Motion Simulation of Cycloidal Gears, Cams, and Other Mechanisms

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1 Motion Simultion of Cyloidl Gers, Cms, nd Other Mehnisms Shih-Ling (Sid) Wng Deprtment of Mehnil Engineering North Crolin A&T Stte University Greensboro, NC Tel (336) , Fx (336) , Introdution Cyloids re urves tred by point on the irumferene of irle tht rolls on stright line or nother irle. The ltter tegory is often referred s trohoids. Mehnisms with yloidl geometry inlude ms, gers, ger trins, rotry engines, nd blowers. Cyloidl gers, whose teeth hve yloidl profiles, re now lmost obsolete, repled by involute gers beuse of mnufturing osts. The Wnkel engine, with its rotor nd hmber bsed on yloidl urves, hs lekge problems nd hs not gined wide eptne. Cms with yloidl displement re being repled by those with polynomil funtion displement. Consequently, most textbooks in kinemtis do not over these topis or over them with limited sopes. Students nd prtiing engineers therefore do not hve proper exposure of these subjets. With new wys of ger mnufturing like injetion molding, yloidl gers re reemerged s n option. Additionlly, yloidl gers should lso ply n importnt role in the emerging field in Miro Eletro-Mehnil Systems (MEMS) [1]. Moreover, there is renewed interest [2] in the Wnkel engine for hybrid vehiles with hydrogen fuel. To help students understnd nd visulize the motion of these mehnisms, the uthor hs developed oursewre on yloidl gers, ms, nd other mehnisms, with simultion files generted from MATLAB, Working Model 2D, nd visulnstrn 4D. Animtions of different yloidl urves n be found t Eri Weisstein s World of Mthemtis, Wolfrm Web Resoure [3,4,5]. An intertive Jv progrm Spirogrph [6] is vilble on the web. However, omputer simultion of yloidl mehnisms nnot be found. All urves shown in this pper re generted with MATLAB, nd eh MATLAB file n genertes nimted simultion. Working Model 2D files, bsed on the geometry generted from these MATLAB files, help students visulize the motion of these yloidl mehnisms. The simultion files re hyperlinked with text files ontining bkground informtion nd yloidl equtions. Cyloids, Trohoids, nd Spirogrph Cyloid is urve tred by ny point rigidly tthed to irle of rdius, t distne b from the enter, when this irle rolls on stright line. The eqution is: Pge

2 x= θ bsinθ (1) y = bosθ where θ defines the ngle of the moving rdius. The urve is lled prolte or urtte if b < or b >, s in Figure 1 nd 2 respetively. When b =, s in Figure 3, it is the speil se of the yloid. Cyloid is subset of trohoids, nd sometimes is treted s synonym of trohoid. Trohoids [7,8] re urves generted by tring the pth of one point on the rdius of one irle (the driven irle) s the irle rolls on nother irle (the bse irle). It should be noted tht the point lies on the sme rigid body s the irle, but is not onfined to the irumferene of the irle, nd sometimes even lies outside the extents of the irle. The epitrohoid (sometimes referred s epiyloid) is urve tred by ny point rigidly tthed to irle of rdius, t distne b from the enter, when this irle rolls without slipping on outside of fixed irle of rdius. The epitrohoid equtions is: x= ( + )os( θ ) bos[(1 + ) θ ] (2) y = ( + )sin( θ ) bsin[(1 + ) θ ] where θ defines the ngle whih the moving rdius mkes with the line of enters. The urve is lled prolte or urtte if b < or b >, s in Figure 4 nd 5 respetively. When b =, s in Figure 6, it is the speil se of the epitrohoid. The hypotrohoid (sometimes referred s hypoyloid) is urve tred by ny point rigidly tthed to irle of rdius, t distne b from the enter, when this irle rolls without slipping on inside of fixed irle of rdius. The hypotrohoid equtions is: x= ( )os( θ ) bos[(1 + ) θ ] (3) y = ( )sin( θ ) bsin[(1 + ) θ ] where θ defines the ngle whih the moving rdius mkes with the line of enters. The urve is lled prolte or urtte if b < or b >, s in Figure 7 nd 8 respetively. When b =, s in Figure 9, it is the speil se of the hypotrohoid. Figure 10 shows the epitrohoid nd hypotrohoid in eh of the three ses. Spirogrph, s shown in Figure 11, is populr toy bsed on yloids. A set of ridged plsti irles with ridged edges (like ger teeth) retes the intrite urves when pen tred the pth of the smll irle s it rolled long inside or outside of bigger irle, s shown in Figures 12 nd 13. Wnkel Engine Wnkel engine, rotry engine, hs rotor shped like n equilterl tringle with urved side nd moves within sttor, s shown in Figure 14. This mehnism llows for volume hnges within the ombustion hmber. In theory, rotry engine should be 2 Pge

