SWARM INTELLIGENCE OPTIMIZATION OF WORM AND WORM WHEEL DESIGN

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1 VOL. 0, NO. 3, JULY 05 ISSN Asia Research Publishig Network (ARPN). All rights reserved. SWARM INTELLIGENCE OPTIMIZATION OF WORM AND WORM WHEEL DESIGN M. Chadrasekara, Padmaabha S. ad V. Sriivasa Rama 3 Faculty of Mechaical Egieerig, Vels Uiversity, Cheai, Idia Departmet of Mechaical ad Productio Egieerig, Sathyabama Uiversity, Cheai, Idia 3 Departmet of Mechaical Egieerig, R.V.S. College of Egieerig ad Techology, Didigul, Idia ch_sekara@yahoo.com ABSTRACT The problem of optimizatio of gears is difficult to solve. Ideed, the objectives are may ad they require a multicriteria optimizatio. The compromise betwee the various objectives is ot easy to fid because the optimizatio criteria are ofte cotradictory. I additio, there are a large umber of desig variables, ad the majority of these variables takes discrete values. Worm ad worm wheel drives are ormally used for o-parallel, o-itersectig, right agle gear drive system where high reductio ratios are ivolved, though they are also employed with low to medium speed reducers i may applicatios. I this paper a attempt has bee made to optimize worm ad worm wheel desig usig covetioal tool LINGO ad heuristics as Particle Swarm Optimizatio (PSO) algorithms. A combied objective fuctio which maimizes the Power, Efficiecy ad miimizes the overall Weight, Cetre distace has bee cosidered i this model. Keywords: worm ad worm wheel, gear pair desig, LINGO, particle swarm optimizatio.. INTRODUCTION Most real world optimizatio problems ivolve the optimizatio task of more tha sigle objective fuctio, which, i tur, requires a sigificat computatioal effort. I this cotet, the use of evolutioary algorithms for multi-objective optimizatio has sigificatly grow for a decade, givig rise to a wide variety of meta-heuristic algorithms. The mai research directio is to improve both the computatioal efficiecy of the algorithm ad distributio of the obtaied odomiated optimal solutios. Caroll ad Johso [] have preseted a optimal desig techique to obtai compact ad stadard spur gear meshes with a objective of miimizig cetre distace. Savage et al [] have described the optimal desig of a eclosed parallel shaft spur gear reductio. The object of the desig was to determie a small, light weight trasmissio with a log service life. Deb [3] has proposed a No-Sorted Geetic Algorithm II for optimizig multi speed gear bo which cosider multi objectives such as maimizig the power ad miimizig the total volume of the gear. Tsay ad Tseg [] have applied a multiple optimizatio method to reduce the level of kiematical errors of helical gear trai ad ivestigated a optimal gear tooth modificatios. Prayoorat et al [5] have preseted a algorithm to desig ad optimize multi spidle gear trais for the o speed-chage type i which the desiger may choose miimum overall cetre distace, miimum overall size, miimum gear volume, or other desirable criteria, such as maimum cotact or overlap ratios to optimize gear trais. Yoo ad Rao [6] have preseted a ovel method to miimize the static trasmissio error usig cubic splies for gear tooth profiles. Ioceti [7] has proposed a ew approach to the efficiecy evaluatio of sigle or multidegree freedom of gear trais. Rosic [8] has proposed a aalytical ad computer aided procedure for the multi criteria desig of gear trai trasmissio system. The problem of worm ad worm wheel desig optimizatio is difficult to solve because it ivolves multiple objectives ad large umber of variables. Therefore a reliable ad robust optimizatio techique will be helpful i obtaiig optimal solutio for the problems. This paper has made a attempt to use the potetial of LINGO ad PSO to solve the worm ad worm wheel desig