimum design of hollow shaft and opti-mal design of belt-pulley drive are discussedThese problemsare adopted from [9] which uses GA as optimization tool

D

d

Free length

Figure 1 Schematic representation of a closed coil helical spring

Case 1 (Closed Coil Helical Spring) The helical spring ismade up of a wire coiled in the form of a helix which is pri-marily intended for compressive and tensile load (Figure 1)The cross-section of the wire from which the spring ismade may be circular square or rectangular Two forms ofhelical springs are used namely compression helical springand tensile spring The helical spring is said to be closelycoiled when the spring wire is coiled so close that the planecontaining each turn is nearly at right angles to the axis ofthe helix and the wire is subjected to torsion (Figure 1) Shearstress is produced in the helical spring due to twisting Theload applied is parallel to or along the axis of the spring

The optimization criterion is to minimize the volume of aclosed coil helical spring under several constraints (Figure 1)The problem may be stated mathematically as follows

The volume of the spring (119880) can be minimized subjectto the constraints discussed below Consider

119880 =1205872

4(119873119888+ 2)119863119889

2 (1)

Stress Constraint The shear stress must be less than thespecified value and can be represented as

119878 minus 8119862119891119865max

119863

1205871198893ge 0 (2)

where

119862119891

=4119862 minus 1

4119862 minus 4+

0615

119862 119862 =

119863

119889 (3)

Here maximum working load (119865max) and allowable shearstress (119878) are set to be 4536 kg and 1328802 kgfcm2 respec-tively

Configuration Constraint The free length of the spring mustbe less than the maximum specified value The spring con-stant (119870) can be determined using the following expression

119870 =1198661198894

81198731198881198633

(4)

where shear modulus 119866 is equal to 8085436 kgfcm2

Mathematical Problems in Engineering 3

The deflection under maximum working load is given by

120575119897=

119865max119870

(5)

It is assumed that the spring length under 119865max is 105times the solid length Thus the free length is given by theexpression

119897119891

= 120575119897+ 105 (119873

119888+ 2) 119889 (6)

Thus the constraint is given by

119897max minus 119897119891

ge 0 (7)

where 119897max is set equal to 3556 cmThe wire dia must exceed the specified minimum value

and it should satisfy the following condition

119889 minus 119889min ge 0 (8)

where 119889min is equal to 0508 cmThe outside dia of the coil must be less than themaximum

specified and it is

119863max minus (119863 + 119889) ge 0 (9)

where 119863max is equal to 762 cmThe mean coil dia must be at least three times the wire

dia to ensure that the spring is not tightly wound and it isrepresented as

119862 minus 3 ge 0 (10)

The deflection under preloadmust be less than themaximumspecified The deflection under preload is expressed as

120575119901

=119865119901

119870 (11)

where 119865119901is equal to 13608 kg

The constraint is given by the expression

120575119901119898

minus 120575119901

ge 0 (12)

where 120575119901119898

is equal to 1524 cmThe combined deflection must be consistent with the

length and the same can be represented as

119897119891

minus 120575119901

=119865max minus 119865

119901

119870minus 105 (119873

119888+ 2) 119889 ge 0 (13)

Truly speaking this constraint should be an equality It isintuitively clear that at convergence the constraint functionwill always be zero

The deflection from preload to maximum load must beequal to the specified value These two made an inequalityconstraint since it should always converge to zero It can berepresented as

119865max minus 119865119901

119870minus 120575119908

ge 0 (14)

where 120575119908is made equal to 3175 cm

bb

bb

di d2

d1

d9984002

Figure 2 Schematic representation of a hollow shaft

During optimization the ranges for different variables arekept as follows

0508 le 119889 le 1016

1270 le 119863 le 7620

15 le 119873119888le 25

(15)

Therefore the above-mentioned problem is a constrainedoptimization problem with a single objective function sub-jected to eight constraints

Case 2 (OptimumDesign ofHollow Shaft) A shaft is rotatingmember which transmits power from one point to another(Figure 2) It may be divided into two groups namely (i)transmission shaft and (ii) line shaft Shafts which are usedto transmit power between the source and the machinesabsorbing power are called transmission shaft Machineshafts are those which form an integral part of the machineitselfThe common example of machine shaft is a crank shaftFigure 2 shows the schematic representation of a hollow shaft

The objective of this study is to minimize the weight of ahollow shaft which is given by the expression

119882119904= cross sectional area times length times density

=120587

4(1198892

0minus 1198892

1) 119871120588

(16)

Substituting the values of 119871 120588 as 50 cm and 00083 kgcm3respectively one finds the weight of the shaft (119882

119904) and it is

given by

119882119904= 0326119889

2

0(1 minus 119896

2) (17)

It is subjected to the following constraintsThe twisting failure can be calculated from the torsion

formula as given below

119879

119869=

119866120579

119871(18)

or

120579 =119879119871

119866119869 (19)

Now 120579 applied should be greater than 119879119871119866119869 that is120579 ge 119879119871119866119869


I

A

A

di d0

Figure 3 Schematic representation of a belt-pulley drive

Substituting the values of 120579 119879 119866 119869 as 2120587180 perm length 10 times 105 kg-cm 084 times 106 kgcm2 and[(12058732)1198894

