Optimization of Gear Pairs Using Genetic Algorithm

Transcription

1 IOSR Joural of Egieerig (IOSRJEN) ISSN: ISBN: PP Natioal Symposium o egieerig ad Research Optimizatio of Gear Pairs Usig Geetic Algorithm 1 Y.K.Mogal, 2 D.D.Palade, 3 V.D.Wakchaure Departmet of Mechaical Egieerig, MCERC,Nasik Departmet of Mechaical Egieerig, AVCOE,Sagamer Abstract: - Up to ow may optimizatio techiques have bee developed & used for optimizatio of egieerig problems to fid optimum desig. Solvig egieerig problems ca be complex & time cosumig whe there are large umber of desig variables & costraits. A Gear desig require the desiger to compromise may desig variables; i.e. cotiuous, discrete & iteger variables i order to determie best performace of gear set. Therefore a covetioal optimizatio techique has difficulty i solvig those kids of problem. I this paper Geetic algorithm is itroduced for, 1] Miimizatio of power loss of worm gear mechaism with respect to specified set of costraits. 2] Miimizatio of volume of two-stage gear trai. Key words: - Optimizatio, Geetic Algorithm ad Gear 1. INTRODUCTION Desig optimizatio for may egieerig applicatios is multiobjective i ature. Due to coflicts amog objectives, it is impossible to obtai a sigle desig that correspods to optima of all the objectives. A multiobjective optimizatio problem ca be solved usig sigle-objective optimizatio methods, if all the objectives ca be combied ito a sigle objective fuctio by assigig each with a weight coefficiet or if all but oe objective ca be coverted ito costraits [1]. Solvig egieerig problems ca be complex ad a time cosumig process whe there are large umbers of desig variables ad costraits. Hece, there is a eed for more efficiet ad reliable algorithms that solve such problems. The improvemet of faster computer has give chace for more robust ad efficiet optimizatio methods. Geetic algorithm is oe of these methods.[3] Origially developed by Hollad, a geetic algorithm (GA) is a robust techique, based o the atural selectio ad geetic productio mechaism. The geetic algorithm works with a group of possible solutios withi a search space istead of workig with a sigle solutio as is see i gradiet optimizatio methods. [2]. The flow chart of geetic algorithm is show i Fig.1. A iitial populatio is chose radomly at the begiig, ad fitess of iitial populatio idividuals are evaluated. The a iterative process starts util the termiatio criteria have bee ru across. After the evaluatio of idividual fitess i the populatio, the geetic operators, selectio, crossover ad mutatio are applied to breed a ew geeratio. The ewly created idividuals replace the existig geeratio ad reevaluatio is started for fitess of ew idividuals. The loop is repeated util acceptable solutio is foud.[3] Fig.1.Flow chart for GA The desig of gear trai is a kid of mixed problems which have to determie various types of desig variables; i,e.,cotiuous, discrete, ad iteger variables.. I this study, the Geetic Algorithm is itroduced for geometrical volume (size) miimizatio problem of the two-stage gear trai ad the simple plaetary gear trai to show that geetic algorithm is better tha the covetioal algorithms for solvig the problems that have [4] Fiite elemet aalysis (FEA) has proved to be a useful method to aalyze the stregth of worm gears; however, it is usually time cosumig ad difficult for parametric desig, ad, hece, it is difficult to apply FEA for desig optimizatio. To overcome the problem, the authors developed a method, which combies ANN ad GA to optimize worm gear desig based o FEA results. [6] The reductio of vibratio ad oise is owadays a very importat issue i the desig of gear power trasmissios. May authors uderlied 22 P a g e

