Design Optimization of Mechanical Components

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; [email protected]

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

Academic Editor: Albert Victoire

Copyright 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. M.Reddy 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 [13].

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 [46]. 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 pump,the 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[912]. Thus, optimization techniques can effectively be used

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 309327, 10 pageshttp://dx.doi.org/10.1155/2014/309327

2 Mathematical Problems in Engineering

to ensure optimal production rate.There 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 necessary,whereas 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 (e.g., 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 [1316].

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 algorithm.The same is the case with PSO,which 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, 1820]. 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-PSO.This 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 discussed.These 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 () can be minimized subjectto the constraints discussed below. Consider

=2

4(+ 2)

2. (1)

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

8max

3 0, (2)

where

=4 1

4 4+0.615

, =

. (3)

Here, maximum working load (max) and allowable shearstress () are set to be 453.6 kg and 13288.02 kgf/cm2, respec-tively.

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

=4

83, (4)

where shear modulus is equal to 808543.6 kgf/cm2.

Mathematical Problems in Engineering 3

The deflection under maximum working load is given by

=max. (5)

It is assumed that the spring length under max is 1.05times the solid length. Thus, the free length is given by theexpression

= + 1.05 (

+ 2) . (6)

Thus, the constraint is given by

max 0, (7)

where max is set equal to 35.56 cm.The wire dia must exceed the specified minimum value

and it should satisfy the following condition:

min 0, (8)

where min is equal to 0.508 cm.The outside dia of the coil must be less than themaximum

specified and it is

max ( + ) 0, (9)

wheremax is equal to 7.62 cm.The 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

3 0. (10)

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

=

, (11)

where is equal to 136.08 kg.

The constraint is given by the expression

0, (12)

where

is equal to 15.24 cm.The combined deflection must be consistent with the

length and the same can be represented as

=max

1.05 (

+ 2) 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

max

0, (14)

where is made equal to 3.175 cm.

bb

bb

di d2

d1

d2

Figure 2: Schematic representation of a hollow shaft.

During optimization, the ranges for different variables arekept as follows:

0.508 1.016,

1.270 7.620,

15 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 machines,absorbing power, are called transmission shaft. Machineshafts are those which form an integral part of the machineitself.The common example of machine shaft is a crank shaft.Figure 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

= cross sectional area length density

=

4(2

0 2

1) .

(16)

Substituting the values of , as 50 cm and 0.0083 kg/cm3,respectively, one finds the weight of the shaft (

) and it is

given by

= 0.326

2

0(1

2) . (17)

It is subjected to the following constraints.The twisting failure can be calculated from the torsion

formula as given below:

=

(18)

or

=

. (19)

Now, applied should be greater than /; that is, /.


I

A

A

di d0

Figure 3: Schematic representation of a belt-pulley drive.

Substituting the values of , , , as 2/180 perm length, 1.0 105 kg-cm, 0.84 106 kg/cm2, and[(/32)4

0(1 4)], respectively, one gets the constraints as

4

0(1

4) 1736.93 0. (20)

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

cr 30(1 )

2.5

122(1 2)0.75. (21)

Substituting the values of cr, , and to 1.0 105 kg-cm, 0.3,

and 2.0 105 kg/cm2, respectively, the constraint is expressedas

3

0(1 )

2.5 0.4793 0. (22)

The ranges of variables are mentioned as follows:

7 0 25,

0.7 0.97.(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

= [

11+ 22+ 1

11

1+ 1

21

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 2.3397870400 2.31140000Wire dia, cm 0.6700824800 0.66802000Volume of spring wire, cm3 46.6653438304 46.53926176

Assuming 1= 0.1

1, 2= 0.1

2, 11= 0.11

1, and 1

2=

0.112and replacing

1, 2, 11, and 1

2by

1, 2, 11, and

12, respectively, and also substituting the values of

1, 2,

11, and 1

2, (to 1000, 250, 500, 500) 7.2 103 kg/cm3,

respectively, the objective function can be written as

= 0.113047

2

1+ 0.0028274

2

2. (25)

It is subjected to the following constraints.The transmitted power () can be represented as

=(1 2)

75. (26)

Substituting the expression for in the above equation, onegets

= (1 2)

75 60 100, (27)

= 1(1 2

1

)

75 60 100. (28)

Assuming 2/1= 1/2, = 10 hp and substituting the values

of 2/1and , one gets

10 = 1(1 1

2)

75 60 100(29)

or

1=286478

. (30)

Assuming

22< 11,

1< .

