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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725717 | P a g eOptimal Gear Design By Using Box And Random SearchMethodsAnitha Santhoshi.M*, Durga Devi.G***(Asst.prof, Department of Mechanical Engg, SVCET, Etcherla)** (Asst.prof, Department of Mechanical Engg, SVCET, Etcherla)ABSTRACTThe development of evolutionary algorithmsplays a major role, in recent days, for optimaldesign of gears, so as to reduce the weight. In thisstudy an optimal weight design (OWD) problemof gear is formulated for constrained bendingstrength of gear, tortional strength of shafts andeach gear dimension as a NIP problem and solvedit directly by keeping nonlinear constraint usingBox and Random search methods, such that thenumber of decision {design} variables does notincrease and easily get the best compromisedsolution. An extensive computer program in Javahas been written exclusively for their purposeand is successfully used to obtain the optimal geardesign.Keywords – optimal weight design (OWD), NIPproblem, Box Method, Random search method,Decision {design} variables.1. INTRODUCTIONThe most important problem that confrontspractical engineers is the mechanical design, a fieldof creativity. In case of gear design, an infinitenumber of possible design solutions are found withinthe overall objective. Any one of these solutions isadequate because it represents a synthesis, whichmerely satisfies the functional requirements [1].Here lies a conductive environment for applying cutand try technique to obtain an optimal designsolution among the available solutions. Theapproach to solve certain design problem has sorelied on the trail-and-cut methods which, because oftheir methodology, take considerable time to obtainthe optimal solution.In this study an optimal weight design(OWD) problem of gear is formulated forconstrained bending strength of gear, tortionalstrength of shafts and each gear dimension as a NIPproblem and solves it directly by keeping nonlinearconstraint by using Box and Random SearchMethods.As a result, the number of decision (design)variables does not increase and easily get the bestcompromised solution. An extensive computerprogram in Java has been written exclusively fortheir purpose and is successfully used to obtain theoptimal gear design.2. ENGINEERING OPTIMIZATIONOptimization is the act of obtaining the bestresult under given circumstances. In design,construction and maintenance of any engineeringsystem, engineers have to take many technologicaland managerial decisions at several stages. Theultimate goal of all such decisions is either tominimize the effort required or to maximize thedesired benefit.2.1 Optimization Algorithms2.1.1 Single variable optimization algorithmsThese algorithms provide a goodunderstanding of the properties of the minimum andmaximum points in a function and how optimizationalgorithms work iteratively to find the optimumpoint in a problem. The algorithms are classified intotwo categories, they are direct methods and gradientbased methods. Direct methods do not use anyderivative information of the objective function:only objective function values are used to guide thesearch process. However, gradient based methodsuse derivative information (first and/ or secondorder) to guide search process.2.1.2 Multi – variable optimization algorithmsA number of algorithms for unconstrained,multi-variable optimization problems are present.These algorithms demonstrate how this search foroptimum points progress in multiple dimensions.2.1.3 Constrained optimization algorithmsConstrained optimization algorithms usedin single variable and multi variable optimizationalgorithms repeatedly and simultaneouslymaintained the search effort inside the feasiblesearch region. These algorithms are mostly used inengineering optimization problems. Thesealgorithms are divided into two broad categories;they are direct search methods and gradient-basedmethods. In constraint optimization problem,equality constraints make the search process slowand difficult to converge.
