Generalized Cycloid Drive

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EpicycloidGeneralized Cycloid Drive EquationsBrian McMillin 12/13/2010From the paper "Gear geometry of cycloid drives," by CHEN BingKui, FANG TingTing, LI ChaoYang & WANG ShuYanhttp://www.scichina.com:8082/sciEe/fileup/PDF/08ye0598.pdfWell, the formulas create a full 360 degree epicycloidal, so why mess around with max.VariableSample Epi ValueSample Hypo ValueSample Hypo ValueUnitsDescription:R9080mmRadius of ring gear, which consist of many cylindrical rollers._r77mmRadius of one cylindrical roller of the ring gear.b (N)1211-Tooth number of the ring gear. ("pin wheel") (number of rollers)g1112-Tooth number of the cyloid gear. (number of lobes)e44mmEccentricity of CrankShaft. (distance between ring gear and cycloidal gear centers)Rz/b7.57.2727272727ratioKeep e < (Rz/b) to avoid interference of teeth. (Rz/eb > 1 ) (From Joong paper) = 'Rotational angle parameter, (0 max), used to generate profile. ( (0 2)tUsed to generate parameter (may be thought of as time of motion for a particle.)

Generalized equations for the cycloid gear ()X =R*SIN((') - (e)*SIN((b)(')/(b - g)) + (_r)*COS()Y =R*COS((') - (e)*COS((b)(')/(b - g)) - (_r)*SIN()whereCOS()=+/- (K*SIN((b)(')/(b - g)) - SIN('))/SQRT(1+K*K - (2)*(K)*COS((g)(')/(b - g))SIN() =+/- ((-K)*COS((b)(')/(b - g)) + COS('))/SQRT(1+K*K - (2)*(K)*COS((g)(')/(b - g))K = =(e*b)/(R*(b-g))"curtate coefficient"Equation of symmetrical axis:y = (ks)x = x*cotangent(/g)Equation 16: Equation of contacting line:Contact ratio:Note 1: For (b - g) (1,2,3), the "Epicycloidal" gear is generated. For (b - g) = (-1) the "Hypocycloidal" outer gear is generated.Note 2: The plus or minus notation: For Epi, INNER profile use plus. For curtate hypocycloidal OUTER profile use MINUS.Note 3: The "Epicycloidal gear-set has a matching external ring gear with cylindrical rollers.Note 4: The "Hypocycloidal" gear-set has a matching inner ring gear with cylindrical rollers.Note 5: The Epi and Hypo gears are NOT a gear-set ??, although it may be possible to generate a different Epi gear, from theHypo gear, to replace the Hypo's inner ring gear, for a matched gear-set, with Epi inner gear, and Hypo outer gear, usingequations modified from the Nikolov and Dolchinkov papers. (?)Note: The lobes of the "epi" generated below, likely do not mesh with the internal valley of the "hypo" unless..?Note: The R of the Hypo, to match diameters, can be calculated by formula R = ((R -r) -r) + 2e. ?Inner Diameter of Epi =(R -r) -e. ?Inner Diameter of Hypo = (R + r) -2e. ?Inner Diameter of Epi, plus e = Inner Diameter of Hypo. ? dePuzzler Note: "tg" means TANgent. "ctg" means Cotangent. Trig Identity: TAN((/2) - ) = Cotangent() for an arbitrary angle. Let = 'CASE: (b - g) =1 "Epi"cycloidal profile for inner wheel.CASE: (b - g) =(-1) "Hypo"cycloidal profile for outer cycloid.X =R*SIN(() - (e)*SIN((b)*()) + (_r)*COS()X =R*SIN(() - (e)*SIN((b)*()*(-1)) + (_r)*COS()Y =R*COS(() - (e)*COS((b)*()) - (_r)*SIN()Y =R*COS(() - (e)*COS((b)*()*(-1)) - (_r)*SIN()COS() =(K*SIN((b)*()) - SIN())/SQRT(1+K*K - (2)*(K)*COS((g)*())COS() =((-1)*K*SIN((b)*()(-1)) - SIN())/SQRT(1+K*K - (2)*(K)*COS((g)*()(-1))SIN() =((-K)*COS((b)*()) + COS())/SQRT(1+K*K - (2)*(K)*COS((g)*())SIN() =(((-1)*(-1)*K)*COS((b)*()(-1)) + COS())/SQRT(1+K*K - (2)*(K)*COS((g)*()(-1))K=K= =0.5333333333=(e*b)/((R)*(b-g)K=K= =-0.55=(e*b)/((R)*(b-g)NOTE: Lots of minus signs.VariableSample Epi ValueVariableSample Hypo ValueR90R80_r7_r7b12b11g11g12e4e4Rz/b7.5Rz/b7.2727272727