(31)

And considering (26) to (28), one gets

2864789

22

. (32)

Substituting = 30 kg/cm2

= 1 cm,

2= 250 rpm in the

above equation, one gets

30 1.0 28864789

2250

(33)

or

381.97

2

(34)

or

2 81.97 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

0.251 (36)

or

1

4 1 0. (37)

The ranges of the variables are mentioned as follows:

15 1 25,

70 2 80,

4 10.

(38)

3. Optimization Procedure

Classical search and optimization techniques demonstrate anumber of difficulties when faced with complex problems.The 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 environment.Most 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 attention.Thesemeth-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 solution.In 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 () within the

population represents a single possible solution to a particularoptimization problem.

is a real-valued vector with

elements, where is the dimension of the problem and isused to represent the number of subjects that an individ-ual, either student or teacher, enrolls to learn/teach 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 ifhis/her new solution is better than his/her previous one. Thealgorithm will repeat until it reaches the maximum numberof generations.

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

) by moving their position

towards the position of theteacher by referring to the currentmean value of the individuals (mean). 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 teachers knowledge and the qualities of allstudents. For stochastic purposes, two randomly generatedparameters are applied within the equation: ranges between0 and 1;

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

emphasizing the importance of student quality:

new = + (teacher ( mean)) . (39)

During the Learner Phase, student () tries to improve

his/her knowledge by peer learning from an arbitrary student, where is unequal to . In the case that

is better than

, moves towards

(40). Otherwise, it is moved away

from (41). If student new performs better by following

(40) or (41), he/she will be accepted into the population.The algorithm will continue its iterations until reaching themaximum number of generations. Consider

new = + ( ) , (40)

new = + ( ) . (41)

Additionally infeasible individuals must be appropriatelyhandled, to determine whether one individual is better thananother, when applied to constrained optimization problems.For comparing two individuals, the TLBO algorithm, accord-ing to [1417], utilizesDebs 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 Operator.All 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 ZjZ11 Z12 Z13 Z14 Z15 Z16

Z11 Z15

Z21 Z22 Z23 Z24 Z25 Z26 Z21 Z23

Z31 Z32 Z33 Z34 Z35 Z36 Z31 Z32

Z41 Z42 Z43 Z44 Z45 Z46 Z41 Z46

Z51

Z11

Z21

Z31

Z41

Z51 Z52 Z53 Z54 Z55 Z56 Z51 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 astudents 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 words,each 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

= (1 2 3 ) (1 2 3 ) ,

(42)

where

1is the first element in the dimension vector

;

is the th element in the dimension vector

;

1is the first element in the dimension vector

;

is the random integer generated separately for each, from 1 to , but = .

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 size,which is the population size, is 6 during the progress of thesearch. Once the neighbor students are identified using thesensing distance, the th 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 46.5392 NA NA NAGA 46.6932 46.6653 46.6821 3.2 3PSO 46.6752 46.5212 46.6254 1.8 1.7ABS 46.6241 46.5115 46.6033 2.5 2.3TLBO 46.5214 46.3221 46.4998 2.2 2DTLBO 46.4322 46.3012 46.3192 2.4 2.2

(10) Repeat (test equal to the number of students) step 8,until 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 methods,namely, 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 0.80; mutation probability is 0.010; number ofgenerations is 3000.