problem.. PROBLEM DESCRIPTION This sectio describes about the desig objectives, costraits, cosidered i this work. A test problem has bee cosidered i this work. This problem deals with the optimal desig of worm ad worm wheel. This work uses a combied objective fuctio, which miimizes the volume ad cetre distace ad maimizes power ad efficiecy. a) Desig of worm ad worm wheel A worm ad worm wheel desig has to be optimized cosiderig the followig data, Power =.5kW Speed of the piio = 0 rpm Speed ratio (or) Gear ratio = i) Objective fuctios for worm ad worm wheel The followig Objective fuctios have bee cosidered i this model. The equatio,, 3 ad represet the maimizatio of Power, miimizatio of Weight, maimizatio of Efficiecy ad miimizatio of Cetre distace. The efficiecy equatio (3) has bee adopted from Dudley. Maimize f = P where, P (L) P P (U) () 55

2 VOL. 0, NO. 3, JULY 05 ISSN Asia Research Publishig Network (ARPN). All rights reserved. Miimize, f = Weight = qm Z m + Z m b () Maimize, f 3 cos f ta 00 (3) cos f cot f = coefficiet of frictio = 0.08 d Miimize, f = d qm Z m ii) Desig costraits for worm ad worm wheel Bedig stress: The tooth breakage is caused by fatigue due to repeated bedig stresses. To safeguard the tooth agaist the breakage, the gear should have adequate bedig stregth. i.e., the iduced bedig stress whe trasmittig a torque should be lesser tha the allowable bedig stress. Crushig stress: To avoid the surface failures of the tooth profile like Pittig, Surface Abrasio, Seizure etc., ad to have the satisfactory life, gear should have the wear resistace. i.e., the iduced crushig stress should be lesser tha the allowable crushig stress. Gear ratio: The gear ratio should be a costat ad it should be equal to the ratio betwee the umber of teeth i gear ad the umber of teeth i piio. The eq. (5) represets this costrait [0]: i = =Z /Z (or) d /d (5) Cetre distace betwee the shafts: Cetre distace betwee the shafts a should be greater tha the miimum cetre distace a mi to assure the required clearace betwee the tip of the piio tooth ad the root of the gear tooth ad vice versa. This costrait is represeted by the eq. (6) a a mi (6) Module: The module m obtaied through the optimizatio process should be greater tha the miimum module to assure the proper trasmissio of rotatioal motio. This costrait is represeted by equatio (7) m m (7) Variable bouds: Aial module is varied from to 5mm, 5 to 6mm, 6 to 7mm ad 7 to 8mm i the rage of 0.00mm. Number of teeth i worm wheel-,, 7, 96. Power is varied from.5 to.5 kw i the rage of 0.00mm. b) Proposed desig objective fuctio The gear pair desig problem has four differet parameters i the objectives cosidered i this work. i.e., () power, weight of material, efficiecy ad ceter distace. Sice all these parameters are o differet scales, these factors are to be ormalized to the same scale. For maimizig criterio value, the values are ormalized by dividig its value with the ormalizig factor, ma i, which is the maimum value of this criterio obtaied from the solutios that have bee eplored so far ad for a miimizig criterio value, it is ormalized by dividig the ormalizig factor, mi i,with its value. The maimum ad miimum value of the criterio will be updated wheever the proposed algorithm fids aother feasible solutio. I additio, to esure the overall objective value to fall betwee 0 ad, the weight of each criterio is also ormalized. The ormalized objective fuctio is obtaied as follows: COF = NW i * N (X i) Where i COF = Combied objective fuctio W i = Pre ormalized weight of criterio i. NW i = Normalized weight of criterio i., Where NWi = W W i i i N (X i) = ormalized value of criterio i of solutio X. Where, N (X i) = Xi / ma i for maimizig criterio. N (X i) = mi i / X i for miimizig criterio. X i = Pre ormalized value