0(1 minus 1198964)] respectively one gets the constraints as

1198894

0(1 minus 119896

4) minus 173693 ge 0 (20)

The critical buckling load (119879cr) is given by the followingexpression

119879cr le12058711988930119864(1 minus 119896)

25

12radic2(1 minus 1205742)075

(21)

Substituting the values of 119879cr 120574 and 119864 to 10 times 105 kg-cm 03and 20 times 105 kgcm2 respectively the constraint is expressedas

1198893

0119864(1 minus 119896)

25minus 04793 le 0 (22)

The ranges of variables are mentioned as follows

7 le 1198890le 25

07 le 119896 le 097(23)

Case 3 (OptimumDesign of Belt-Pulley Drive) The belts areused to transmit power from one shaft to another by meansof pulleys which rotate at the same speed or at differentspeeds (Figure 3)The stepped flat belt drives are mostly usedin factories and workshops where the moderate amount ofpower is to be transmitted Generally the weight of pulleyacts on the shaft and bearing The shaft failure is mostcommon due to weight of the pulleys (Table 1) In order toprevent the shaft and bearing failure weight minimization offlat belt drive is very essential The schematic representationof a belt-pulley drive is shown in Figure 3

Objective Function The weight of the pulley is considered asobjective function which is to be minimized as

119882119901

= 120587120588119887 [11988911199051+ 11988921199052+ 1198891

11199051

1+ 1198891

21199051

2] (24)

Table 1 Comparison of the results obtained by GA with thepublished results (Case 1)

Optimal values Results obtainedby GA Published result

Coil mean dia cm 23397870400 231140000Wire dia cm 06700824800 066802000Volume of spring wire cm3 466653438304 4653926176

Assuming 1199051

= 011198891 1199052

= 011198892 11990511

= 0111988911 and 1199051

2=

0111988912and replacing 119889

1 1198892 11988911 and 1198891

2by 1198731 1198732 11987311 and

11987312 respectively and also substituting the values of 119873

1 1198732

11987311 and 1198731

2 120588 (to 1000 250 500 500) 72 times 10minus3 kgcm3

respectively the objective function can be written as

119882119901

= 01130471198892

1+ 00028274119889

2

2 (25)

It is subjected to the following constraintsThe transmitted power (119875) can be represented as

119875 =(1198791minus 1198792)

75119881 (26)

Substituting the expression for 119881 in the above equation onegets

119875 = (1198791minus 1198792)

120587119889119901119873119901

75 times 60 times 100 (27)

119875 = 1198791(1 minus

1198792

1198791

)120587119889119901119873119901

75 times 60 times 100 (28)

Assuming 11987921198791= 12 119875 = 10 hp and substituting the values

of 11987921198791and 119875 one gets

10 = 1198791(1 minus

1

2)

120587119889119901119873119901

75 times 60 times 100(29)

or

1198791=

286478

119889119901119873119901

(30)

Assuming

11988921198732lt 11988911198731

1198791lt 120590119887119887119905119887

(31)

And considering (26) to (28) one gets

120590119887119887119905119887ge

2864789

11988921198732

(32)

Substituting 120590119887

= 30 kgcm2 119905119887

= 1 cm 1198732

= 250 rpm in theabove equation one gets

30119887 times 10 ge28864789

1198892250

(33)

or

119887 ge38197

1198892

(34)

or

1198871198892minus 8197 ge 0 (35)


Assuming that width of the pulley is either less than or equalto one-fourth of the dia of the first pulley the constraint isexpressed as

119887 le 0251198891 (36)

or

1198891

4119887minus 1 ge 0 (37)

The ranges of the variables are mentioned as follows

15 le 1198891le 25

70 le 1198892le 80

4 le 119887 le 10

(38)

3 Optimization Procedure

Classical search and optimization techniques demonstrate anumber of difficulties when faced with complex problemsThe major difficulties arise when one algorithm is appliedto solve a number of different problems This is becauseclassical method is designed to solve only a particular classof problems efficiently Thus this method does not have thebreadth to solve different types of problem Moreover mostof the classical methods do not have the global perspectiveand often get converged to a locally optimal solution anotherdifficulty that exist in classical methods includes they cannotbe efficiently utilized in parallel computing environmentMost classical algorithms are serial in nature and hence moreadvantages cannot be derived with these algorithms

Over few years a number of search and optimizationtechniques drastically different in principle from classicalmethod are increasingly gettingmore attentionThesemeth-ods mimic a particular natural phenomenon to solve an opti-mization problem Genetic Algorithm Simulated Annealingare few among the nature inspired techniques

4 Teaching-Learning Based Optimization

Teaching-learning based optimization (TLBO) is an opti-mization technique developed by Ragsdell and Phillips andDavid Edward [14 15] based on teaching-learning processin a class among the teacher and the students Like othernature-inspired algorithms TLBO is also a population-basedtechnique with a predefined population size that uses thepopulation of solutions to arrive at the optimal solutionIn this method populations are the students that exist in aclass and design variables are the subjects taken up by thestudents Each candidate solution comprises design variablesresponsible for the knowledge scale of a student and theobjective function value symbolizes the knowledge of aparticular student The solution having best fitness in thepopulation (among all students) is considered as the teacher