Particle Swarm Optimization. Particle size is 30; max = 1.1,min = 0.7, and 1 = 2 = 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 50;number 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 6.These 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 11.0928360 10.9000Ratio of inner dia to outer dia 0.9699000 0.9685Weight of hallow shaft, kg 2.3704290 2.4017

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



Method Maximum Minimum Mean Average time (min) Minimum time (min)Conventional NA 46.5392 NA NA NAGA 46.6932 46.6653 46.6821 3.2 3PSO 46.6752 46.5212 46.6254 1.8 1.7ABS 46.6241 46.5115 46.6033 2.5 2.3TLBO 46.5214 46.3221 46.4998 2.2 2DTLBO 46.4322 46.3012 46.3192 2.4 2.2

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

2.5

3

3.5

4

4.5

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 (1), cm 20.957056 21.12

Pulley dia (2), cm 72.906562 73.25

Pulley dia (11), cm 42.370429 42.25

Pulley dia (12), cm 36.453281 36.60

Pulley width (), cm 05.239177 05.21Pulley weight, kg 104.533508 105120

Nomenclature

: Width of the pulley, cm: Ratio of mean coil dia to wire dia: Dia of spring wire, cm: Dia of any pulley, cm1: Dia of the first pulley, cm11: Dia of the third pulley, cm2: Dia of the second pulley, cm12: Dia of the fourth pulley, cm: Inner dia of hollow shaft, cm0: Outer dia of hollow shaft, cmmin: Minimum wire dia, cm: Mean coil dia of spring, cmmax: Maximum outside dia of spring, cm: Youngs modulus, kgf/cm

2



Method Maximum Minimum Mean Average time (min) Minimum time (min)Conventional NA 105.12 NA NA NAGA 104.6521 104.5335 104.5441 4.6 4.2PSO 104.4651 104.4215 104.4456 2.1 1.9ABS 104.5002 104.4119 104.4456 3.1 2.9TLBO 104.4224 104.3987 104.4222 2.9 2.8DTLBO 104.3992 104.3886 104.3912 3.3 3.1

0 5 10 15 20 25 30 35 40 45 5046

46.1

46.2

46.3

46.4

46.5

46.6

Trial runs

(cm

3)

Volu

me o

f the

sprin

g

(a) Case 1

0 5 10 15 20 25 30 35 40 45 502.35

2.355

2.36

2.365

2.37

2.375

Trial runs

Wei

ght o

f hal

low

shaft

(kg)

(b) Case 2

0 5 10 15 20 25 30 35 40 45 50104.36

104.37

104.38

104.39

104.4

104.41

104.42

104.43

104.44

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.

2: rpm of the second pulley12: rpm of the fourth pulley: Number of active coils: rpm of any pulley: Power transmitted by belt-pulley drive, hp: Any nonnegative real number: Allowable shear stress, kgf/cm2: Thickness of the belt, cm1: Thickness of the first pulley, cm11: Thickness of the third pulley, cm2: Thickness of the second pulley, cm12: Thickness of the fourth pulley, cm: Twisting moment on shaft, kgf-cmmax: Maximum working load, kgf: Preload compressive force, kgf

: Shear modulus, kgf/cm: Polar moment of inertia, cm4: Ratio of inner dia to outer dia: Spring stiffness, kgf/cm: Free length, cmmax: Maximum free length, cm: Length of shaft, cm1: rpm of the first pulley11: rpm of the third pulley: Weight of shaft, kg: Weight of pulleys, kg

: Tangential velocity of pulley, cm/s: Volume of spring wire, cm3: A random number1: Tension at the tight side, kgf


2: Tension at the slack side, kgfcr: Critical twisting moment, kgf-cm.

Greek Symbols

: Spread factor: Poissons ratio1: Cumulative probability: Perturbance factor: Deflection under preload, cmmax: Maximum perturbance factor

: Allowable maximum deflection under preload, cm: Deflection from preload to maximum load, cm1: Deflection under maximum working load, cm: Angle of twist, degree: Density of shaft material, kg/cm3: Allowable tensile stress of belt material, kg/cm3.

Conflict of Interests

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

References

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