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725718 | P a g e2.1.4 Specialized optimization algorithmsThere exist a number of structuredalgorithms, which are ideal for only a certain class ofoptimization problems. Two of these algorithms areinteger programming and geometric programming.These are often used in engineering designproblems. Integer programming methods can solveoptimization problems with integer design variables.Geometric programming methods solve optimizationproblems with objective functions and constraintswritten in a special form.3. BOX METHODThe Box method is similar to the simplexmethod of unconstrained search except that theconstraints are handled in the former method. Thismethod was developed by M.J.Box in 1965; [2], thealgorithm begins with a number of feasible pointscreated at random. If a point is found to beinfeasible, a new point is created using thepreviously – generated feasible points. Usually, theinfeasible point is pushed towards the centroid of thepreviously found feasible points. Once a set offeasible points is found, the worst point is reflectedabout the centroid of rest of the points to find a newpoint, Depending on the feasibility and functionvalue of the new point, the point is further modifiedor accepted. If the new point falls outside thevariable boundaries, the point is modified to fall onthe violated boundary. If the new point is infeasible,the point is retracted to towards the feasible points.The worst point in the simplex is replaced by thisnew feasible point and the algorithm continues forthe next iteration. The Box Method is also called as“Complex Search Method”.3.1 BOX [Complex Search] Algorithm [2]Step 1: Assume a bound in x (x (L), x (U)), areflection parameter α.Step 2: Generate an initial set of P (usually 2n)feasible points. For each point(a) Sample n times to determine the point)( pix in the given bound.(b) If x (p)is infeasible, calculate x(centroid) of current set of points and reset)(21 )()()( pppxxxx Until)( px is feasible;Else if)( px is feasible, continue with (a)until P points are created(c) Evaluate )( )( pxf for p = 0, 1, 2…, (P-1)Step 3: Carry out the reflection step:(a) Select xRsuch thatf (xR) = max f(x (p)) = Fmax(b) Calculate the centroi x d (of pointsexcept xR) and the next point)( Rmxxxx  (c) If xmis feasible and f (xm) > Fmaxretract half the distance to thecentroid x . Continue until f(xm) < FmaxElse if xmis feasible and f (xm) < Fmax,go to Step 5.Else if xmis infeasible, go to Step 4.Step 4: Check for feasibility of the solution(a) For all i, reset violated variablebounds:If)()( LimiLimi xxsetxx If)()( UimiUimi xxsetxx (b) If the resulting xmis infeasible, retracthalf the distance to the centroid.Continue until xmis feasible. Go toStep 3(c).Step 5: Replace xRby xm. Check for termination.Calculate ppxfPf )(1 )(and x =ppxP)(1
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725719 | P a g eFig.3.1 Flow chart of Box Methodq=q+1constraintsatisfactionNoinfeasible feasibleinfeasible feasibleYesSet xl= (20,10,30,18), xr= (12,20,10,7)n=8,p=1q=1,P=8,Q=500,α =1.3StartPrint lowest of f (xp) = w and related xpCalculate Xi(P)= Xil+ riXirSelect xR such that f (xR) and f (xp) =Xm = + α ( -xr) x is centriod (expect xR)xp= xp+1/2( - xp ) Save xp, f (xp) =w. setp= p+1 until P is createdTerminateIf q>QCalculate f (xm)If xm < Xilthen xm =Xiland xm < Xiu thenxm = XiuRetract half distance to thecentriodReplace xR by xm calculate f bar,IF xmf (xm) ≥ Fmax retracthalf of distance tocentriod.until f (xm) <Fmaxf (xm) < FmaxStopGenerate the random numbersri=(i=1, 2,…8) b/w limit 0 to 1