More specifically an individual student (119883119894) within the

population represents a single possible solution to a particularoptimization problem 119883

119894is a real-valued vector with 119863

elements where 119863 is the dimension of the problem and isused to represent the number of subjects that an individ-ual either student or teacher enrolls to learnteach in theTLBO context The algorithm then tries to improve certainindividuals by changing these individuals during the Teacherand Learner Phases where an individual is only replaced ifhisher new solution is better than hisher previous one Thealgorithm will repeat until it reaches the maximum numberof generations

During the Teacher Phases the teaching role is assignedto the best individual (119883teacher) The algorithm attempts toimprove other individuals (119883

119894) by moving their position

towards the position of the119883teacher by referring to the currentmean value of the individuals (119883mean) This is constructedusing themean values for each parameter within the problemspace (dimension) and represents the qualities of all studentsfrom the current generation Equation (39) simulates howstudent improvement may be influenced by the differencebetween the teacherrsquos knowledge and the qualities of allstudents For stochastic purposes two randomly generatedparameters are applied within the equation 119903 ranges between0 and 1 119879

119865is a teaching factor which can be either 1 or 2 thus

emphasizing the importance of student quality

119883new = 119883119894+ 119903 sdot (119883teacher minus (119879

119865sdot 119883mean)) (39)

During the Learner Phase student (119883119894) tries to improve

hisher knowledge by peer learning from an arbitrary student119883119894119894 where 119894 is unequal to 119894119894 In the case that 119883

119894119894is better than

119883119894 119883119894moves towards 119883

119894119894(40) Otherwise it is moved away

from 119883119894119894(41) If student 119883new performs better by following

(40) or (41) heshe will be accepted into the populationThe algorithm will continue its iterations until reaching themaximum number of generations Consider

119883new = 119883119894+ 119903 sdot (119883

119894119894minus 119883119894) (40)

119883new = 119883119894+ 119903 sdot (119883

119894minus 119883119894119894) (41)

Additionally infeasible individuals must be appropriatelyhandled to determine whether one individual is better thananother when applied to constrained optimization problemsFor comparing two individuals the TLBO algorithm accord-ing to [14ndash17] utilizesDebrsquos constrained handlingmethod [4]

(i) If both individuals are feasible the fitter individual(with the better value of fitness function) is preferred

(ii) If one individual is feasible and the other one infeasi-ble the feasible individual is preferred

(iii) If both individuals are infeasible the individual hav-ing the smaller number of violations (this value isobtained by summing all the normalized constraintviolations) is preferred

Differential OperatorAll students can generate new positionsin the search space using the information derived fromdifferent students using best information To ensure that astudent learns fromgood exemplars and tominimize the timewasted on poor directions we allow the student to learn fromthe exemplars until the student ceases improving for a certain


Dim

ensio

n

ith individual jth individual Zi minus Zj

Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54

Figure 4 Differential operator illustrated

number of generations called the refreshing gap We observethree main differences between the DTLBO algorithm andthe original TLBO [4]

(1) Once the sensing distance is used to identify theneighboring members of each student as exemplarsto update the position this mechanism utilizes thepotentials of all students as exemplars to guide astudentrsquos new position

(2) Instead of learning from the same exemplar studentsfor all dimensions each dimension of a student ingeneral can learn from different students for differentdimensions to update its position In other wordseach dimension of a student may learn from thecorresponding dimension of different student basedon the proposed equation (42)

(3) Finding the neighbor for different dimensions toupdate a student position is done randomly (with avigil that repetitions are avoided) This improves thethorough exploration capability of the original TLBOwith large possibility to avoid premature convergencein complex optimization problems

Compared to the original TLBO DTLBO algorithm searchesmore promising regions to find the global optimum Thedifference between TLBO and DTLBO is that the differentialoperator applied accepts the basic TLBO which generatesbetter solution for each student instead of accepting all thestudents to get updated as in KH This is rather greedy Theoriginal TLBO is very efficient and powerful but highly proneto premature convergence Therefore to evade prematureconvergence and further improve the exploration ability ofthe original TLBO a differential guidance is used to tap usefulinformation in all the students to update the position of aparticular student Equation (42) expresses the differentialmechanism Consider

119885119894minus 119885119895= (1199111198941 119911

11989421199111198943

sdot sdot sdot 119911119894119899) minus (1199111205881 119911

12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where

1199111198941is the first element in the 119899 dimension vector 119885

119894

119911119894119899is the 119899th element in the 119899 dimension vector 119885

119894

1199111205881

is the first element in the 119899 dimension vector 119885119901

120588 is the random integer generated separately for each119911 from 1 to 119899 but 120588 = 119894

Figure 4 shows the differential mechanism for choosingthe neighbor student for (34) This assumes that the dimen-sion of the considered problem is 5 and the student sizewhich is the population size is 6 during the progress of thesearch Once the neighbor students are identified using thesensing distance the 119894th individual position will be updated(with all neighbor students) as shown in Figure 4 This is inan effort to avoid premature convergence and explore a largepromising region in the prior run phase to search the wholespace extensively

