Generalized Cycloid Drive Equations Brian McMillin 12/13/2010

From the paper "Gear geometry of cycloid drives," by CHEN BingKui, FANG TingTing, LI ChaoYang & WANG ShuYanhttp://www.scichina.com:8082/sciEe/fileup/PDF/08ye0598.pdf

Variable Units Description:90 80 mm Radius of ring gear, which consist of many cylindrical rollers.7 7 mm Radius of one cylindrical roller of the ring gear.

Ƶb (N) 12 11 - Tooth number of the ring gear. ("pin wheel") (number of rollers)Ƶg 11 12 - Tooth number of the cyloid gear. (number of lobes)e 4 4 mm Eccentricity of CrankShaft. (distance between ring gear and cycloidal gear centers)

7.5 7.272727272727 ratioɸ = ϕƀ't

Rƶ*SIN((ϕƀ') - (e)*SIN((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) + (_rƶ)*COS(β)Rƶ*COS((ϕƀ') - (e)*COS((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) - (_rƶ)*SIN(β)

where

+/- (K₁*SIN((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) - SIN(ϕƀ'))/SQRT(1+K₁*K₁ - (2)*(K₁)*COS((Ƶg)(ϕƀ')/(Ƶb - Ƶg))+/- ((-K₁)*COS((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) + COS(ϕƀ'))/SQRT(1+K₁*K₁ - (2)*(K₁)*COS((Ƶg)(ϕƀ')/(Ƶb - Ƶg))

"curtate coefficient"Equation of symmetrical axis:Equation 16: Equation of contacting line:Contact ratio:

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 the

Hypo gear, to replace the Hypo's inner ring gear, for a matched gear-set, with Epi inner gear, and Hypo outer gear, using

equations 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…..?

Inner Diameter of Hypo = (RƵ₂ + rƵ₂) -2e. ?

Inner Diameter of Epi, plus e = Inner Diameter of Hypo. ?

Well, the formulas create a full 360 degree epicycloidal, so why mess around with ϕmax.Sample Epi Value

Sample Hypo Value

Sample Hypo Value

Rƶ_rƶ

Rz/Ƶb Keep e < (Rz/Ƶb) to avoid interference of teeth. (Rz/eƵb > 1 ) (From Joong paper)Rotational angle parameter, (0 ≤ ϕ ≤ ϕmax), used to generate profile. ( (0 ≤ ɸ ≤ 2π)Used to generate ϕ parameter (may be thought of as time of motion for a particle.)

Generalized equations for the cycloid gear (∑²)

X₂ =Y₂ =

COS(β)=SIN(β) =K₁ =λ = (e*Ƶb)/(Rƶ*(Ƶb-Ƶg))

y = (ks)x = x*cotangent(π/Ƶg)

Note 1: For (Ƶb - Ƶg) ϵ (1,2,3), the "Epicycloidal" gear is generated. For (Ƶb - Ƶg) = (-1) the "Hypocycloidal" outer gear is generated.

Note: The RƵ₂ of the Hypo, to match diameters, can be calculated by formula RƵ₂ = ((RƵ -rƵ) -rƵ₂) + 2e. ?

Inner Diameter of Epi =(RƵ -rƵ) -e. ?

Rƶ*SIN((ɸ) - (e)*SIN((Ƶb)*(ɸ)) + (_rƶ)*COS(β)Rƶ*COS((ɸ) - (e)*COS((Ƶb)*(ɸ)) - (_rƶ)*SIN(β)(K*SIN((Ƶb)*(ɸ)) - SIN(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ))

((-K)*COS((Ƶb)*(ɸ)) + COS(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ))

0.533333

Variable907

Ƶb 12Ƶg 11e 4

7.5

t X Y X;Y concatenate

0 0 0.00 79.00 0;79

0.01 0.0314159265 3.53292893 79.58409265 3.53292893315276;79.5840926533631

0.02 0.0628318531 6.02433264 80.63601672 6.02433263669055;80.6360167200273

0.03 0.0942477796 8.00839027 81.65027228 8.00839026643731;81.6502722773689

0.04 0.1256637061 10.02320085 82.59571147 10.0232008474502;82.5957114726157

0.05 0.1570796327 12.31319640 83.43135909 12.3131964042415;83.4313590856471

0.06 0.1884955592 14.94922937 84.05350571 14.9492293715541;84.0535057144501

0.07 0.2199114858 17.89395299 84.34026236 17.8939529875138;84.3402623626725

0.08 0.2513274123 21.03704379 84.19197589 21.0370437879147;84.1919758895386

0.09 0.2827433388 24.22458785 83.55777068 24.2245878545102;83.5577706767399

0.1 0.3141592654 27.28872435 82.44833113 27.2887243498562;82.4483311260234

0.11 0.3455751919 30.07678407 80.93537994 30.0767840670057;80.9353799414315

0.12 0.3769911184 32.47703936 79.13855052 32.4770393559845;79.1385505151708

0.13 0.408407045 34.43811321 77.20063649 34.4381132062983;77.2006364946683

0.14 0.4398229715 35.98063512 75.25156678 35.9806351216725;75.2515667778519

dePuzzler Note: "tg" means TANgent. "ctg" means Cotangent. Trig Identity: TAN((π/2) - φ₀) = Cotangent(φ₀) for φ₀ an arbitrary angle.