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725720 | P a g e4. RANDOM SEARCH METHODLike the Box method, the random searchmethod also works with a population of points. Butinstead of replacing the next point in the populationby a point created in a structured manner points arecreated either at random or by performing aunidirectional search along the random searchdirections. Here, we describe one such method.Since there is no specific search direction used in themethod, random search methods work equallyefficiently to many problems. In the Luus andjaakola method 1973; an initial point and an initialinterval are chosen at random. Depending on thefunction values at a number of random points in theinterval, the search interval is reduced at everyiteration by a constant factor. Then it is increased. Inthe following algorithm, P points are considered ateach iteration and Q such iterations are performed.Thus, if the initial interval in one variable is do andat every iteration the interval is reduced by a factor€, the final accuracy in the solution in that variablebecomes (1-€) ^Q do and the required number offunction evaluations is P X Q.4.1. Random Search Algorithm [2]Step 1 Given an initial feasible point x0,an initial range zosuch that the minimum, x*, lies in)21,21( 0000zxzx  Choose the Parameter0< 1 For each of Q blocks, initially set q = 1 &p = 1.Step 2 For i = 1, 2 …N, create pointsusing a uniform distribution of r in the range (-0.5,0.5). Set11)(  qiqipi rzxxStep 3 If)( px is infeasible and p < P, repeatStep-2. If x(p)is feasible, save x(p)and f(x(p)),increment p and repeat Step-2;Else if p = P, set xqto be the point that has thelowest f(x(p)), overall feasible x(p)including xq-1andreset p = 1.Step 4 Reduce the rangevia1)1(  qiqi zz .Step 5 If q > Q, Terminate;Else increment q and continue with Step-2.The suggested values of parameters are €=0.05, P= 5(depending upon the design variables),and Q is related to the desired accuracy in thesolution. It is to be noted that the obtained solution isnot guaranteed to be the true optimum.5. PROBLEM DESCRIPTIONIn the present work an OWD of a gear witha minimum weight is considered in fig above. Inputpower of 7.5 KW, the speed of crank shaft gear(pinion) is considered to be 1500 rpm and the gearratio is 4. Necessary conditions required fordeveloping a mathematical model for gear design arediscussed in this section as given in [4].Preliminary Gear considerations: The followingare input parameters required for preliminary geardesign [6].1. Power to be transmitted (H), KW.2. Speed of the pinion (N1), rpm.3. Gear ratio (a)
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725721 | P a g eFig4.1 Flow chart of Random Search MethodSet x0= (20,10,30,18), z0= (12,20,10,7)n=5,p=1q=1,P=5,Q=500,ε =0.05Generate the random numbersri=(i=1,2,3,4) b/w limit -0.5 to +0.5uniformly distributedTerminateq=q+1If q>QXi(P)= Xiq-1+ ri Ziq-1If p=P take lowest value of f (xp) and xp reset p=1Ziq= (1-ε) Ziq-1p<P Save xp, f (xp) =w.set p=p+1.until P iscreatedConstraintsatisfactionNo Yesinfeasible feasiblePrint lowest of f (xp) = w and relate xpStopStart
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725722 | P a g eThe objective function which is used to minimizethe weight of gear considered can be expressed asmin bddbwndbdDazbmW pwoi )()1()()1(1000422212222212The decision variables in this respect areface width (b), diameter of pinion ( ), diameter ofgear ( ), number of teeth on pinion ( ).Following are the constraints of the objectivefunction. Satisfied bending strength of spur gear,tortional strength of shafts and each gear dimensionwith a minimized gear weight F ( b, , , , m ).It is the following NIP problem:min. F (b, , , , m) =W (1)subjected to. (b, m) = ≥ (2)(b, m, ) = / ≥ (3)( ) = ≥ (4)( ) = ≥(5)(∝, ) = (1+a) m /2 ≤ (6)= m (a -2.5), = 2.5m, = -2 ,= 3.5m, = +25, = 0.25( - ),= π σbmy, = b (2 / + ),= m , = am , = /a, =/ , ν = π /60, =102H/ν,= 4.97× H/ ×γ, = 4.97× H/×γ.Table 5.1: The range of Variable and coefficientvaluesWhere F (b, , , ) is the weightfunction, b and / are face width and diameterof pinion/gear shaft, respectively. and m arenumber of teeth in pinion and module, respectively,W is weight of gears. σ is allowable stress of gear„a‟ is gear ratio. is bending strength of teeth ( :Lewis formula, pinion). is surface durability(k). is wear load.