5 The Pseudocode of the Proposed RefinedTLBO Algorithm

The following steps enumerate the step-by-step procedure ofthe teaching-learning based optimization algorithm refinedusing the differential operator scheme

(1) Initialize the number of students (population) rangeof design variables iteration count and terminationcriterion

(2) Randomly generate the students using the designvariables

(3) Evaluate the fitness function using the generated(new) students

Teacher Phase

(4) Calculate the mean of each design variable in theproblem

(5) Identify the best solution as teacher amongst thestudents based on their fitness value Use differentialoperator scheme to fine-tune the teacher

(6) Modify all other students with reference to the meanof the teacher identified in step 4

Learner Phase

(7) Evaluate the fitness function using the modifiedstudents in step 6

(8) Randomly select any two students and compare theirfitness Modify the student whose fitness value isbetter than the other and use again the differentialoperator scheme Reject the unfit student

(9) Replace the student fitness and its correspondingdesign variable


Table 2 Best worst and mean production cost produced by the various methods for Case 1

Method Maximum Minimum Mean Average time (min) Minimum time (min)Conventional NA 465392 NA NA NAGA 466932 466653 466821 32 3PSO 466752 465212 466254 18 17ABS 466241 465115 466033 25 23TLBO 465214 463221 464998 22 2DTLBO 464322 463012 463192 24 22

(10) Repeat (test equal to the number of students) step 8until all the students participate in the test ensuringthat no two students (pair) repeat the test

(11) Ensure that the final modified students strengthequals the original strength ensuring there is noduplication of the candidates

(12) Check for termination criterion and repeat from step4

6 Results and Discussions

In this section simulation experiments for the above threeoptimization problems are done For the sake of comparisonof results of the proposed TLBO based method this researchalso adopts the four nature-inspired optimization methodsnamely GA [11] PSO [12] ABC [13] and TLBO [14]methods All the four methods are original versions withoutany modification

Parameter Settings of the Algorithms

Genetic Algorithm Population size is 100 crossoverprobability is 080 mutation probability is 0010 number ofgenerations is 3000

Particle Swarm Optimization Particle size is 30 119908max = 11119908min = 07 and 119888

1= 1198882= 2 number of generations is 3000

Artificial Bee Colony Population size is 50 employed beesare 50 onlooker bees are 50 number of generations is 3000

Teaching-Learning Based Optimization Population size is 50number of generations is 3000

For the proposed TLBO based algorithm also the sameparameter values are used as above (Tables 2 and 3) Asdetailed above these optimization methods require algo-rithm parameters that affect the performance of the algo-rithm GA requires crossover probability mutation rate andselection method PSO requires learning factors variationof weight and maximum value of velocity ABC requiresnumber of employed bees onlooker bees On the otherhand the TLBO requires only the number of individuals anditeration number (Figures 5 6 7 and 8)

A comparison is made of the results obtained by GA withthe published results and is given in Table 6These results aresummarized based on the 50 independent trial runs of eachtechnique It is observed that optimal values obtained by GA



Outer dia hallow shaft cm 110928360 109000Ratio of inner dia to outer dia 09699000 09685Weight of hallow shaft kg 23704290 24017

0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue

Figure 5 Convergence plot of the various methods for Case 1

are slightly better as compared to the published results It isimportant to highlight that the performance of a GA dependsupon its parameter selection Thus there is still a chance forfurther improvement of results although the GA parametersare selected after a careful study (Tables 4 and 5)

Figure 9 shows the 50 different trial run results of theproposed technique for all the three cases to show therobustness in producing the optimum values

7 Conclusion

In the paper three different mechanical component opti-mization problems namely weight minimization of a hollowshaft weight minimization of a belt-pulley drive and volumeminimization of a closed coil helical spring have been inves-tigated A new teaching-learning based optimization (TLBO)algorithm with differential operator is proposed to solvethe above problems and checked for different performancecriteria such as best fitness mean solution average number




0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO

Figure 6 Convergence (magnified) plot of the various methods forCase 1

0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


of function evaluations required and convergence rate Theresults show the better performance of the proposed TLBObased algorithm over other nature-inspired optimizationmethods for the design problems considered Although thisresearch focuses on three typical mechanical componentoptimization problems that too with minimum number ofconstraints this proposed method can be extended for theoptimization of other engineering design problems whichwill be considered in a future work

0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660

Pulley width (119887) cm 05239177 0521Pulley weight kg 104533508 105120

Nomenclature

119887 Width of the pulley cm119862 Ratio of mean coil dia to wire dia119889 Dia of spring wire cm119889119901 Dia of any pulley cm

1198891 Dia of the first pulley cm

11988911 Dia of the third pulley cm

1198892 Dia of the second pulley cm

11988912 Dia of the fourth pulley cm

119889119894 Inner dia of hollow shaft cm

1198890 Outer dia of hollow shaft cm

119889min Minimum wire dia cm119863 Mean coil dia of spring cm119863max Maximum outside dia of spring cm119864 Youngrsquos modulus kgfcm

2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3

Figure 9 Final cost of the optimization obtained for all test cases using DTLBO method