Let ɸ = ϕƀ'

CASE: (Ƶb - Ƶg) =1 "Epi"cycloidal profile for inner wheel.X₂ =Y₂ =COS(β) =

SIN(β) =

K₁=K=λ = =(e*Ƶb)/((Rƶ)*(Ƶb-Ƶg)

Sample Epi Value

Rƶ_rƶ

Rz/Ƶb

ɸ=t*π

0.15 0.471238898 37.20454177 73.35854418 37.2045417651464;73.358544177719

0.16 0.5026548246 38.31186221 71.45637654 38.3118622137986;71.4563765365539

0.17 0.5340707511 39.71219035 69.29859582 39.7121903457478;69.2985958222496

0.18 0.5654866776 42.13409510 66.85694986 42.134095103624;66.8569498555227

0.19 0.5969026042 45.43106385 65.20198037 45.4310638527824;65.2019803715814

0.2 0.6283185307 48.22789437 64.62955503 48.2278943672161;64.6295550277115

0.21 0.6597344573 50.48756019 64.39841239 50.4875601940533;64.3984123925903

0.22 0.6911503838 52.67514929 64.13214450 52.6751492922082;64.1321445011101

0.23 0.7225663103 55.01784327 63.65126906 55.0178432707259;63.6512690550082

0.24 0.7539822369 57.54490654 62.82379116 57.5449065360773;62.8237911628808

0.25 0.7853981634 60.17298209 61.55828736 60.1729820940222;61.558287355782

0.26 0.8168140899 62.76120966 59.81918402 62.7612096646103;59.8191840234854

0.27 0.8482300165 65.15190800 57.63519631 65.1519080025135;57.635196310938

0.28 0.879645943 67.20367228 55.09618623 67.2036722808004;55.0961862262374

0.29 0.9110618695 68.81712564 52.33882075 68.8171256391841;52.3388207497111

0.3 0.9424777961 69.95140335 49.52301428 69.9514033513346;49.5230142793735

0.31 0.9738937226 70.62932399 46.80161777 70.6293239890026;46.8016177666445

0.32 1.0053096491 70.93001588 44.28484797 70.9300158791468;44.2848479739735

0.33 1.0367255757 70.96975600 41.99681462 70.9697559992324;41.9968146217973

0.34 1.0681415022 70.88050378 39.81092898 70.8805037794425;39.8109289792114

0.35 1.0995574288 70.84475957 37.35038780 70.8447595701386;37.3503877973459

0.36 1.1309733553 71.36730347 34.11617032 71.3673034747897;34.1161703183515

0.37 1.1623892818 73.08111757 30.77736813 73.0811175656185;30.777368132234

0.38 1.1938052084 75.16469282 28.59223620 75.1646928167604;28.5922361962135

0.39 1.2252211349 76.97783681 27.12499655 76.9778368103568;27.1249965549959

0.4 1.2566370614 78.67659394 25.74294856 78.6765939394917;25.7429485561974

0.41 1.288052988 80.38505710 24.13258625 80.3850571043691;24.1325862537542

0.42 1.3194689145 82.07705464 22.14623584 82.0770546393898;22.1462358405696

0.43 1.350884841 83.64166356 19.73638468 83.641663561829;19.7363846849422

0.44 1.3823007676 84.94117171 16.93605620 84.9411717105828;16.9360561989882

0.45 1.4137166941 85.85240872 13.84451026 85.8524087206031;13.844510255408

0.46 1.4451326207 86.29510080 10.60784181 86.2951007975912;10.6078418053952

0.47 1.4765485472 86.24803941 7.39382216 86.2480394082622;7.3938221582582

0.48 1.5079644737 85.75257521 4.36342469 85.7525752095693;4.36342469496191

0.49 1.5393804003 84.90298981 1.64224301 84.9029898119096;1.6422430149688

0.5 1.5707963268 83.82352941 -0.70588235 83.8235294117647;-0.705882352941217

0.51 1.6022122533 82.63145529 -2.70441508 82.6314552923682;-2.70441508235225

0.52 1.6336281799 81.38585836 -4.50146998 81.3858583597957;-4.50146997986075

0.53 1.6650441064 80.04826289 -6.45810518 80.0482628937388;-6.45810517924765

0.54 1.6964600329 78.68175085 -9.24589901 78.681750851294;-9.24589901486189

0.55 1.7278759595 78.08878772 -12.96502844 78.0887877171821;-12.9650284373174

0.56 1.759291886 78.56080079 -16.12068082 78.5608007905648;-16.1206808185154

0.57 1.7907078125 79.29156480 -18.42024917 79.2915647997619;-18.4202491690663

0.58 1.8221237391 79.98693761 -20.49240080 79.9869376050948;-20.4924007976811

0.59 1.8535396656 80.58094823 -22.72183540 80.58094823239;-22.7218354020181

0.6 1.8849555922 80.97882161 -25.25018780 80.978821614091;-25.2501878008583

0.61 1.9163715187 81.06164824 -28.07680320 81.0616482412385;-28.0768032031265

0.62 1.9477874452 80.72520475 -31.11167967 80.725204745292;-31.1116796653298

0.63 1.9792033718 79.90940769 -34.21103622 79.9094076904069;-34.2110362195229

0.64 2.0106192983 78.61416357 -37.20922051 78.6141635712765;-37.2092205051982

0.65 2.0420352248 76.90174731 -39.94905621 76.9017473148365;-39.949056206959

0.66 2.0734511514 74.88651970 -42.30864772 74.8865197030664;-42.308647715064

0.67 2.1048670779 72.71328692 -44.22169043 72.7132869192996;-44.2216904337022

0.68 2.1362830044 70.52553000 -45.68895510 70.52553000361;-45.6889550985749

0.69 2.167698931 68.42287947 -46.78098224 68.4228794667059;-46.7809822354439

0.7 2.1991148575 66.40199451 -47.63963629 66.4019945106193;-47.639636294783

0.71 2.230530784 64.27560674 -48.51524544 64.2756067415439;-48.5152454419838

0.72 2.2619467106 61.70609309 -49.93501195 61.706093086079;-49.9350119529239

0.73 2.2933626371 59.05828315 -52.55798324 59.0582831483721;-52.5579832409292

0.74 2.3247785637 57.53638344 -55.55293323 57.5363834409148;-55.5529332311513

0.75 2.3561944902 56.84187386 -57.96904777 56.8418738598486;-57.9690477674228

0.76 2.3876104167 56.31463711 -60.09744243 56.3146371079125;-60.0974424287818

0.77 2.4190263433 55.65344382 -62.26971488 55.6534438223295;-62.2697148763291