(i=1 ……5) are constraint quantity each other.is dedendum circle, , and l are insidediameter of rim, out diameter of boss and length ofboss. , and n are thickness of web, drill holldiameter and number of . ρ is density of gear./ is velocity/load factor. y is form factor./ is cubic diameter of pinion/gear shaft,respectively. is distance between the axes©.( , , ) and ( , , ) are lower/upperlimit values of the design variable, respectively./ and / are pitch diameter ofpinion/gear and speed of pinion/gear. is numberof teeth on gear. ν, H and γ are pitch line velocity,input power and allowable shearing stress of shafts,respectively.5.1. Application of Box [Complex Search]MethodHere first created random numbersdepending upon the design variables (P=2n). „n‟ isthe number of design variables. Here 8 designvariables are r1, r2.....r8.The random numbers arecreated between 0 to 1, after that determined)( pixin the given bound. Initial point must be feasibleand then calculated)( pix . It must satisfy allconstraints. If it is infeasible and then calculatecentroid of current set of points andreset )(21 )()()( pppxxxx  Until)( px isfeasible; Calculate “W” for P points, and takeminimum of “W” of these points. Take themaximum” value of above set of points it is markedas , and then calculate )( Rmxxxx   .xRmeans the worst points related to the maximum“W” value. Keep this xmin “W”, it should be lessthan maximum “W”, if it is greater than “W”retract the half distance to the centroid until it isless than “W”. If xmfeasible calculate “W” relatedto “x=[b,d1,d2,z1]”.In case if xmis infeasible checkfor feasibility of the solution if the design variablesare out of the boundary set for with in theboundary limits. If the resulting xmis alsoinfeasible retract half the distance to the centroid.Continue until xmif feasible and keep this xmin“W” and take least “W” value, this completes oneiteration. Else set k=k+1 by doing 500 iterationstake least “W” of this iterations related to “x”value.20≤ b ≤32 10≤ ≤3030≤ ≤40 m = 2.75,3,3.518≤ ≤25 a =4H=7.5 = 1500σ =30 γ =2= 0.193 = 0.389= 0.8 y = 0.102ρ =8 n =6
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725723 | P a g e5.2. Application of Random Search Method[RSM]Here first create random numbersdepending upon the decision variables .Here 4design variables so random numbers (r1, r2, r3,r4).The random numbers are created between(-0.5to+0.5)limit. It is uniformly distributed. Afterthat we have to find11)(  qiqipi rzxx.where(i=1, 2…4).If)( px is infeasible and p < P, repeatfinding11)(  qiqipi rzxx.If x (p)is feasible,save x (p)and f(x (p)), increment p and repeat sameprocedure until p=P. Take minimum of value of„W‟ (i.e. f(x (p)) and reset p=1. Reduce the rangevia1)1(  qiqi zz .Repeat the procedure untilq>Q. Terminate; Else increment q and continue theprocedure and calculate „W‟ value by doing 500iterations we will take least value of „W‟ andcorresponding x = [b, , , ].6. DISCUSSION ON THE RESULTSHere the main objective is to minimize theweight. For that Box and Random Search Methodsare used. The gear module (m) values consideredare 2.75mm; 3mm and 3.5mm.These have beencompared with those of literature and incorporatedin tables 7.1 to 7.3. By observing the tabulatedvalues it is found that Random search method givesbetter results than the Box method.Box method and Random search method[RSM] are applied to the OWD problem of thegear. The results obtained by both the methods arecompared with that available in literature [4].Among the three methods the Random searchmethod is found to be giving good results for theproblem considered can be effectively applied forsingle stage gear design problem. From the tables7.1 to 7.3 even though the results of [4] giveminimum values the variables are violated theconstraints. Therefore the solutions presented arenot feasible solutions. Hence RSM is found to bebest method.7. COMPARISON OF RESULTSFor Module m=2.75** indicates constrained violationTable: 7.1For Module m=3Thicknessof web:9.625 9.63 9.63Outsidediameter ofboss:65 64.99 55Drill holls:36.09 28.10 30.6BY BOXMETHODBYRANDOMSEARCHMETHODBYLITERATUREWEIGHT7560.98 7077.23 3512.6Face width: b 26.69 23.94 24Diameter ofpinion shaft: 30.0 29.88 30Diameter ofgear shaft: 40.0 39.99 30**Number ofteeth(pinion):20.91=(21) 18 18Number ofteeth(gear):83.64=(84) 72 72Module: m 2.75 2.75 2.75Pitchcircle(pinion): 57.50 49.5 49.5Pitchcircle(gear):230.01 198 198Between theaxes: C 143.75 123.75 123.75Surfacedurability: k 0.287 0.3747 0.374Dedendumcircle(gear): 223.135 191.1 191.1Insidediameter ofrim:209.385 177.4 177.4
