07 4th Intrnational Matrials, Machinry and Civil Enginring Confrnc(MATMCE 07) Simulatd Analysis of Tooth Profil Error of Cycloid Stl Ball Plantary Transmission Ruixu Hu,a, Yuquan Zhang,b,*, Zhanliang Zhao,c, Liwn Zhang,d Dpartmnt of Automobil Enginring, Hbi Jiaotong Vocational and Tchnical Collg, No.9, Zhujiang Strt, Shijiazhuang, China. Norndar Intrnational LTD.,No.55, Yuhua East Road,Shijiazhuang, China a 40739@qq.com, b 3789063@qq.com, c 9993949@qq.com,d 46440@qq.com *corrsponding author ywords: Cycloid, plantary transmission, tooth profil rror, paramtr, limousin Abstract: Cycloid stl ball plantary transmission mchanism has good prospct whn bing usd as frquntly rciprocating transmission mchanisms of prcision machinry such as th windshild wipr driving mchanism of limousin. But manufactur/installation rror and tooth profil rror gratly affct th rror of cycloid stl ball transmission. Th quantitativ study of transmission accuracy affctd by manufactur/ procssing rror can control th vry rror in purpos and with spcial mphasis, and assur th transmission to rach xpctd accuracy. It is also vry important to study th tooth profil accuracy of cycloid plat, which is th ky transmission part of cycloid stl plantary transmission, and th factors affcting th accuracy as wll. This papr analyzs th rlationship btwn th transmission mchanism s structural paramtrs and cycloid tooth profil rror.. Introduction With th dvlopmnt of modrn industry and th us of modrn industrial products from arospac industry and robotic industry, tc., th mchanical transmission mchanism has dvlopd rapidly toward th high prformanc dirction such as high prcision, high fficincy and high spd; morovr, th prcision machinry rquir highr in both srvo transmission mchanism s accuracy and othr transmission prformancs. Cycloid stl ball plantary transmission mchanism is a nw typ of mchanical transmission mchanism dvlopd in rcnt yars, and has good application faturs such as no-backlash, good carrying capacity, high mshing fficincy, larg transmission ratio and wid rang, compact structur, and low nois, tc. It has good prospct whn bing usd as frquntly rciprocating transmission mchanisms of prcision machinry such as robots, manipulators, machin indxing mchanisms and sdan limousin. Ths prcision transmission machinry rquirs highr in transmission accuracy, whil th mshing rror causd by tooth profil rror of cycloid stl ball plantary transmission will affct th transmission accuracy obviously, thrfor, th study of tooth profil rror is important for th dvlopmnt of this kind of transmission. This papr mainly focuss on th study and analysis of th rlationship btwn th transmission mchanism s structural paramtrs and cycloid tooth profil rror, thus providing thortical basis for th analysis of th manufactur and transmission accuracy of th cycloid stl ball plantary transmission.. Structur and transmission principl Th structur of th cycloid stl ball plantary transmission mchanism is as shown in Figur. Cntr plat (3) is fixd to fram (9), and its lft nd fac is machind with outr cycloid nclosd Copyright (07) Francis Acadmic Prss, U 70
groov whos tooth numbr (wav numbr) is Z. Th plantary plat s right nd fac is machind with innr cycloid nclosd groov whos tooth numbr (wav numbr) is Z. Stl ball (5) is placd insid th staggrd ara of th abov two groovs. Th right nd fac of plat (8) (i.. output shaft) and lft nd fac of plat (6) ar providd with ring groovs whos numbrs ar th sam. Stl balls (7) ar placd insid th staggrd ara of th ring groovs; and th numbr of balls should b th sam as that of groovs. Th driv principl is as follow: Whn th driving ccntric shaft () prforms constant-spd rotation and drivs plat (6) for plantary motion, th disk (6)pushs stl ball (5) to mov. Th ball (5) is rstraind by th nclosd groov on plat(3) and pushs back th plat (6) for slow spin motion with an angl spd output by output shaft (8) through th stl ball constant-spd output mchanism (W mchanism) consisting of ring groovs on plats (6) and (8), and stl balls (7). In anothr word, for this transmission, th chang of its moving spd is ralizd through th constant-spd output mchanism and mshing motion btwn plantary plat, cntr plat and stl balls. 3 0 9 8 7 6 5 4 3 Figur Structur of cycloid stl ball plantary transmission 3. Tooth profil quation and analysis of causs of profil rror Sinc th formation principl of outr cycloid and factors affcting it ar th sam as thos of innr cycloid, this papr taks th outr cycloid as instanc to study and analyz th tooth profil rror affctd by th rror of structural paramtrs. 