1198732 rpm of the second pulley

11987312 rpm of the fourth pulley

119873119888 Number of active coils

119873119901 rpm of any pulley

119875 Power transmitted by belt-pulley drive hp119902 Any nonnegative real number119878 Allowable shear stress kgfcm2119905119887 Thickness of the belt cm

1199051 Thickness of the first pulley cm

11990511 Thickness of the third pulley cm

1199052 Thickness of the second pulley cm

11990512 Thickness of the fourth pulley cm

119879 Twisting moment on shaft kgf-cm119865max Maximum working load kgf119865119901 Preload compressive force kgf

119866 Shear modulus kgfcm119869 Polar moment of inertia cm4119896 Ratio of inner dia to outer dia119870 Spring stiffness kgfcm119897119891 Free length cm

119897max Maximum free length cm119871 Length of shaft cm1198731 rpm of the first pulley

11987311 rpm of the third pulley

119882119904 Weight of shaft kg

119882119901 Weight of pulleys kg

119881 Tangential velocity of pulley cms119880 Volume of spring wire cm3119906 A random number1198791 Tension at the tight side kgf


1198792 Tension at the slack side kgf

119879cr Critical twisting moment kgf-cm

Greek Symbols

120573 Spread factor120574 Poissonrsquos ratio1205741 Cumulative probability

120575 Perturbance factor120575119901 Deflection under preload cm

120575max Maximum perturbance factor120575119901119898

Allowable maximum deflection under preload cm120575119908 Deflection from preload to maximum load cm

1205751 Deflection under maximum working load cm

120579 Angle of twist degree120588 Density of shaft material kgcm3120590 Allowable tensile stress of belt material kgcm3

Conflict of Interests

The authors do not have any conflict of interests in thisresearch work

References

[1] J N Siddall Optimal Engineering Design Principles and Appli-cation Marcel Dekker New York NY USA 1982

[2] Y V M Reddy and B S Reddy ldquoOptimum design of hollowshaft using graphical techniquesrdquo Journal of Industrial Engineer-ing vol 7 p 10 1997

[3] Y V M Reddy ldquoOptimal design of belt drive using geometricprogrammingrdquo Journal of Industrial Engineering vol 3 p 211996

[4] K Deb and B R Agarwal ldquoSimulated binary cross-over forcontinuous search spacerdquo Complex System vol 9 no 2 p 1151995

[5] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization a novelmethod for constrainedmechanicaldesign optimization problemsrdquoComputer Aided Design vol 43no 3 pp 303ndash315 2011

[6] S Rajeev and C S Krishnamoorthy ldquoDiscrete optimizationof structures using genetic algorithmsrdquo Journal of StructuralEngineering vol 118 no 5 pp 1233ndash1250 1992

[7] R V Rao and V J Savsani Mechanical Design OptimizationUsing Advanced Optimization Techniques Springer LondonUK 2012

[8] A N Haq K Sivakumar R Saravanan and VMuthiah ldquoToler-ance design optimization of machine elements using geneticalgorithmrdquo The International Journal of Advanced Manufactur-ing Technology vol 25 no 3-4 pp 385ndash391 2005

[9] A K Das and D K Pratihar ldquoOptimal design of machineelements using a genetic algorithmrdquo Journal of the Institutionof Engineers vol 83 no 3 pp 97ndash104 2002

[10] Y Peng S Wang J Zhou and S Lei ldquoStructural designnumerical simulation and control system of a machine toolfor stranded wire helical springsrdquo Journal of ManufacturingSystems vol 31 no 1 pp 34ndash41 2012

[11] S Lee M Jeong and B Kim ldquoDie shape design of tubedrawing process using FE analysis and optimization methodrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 381ndash392 2013

[12] HWang and Z Zhu ldquoOptimization of the parameters of hollowaxle for high-speed passenger car by orthogonal designrdquo Journalof Soochow University Engineering Science Edition vol 31 no 6pp 6ndash9 2011

[13] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[14] K M Ragsdell and D T Phillips ldquoOptimal design of a classof welded structures using geometric programmingrdquo Journal ofEngineering for IndustrymdashTransactions of the ASME vol 98 no3 pp 1021ndash1025 1976

[15] G David Edward Genetic Algorithms in Search Optimizationand Machine Learning vol 412 Addison-Wesley ReadingMass USA 1989

[16] R C Eberhart and Y Shi ldquoParticle swarm optimizationdevelopments applications and resourcesrdquo in Proceedings of theCongress on Evolutionary Computation vol 1 pp 81ndash86 SeoulRepublic of Korea 2001

[17] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008

[18] R V Rao V J Savsani and D P Vakharia ldquoTeaching-learning-based optimization an optimization method for continuousnon-linear large scale problemsrdquo Information Sciences vol 183no 1 pp 1ndash15 2012

[19] T Niknam R Azizipanah-Abarghooee and J Aghaei ldquoA newmodified teaching-learning algorithm for reserve constraineddynamic economic dispatchrdquo IEEE Transactions on PowerSystems vol 28 no 2 pp 749ndash763 2012