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725724 | P a g ePitchcircle(gear):230.01 198 198Between theaxes: C143.75 123.75 123.75Surfacedurability: k0.287 0.3747 0.374Dedendumcircle(gear):223.135 191.1 191.1Inside diameterof rim:209.385 177.4 177.4Thickness ofweb:9.625 9.63 9.63Outside diameterof boss:65 64.99 55Drill holls: 36.09 28.10 30.6** indicates constrained violationTable: 7.2For Module m=3.5BY BOXMETHODBYRANDOMSEARCHMETHODWEIGHT 10033.21 7111.95Face width: b 25.82 23.94Diameter ofpinion shaft: 30.0 26.55Diameter ofgear shaft: 40.0 39.52Number ofteeth(pinion):21.98=(22) 18Number ofteeth(gear):87.92=(88) 72Module: m 3.5 3.5Pitchcircle(pinion): 76.93 63Pitchcircle(gear):307.72 252Between theaxes: C192.325 157.5Surfacedurability: k0.273 0.333Dedendumcircle(gear):298.97 243.25Inside diameterof rim:281.47 225.75Thickness ofweb:12.25 63Outside diameterof boss:65 64.52Drill holls: 54.11 40.30Table: 7.38. CONCLUSIONS AND SUGGESTIONSFOR FURTHER WORKThe gear is one of the machine elements.It transmits power with accuracy to parallel shafts,skew shafts and intermittent action gear etc.Therefore it has various uses in industrialproduction. When designing a gear usually the trailand cut methods are used to determine factors suchas input power, rotation frequency, transmissionratio, bending strength of the gear, tortionalstrength of shafts and each gear dimension.However, this method does not include the methodof optimal weight design [4]. The mathematicalmodel of an optimal weight design problem of gearfor minimizing objective functions includes theabove mentioned design factors.Box method and Random search method[RSM] are applied to the OWD problem of thegear. Example taken in this study is a spur gear.The results obtained by both the methods arecompared with that available in literature [4].Among the three methods the Random searchmethod is found to be giving good results for theproblem considered can be effectively applied forsingle stage gear design problem. From the tables7.1 to 7.3 even though the results of [4] giveminimum values the variables are violated theconstraints. Therefore the solutions presented areBY BOXMETHODBYRANDOMSEARCHMETHODBYLITERATUREWEIGHT 7560.98 7077.23 3512.6Face width: b 26.69 23.94 24Diameter ofpinion shaft:30.0 29.88 30Diameter of gearshaft:40.0 39.99 30**Number ofteeth(pinion):20.91=(21) 18 18Number ofteeth(gear):83.64=(84) 72 72Module: m 2.75 2.75 2.75Pitchcircle(pinion): 57.50 49.5 49.5
    • Anitha Santhoshi.M, Durga Devi.G / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.717-725725 | P a g enot feasible solutions. Hence RSM is found to bebest method. As a result the minimum weight ofthe gear considered using RSM is 7077.23.This study can be extended using othermethods like cutting plane method and feasibledirection method to get faster and better values.And also the BOX and RSM Algorithms can beapplied for designing optimization of themechanical elements.BIBLOGRAPHY1. Jhonson, C.R., 1961, “Optimal Design ofMechanical Elements”, John Wiley andSons Inc, New York2. Deb, K., 1996, “Optimization forEngineering Design”, Prentice Hall ofIndia, New Delhi.3. Rao, S.S., 1984, “Optimization theory andApplications”, Wiley Eastern, New Delhi.4. TakaoYokota, Takeaki Taguchi, andMitsuo Gen, 1998, “A Solution Methodfor optimal Weight Design Problem ofGear Using Genetic Algorithm”,Computer and Industrial EngineeringVol.35 (3-4), pp.523-526.5. Department of Mechanical Engineering,PSG College of Technology, Coimbatore641004, 1983, “Design Data”.6. Dudley, D.W., 1962, “Gear Hand Book”,The Design, Manufacture, and Applicationof Gears”, Mc Graw Hill Book Co, NewYork.