2 the effect of modificatios o the dyamics of gears: Kahrama ad Blakeship experimetally measured the effect of the ivolutes tip relief o the dyamic respose, while Boori et al. simulated the effect of differet profile modificatios ad maufacturig errors o gear oise. A key poit o the desig process of spur gears is the choice of the best profile modificatio; it is ot trivial difficult to defie desig guidelies for profile modificatios.[7] 2. APPLICATION OF GA FOR POWER LOSS OPTIMIZATION I this study, a o-covetioal algorithm amely geetic algorithm is preseted for miimizatio of power-loss of worm gear mechaism with respect to specified set of costraits. Number of gear tooth, frictio coefficiet, ad helix (thread) agle of worm are used as desig variables ad liear pressure, bedig stregth of tooth, ad deformatio of worm are set as costraits. The results for miimizatio of power-loss of worm gear mechaism are preseted to provide a compariso with aalytical method I the desig of gear set, the shafts agle is 90 o. Trasmissio rate is 1/15; ad three Thread worm. I the worm gear system because of frictio, missig power is tured to heat. The objective fuctio is the power loss with respect to costraits such as liear pressure worm gear tooth, bedig stress of gear tooth for acceptable deflectio, acceptable deflectio of worm shaft.[3] Power loss, ΔP, formulated as: ΔP = P i P o P i : iput power, P o : output power P o =F (cos α cos γ -µ si γ ) * F : F t1 : F : Normal force α :Pressure agle γ :Helix agle µ: Frictio coefficiet m a : Module z g : Number of gear tooth w w : Agular velocity of worm i: Trasmissio rate. 2.1 Objective Fuctio It is desired to obtai the lowest power loss of worm gear mechaism subject to liear Pressure worm gear, bedig stress of gear tooth ad deflectio of worm shaft uder the Load. The objective fuctio is formulated as: F obj =F(z g, γ, µ)=p i - F (cos α cos γ -µ si γ ) * 2.2 Desig Variables ad Parameters Gear tooth umber 21 Z g 80 Frictio coefficiet 0.03 µ 0.05 Helix agle 15 0 γ 25 0 Table 1: Coefficiets ad iput values for sample desig practice Defiitio Symbol Uit Values Iput power P o KW 11 Number of Rpm 720 revolutio tour of worm Trasmissio i rate Ceter distace a mm 200 of worm gear pair Distace L mm 330 betwee of worm shaft bearigs Module m a mm 7 Number of z w - 3 worm teeth Worm diameter do w mm 71 Bottom of teeth df w mm 55 diameter of worm Pressure agle α Degree 22.5 Elasticity E N/mm module Iertia I mm Costraits Costraits are coditios that must be met i the optimum desig ad iclude restrictios o the desig variables. These costraits defie the boudaries of the feasible ad ifeasible desig space domai. The costraits cosidered for the optimum desig of the power loss of worm gear are the followig g j =(Z g, γ, µ) 0 j= 1,. No. of costraits g 1 (x)= * F t1 : Tagetial force : Number of revolutio of worm do w : worm diameter g 2 (x)= P a g e

3 Number of Radomized biary strig Idividual g 3 (x)= 0 g 1 x: Liear pressure worm gear tooth g 2 x: Bedig stress of gear tooth g 3 x :Acceptable deflectio of worm shaft F t2 = F (cos α cos γ -µ si γ ) bo g =0.45(do 1 +6m a ) F tr1 : F r1 : Radial force represeted with Equatio F r1 : F si α Fittess Fuctio =F-[F(Z g, γ, µ)+pf] PF= 2 Cocaated variables head-to-tail *l Populatio Size =165.2 For a strig legth of 27bits, a optimal populatio size of 85 maybe used. Table 4. A set of startig populatio The settig parameters of geetic algorithm for this study are chose as follows: Chromosome legth = 27, Populatio size = 85, Number of geeratio = 100, Crossover Probability = 0.5, Mutatio probability = 0.005, 2.4 Results & Discussio Fig 2 shows the 3-D plots of desig variables values durig the workig of GA. Desig variables have bee got differet values ad take o miimum value of objective fuctio at 69-th geeratio show i Fig 