3.. Tooth profil quation of cycloid stl ball transmission Th tooth profil quation will b takn as basis for study of th tooth profil rror of cycloid stl ball transmission. Th thortical tooth profil quation of outr cycloid groov is shown as in: x y ( Z ( Z + ) cosθ cos[( + Z) θ] + ) sinθ sin[( + ) ] Z θ () Whr, is ccntricity; θ is rotation paramtr of curv. Figur (a) is th sctional viw at rotation axis, in which angl β is th groovd angl of cycloid nclosd groov (i.. th con angl of cutting tool); th A and A point paths ar rspctivly th actual tooth profils of plat (3) innr and outr sids; B and B point paths ar rspctivly th actual tooth profils of plat (6) innr and outr sids, and C and C point paths ar th thortical tooth profils of plat (3) and plat (6). Figur (b) is an nd sctional viw at A A and whr it is prpndicular to rotation shaft. Sinc th actual tooth profil of th cycloid groov is th quidistant lin of its thortical tooth profil, th actual tooth profil quations of th outr (innr) sid of th outr cycloid groov can b dducd rspctivly as follows: 7
x ( Z y ( Z + ) cosθ cos[( + Z) θ] ± r cos β cosϕ () + ) sinθ + ± sin[( Z) θ] r cos β sinϕ Whr, "+", "-" usd rspctivly for outr and innr tooth profils of outr cycloid groov; r stl ball radius. β B r β C C B A A n A C A n (a) (b) 3.. Analysis of rror causs Figur Sctional viw of mshing countrshaft From th abov quation (), w can s that th tooth profil rror is mainly causd by: rror of th ccntricity, of th cycloid s curat cofficint, rror of stl ball radius r (in th cycloid groov), and rror of groovd angl β,othr factors lik th vibration during procssing may caus tooth profil rror as wll. Th actual tooth profil rror is formd by th combination of various paramtr rrors, which is a highly nonlinar problm. As basic rror analysis, this papr focuss on th tooth profil rror only causd by structural paramtrs. 4. Analysis of tooth profil rror affctd by structural paramtr rror Th tooth profil rror dnots th distanc dviation btwn th actual tooth profil and th dsign tooth profil du to th structural paramtr rror in procssing. Its valu is dfind as th dviation at th tooth profil s normal dirction. According to th dfinition, th mathmatical modl of th tooth profil s normal rror L is stablishd as blow: L x cosα + y sinα (3) Whr, x, y -- ar rspctivly th actual tooth profil rrors at x, y coordinat dirctions; α -- angl btwn tooth profil s normal and axis x. From th formula (), tooth profil normal slop tαn α can b obtaind, and th rlationship btwn tαn α and sin α, cosα can b dducd rspctivly: sin α sinθ / sin[( + Z) θ] (4) + / cos[( + Z ) θ ] / cosα cosθ / cos[( + Z) θ] (5) + / cos[( + Z ) θ ] / Th influnc of structural paramtr on tooth profil rror is analyzd as blow. 7
4.. Tooth profil rror L causd by ccntricity rror Basd on ccntricity in th quation (), th tooth profil rrors at x, y coordinat dirctions ar obtaind by partial diffrntial mthod: x x {( Z + ) cosθ / cos[( Z + ) θ]} y y {( Z + )sinθ / sin[( Z + ) θ]} According to x and y, th tooth profil normal rror L can b obtaind whn th ccntricity is th singl rror: L x cosα + y sin α ( + Z) cos ( + Z)cosθ cos[( + Z) θ] ( + Z)sinθ sinθ θ + cos [( + Z) θ] + { sin[( + Z)]} sin[( + Z) θ] (6) + / cos[( + Z ) θ ]/ Tak th paramtrs of th sampl cycloid stl ball plantary transmission as instanc, and mak Z, Z 4, r 4 mm, k 0.55,.5 mm, and β 45.Tak on tooth profil of th outr cycloid plat for study; whn diffrnt positiv valus ar takn rspctivly as, a group of tooth profil normal rrors L can b obtaind through calculation, whos curv is shown as in Figur 3. Tak th curv obtaind whn 0.0 mm as xampl and analysis th influnc of on tooth profil rror. According to th Fig., th tooth profil normal rror is smallr at th ddndum (b and c in Fig.), whil th normal rror is largr at th addndum (a in Fig.), i.. th amplitud of actual tooth profil bcoms largr. Convrsly, ngativ valus will affct in th opposit way. Figur 3 L chang curv 4.. Th rror of tooth profil L causd by curat cofficint rror k Similarly, basing on curtat cofficint in th quation () to obtain th actual tooth profil rrors x and y at x, y coordinat dirctions by partial diffrntial mthod. Thn th tooth profil normal rror can b dducd whn is th only rror. L ( Z + ) cos θ sinθ {cosθ cos[( + Z) θ] sinθ sin[( Z + ) θ]} (7) + / cos[( + Z ) θ ] / Whn diffrnt positiv valus ar takn rspctivly as k, a group of tooth profil normal 73