[20] P K Roy and S Bhui ldquoMulti-objective quasi-oppositionalteaching learning based optimization for economic emissionload dispatch problemrdquo International Journal of Electrical Powerand Energy Systems vol 53 pp 937ndash948 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to ensure optimal production rateThere are several methodsavailable in the literature of optimization Some of them aredirect search methods and others are gradient methods Indirect search method only the function value is necessarywhereas the gradient based methods require gradient infor-mation to determine the search direction However thereare some difficulties with most of the traditional methods ofoptimization and these are given below

A large body of literature is available on traditional meth-ods for solving the above problems Perhaps the traditionaltechniques have a variety of drawbacks paving the way foradvent of new and versatile methodologies to solve suchoptimization problems Such problems cannot be handledby classical methods (eg gradient methods) that onlycompute local optima So there remains a need for efficientand effective optimization methods for mechanical designproblems Continuous research is being conducted in thisfield and nature-inspired heuristic optimization methods areproving to be better than deterministic methods and thus arewidely used [13ndash16]

The most commonly used evolutionary optimizationtechnique is the genetic algorithm (GA) However GA pro-vides a near optimal solution for a complex problem havinglarge number of variables and constraints This is mainly dueto the difficulty in determining the optimum controllingparameters such as population size crossover rate andmuta-tion rate A change in the algorithm parameters changes theeffectiveness of the algorithmThe same is the case with PSOwhich uses inertia weight and social and cognitive param-eters Similarly ABC [17] requires optimum controllingparameters of number of bees (employed scout and onlook-ers) limit and so forth HS requires the harmony memoryconsideration rate the pitch adjusting rate and the numberof improvisations Therefore the efforts must be continuedto develop a new optimization technique which is free fromthe algorithm parameters that is no algorithm parametersare required for the working of the algorithm This aspect isconsidered in the present work

Recently a new optimization technique known asteaching-learning based optimization (TLBO) has beendeveloped byRao et al [5 18ndash20] It is one of the recent evolu-tionary algorithms and is based on the natural phenomenonof teaching and learning process It has already proved itssuperiority over other existing optimization techniques suchas GA ABC PSO harmony search (HS) DE and hybrid-PSOThis research also proposes a hybridmethod combiningthe teaching-learning based optimization (TLBO) and adifferential mechanism Here TLBO will be performed asa base level search procedure which makes a decision todirect the search towards the optimal region Later the exactmethod (SQP) will be used to fine-tune that region to get thefinal solution

2 Mathematical Formulation

In this section the detailed design considerations of closedcoil helical spring optimum design of hollow shaft and opti-mal design of belt-pulley drive are discussedThese problemsare adopted from [9] which uses GA as optimization tool

D

d

Free length

Figure 1 Schematic representation of a closed coil helical spring

Case 1 (Closed Coil Helical Spring) The helical spring ismade up of a wire coiled in the form of a helix which is pri-marily intended for compressive and tensile load (Figure 1)The cross-section of the wire from which the spring ismade may be circular square or rectangular Two forms ofhelical springs are used namely compression helical springand tensile spring The helical spring is said to be closelycoiled when the spring wire is coiled so close that the planecontaining each turn is nearly at right angles to the axis ofthe helix and the wire is subjected to torsion (Figure 1) Shearstress is produced in the helical spring due to twisting Theload applied is parallel to or along the axis of the spring

The optimization criterion is to minimize the volume of aclosed coil helical spring under several constraints (Figure 1)The problem may be stated mathematically as follows

The volume of the spring (119880) can be minimized subjectto the constraints discussed below Consider

119880 =1205872

4(119873119888+ 2)119863119889

2 (1)

Stress Constraint The shear stress must be less than thespecified value and can be represented as

119878 minus 8119862119891119865max

119863

1205871198893ge 0 (2)

where

119862119891

=4119862 minus 1

4119862 minus 4+

0615

119862 119862 =

119863

119889 (3)

Here maximum working load (119865max) and allowable shearstress (119878) are set to be 4536 kg and 1328802 kgfcm2 respec-tively

Configuration Constraint The free length of the spring mustbe less than the maximum specified value The spring con-stant (119870) can be determined using the following expression

119870 =1198661198894

81198731198881198633

(4)

where shear modulus 119866 is equal to 8085436 kgfcm2



120575119897=

119865max119870

(5)


119897119891

= 120575119897+ 105 (119873

119888+ 2) 119889 (6)


119897max minus 119897119891

ge 0 (7)



119889 minus 119889min ge 0 (8)


specified and it is

119863max minus (119863 + 119889) ge 0 (9)



119862 minus 3 ge 0 (10)


120575119901

=119865119901

119870 (11)



120575119901119898

minus 120575119901

ge 0 (12)

where 120575119901119898



119897119891

minus 120575119901

=119865max minus 119865

119901

119870minus 105 (119873

119888+ 2) 119889 ge 0 (13)



119865max minus 119865119901

119870minus 120575119908

ge 0 (14)


bb

bb

di d2

d1

d9984002



0508 le 119889 le 1016

1270 le 119863 le 7620

15 le 119873119888le 25

(15)





=120587

4(1198892

0minus 1198892

1) 119871120588

(16)


119904) and it is

given by

119882119904= 0326119889

2

0(1 minus 119896

2) (17)