0.78 2.4504422698 54.68965490 -64.58733269 54.6896548981971;-64.5873326930876

0.79 2.4818581963 53.31345239 -67.00390056 53.3134523878804;-67.0039005617248

0.8 2.5132741229 51.47416996 -69.39577244 51.4741699603443;-69.3957724418494

0.81 2.5446900494 49.18693298 -71.61050997 49.1869329750871;-71.6105099708268

0.82 2.5761059759 46.53067388 -73.50390098 46.5306738788883;-73.5039009834016

0.83 2.6075219025 43.63555013 -74.96838536 43.6355501302627;-74.96838535561

0.84 2.638937829 40.66120214 -75.95196758 40.6612021412897;-75.9519675762619

0.85 2.6703537556 37.76846219 -76.46583064 37.7684621949193;-76.4658306367128

0.86 2.7017696821 35.08700892 -76.57922700 35.0870089216036;-76.5792269976963

0.87 2.7331856086 32.67922690 -76.40110000 32.6792268988867;-76.4010999962816

0.88 2.7646015352 30.49341814 -76.05050411 30.4934181364367;-76.0505041074924

0.89 2.7960174617 28.28433965 -75.63532333 28.2843396515394;-75.6353233313111

0.9 2.8274333882 25.52058732 -75.35927734 25.5205873194052;-75.3592773438298

0.91 2.8588493148 21.92174061 -75.90438747 21.9217406122161;-75.9043874738919

0.92 2.8902652413 18.80073714 -77.51793056 18.8007371369348;-77.517930555776

0.93 2.9216811678 16.78137364 -79.21204731 16.7813736442613;-79.2120473053236

0.94 2.9530970944 15.17519277 -80.73406176 15.1751927689411;-80.7340617637349

0.95 2.9845130209 13.48885098 -82.20716058 13.4888509798392;-82.2071605843305

0.96 3.0159289474 11.49371702 -83.64879161 11.4937170239768;-83.6487916107168

0.97 3.047344874 9.10287921 -84.97260042 9.10287920904472;-84.9726004151831

0.98 3.0787608005 6.32557297 -86.04941727 6.32557297246048;-86.0494172691328

0.99 3.1101767271 3.24553622 -86.75449183 3.24553622077201;-86.7544918341167

1 3.1415926536 0.00000000 -87.00000000 -2.21421301809992E-13;-87

1.01 3.1730085801 -3.24553622 -86.75449183 -3.2455362207724;-86.7544918341167

1.02 3.2044245067 -6.32557297 -86.04941727 -6.32557297246087;-86.0494172691327

1.03 3.2358404332 -9.10287921 -84.97260042 -9.10287920904503;-84.9726004151829

1.04 3.2672563597 -11.49371702 -83.64879161 -11.4937170239771;-83.6487916107167

1.05 3.2986722863 -13.48885098 -82.20716058 -13.4888509798394;-82.2071605843303

1.06 3.3300882128 -15.17519277 -80.73406176 -15.1751927689413;-80.7340617637348

1.07 3.3615041393 -16.78137364 -79.21204731 -16.7813736442615;-79.2120473053234

1.08 3.3929200659 -18.80073714 -77.51793056 -18.8007371369351;-77.5179305557758

1.09 3.4243359924 -21.92174061 -75.90438747 -21.9217406122166;-75.9043874738917

1.1 3.4557519189 -25.52058732 -75.35927734 -25.5205873194056;-75.3592773438298

1.11 3.4871678455 -28.28433965 -75.63532333 -28.2843396515397;-75.6353233313111

1.12 3.518583772 -30.49341814 -76.05050411 -30.4934181364369;-76.0505041074924

1.13 3.5499996986 -32.67922690 -76.40110000 -32.679226898887;-76.4010999962816

1.14 3.5814156251 -35.08700892 -76.57922700 -35.0870089216039;-76.5792269976963

1.15 3.6128315516 -37.76846219 -76.46583064 -37.7684621949197;-76.4658306367128

1.16 3.6442474782 -40.66120214 -75.95196758 -40.66120214129;-75.9519675762618

1.17 3.6756634047 -43.63555013 -74.96838536 -43.6355501302631;-74.9683853556098

1.18 3.7070793312 -46.53067388 -73.50390098 -46.5306738788886;-73.5039009834015

1.19 3.7384952578 -49.18693298 -71.61050997 -49.1869329750874;-71.6105099708265

1.2 3.7699111843 -51.47416996 -69.39577244 -51.4741699603445;-69.3957724418491

1.21 3.8013271108 -53.31345239 -67.00390056 -53.3134523878806;-67.0039005617246

1.22 3.8327430374 -54.68965490 -64.58733269 -54.6896548981973;-64.5873326930873

1.23 3.8641589639 -55.65344382 -62.26971488 -55.6534438223296;-62.2697148763288

1.24 3.8955748905 -56.31463711 -60.09744243 -56.3146371079126;-60.0974424287815

1.25 3.926990817 -56.84187386 -57.96904777 -56.8418738598487;-57.9690477674225

1.26 3.9584067435 -57.53638344 -55.55293323 -57.5363834409149;-55.5529332311509

1.27 3.9898226701 -59.05828315 -52.55798324 -59.0582831483724;-52.5579832409288

1.28 4.0212385966 -61.70609309 -49.93501195 -61.7060930860793;-49.9350119529236

1.29 4.0526545231 -64.27560674 -48.51524544 -64.2756067415442;-48.5152454419837

1.3 4.0840704497 -66.40199451 -47.63963629 -66.4019945106195;-47.6396362947829

1.31 4.1154863762 -68.42287947 -46.78098224 -68.4228794667061;-46.7809822354438

1.32 4.1469023027 -70.52553000 -45.68895510 -70.5255300036103;-45.6889550985748

1.33 4.1783182293 -72.71328692 -44.22169043 -72.7132869192999;-44.221690433702

1.34 4.2097341558 -74.88651970 -42.30864772 -74.8865197030667;-42.3086477150638

1.35 4.2411500823 -76.90174731 -39.94905621 -76.9017473148368;-39.9490562069587

1.36 4.2725660089 -78.61416357 -37.20922051 -78.6141635712767;-37.2092205051978

1.37 4.3039819354 -79.90940769 -34.21103622 -79.9094076904071;-34.2110362195225

1.38 4.335397862 -80.72520475 -31.11167967 -80.7252047452921;-31.1116796653295

1.39 4.3668137885 -81.06164824 -28.07680320 -81.0616482412386;-28.0768032031261

1.4 4.398229715 -80.97882161 -25.25018780 -80.978821614091;-25.2501878008579