rrors L can b obtaind through calculation, whos curv is shown as in Figur 4. Tak th curv obtaind whn k 0.0 as xampl and analysis th influnc of k on tooth profil rror. According to th Fig., th tooth profil normal rror is smallr at th ddndum (b and c in Fig.), whil th normal rror is largr at th addndum (a in Fig.). Sinc th vry rror is ngativ valu, th amplitud of actual cycloid tooth profil bcoms smallr and th profil bcoms gntl. Convrsly, ngativ valus k will affct in th opposit way. Figur 4 L chang curv 4.3. Tooth profil rror Lr causd by stl ball radius rror r Basing on th stl ball radius r in th quation () to obtain th actual tooth profil rrors x r and yr at x, y coordinat dirctions by partial diffrntial mthod. Thn th tooth profil normal rror can b dducd whn r is th only rror. L r cosϕ {cosθ / cos[( + Z) θ]} + sinϕ sinθ / sin[( + Z) θ] cos β r (8) + / cos[( + Z ) θ ]/ Whn diffrnt positiv valus ar takn rspctivly as r, a group of tooth profil normal rrors Lr can b obtaind through calculation, whos curv is shown as in Figur 5. Tak th curv obtaind whn r 0.0 mm as xampl and analysis th influnc of r on tooth profil rror. According to th Fig., th tooth profil normal rror is smallr at th ddndum (b and c in Fig.), whil th normal rror is largr at th addndum (a in Fig.), i.. th amplitud of actual cycloid tooth profil bcoms largr and th profil bcoms bnt. Convrsly, ngativ valus r will affct in th opposit way. Figur 5 Lr chang curv 74
4.4. Tooth profil rror Lβ causd by groovd angl β Basing on th groovd anglβin th quation () to obtain th actual tooth profil rrors xr and y r at x, y coordinat dirctions by partial diffrntial mthod. Thn th tooth profil normal rror can b dducd whn β is th only rror. Whn diffrnt valus ar takn rspctivly as cosϕ{cos[( + Z) θ] cosθ / } sinϕ sinθ / sin[( + Z) θ] L β r sin β β (9) + / cos[( + Z ) θ ]/ β, a group of tooth profil normal rrors can b obtaind through calculation, whos curv is shown as in Fig. 6. Tak th curv L β obtaind whn β 0.0 radas xampl and analysis th influnc of β on tooth profil rror. According to th Fig., th tooth profil normal rror is smallr at th ddndum (b and c in Fig.), whil th normal rror is largr at th addndum (a in Fig.) and is of ngativ valu, i.. th amplitud of actual cycloid tooth profil bcoms smallr. Convrsly, ngativ valus β will affct in th opposit way. Fig.6 Lβ chang curv In summary, Fig. 3 ~ 6 shows th influnc dgr of outr cycloid groov s structural paramtr dviation on tooth profil rror, and influnc law of that to th shap changing of actual tooth profil. Fig. 3 and 5 show that th curvs of influnc of and r on tooth profil rror ar similar in thir shaps. Fig. 4 and 6 show that th curvs of influnc of and β on tooth profil rror ar similar in thir shaps. Whn th paramtr rrors ar th sam in valus, th amplitud of variation ΔL causd by and β will b largr, and th whol tooth profil will hav largr rror, thn th amplitud of variation ΔL causd by will b largr than that causd by r. Thrfor, th structural paramtrs, β and k ar th most snsitiv ons, thir dviations must b strictly controlld in th dsign of cycloid stl ball plantary transmission, whil th paramtr r is allowd to b fr from this kind of strictly control. For innr cycloid groov, similar rror changing law can b obtaind through calculation and drawing. 5. Conclusion () Equations (6) ~ (9) ar applicabl to rror analysis of cycloid stl ball plantary transmission. () It can b larnd from Fig. 3~6 that th tooth profil rror can b gratly affctd by,, β, r. (3) Eccntricity, stl ball radius r, curtat cofficint k, groovd angl β ar important paramtrs affcting tooth profil rror; and dviations of, β, k should b particularly controlld as 75
possibl in th dsign and procssing of cycloid plat. Rfrncs [] QuJifang. (993),Movabl tooth transmission thory, China Machin Prss. [] An Zijun, tc. (996), Comprhnsiv study of tooth profil of cycloid stl ball transmission, Chins Journal of Mchanical Enginring. [3] QuJifang,AnZijun. (994),Rsarch on zro claranc cycloid ball transmission. Chins Journal of Mchanical Enginring(English Edition), 7():7~ [4] Zhang Shuying, Liang Guoming. (986),Comprhnsiv analysis of procssing rror of cycloid pinwhl mshing gars, Journal of Mchanical Enginring [5] Mao Yingtai. (98),Dviation thory and accuracy Analysis,Dfns Industry Prss. 76