119879

119869=

119866120579

119871(18)

or

120579 =119879119871

119866119869 (19)



I

A

A

di d0




1198894

0(1 minus 119896

4) minus 173693 ge 0 (20)


119879cr le12058711988930119864(1 minus 119896)

25

12radic2(1 minus 1205742)075

(21)


1198893

0119864(1 minus 119896)

25minus 04793 le 0 (22)


7 le 1198890le 25

07 le 119896 le 097(23)



119882119901

= 120587120588119887 [11988911199051+ 11988921199052+ 1198891

11199051

1+ 1198891

21199051

2] (24)




Assuming 1199051

= 011198891 1199052

= 011198892 11990511

= 0111988911 and 1199051

2=


1 1198892 11988911 and 1198891

2by 1198731 1198732 11987311 and


1 1198732

11987311 and 1198731



119882119901

= 01130471198892

1+ 00028274119889

2

2 (25)


119875 =(1198791minus 1198792)

75119881 (26)


119875 = (1198791minus 1198792)

120587119889119901119873119901

75 times 60 times 100 (27)

119875 = 1198791(1 minus

1198792

1198791

)120587119889119901119873119901

75 times 60 times 100 (28)


of 11987921198791and 119875 one gets

10 = 1198791(1 minus

1

2)

120587119889119901119873119901


or

1198791=

286478

119889119901119873119901

(30)

Assuming

11988921198732lt 11988911198731

1198791lt 120590119887119887119905119887

(31)


120590119887119887119905119887ge

2864789

11988921198732

(32)


= 30 kgcm2 119905119887

= 1 cm 1198732


30119887 times 10 ge28864789

1198892250

(33)

or

119887 ge38197

1198892

(34)

or

1198871198892minus 8197 ge 0 (35)



119887 le 0251198891 (36)

or

1198891

4119887minus 1 ge 0 (37)


15 le 1198891le 25

70 le 1198892le 80

4 le 119887 le 10

(38)
















119865sdot 119883mean)) (39)




119883119894 119883119894moves towards 119883




119883new = 119883119894+ 119903 sdot (119883

119894119894minus 119883119894) (40)

119883new = 119883119894+ 119903 sdot (119883

119894minus 119883119894119894) (41)







Dim

ensio

n


Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54







119885119894minus 119885119895= (1199111198941 119911

11989421199111198943


12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where


119894


119894

1199111205881









Teacher Phase




Learner Phase























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120575119897=

119865max119870

(5)


119897119891

= 120575119897+ 105 (119873

119888+ 2) 119889 (6)


119897max minus 119897119891

ge 0 (7)



119889 minus 119889min ge 0 (8)


specified and it is

119863max minus (119863 + 119889) ge 0 (9)



119862 minus 3 ge 0 (10)


120575119901

=119865119901

119870 (11)



120575119901119898

minus 120575119901

ge 0 (12)

where 120575119901119898



119897119891

minus 120575119901

=119865max minus 119865

119901

119870minus 105 (119873

119888+ 2) 119889 ge 0 (13)



119865max minus 119865119901

119870minus 120575119908

ge 0 (14)


bb

bb

di d2

d1

d9984002



0508 le 119889 le 1016

1270 le 119863 le 7620

15 le 119873119888le 25

(15)





=120587

4(1198892

0minus 1198892

1) 119871120588

(16)


119904) and it is

given by

119882119904= 0326119889

2

0(1 minus 119896

2) (17)



119879

119869=

119866120579

119871(18)

or

120579 =119879119871

119866119869 (19)



I

A

A

di d0




1198894

0(1 minus 119896

4) minus 173693 ge 0 (20)


119879cr le12058711988930119864(1 minus 119896)

25

12radic2(1 minus 1205742)075

(21)


1198893

0119864(1 minus 119896)

25minus 04793 le 0 (22)


7 le 1198890le 25

07 le 119896 le 097(23)



119882119901

= 120587120588119887 [11988911199051+ 11988921199052+ 1198891

11199051

1+ 1198891

21199051

2] (24)




Assuming 1199051

= 011198891 1199052

= 011198892 11990511

= 0111988911 and 1199051

2=


1 1198892 11988911 and 1198891

2by 1198731 1198732 11987311 and


1 1198732

11987311 and 1198731



119882119901

= 01130471198892

1+ 00028274119889

2

2 (25)


119875 =(1198791minus 1198792)

75119881 (26)


119875 = (1198791minus 1198792)

120587119889119901119873119901

75 times 60 times 100 (27)

119875 = 1198791(1 minus

1198792

1198791

)120587119889119901119873119901

75 times 60 times 100 (28)


of 11987921198791and 119875 one gets

10 = 1198791(1 minus

1

2)

120587119889119901119873119901


or

1198791=

286478

119889119901119873119901

(30)

Assuming

11988921198732lt 11988911198731

1198791lt 120590119887119887119905119887

(31)


120590119887119887119905119887ge

2864789

11988921198732

(32)


= 30 kgcm2 119905119887

= 1 cm 1198732


30119887 times 10 ge28864789

1198892250

(33)

or

119887 ge38197

1198892

(34)

or

1198871198892minus 8197 ge 0 (35)