1.41 4.4296456416 -80.58094823 -22.72183540 -80.58094823239;-22.7218354020178

1.42 4.4610615681 -79.98693761 -20.49240080 -79.9869376050947;-20.4924007976808

1.43 4.4924774946 -79.29156480 -18.42024917 -79.2915647997618;-18.420249169066

1.44 4.5238934212 -78.56080079 -16.12068082 -78.5608007905647;-16.120680818515

1.45 4.5553093477 -78.08878772 -12.96502844 -78.0887877171821;-12.965028437317

1.46 4.5867252742 -78.68175085 -9.24589901 -78.6817508512941;-9.24589901486145

1.47 4.6181412008 -80.04826289 -6.45810518 -80.0482628937389;-6.45810517924739

1.48 4.6495571273 -81.38585836 -4.50146998 -81.3858583597958;-4.50146997986054

1.49 4.6809730538 -82.63145529 -2.70441508 -82.6314552923684;-2.704415082352

1.5 4.7123889804 -83.82352941 -0.70588235 -83.8235294117648;-0.705882352940942

1.51 4.7438049069 -84.90298981 1.64224301 -84.9029898119097;1.64224301496912

1.52 4.7752208335 -85.75257521 4.36342469 -85.7525752095694;4.36342469496226

1.53 4.80663676 -86.24803941 7.39382216 -86.2480394082622;7.39382215825864

1.54 4.8380526865 -86.29510080 10.60784181 -86.2951007975911;10.6078418053957

1.55 4.8694686131 -85.85240872 13.84451026 -85.852408720603;13.8445102554084

1.56 4.9008845396 -84.94117171 16.93605620 -84.9411717105826;16.9360561989886

1.57 4.9323004661 -83.64166356 19.73638468 -83.6416635618288;19.7363846849425

1.58 4.9637163927 -82.07705464 22.14623584 -82.0770546393896;22.1462358405699

1.59 4.9951323192 -80.38505710 24.13258625 -80.3850571043689;24.1325862537544

1.6 5.0265482457 -78.67659394 25.74294856 -78.6765939394915;25.7429485561976

1.61 5.0579641723 -76.97783681 27.12499655 -76.9778368103565;27.124996554996

1.62 5.0893800988 -75.16469282 28.59223620 -75.1646928167601;28.5922361962137

1.63 5.1207960254 -73.08111757 30.77736813 -73.0811175656182;30.7773681322344

1.64 5.1522119519 -71.36730347 34.11617032 -71.3673034747895;34.1161703183519

1.65 5.1836278784 -70.84475957 37.35038780 -70.8447595701386;37.3503877973463

1.66 5.215043805 -70.88050378 39.81092898 -70.8805037794425;39.8109289792117

1.67 5.2464597315 -70.96975600 41.99681462 -70.9697559992324;41.9968146217976

1.68 5.277875658 -70.93001588 44.28484797 -70.9300158791467;44.2848479739738

1.69 5.3092915846 -70.62932399 46.80161777 -70.6293239890025;46.8016177666448

1.7 5.3407075111 -69.95140335 49.52301428 -69.9514033513344;49.5230142793739

1.71 5.3721234376 -68.81712564 52.33882075 -68.8171256391839;52.3388207497115

1.72 5.4035393642 -67.20367228 55.09618623 -67.2036722808002;55.0961862262377

1.73 5.4349552907 -65.15190800 57.63519631 -65.1519080025132;57.6351963109383

1.74 5.4663712172 -62.76120966 59.81918402 -62.76120966461;59.8191840234856

1.75 5.4977871438 -60.17298209 61.55828736 -60.1729820940219;61.5582873557822

1.76 5.5292030703 -57.54490654 62.82379116 -57.544906536077;62.8237911628809

1.77 5.5606189969 -55.01784327 63.65126906 -55.0178432707256;63.6512690550083

1.78 5.5920349234 -52.67514929 64.13214450 -52.6751492922079;64.1321445011101

1.79 5.6234508499 -50.48756019 64.39841239 -50.487560194053;64.3984123925903

1.8 5.6548667765 -48.22789437 64.62955503 -48.2278943672158;64.6295550277116

1.81 5.686282703 -45.43106385 65.20198037 -45.431063852782;65.2019803715815

1.82 5.7176986295 -42.13409510 66.85694986 -42.1340951036236;66.856949855523

1.83 5.7491145561 -39.71219035 69.29859582 -39.7121903457475;69.29859582225

1.84 5.7805304826 -38.31186221 71.45637654 -38.3118622137985;71.4563765365542

1.85 5.8119464091 -37.20454177 73.35854418 -37.2045417651463;73.3585441777192

1.86 5.8433623357 -35.98063512 75.25156678 -35.9806351216723;75.2515667778521

1.87 5.8747782622 -34.43811321 77.20063649 -34.4381132062981;77.2006364946686

1.88 5.9061941887 -32.47703936 79.13855052 -32.4770393559842;79.138550515171

1.89 5.9376101153 -30.07678407 80.93537994 -30.0767840670053;80.9353799414317

1.9 5.9690260418 -27.28872435 82.44833113 -27.2887243498558;82.4483311260236

1.91 6.0004419684 -24.22458785 83.55777068 -24.2245878545098;83.55777067674

1.92 6.0318578949 -21.03704379 84.19197589 -21.0370437879142;84.1919758895387

1.93 6.0632738214 -17.89395299 84.34026236 -17.8939529875133;84.3402623626725

1.94 6.094689748 -14.94922937 84.05350571 -14.9492293715537;84.05350571445

1.95 6.1261056745 -12.31319640 83.43135909 -12.3131964042411;83.431359085647

1.96 6.157521601 -10.02320085 82.59571147 -10.0232008474499;82.5957114726155

1.97 6.1889375276 -8.00839027 81.65027228 -8.00839026643702;81.6502722773687

1.98 6.2203534541 -6.02433264 80.63601672 -6.02433263669025;80.6360167200272

1.99 6.2517693806 -3.53292893 79.58409265 -3.53292893315235;79.584092653363

2 6.2831853072 0.00000000 79.00000000 0;79 equal start #'s

Usage:Copy the X;Y column data into new text document on desktop. (For full epicycloidal) (similarly for full Hypocycloidal.)Import the text document into Alibre using Renner Alibre Point Import Addin, 2D, with Spline (and can do points also), not closed.(With points, can see relatively uniform interval of points, in sketch mode Alibre)Closed Spline can be extruded!!!