2. It has bee show that desig variables take o values as: umber of gear teeth is 44, frictio coefficiet is , ad helix agle is 5.246o.Fig 2 shows the plots of the helix agle ad umber of worm gear teeth i each geeratio as optimizatio proceed. The overall results show that the best desig coverge 69-th geeratio ad refie the desig over remaiig geeratios,. The results compared with results of aalytical method, show i Table 5. Table 2. Codig of biary desig variable vectors ito biary digits Desig variables Lower limit Upper limit Precisio Strig legth vector Z g γ µ To start the algorithm, a iitial populatio set is radomly assiged. This set of iitialized populatio is potetial solutio to the problem. The biary strig represetatio for the desig variables (Z g, γ, µ) i Table 3 gives a example of a chromosome that represets desig variables accordigly. This desig strig is composed of 27 oes ad zeros. Table 3: The biary strig represetatio of the Gear tooth umber (Z g ) variables Desig Variables Frictio Helix coefficiet (µ ) agle(γ ) Fig.2. Variatio of variables through geeratios Table 5 Compariso of the results Desig Variables Aalytical method Geetic algorithm Number of teeth of worm gear (Z g ) Frictio coefficiet (, µ) Helix agle (, γ, ) o o Miimum power loss KW KW 24 P a g e

4 The startig poit of aalytical method Z g = 45, µ=0.046, γ = : 22.5 o The programme, developed i MATLAB 7.0 for aalytical method has bee ru several time for differet values of desig variables. The results obtaied are give i Table 5. As ca be see from the results, the geetic algorithm produced much better results tha aalytical method. 3. APPLICATION OF GA FOR GEOMETRICAL VOLUME & GEAR RATIO OPTIMIZATION The objectives of the desig are geometrical volume (size) miimizatio of 2-stage gear trai ad that of simple plaetary gear trai that has very valuable aspects i lightweight, small size ad high stregth. With regard to these desig objects, we determie the umber of teeth, module, face width, ad helix agle, cosiderig costraits such as stregth (durability), iterferece, cotact ratio ad other factors based o AGMA stadards4 so that these gear trais ca perform the tasks required i desig specificatio.[4] 3.1Volume Miimizatio of 2-Stage Gear Trai The objective fuctio ad costraits ca be writte as, F objective =w 1 F obj1+ w 2 F obj2 + F obj1 = µ- F obj2 =b s1 ( ) 2 (z 2 1s1+z 2 2s1) +b s2 ( ) 2 (z 2 1s2 + z 2 2s2) G 1 =1.2 σ HS1 -σ H lim G 2 =1.2 σ HS2 -σ H lim G 3 =1.15σ Fs1 - σ F lim σ F lim G 4 =1.15σ Fs1 - G 5 =v ts1 -v tmax G 6 = v ts2 -v tmax G 7 =1-ε αs1 G 8 = ε αs1-2.5 G 9 =1-ε αs2 G 10 = ε αs2-2.5 G 11 =0.8- ε βs1 G 12 = ε βs1-6 G 13 =0.8- ε βs2 G 14 = ε βs2-6 G 15 =b s1cos β s1-2z 1s1 m s1 G 16 =0.5 z 1s1 m s1 - b s1cos β s1 G 17 = b s2cos β s2-2z 1s2 m s2 G 18 = 0.5 z 1s2 m s2 - b s2cos β s2 G 19 = z 1s1 lim - z 1s1 G 20 = z 1s2 lim - z 1s2 G 21 = - G 22 = - where, F obj1 is the objective fuctio for volume miimizatio, F obj2 is the objective fuctio 2 for reductio gear ratio, w 1 ad w 2 are weight factors for F obj1 ad F obj2 respectively, γ p is the pealty coefficiet, ad G j is the violet value of costraits.1 G 1 ~ G 4 represet the costraits for the bedig stregth ad pittig resistace o the 1st ad 2d stage gears cosiderig factor of safety. G 5 ~ G 6 represet the costraits for pitch lie velocity. G 7 ~ G 14 Represet the costraits for cotact ratios. G 15 ~ G 18 represet the costraits for aspect ratios of piio of 1st ad 2d stage. G 19 ~ G 20 represet the costraits for udercut. G 21 ~ G 22 represet costraits for reductio gear ratio ad pitch diameter of gear respectively. Fitess fuctio is costructed by subtractig costructed sigle objective from Cmax, what is a adequate value which prevet the fitess value from beig egative F fitess = Cmax F