3 smooth nd muh more effiient thn onventionl reiproting piston engine. Furthermore, the number of moving omponents is gretly redued in omprison to onventionl engine. This results in onsiderble redution in weight, n improvement in effiieny nd redution in mnufturing osts. The rotor hs n internl ger ut into it, nd s the rotor moves round the hmber, its ring ger drives pinion on eentri output shft. Thus the pinion nd ring ger beome the bse nd driven irles (respetively) in the peritrohoid s shown in Figure 15. The peritrohoid urve, s shown in Figure 16, is used s the bore of the Wnkel engine. The peritrohoid is urve tred by ny point rigidly tthed to irle of rdius, t distne b from the enter, when this irle rolls without slipping on outside of fixed irle of rdius. The peritrohoid [9] is similr to the epitrohoid exept tht in peritrohoid, the smller bse irle is inside the driven irle. The peritrohoid eqution is: x= ( )os[(1 + ) θ ] bos( θ ) (4) y = ( )sin([(1 + ) θ ] bsin( θ ) where θ is the ngle whih the moving rdius mkes with the line of enters. is the rdius of moving irle; b is the distne from the tred point to the enter of moving irle; is the rdius of bse irle. Another form of the eqution n be represented: x= ( )os( λθ) + bosθ (5) y = ( )sin( λθ) + bsinθ θ in this eqution is defined s the ngle whih the fixed rdius mkes with the line of enters. Notie tht to get losed urve, one must mintin ertin reltionship between the two irles. Nmely, the rdii of the bse nd the driven irles re onstrined s: λ = ( λ 1) λ 1, λ N (6) Peritrohoidl urves hve lobes, with the number of lobes in the urve being equl to γ. Although Wnkel engines n be produed from ny trohoid, the two-lobed epitrohoid is the preferred profile for the outer hmber. To find the interior envelope of given trohoidl bore, one n invert the Wnkel engine mehnism. When the pinion rolls inside the ring ger, every point on the peritrohoid (whih is still rigidly fixed to the pinion) tres out some urve in spe. The olletion of ll these urves hs both n outer nd n inner profile. Figure 17 illustrtes the lous of urves generted by rotting two-lobed peritrohoid. The eqution for ontour of these urves is: 3 Pge

4 2 ( + )( ) 2 x=± 2( ) 1 sin (1 + ) η os(2 η) os (1 + ) η b 2 ( + )( ) b os(2 η) + sin 2(1 ) η os(2 η) b + y =± + η b 2 ( + )( ) 2 2( ) 1 sin (1 ) sin(2 )os 2 ( + )( ) bsin(2 η) sin 2(1 ) η os(2 η) b + η (7) (1 + ) η where is the rdius of moving irle; b is the distne from the tred point to the enter of moving irle; is the rdius of bse irle, η is from 0 to 360 s shown in Figure 18. The rotor profile, s in Figure 19, is the interior portion of the ontour in Figure 18. It is obtined by plotting the positive roots over the rnge η = [30, 90 ], [150, 210 ] nd [270, 330 ]. Figure 20 shows both the rotor (Figure 19) nd hmber (Figure 16). This geometry is exported to Working Model 2D to produe the geometry for motion simultion, s show in Figure 14. Cms with Cyloidl Displement The displement s of the m follower is the projetion of point of yloidl urve, whih is generted by rolling irle on line, to the s-xis (y-xis), s shown in Fig 21. The eqution of yloidl displement is: s h θ 1 θ = sin 2 π (8) β 2π β where h is the totl rise or lift; θ is the mshft ngle; β is the totl ngle of the rise intervl. Note tht Eqution (8) is in the sme form s Eqution (1). The veloity, elertion nd jerk equtions re: h θ v = 1 os 2π β β h θ = 2π sin 2π 2 β β 2 h θ j = 4π sin 2π 3 β β The s v j digrms re shown in Fig. 22. The m profile reted is then shown in Figure Pge