Brian McMillin 12/13/2010

From the paper "Gear geometry of cycloid drives," by CHEN BingKui, FANG TingTing, LI ChaoYang & WANG ShuYan

Description:Radius of ring gear, which consist of many cylindrical rollers.Radius of one cylindrical roller of the ring gear.Tooth number of the ring gear. ("pin wheel") (number of rollers)Tooth number of the cyloid gear. (number of lobes)Eccentricity of CrankShaft. (distance between ring gear and cycloidal gear centers)

+/- (K₁*SIN((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) - SIN(ϕƀ'))/SQRT(1+K₁*K₁ - (2)*(K₁)*COS((Ƶg)(ϕƀ')/(Ƶb - Ƶg))+/- ((-K₁)*COS((Ƶb)(ϕƀ')/(Ƶb - Ƶg)) + COS(ϕƀ'))/SQRT(1+K₁*K₁ - (2)*(K₁)*COS((Ƶg)(ϕƀ')/(Ƶb - Ƶg))

"curtate coefficient"

Note 2: The plus or minus notation: For Epi, INNER profile use plus. For curtate hypocycloidal OUTER profile use MINUS.

Note 5: The Epi and Hypo gears are NOT a gear-set ??, although it may be possible to generate a different Epi gear, from the

Hypo gear, to replace the Hypo's inner ring gear, for a matched gear-set, with Epi inner gear, and Hypo outer gear, using

Note: The lobes of the "epi" generated below, likely do not mesh with the internal valley of the "hypo" unless…..?

Well, the formulas create a full 360 degree epicycloidal, so why mess around with ϕmax.

Keep e < (Rz/Ƶb) to avoid interference of teeth. (Rz/eƵb > 1 ) (From Joong paper)

)x = x*cotangent(π/Ƶg)

Ƶb - Ƶg) ϵ (1,2,3), the "Epicycloidal" gear is generated. For (Ƶb - Ƶg) = (-1) the "Hypocycloidal" outer gear is generated.

Rƶ*SIN((ɸ) - (e)*SIN((Ƶb)*(ɸ)*(-1)) + (_rƶ)*COS(β)Rƶ*COS((ɸ) - (e)*COS((Ƶb)*(ɸ)*(-1)) - (_rƶ)*SIN(β)((-1)*K*SIN((Ƶb)*(ɸ)(-1)) - SIN(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ)(-1))

(((-1)*(-1)*K)*COS((Ƶb)*(ɸ)(-1)) + COS(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ)(-1))

-0.55NOTE: Lots of minus signs.

Variable807

Ƶb₂ 11Ƶg₂ 12

47.272727272727

X Y X;Y concatenate

0.00 83.00 0;83

3.15676182 83.16079433 3.15676182151284;83.1607943310928

6.18420953 83.62094783 6.18420952726459;83.6209478260832

8.97642194 84.31842131 8.97642193617868;84.3184213101208

11.47576895 85.16559676 11.4757689489185;85.1655967564881

13.70582759 86.07385647 13.7058275905789;86.0738564737556

15.83651103 86.98522019 15.8365110325232;86.9852201906955

18.34534804 87.85115512 18.3453480350256;87.8511551195625

22.07176744 88.17025981 22.0717674377427;88.1702598075317

26.20762794 86.72424213 26.207627943204;86.7242421310964

28.70513323 84.55278565 28.7051332305452;84.5527856474624

30.24972064 82.62799281 30.2497206430678;82.6279928078665

31.63815245 80.77103677 31.6381524537805;80.7710367734246

33.19058915 78.83956734 33.1905891466359;78.8395673416726

35.03773023 76.84047346 35.0377302316489;76.84047345944

π/2) - φ₀) = Cotangent(φ₀) for φ₀ an arbitrary angle.

CASE: (Ƶb - Ƶg) =(-1) "Hypo"cycloidal profile for outer cycloid.X₂ =Y₂ =COS(β) =

SIN(β) =

K₁=K=λ = =(e*Ƶb)/((Rƶ)*(Ƶb-Ƶg)