objective As for parameters used i geetic algorithm, umber of idividuals is 30, the probability of crossover is 0.8,the probability of mutatio is 0.3, ad the algorithm is set to termiate whe the umber of a shift i a geeratio reaches Table 6. Bouds of desig variables for 2-stage gear trai for escalator Desig Bouds variable Z 1s Z 2s Z 1s Z 2s Normal 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, module 0.8,1, 1.25, 1.5, 2, 2.5, 3, 4, (mm) 5, 6, 8,10, 12, 16, 20, 25, 32, 40, 50 b s (mm) b s (mm) β s β s Table 7 Desig specificatio of 2-stage gear trai for escalator Item 1-stage 2-stage Piio Gea r Piio Gear Normal module [mm] Number of teeth Total gear ratio Helix agle [ ] Addedum modificatio coefficiet P a g e

5 Effective face width [mm] Pressure agle [ ] 20 Rotatioal speed of 1750 iput driver [rpm] Trasmitted power 7.5 [kw] Grade (AGMA) 9 Material SCM415 Heat treatmet Carburized & case hardeed Surface hardess HRC 60 Volume (pitch circle) [mm3] Table 8. Desig result of optimizatio Item 1-stage 2-stage Piio Gea r Piio Gear Normal module [mm] Number of teeth Total gear ratio Helix agle [ ] Addedum modificatio coefficiet Effective face width [mm] Pressure agle [ ] 20 Rotatioal speed of 1750 iput driver [rpm] Cotact stress [MPa] Allowable cotact stress [MPa] Bedig Stress [MPa] Allowable bedig stress [MPa] Volume (pitch circle) [mm3] 3.2 Result & Discussio Table 8 shows desig results that are obtaied at 8664 th geeratio, ad from this, we ca easily otice optimum values of iteger ad discrete variables are foud adequately satisfyig costraits. I this desig results, the correctio of solutio accordig to the types of variables after desig is ot eeded. Therefore geetic algorithm is the effective method i desigig gear trais. Fig.3 shows the compariso betwee the model of existig product ad the model havig dimesios foud by optimizatio i this research. I this result, the volume of pitch diameter ad face width is reduced about 40%, ad the error of reductio gear ratio betwee objective ad result are about 3%. 4. CONCLUSION By studyig above cases we cocluded that, while optimizig the complex problems of mechaical system a Geetic Algorithm is importat tool. So optimizatio of gear pair cosistig of various parameters, objectives ad costraits ca be doe easily usig No covetioal optimizatio techique i.e. Geetic Algorithm as compared to covetioal techiques. 5. FUTURE SCOPE By usig GA we ca optimize gearbox by cosiderig differet objective fuctios ad costraits. REFERENCES [1] Hog big Fag, Qia Wag, Multiobjective desig of a vehicular structure usig metamodellig ad a efficiet geetic algorithm, Iteratioal joural of desig Egieerig, Vol. 1,pp.41-55, [2] Khadiza Tahera, Raafat N. Ibrahim, Paul B. Lochert, GADYM - A Novel Geetic Algorithm i Mechaical Desig Problems, Joural of Uiversal Computer Sciece, vol. 14, o. 15,pp ,2008. [3] Muhammet Yama, Hamit Saruha, Faruk Med, Power Loss Optimizatio Of A Worm Gear Mechaism By Usig Geetic Algorithm Techical Educatio Faculty, Machie Desig Ad Costructio Departmet, Kouralp Yerleskes, [4] Tae Hyog Chog* ad Joug Sag Lee, A Desig Method of Gear Trais Usig a Geetic Algorithm, Iteratioal Joural of the Korea Society of Precisio Egieerig, Vol. 1, No. 1, Jue [5] Marco Barbieri, Giorgio Scagliarii, Giorgio Boori, Fracesco Pellicao, Gabriele Bertacchi, Optimizatio Methods For Spur Gear Dyamics, Sait Petersburg, Russia, Jue, 30 July, [6] Daizhog Su Ad Wejie Peg Optimum Desig Of Worm Gears With Multiple Computer Aided Techiques, ICCES, Vol.6, No.4, Pp ,2008. [7] M.Barbieri,G.Boori,G.Scagliarii F.Pellicao, Gear Vibratio Reductio Usig Geetic Algorithms, 12 th IFToMM World Cogress, Besaço (Frace), Jue18-21, 2007 (a) Old model (b) New model Fig.3 Model of 2-stage gear trais of escalator 26 P a g e