of criterio X. ma i = Pre ormalized maimum value of criterio i amog all solutios eplored so far. mi i = Pre ormalized miimum value of criterio i amog all solutios eplored so far. N = umber of criteria. Hece the COF for this problem is, COF= power mi. weight ma. power weight efficiecy mi. cet. dist 3 ma. efficiecy cet. dist Where NW, NW, NW 3 ad NW = SOLUTION METHODOLOGIES (7) May researchers have show eormous iterests o o-traditioal optimizatio techiques such as Geetic Algorithm, Simulated Aealig ad applied the same i various egieerig fields. I this paper, a attempt is made to brig out the best of this algorithm for the desig of worm ad worm wheel. (8) (9) a) LINGO software LINGO is a software tool desiged to efficietly build ad solve liear, oliear, ad iteger optimizatio 555

3 VOL. 0, NO. 3, JULY 05 ISSN Asia Research Publishig Network (ARPN). All rights reserved. models. I this work four models have bee created i LINGO to solve the four objective fuctios. As these models have bee solved separately, these values ca ot be take as optimum values. Hece a fifth model has bee created i LINGO which combies all the four objective fuctios. The miimum values of weight ad cetre distace ad maimum values of power ad efficiecy obtaied from the previous models have bee give as iputs for the fifth model ad COF value ad correspodig optimal results have bee foud out. b) Particle swarm optimizatio PSO is a evolutioary computatio techique ispired by social behavior of bird flockig or fish schoolig. Similar to other otraditioal techiques, PSO is a populatio based optimizatio techique. i) Adaptatio of PSO to worm ad worm wheel desig I PSO the iitial populatio is radomly geerated ad COF value is computed for each particle. The velocity of each particle is calculated accordig to the respective equatio. The populatio best ad global best particles are computed ad the global best is updated cotiuously. The velocity of each particle is calculated accordig to the followig equatio. v[] = ω*v[] + C l * rad( ) * (pbest []- preset[] ) + C * rad ( ) * (gbest []- preset[] ) (0) Where, v [] : The velocity for the i th particle, represets the distace to be traveled by this particle from the curret positio. ω iertia weights rages from 0.8 to 0.9. rad ( ) is a radom umber betwee (0,) C ad C are learig factors. C = C =. Preset []: The locatio of the i th particle i.e., particle positio. Pbest []: The best previous positio of the i th particle is recorded ad represeted as pbest[] Gbest []: The ide of the best particle amog all the particles i the populatio is represeted by gbest []. The ew particle for the et iteratio is geerated by addig the velocity with the Preset Particle. This is give i the followig equatio. Preset [] = preset [] + v[] () The COF value is computed for the curret particle ad it is compared with preset COF. If the COF value of curret particle is better tha the COF of the previous oe, the curret particle is set as ew P best. The procedure is carried out for the required umber of iteratios to obtai the optimum value. Figure-. The algorithm for particle swarm optimizatio.. COMPUTATIONAL RESULTS The efficiecy of the proposed algorithms are validated through the test problem. The proposed algorithms have bee eecuted i MATLAB. The Iput parameters cosidered for the worm ad worm wheel desig is give the Table- below. Table-. Iput parameters. 556