Sample Hypo Value

Rƶ₂_rƶ₂

e₂Rz₂/Ƶb₂

37.21823117 74.85728436 37.2182311654925;74.8572843617527

39.70627522 72.99877683 39.7062752215587;72.998776830667

42.42602926 71.36631214 42.4260292620762;71.3663121355909

45.26699938 70.03230598 45.2669993825271;70.0323059799463

48.10343948 69.02723717 48.1034394764969;69.0272371692255

50.81869322 68.33457981 50.8186932160987;68.3345798114435

53.33682575 67.89285563 53.3368257489347;67.8928556259621

55.67073883 67.60014878 55.6707388319104;67.6001487790669

58.01257836 67.29426353 58.0125783622845;67.2942635317811

60.85655120 66.56903647 60.8565512043432;66.5690364691566

64.34671709 64.34671709 64.3467170879758;64.3467170879758

66.56903647 60.85655120 66.5690364691566;60.8565512043432

67.29426353 58.01257836 67.2942635317811;58.0125783622845

67.60014878 55.67073883 67.6001487790669;55.6707388319104

67.89285563 53.33682575 67.8928556259621;53.3368257489347

68.33457981 50.81869322 68.3345798114435;50.8186932160987

69.02723717 48.10343948 69.0272371692255;48.1034394764969

70.03230598 45.26699938 70.0323059799463;45.2669993825271

71.36631214 42.42602926 71.3663121355909;42.4260292620762

72.99877683 39.70627522 72.9987768306671;39.7062752215586

74.85728436 37.21823117 74.8572843617527;37.2182311654924

76.84047346 35.03773023 76.8404734594401;35.0377302316489

78.83956734 33.19058915 78.8395673416726;33.1905891466359

80.77103677 31.63815245 80.7710367734246;31.6381524537805

82.62799281 30.24972064 82.6279928078665;30.2497206430677

84.55278565 28.70513323 84.5527856474625;28.7051332305451

86.72424213 26.20762794 86.7242421310965;26.207627943204

88.17025981 22.07176744 88.1702598075317;22.0717674377426

87.85115512 18.34534804 87.8511551195625;18.3453480350256

86.98522019 15.83651103 86.9852201906955;15.8365110325232

86.07385647 13.70582759 86.0738564737556;13.7058275905788

85.16559676 11.47576895 85.1655967564881;11.4757689489185

84.31842131 8.97642194 84.3184213101208;8.97642193617863

83.62094783 6.18420953 83.6209478260832;6.18420952726452

83.16079433 3.15676182 83.1607943310928;3.15676182151276

83.00000000 0.00000000 83;-6.3566126348948E-14

83.16079433 -3.15676182 83.1607943310928;-3.1567618215129

83.62094783 -6.18420953 83.6209478260832;-6.18420952726466

84.31842131 -8.97642194 84.3184213101209;-8.97642193617875

85.16559676 -11.47576895 85.1655967564881;-11.4757689489186

86.07385647 -13.70582759 86.0738564737556;-13.7058275905789

86.98522019 -15.83651103 86.9852201906955;-15.8365110325233

87.85115512 -18.34534804 87.8511551195625;-18.3453480350257

88.17025981 -22.07176744 88.1702598075317;-22.0717674377429

86.72424213 -26.20762794 86.7242421310964;-26.2076279432041

84.55278565 -28.70513323 84.5527856474624;-28.7051332305452

82.62799281 -30.24972064 82.6279928078664;-30.2497206430678

80.77103677 -31.63815245 80.7710367734245;-31.6381524537805

78.83956734 -33.19058915 78.8395673416725;-33.190589146636

76.84047346 -35.03773023 76.84047345944;-35.037730231649

74.85728436 -37.21823117 74.8572843617526;-37.2182311654926

72.99877683 -39.70627522 72.998776830667;-39.7062752215587

71.36631214 -42.42602926 71.3663121355909;-42.4260292620763

70.03230598 -45.26699938 70.0323059799463;-45.2669993825272

69.02723717 -48.10343948 69.0272371692254;-48.103439476497

68.33457981 -50.81869322 68.3345798114434;-50.8186932160988

67.89285563 -53.33682575 67.8928556259621;-53.3368257489348

67.60014878 -55.67073883 67.6001487790669;-55.6707388319105

67.29426353 -58.01257836 67.2942635317811;-58.0125783622846

66.56903647 -60.85655120 66.5690364691566;-60.8565512043434

64.34671709 -64.34671709 64.3467170879757;-64.3467170879759

60.85655120 -66.56903647 60.856551204343;-66.5690364691567

58.01257836 -67.29426353 58.0125783622844;-67.2942635317811

55.67073883 -67.60014878 55.6707388319103;-67.6001487790669

53.33682575 -67.89285563 53.3368257489346;-67.8928556259621

50.81869322 -68.33457981 50.8186932160986;-68.3345798114435

48.10343948 -69.02723717 48.1034394764968;-69.0272371692255

45.26699938 -70.03230598 45.266999382527;-70.0323059799464

42.42602926 -71.36631214 42.4260292620761;-71.366312135591

39.70627522 -72.99877683 39.7062752215585;-72.9987768306671

37.21823117 -74.85728436 37.2182311654923;-74.8572843617528

35.03773023 -76.84047346 35.0377302316488;-76.8404734594401

33.19058915 -78.83956734 33.1905891466359;-78.8395673416727

31.63815245 -80.77103677 31.6381524537804;-80.7710367734247

30.24972064 -82.62799281 30.2497206430677;-82.6279928078666

28.70513323 -84.55278565 28.7051332305451;-84.5527856474625

26.20762794 -86.72424213 26.2076279432038;-86.7242421310966

22.07176744 -88.17025981 22.0717674377425;-88.1702598075317