t=t*XYX;Y concatenateXYX;Y concatenate000.0079.000;790.0083.000;830.010.03141592653.5329289379.584092653.53292893315276;79.58409265336313.1567618283.160794333.15676182151284;83.16079433109280.020.06283185316.0243326480.636016726.02433263669055;80.63601672002736.1842095383.620947836.18420952726459;83.62094782608320.030.09424777968.0083902781.650272288.00839026643731;81.65027227736898.9764219484.318421318.97642193617868;84.31842131012080.040.125663706110.0232008582.5957114710.0232008474502;82.595711472615711.4757689585.1655967611.4757689489185;85.16559675648810.050.157079632712.3131964083.4313590912.3131964042415;83.431359085647113.7058275986.0738564713.7058275905789;86.07385647375560.060.188495559214.9492293784.0535057114.9492293715541;84.053505714450115.8365110386.9852201915.8365110325232;86.98522019069550.070.219911485817.8939529984.3402623617.8939529875138;84.340262362672518.3453480487.8511551218.3453480350256;87.85115511956250.080.251327412321.0370437984.1919758921.0370437879147;84.191975889538622.0717674488.1702598122.0717674377427;88.17025980753170.090.282743338824.2245878583.5577706824.2245878545102;83.557770676739926.2076279486.7242421326.207627943204;86.72424213109640.10.314159265427.2887243582.4483311327.2887243498562;82.448331126023428.7051332384.5527856528.7051332305452;84.55278564746240.110.345575191930.0767840780.9353799430.0767840670057;80.935379941431530.2497206482.6279928130.2497206430678;82.62799280786650.120.376991118432.4770393679.1385505232.4770393559845;79.138550515170831.6381524580.7710367731.6381524537805;80.77103677342460.130.40840704534.4381132177.2006364934.4381132062983;77.200636494668333.1905891578.8395673433.1905891466359;78.83956734167260.140.439822971535.9806351275.2515667835.9806351216725;75.251566777851935.0377302376.8404734635.0377302316489;76.840473459440.150.47123889837.2045417773.3585441837.2045417651464;73.35854417771937.2182311774.8572843637.2182311654925;74.85728436175270.160.502654824638.3118622171.4563765438.3118622137986;71.456376536553939.7062752272.9987768339.7062752215587;72.9987768306670.170.534070751139.7121903569.2985958239.7121903457478;69.298595822249642.4260292671.3663121442.4260292620762;71.36631213559090.180.565486677642.1340951066.8569498642.134095103624;66.856949855522745.2669993870.0323059845.2669993825271;70.03230597994630.190.596902604245.4310638565.2019803745.4310638527824;65.201980371581448.1034394869.0272371748.1034394764969;69.02723716922550.20.628318530748.2278943764.6295550348.2278943672161;64.629555027711550.8186932268.3345798150.8186932160987;68.33457981144350.210.659734457350.4875601964.3984123950.4875601940533;64.398412392590353.3368257567.8928556353.3368257489347;67.89285562596210.220.691150383852.6751492964.1321445052.6751492922082;64.132144501110155.6707388367.6001487855.6707388319104;67.60014877906690.230.722566310355.0178432763.6512690655.0178432707259;63.651269055008258.0125783667.2942635358.0125783622845;67.29426353178110.240.753982236957.5449065462.8237911657.5449065360773;62.823791162880860.8565512066.5690364760.8565512043432;66.56903646915660.250.785398163460.1729820961.5582873660.1729820940222;61.55828