4 VOL. 0, NO. 3, JULY 05 ISSN Asia Research Publishig Network (ARPN). All rights reserved. The Mathematical models for the test problem have bee formulated i terms of desig variable m, z ad P. Iitially, the iput values are geerated radomly with their variable bouds. If the geerated values satisfy the desig costraits, the the values of objective fuctios f, f, f 3 ad f are computed alog with COF. The optimum values of objective fuctio ad desig variables correspodig to the miimum COF value obtaied by the proposed algorithms for the test problem are show i a Table-. Table-. Optimum results for worm ad worm wheel for LINGO& PSO (Module rage: to 5). Power (kw) Weight (Kg) Parameters LINGO PSO Aial module (mm) No. of teeth i worm wheel Power (kw) Weight (kg) Efficiecy (%) Ceter distace (mm) LINGO.895 PSO Figure-. Compariso of power trasmitted LINGO 53.3 PSO Figure-3. Compariso of weight reductio. Table- shows the comparative results of the proposed algorithms for the test problem. The PSO algorithm performed better whe compared with LINGO i terms of Power trasmitted, Weight reductio ad ceter distace. 5. CONCLUSIONS I this paper a attempt has bee made to obtai optimal solutio of Worm ad Worm Wheel desig problem. Withi the various desig variables available for a gear pair desig, the power, weight, efficiecy ad ceter distace have bee cosidered as objective fuctios ad bedig stress, crushig stress as vital costraits to get a efficiet compact ad high power trasmittig drive. Amog the proposed algorithm PSO has produced covicig results for the test problems whe compared with the LINGO. As a future work, miimizatio of vibratio, maimizatio of life ad miimizatio of oise ca also be icluded i the objective fuctio to obtai a more reliable gear pair desig. Notatios ρ = desity of the material of worm i kg/mm 3 ρ = desity of the material of worm wheel i kg/mm 3 q = diameter factor λ = lead agle of the worm i = Gear (or) trasmissio ratio z, z = umber of starts i worm, umber of teeth i worm wheel d,d = Pitch circle diameter of worm, worm wheel i mm E = Youg s modulus i N/mm a = Cetre distace betwee shafts i mm η = Percetage efficiecy y = Form factor f = Average coefficiet of frictio Φ = Normal pressure agle i degrees m = Aial module i mm REFERENCES [] Carrol R. K. ad Johso G. E. 98. Optimal desig of compact spur gear sets. ASME Joural of Mechaisms, Trasmissios ad automatio i desig, Vol. 06, pp [] Savage M., Lattime S. B. ad Kimmel J. A. 99. Optimal Desig of Compact Spur Gear Reductios, Trasactios of the ASME, 690 / Vol. 6, pp [3] Deb K. Jai S Multi-Speed Gearbo Desig Usig Multi-Objective Evolutioary Algorithms. Joural of Mechaical desig, Vol.5, pp [] Tsay C-B Tseg, C-H, Modified Helical Gear Trai, Joural of Mechaical Desig. Vol. 9, pp

5 VOL. 0, NO. 3, JULY 05 ISSN Asia Research Publishig Network (ARPN). All rights reserved. [5] Prayoorat S. Walto D Practical approach to optimum gear trai desig, Mechaical Egieerig Departmet, Uiversity of Birmigham, Birmigham B5 TT, UK, Vol. 0, pp [6] Yoo K. Y. ad Rao S.S Dyamic Load Aalysis of Spur Gears Usig a New Tooth Profile, Joural of Mechaical Desig, Vol. 8, pp. -6. [7] Ioceti C A Framework for Efficiecy Evaluatio of Multi-Degree-of-Freedom Gear Trais, Trasactios of the ASME, 556 / Vol. 8, pp [8] Rosic B. 00. Multicriterio optimizatio of multistage gear trai trasmissio, Mechaical Egieerig, Vol. 8, pp []Keady J. ad Eberhart R. C. 00. Swarm itelligece, Morga Kaufma. [] Rao S. S., Eslampour H. R Multi stage multi objective optimizatio of gear boes, Joural of mechaisms, Trasmissios ad Automatio i Desig, Vol.08, 986, pp [3]Keedy J. ad Eberhart R. Particle swarm optimizatio procedures, Iteratioal coferece o eural etworks, Australia IEEE, [] Marceli J. L. 00. A meta model usig eural etworks ad Geetic Algorithms for a Itegrated Optimal desig of mechaism, Iteratioal Joural of Advaced Maufacturig Techology, Vol., pp [9] Baskar N., Asoka P., Saravaa R. ad Prabhakara G Optimizatio of machiig parameters for millig operatios usig o-covetioal methods, Iteratioal Joural of Advaced Maufacturig Techology, Vol. 5, pp [0] Desig Data Book, Faculty of Mechaical Egieerig, PSG College of Techology, Coimbatore-600, Idia. 558