18.34534804 -87.85115512 18.3453480350255;-87.8511551195624

15.83651103 -86.98522019 15.8365110325231;-86.9852201906955

13.70582759 -86.07385647 13.7058275905788;-86.0738564737556

11.47576895 -85.16559676 11.4757689489184;-85.1655967564881

8.97642194 -84.31842131 8.97642193617853;-84.3184213101208

6.18420953 -83.62094783 6.1842095272644;-83.6209478260832

3.15676182 -83.16079433 3.15676182151266;-83.1607943310927

0.00000000 -83.00000000 -2.12970399324398E-13;-83

-3.15676182 -83.16079433 -3.15676182151303;-83.1607943310928

-6.18420953 -83.62094783 -6.18420952726479;-83.6209478260832

-8.97642194 -84.31842131 -8.97642193617885;-84.3184213101209

-11.47576895 -85.16559676 -11.4757689489187;-85.1655967564882

-13.70582759 -86.07385647 -13.705827590579;-86.0738564737556

-15.83651103 -86.98522019 -15.8365110325234;-86.9852201906956

-18.34534804 -87.85115512 -18.3453480350259;-87.8511551195625

-22.07176744 -88.17025981 -22.071767437743;-88.1702598075317

-26.20762794 -86.72424213 -26.2076279432043;-86.7242421310963

-28.70513323 -84.55278565 -28.7051332305453;-84.5527856474623

-30.24972064 -82.62799281 -30.2497206430679;-82.6279928078663

-31.63815245 -80.77103677 -31.6381524537806;-80.7710367734244

-33.19058915 -78.83956734 -33.1905891466361;-78.8395673416724

-35.03773023 -76.84047346 -35.037730231649;-76.8404734594399

-37.21823117 -74.85728436 -37.2182311654927;-74.8572843617525

-39.70627522 -72.99877683 -39.7062752215588;-72.9987768306669

-42.42602926 -71.36631214 -42.4260292620764;-71.3663121355908

-45.26699938 -70.03230598 -45.2669993825273;-70.0323059799462

-48.10343948 -69.02723717 -48.1034394764972;-69.0272371692254

-50.81869322 -68.33457981 -50.8186932160989;-68.3345798114434

-53.33682575 -67.89285563 -53.3368257489349;-67.8928556259621

-55.67073883 -67.60014878 -55.6707388319106;-67.6001487790668

-58.01257836 -67.29426353 -58.0125783622846;-67.294263531781

-60.85655120 -66.56903647 -60.8565512043435;-66.5690364691565

-64.34671709 -64.34671709 -64.3467170879761;-64.3467170879756

-66.56903647 -60.85655120 -66.5690364691567;-60.8565512043429

-67.29426353 -58.01257836 -67.2942635317811;-58.0125783622843

-67.60014878 -55.67073883 -67.6001487790669;-55.6707388319102

-67.89285563 -53.33682575 -67.8928556259622;-53.3368257489345

-68.33457981 -50.81869322 -68.3345798114435;-50.8186932160985

-69.02723717 -48.10343948 -69.0272371692256;-48.1034394764966

-70.03230598 -45.26699938 -70.0323059799464;-45.2669993825268

-71.36631214 -42.42602926 -71.366312135591;-42.4260292620759

-72.99877683 -39.70627522 -72.9987768306672;-39.7062752215584

-74.85728436 -37.21823117 -74.8572843617528;-37.2182311654923

-76.84047346 -35.03773023 -76.8404734594402;-35.0377302316487

-78.83956734 -33.19058915 -78.8395673416728;-33.1905891466358

-80.77103677 -31.63815245 -80.7710367734248;-31.6381524537804

-82.62799281 -30.24972064 -82.6279928078666;-30.2497206430677

-84.55278565 -28.70513323 -84.5527856474627;-28.705133230545

-86.72424213 -26.20762794 -86.7242421310966;-26.2076279432037

-88.17025981 -22.07176744 -88.1702598075317;-22.0717674377423

-87.85115512 -18.34534804 -87.8511551195624;-18.3453480350254

-86.98522019 -15.83651103 -86.9852201906954;-15.836511032523

-86.07385647 -13.70582759 -86.0738564737555;-13.7058275905787

-85.16559676 -11.47576895 -85.165596756488;-11.4757689489183

-84.31842131 -8.97642194 -84.3184213101208;-8.97642193617843

-83.62094783 -6.18420953 -83.6209478260832;-6.18420952726433

-83.16079433 -3.15676182 -83.1607943310927;-3.15676182151248

-83.00000000 0.00000000 -83;3.38221949467358E-13

-83.16079433 3.15676182 -83.1607943310928;3.15676182151316

-83.62094783 6.18420953 -83.6209478260833;6.18420952726487

-84.31842131 8.97642194 -84.3184213101209;8.97642193617899

-85.16559676 11.47576895 -85.1655967564882;11.4757689489188

-86.07385647 13.70582759 -86.0738564737557;13.7058275905791

-86.98522019 15.83651103 -86.9852201906956;15.8365110325235

-87.85115512 18.34534804 -87.8511551195626;18.345348035026

-88.17025981 22.07176744 -88.1702598075316;22.0717674377433

-86.72424213 26.20762794 -86.7242421310962;26.2076279432044

-84.55278565 28.70513323 -84.5527856474622;28.7051332305454

-82.62799281 30.24972064 -82.6279928078663;30.2497206430679

-80.77103677 31.63815245 -80.7710367734243;31.6381524537807

-78.83956734 33.19058915 -78.8395673416723;33.1905891466362

-76.84047346 35.03773023 -76.8404734594398;35.0377302316491

-74.85728436 37.21823117 -74.8572843617525;37.2182311654927

-72.99877683 39.70627522 -72.9987768306668;39.706275221559

-71.36631214 42.42602926 -71.3663121355907;42.4260292620765

-70.03230598 45.26699938 -70.0323059799462;45.2669993825274

-69.02723717 48.10343948 -69.0272371692254;48.1034394764972