735578264.3467170964.3467170964.3467170879758;64.34671708797580.260.816814089962.7612096659.8191840262.7612096646103;59.819184023485466.5690364760.8565512066.5690364691566;60.85655120434320.270.848230016565.1519080057.6351963165.1519080025135;57.63519631093867.2942635358.0125783667.2942635317811;58.01257836228450.280.87964594367.2036722855.0961862367.2036722808004;55.096186226237467.6001487855.6707388367.6001487790669;55.67073883191040.290.911061869568.8171256452.3388207568.8171256391841;52.338820749711167.8928556353.3368257567.8928556259621;53.33682574893470.30.942477796169.9514033549.5230142869.9514033513346;49.523014279373568.3345798150.8186932268.3345798114435;50.81869321609870.310.973893722670.6293239946.8016177770.6293239890026;46.801617766644569.0272371748.1034394869.0272371692255;48.10343947649690.321.005309649170.9300158844.2848479770.9300158791468;44.284847973973570.0323059845.2669993870.0323059799463;45.26699938252710.331.036725575770.9697560041.9968146270.9697559992324;41.996814621797371.3663121442.4260292671.3663121355909;42.42602926207620.341.068141502270.8805037839.8109289870.8805037794425;39.810928979211472.9987768339.7062752272.9987768306671;39.70627522155860.351.099557428870.8447595737.3503878070.8447595701386;37.350387797345974.8572843637.2182311774.8572843617527;37.21823116549240.361.130973355371.3673034734.1161703271.3673034747897;34.116170318351576.8404734635.0377302376.8404734594401;35.03773023164890.371.162389281873.0811175730.7773681373.0811175656185;30.77736813223478.8395673433.1905891578.8395673416726;33.19058914663590.381.193805208475.1646928228.5922362075.1646928167604;28.592236196213580.7710367731.6381524580.7710367734246;31.63815245378050.391.225221134976.9778368127.1249965576.9778368103568;27.124996554995982.6279928130.2497206482.6279928078665;30.24972064306770.41.256637061478.6765939425.7429485678.6765939394917;25.742948556197484.5527856528.7051332384.5527856474625;28.70513323054510.411.28805298880.3850571024.1325862580.3850571043691;24.132586253754286.7242421326.2076279486.7242421310965;26.2076279432040.421.319468914582.0770546422.1462358482.0770546393898;22.146235840569688.1702598122.0717674488.1702598075317;22.07176743774260.431.35088484183.6416635619.7363846883.641663561829;19.736384684942287.8511551218.3453480487.8511551195625;18.34534803502560.441.382300767684.9411717116.9360562084.9411717105828;16.936056198988286.9852201915.8365110386.9852201906955;15.83651103252320.451.413716694185.8524087213.8445102685.8524087206031;13.84451025540886.0738564713.7058275986.0738564737556;13.70582759057880.461.445132620786.2951008010.6078418186.2951007975912;10.607841805395285.1655967611.4757689585.1655967564881;11.47576894891850.471.476548547286.248039417.3938221686.2480394082622;7.393822158258284.318421318.9764219484.3184213101208;8.976421936178630.481.507964473785.752575214.3634246985.7525752095693;4.3634246949619183.620947836.1842095383.6209478260832;6.184209527264520.491.539380400384.902989811.6422430184.9029898119096;1.642243014968883.160794333.1567618283.1607943310928;3.156761821512760.51.570796326883.82352941-0.7058823583.8235294117647;-0.70588235294121783.00000000-0.0000000083;-6.3566126348948E-140.511.602212253382.63145529-2.7044150882.6314552923682;-2.7044150823522583.16079433-3.1567618283.1607943310928;-3.15676182151290.521.633628179981.38585836-4