5 Cyloidl Gers The yloidl tooth profile [10] of ger is generted by two irles rolling on the inside nd outside of the pith irle for the hypoyloidl flnk nd the epiyloidl fe respetively, s shown in Fig 24. Figure 25 shows the urves superimposed on ger. Note tht in generting one side of tooth, the two generting irles roll in opposite diretions. Note tht these two irles re of the sme size nd the generting irle s rdius is 1/3 of the rdius of the bse irle. The yloidl tooth profile ws extensively used for ger mnufture bout entury go beuse it is esy to form by sting. Involute gers hve ompletely repled yloidl gers for power trnsmission s involute gers n be produed urtely nd heply using hobbing mhines. Nevertheless, yloidl gers re extensively used in wthes, loks, nd ertin instruments in ses where the question of interferene nd strength is prime onsidertion. A yloidl tooth is in generl stronger thn n involute tooth beuse it hs spreding flnks in ontrst to the rdil flnks of n involute tooth. In wthes nd loks, the ger trin from the power soure to the espement inreses its ngulr veloity rtio with the ger driving the pinion. In wth, this step up my be s high s 5000:1. It is therefore neessry to use pinions hving s few s six or seven teeth. In the field of Miro Eletro-Mehnil Systems (MEMS), lrge speed redution is expeted, nd therefore yloidl gers n ply n importnt role. Roots Blowers A Roots blower is positive displement mhine tht uses two or more rotting lobes in speilly shped rotor, usully shped like the figure-8, s shown in Fig. 26. The lobes intermesh with eh other nd re driven by pir of meshing gers of equl size on the bk of the se. As the rotors turn, fixed quntity of ir is drwn in from the opening t the inlet trpped between the rotor nd the sing, nd then fored out the dishrge. There is no tul ompression rtio built into the mhine, it is simply n ir mover. The rotors re yloids, nd the yloidl urves re ombintion of the epiyloidl nd epiyloidl urves, just s in yloidl gers. Note tht both moving irles re of the sme size nd the rdius of the moving irle is ¼ of the rdius of the bse irle. In the modern pplition, the Roots blower hs three lobes on eh rotor, s shown in Fig. 27, nd is used for low-pressure superhrger on diesel engines. Note tht the moving irles re of the sme size nd the rdius of the moving irle is 1/6 of the rdius of the bse irle. Epi-Cyloidl Ger Trins The epiyloidl ger trin, s shown in Fig. 28, is similr to the plnetry ger trin shown in Fig. 29, exept tht the ring ger is repled by seond sun ger. The ger trin is oxil nd offers high redution rtios. 5 Pge

6 Disussion Mthemtil terms of trohoidl urves nd their pplitions in different mehnisms re reviewed in this pper. These topis do not get suffiient overge in typil textbook, nd visulizing them in motion is very hllenging. Eh yloidl mehnism disussed in this pper hs motion simultion file developed using MATLAB, Working Model 2D, nd visulnstrn 4D files. These simultion files re prt of multimedi hndbook of mehnil devies [11] with over 300 simultion files. In this multimedi resoure, hyper-linked text files nd simultion files in MATLAB, Working Model 2D nd visulnstrn 4D will ssist students nd working professionl gin suffiient informtion in just-in-time mode. The multimedi oursewre is now under ontrt with MGrw-Hill for publition in Referenes 1. Hydrogen Powered Rotries, URL: 2. Sndi Ntionl Lbortories, URL: 3. World of Mthemtis, URL: 4. World of Mthemtis, URL: 5. World of Mthemtis, URL: 6. Spirogrph, URL: 7. Korn, G.A. nd Korn T.M., Mthemtil Hndbook for Sientists nd Engineers, MGrw-Hll, Bumeister, T., Avllone, E.A. nd Bumeister III, T., Mrk s Stndrd Hndbook for Mehnil Engineers, eighth edition, MGrw-Hll, Hubert, C.J. nd Kinzel, G.L., "Design nd Mnufture of Wnkel Engine Model - An Edutionl CAD/CAM Lb Exerise" Proeedings of the Sixth Ntionl Applied Mehnisms & Robotis Conferene, AMR , Cininnti, Ohio, Deember 12-15, Albert, C.D. nd Rogers, F.S., Kinemtis of Mhinery, John Wiley & Sons, In., Wng, S-L., A Multimedi Hndbook of Mehnil Devies, in CD Proeedings of 2001 ASEE Annul Conferene, June 24-27, 2001 Albuquerque, NM. Figure 1 Prolte Cyloid Figure 2 Curtte Cyloid Figure 3 Cyloid 6 Pge

7 Figure 4 Prolte Epitrohoid Figure 5 Curtte Epitrohoid Figure 6 Epitrohoid Figure 7 Prolte Hypoyloid Figure 8 Curtte Hypoyloid Figure 9 Hypoyloid Figure 10 Hypoyloid nd Epitrohoid Prolte, Curtte, nd Speil Cse Figure 11 Spirogrph 7 Pge

8 Figure 12 Spirogrph- Epitrohoid Figure 13 Spirogrph- Hypotrohoid Figure 14 Wnkel Engine Figure 15 Bse nd Driven Cirles Figure 16 Two-Lobed Peritrohoid Figure 17 Inversion of the Peritrohoid Figure 18 Contour of the Inversion 8 Pge

9 Figure 19 Interior of the Contour s the Rotor Figure 20 Rotor nd Chmber Figure 21 Cyloidl Displement of Cm 4 s 2 v j Figure 22 s v j digrms 9 Pge

10 Figure 23 A Cm with Cyloidl Displement Figure 24 A Cyloidl Ger Tooth Profile Figure 25 A Cyloidl Ger with Cyloidl Curves Figure 26 A Roots Blower with Two Lobes 10 Pge

11 Figure 27 A Roots Blower with Three Lobes Figure 28 Epi-Cyloidl Ger Trin Figure 29 Plnetry Ger Trin 11 Pge