-68.33457981 50.81869322 -68.3345798114434;50.8186932160991

-67.89285563 53.33682575 -67.8928556259621;53.336825748935

-67.60014878 55.67073883 -67.6001487790668;55.6707388319107

-67.29426353 58.01257836 -67.2942635317811;58.0125783622848

-66.56903647 60.85655120 -66.5690364691564;60.8565512043436

-64.34671709 64.34671709 -64.3467170879754;64.3467170879763

-60.85655120 66.56903647 -60.8565512043428;66.5690364691568

-58.01257836 67.29426353 -58.0125783622842;67.2942635317812

-55.67073883 67.60014878 -55.6707388319101;67.6001487790669

-53.33682575 67.89285563 -53.3368257489344;67.8928556259622

-50.81869322 68.33457981 -50.8186932160984;68.3345798114435

-48.10343948 69.02723717 -48.1034394764966;69.0272371692256

-45.26699938 70.03230598 -45.2669993825267;70.0323059799465

-42.42602926 71.36631214 -42.4260292620758;71.3663121355911

-39.70627522 72.99877683 -39.7062752215583;72.9987768306673

-37.21823117 74.85728436 -37.2182311654921;74.8572843617529

-35.03773023 76.84047346 -35.0377302316486;76.8404734594403

-33.19058915 78.83956734 -33.1905891466357;78.8395673416728

-31.63815245 80.77103677 -31.6381524537803;80.7710367734248

-30.24972064 82.62799281 -30.2497206430676;82.6279928078668

-28.70513323 84.55278565 -28.7051332305449;84.5527856474627

-26.20762794 86.72424213 -26.2076279432036;86.7242421310967

-22.07176744 88.17025981 -22.0717674377421;88.1702598075318

-18.34534804 87.85115512 -18.3453480350252;87.8511551195623

-15.83651103 86.98522019 -15.8365110325229;86.9852201906954

-13.70582759 86.07385647 -13.7058275905786;86.0738564737555

-11.47576895 85.16559676 -11.4757689489181;85.165596756488

-8.97642194 84.31842131 -8.97642193617828;84.3184213101207

-6.18420953 83.62094783 -6.18420952726416;83.6209478260831

-3.15676182 83.16079433 -3.15676182151241;83.1607943310927

0.00000000 83.00000000 0;83 equal start #'s

Copy the X;Y column data into new text document on desktop. (For full epicycloidal) (similarly for full Hypocycloidal.)Import the text document into Alibre using Renner Alibre Point Import Addin, 2D, with Spline (and can do points also), not closed.

((-1)*K*SIN((Ƶb)*(ɸ)(-1)) - SIN(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ)(-1))

(((-1)*(-1)*K)*COS((Ƶb)*(ɸ)(-1)) + COS(ɸ))/SQRT(1+K*K - (2)*(K)*COS((Ƶg)*(ɸ)(-1))

page2 Pin Wheel Epitrochoid meshing Brian McMillin 12/10/2010Based upon "Geometry Design and Analysis for Trochoidal-Type Speed Reducers: with Conjugate Envelopes". Chiu-Fau Hsieh

Variable Value Units Description:r 60 mm Radius of ring gear, which consist of many rollers.R 5 mm Radius of one cylindrical roller of the ring gear. ("R_sub_r" in article)N 16 - Tooth number of the ring gear. ("pin wheel") (number of rollers)m 1 - Tooth number difference. N-m = N-1 = Epitrhochod tooth number. λc 3 mm Eccentricity of CrankShaft. (distance between ring gear and cycloidal gear centers)

MATRIX CALCULATIONS

0 0 0(r₂ᵖ) = 0 (-1)c r

0 0 1 0 0 1 1

0 cosϕ * 0 + (-1)sinϕ * r + 0 * 1(r₂ᵖ) = 0 sinϕ * 0 + cosϕ * r + (-1)c * 1

0 0 1 0 * 0 + 0 * r + 1 * 1

0 (-1)rsinϕ(r₂ᵖ) = 0 (rcosϕ + (-1)c)

0 0 1 1

cos((N/(N-m))ϕ) * (-1)rsinϕ + sin((N/(N-m))ϕ) * (rcosϕ + (-1)c) + 0 * 1(r₂ᵖ) = (-1)sin((N/(N-m))ϕ) * (-1)rsinϕ + cos((N/(N-m))ϕ) * (rcosϕ + (-1)c) + 0 * 1

(r₂ᵖ) = (M₂₁)(r₁ᵖ) = (M₂ᶠ)(Mᶠ₁)(r₁ᶠ)

cos((N/(N-m))ϕ) sin((N/(N-m))ϕ) cosϕ (-1)sinϕ(-1)sin((N/(N-m))ϕ) cos((N/(N-m))ϕ) sinϕ cosϕ

cos((N/(N-m))ϕ) sin((N/(N-m))ϕ)(-1)sin((N/(N-m))ϕ) cos((N/(N-m))ϕ)

cos((N/(N-m))ϕ) sin((N/(N-m))ϕ)(-1)sin((N/(N-m))ϕ) cos((N/(N-m))ϕ)

0 * (-1)rsinϕ + 0 * (rcosϕ + (-1)c) + 1 * 1

cos((N/(N-m))ϕ) * (-1)rsinϕ + sin((N/(N-m))ϕ) * rcosϕ + (-1)csin((N/(N-m))ϕ)(r₂ᵖ) = sin((N/(N-m))ϕ) * rsinϕ + cos((N/(N-m))ϕ) * rcosϕ + (-1)ccos((N/(N-m))ϕ)

1

sin(x-y) = sin(x)cos(y) - cos(x)sin(y) (Substitute Trig Identities) cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

rsin(((N/(N-m))ϕ) - ϕ) + (-1)c*sin((N/(N-m))ϕ)(r₂ᵖ) =

1

rsin((m/(N-m))ϕ) + (-1)c*sin((N/(N-m))ϕ) (AGREES with source Paper)(r₂ᵖ) = =

1 1

1

1

Norm = = = SQRT[[(1/(N-m))[mrcos(m/(N-m)) - cN cos(((N/(N-m))ϕ)]^2 +(speed) [(1/(N-m))[(-1)mrsin(m/(N-m)) + cN sin(((N/(N-m))ϕ)]^2]

rcos(((N/(N-m))ϕ) -ϕ) + (-1)c*cos((N/(N-m))ϕ)

Nϕ/(N-m) - ϕ = (N/(N-m) - 1)ϕ = N/(N-m) - (N-m)/(N-m) = (N-N+m)/(N-m) = m/(N-m)

r x₂ᵖrcos((m/(N-m))ϕ) + (-1)c*cos((N/(N-m))ϕ) r y₂ᵖ

Epicycloidal Wheel Profile: (from r₂ᵖ and derivatives and unit normal vectors, per source Paper..)r x₂ᵖ + ηᵪ₂ᵖ * R

r₂ = r y₂ᵖ + ηᵧ₂ᵖ* R

rsin(((N/(N-m))ϕ) - ϕ) + (-1)c*sin((N/(N-m))ϕ) + [(1/(N-m))[mrcos(m/(N-m)) - cN cos(((N/(N-m))ϕ)]/NORM] * Rr₂ = rcos(((N/(N-m))ϕ) -ϕ) + (-1)c*cos((N/(N-m))ϕ) + [(1/(N-m))[(-1)mrsin(m/(N-m)) + cN sin(((N/(N-m))ϕ)]/NORM] * R

∂ r₂ᵖ SQRT[(Tₓ₂ᵖ)² + (Tᵧ₂ᵖ)²]∂ϕ