119887 le 0251198891 (36)

or

1198891

4119887minus 1 ge 0 (37)


15 le 1198891le 25

70 le 1198892le 80

4 le 119887 le 10

(38)
















119865sdot 119883mean)) (39)




119883119894 119883119894moves towards 119883




119883new = 119883119894+ 119903 sdot (119883

119894119894minus 119883119894) (40)

119883new = 119883119894+ 119903 sdot (119883

119894minus 119883119894119894) (41)







Dim

ensio

n


Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54







119885119894minus 119885119895= (1199111198941 119911

11989421199111198943


12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where


119894


119894

1199111205881









Teacher Phase




Learner Phase























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










I

A

A

di d0




1198894

0(1 minus 119896

4) minus 173693 ge 0 (20)


119879cr le12058711988930119864(1 minus 119896)

25

12radic2(1 minus 1205742)075

(21)


1198893

0119864(1 minus 119896)

25minus 04793 le 0 (22)


7 le 1198890le 25

07 le 119896 le 097(23)



119882119901

= 120587120588119887 [11988911199051+ 11988921199052+ 1198891

11199051

1+ 1198891

21199051

2] (24)




Assuming 1199051

= 011198891 1199052

= 011198892 11990511

= 0111988911 and 1199051

2=


1 1198892 11988911 and 1198891

2by 1198731 1198732 11987311 and


1 1198732

11987311 and 1198731



119882119901

= 01130471198892

1+ 00028274119889

2

2 (25)


119875 =(1198791minus 1198792)

75119881 (26)


119875 = (1198791minus 1198792)

120587119889119901119873119901

75 times 60 times 100 (27)

119875 = 1198791(1 minus

1198792

1198791

)120587119889119901119873119901

75 times 60 times 100 (28)


of 11987921198791and 119875 one gets

10 = 1198791(1 minus

1

2)

120587119889119901119873119901


or

1198791=

286478

119889119901119873119901

(30)

Assuming

11988921198732lt 11988911198731

1198791lt 120590119887119887119905119887

(31)


120590119887119887119905119887ge

2864789

11988921198732

(32)


= 30 kgcm2 119905119887

= 1 cm 1198732


30119887 times 10 ge28864789

1198892250

(33)

or

119887 ge38197

1198892

(34)

or

1198871198892minus 8197 ge 0 (35)



119887 le 0251198891 (36)

or

1198891

4119887minus 1 ge 0 (37)


15 le 1198891le 25

70 le 1198892le 80

4 le 119887 le 10

(38)
















119865sdot 119883mean)) (39)




119883119894 119883119894moves towards 119883




119883new = 119883119894+ 119903 sdot (119883

119894119894minus 119883119894) (40)

119883new = 119883119894+ 119903 sdot (119883

119894minus 119883119894119894) (41)







Dim

ensio

n


Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54







119885119894minus 119885119895= (1199111198941 119911

11989421199111198943


12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where


119894


119894

1199111205881









Teacher Phase




Learner Phase























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119887 le 0251198891 (36)

or

1198891

4119887minus 1 ge 0 (37)


15 le 1198891le 25

70 le 1198892le 80

4 le 119887 le 10

(38)
















119865sdot 119883mean)) (39)




119883119894 119883119894moves towards 119883




119883new = 119883119894+ 119903 sdot (119883

119894119894minus 119883119894) (40)

119883new = 119883119894+ 119903 sdot (119883

119894minus 119883119894119894) (41)







Dim

ensio

n


Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54







119885119894minus 119885119895= (1199111198941 119911

11989421199111198943


12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where


119894


119894

1199111205881









Teacher Phase




Learner Phase























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Dim

ensio

n


Z11 Z12 Z13 Z14 Z15 Z16Z11 minus Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 minus Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 minus Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 minus Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 minus Z54







119885119894minus 119885119895= (1199111198941 119911

11989421199111198943


12058821199111205883

sdot sdot sdot 119911120588119899)

(42)

where


119894


119894

1199111205881









Teacher Phase




Learner Phase























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























0 500

100

90

80

70

60

50

401000 1500 2000 2500 3000

TLBOABC

ETLBOPSO

Iterations

Fitn

ess v

alue




7 Conclusion





0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 500 1000 1500 2000 2500 30004546474849505152535455

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO


0 500 1000 1500 2000 2500 30002

25

3

35

4

45

5

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO



0 500 1000 1500 2000 2500 3000103

104

105

106

107

108

109

110

Iterations

Fitn

ess v

alue

TLBOABC

ETLBOPSO




Pulley dia (1198891) cm 20957056 2112

Pulley dia (1198892) cm 72906562 7325

Pulley dia (11988911) cm 42370429 4225

Pulley dia (11988912) cm 36453281 3660


Nomenclature









2




0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 5 10 15 20 25 30 35 40 45 5046

461

462

463

464

465

466

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 50235

2355

236

2365

237

2375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 5010436

10437

10438

10439

1044

10441

10442

10443

10444

Trial runs

Pulle

y w

eigh

t (kg

)

(c) Case 3





















Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Greek Symbols









References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Design Optimization of Mechanical Components … · 2019. 7. 31. · Design Optimization of Mechanical Components Using an ... to develop a new optimization technique