Analysis on the Influence of Backlash and Motor Input Voltage in Geared Servo System

Transcription

1 - Analysis on he Influence of Backlash an Moo Inpu Volage in Geae Sevo Syse J. H. Baek Y. K. Kwak an S. H. Ki Absac his pape analyzes how he inpu volage of he oo an he agniue of he oal backlash of a sevo syse wih a gea euce affec he fequency esponse chaaceisic of he sevo syse. he banwih of he syse is efine as he ani-esonance fequency which appeas in he fequency esponse chaaceisic. I is foun ha he aoun of syse s banwih eucion ue o he backlash changes gealy wih oo inpu volage. I is also shown ha when he oo inpu volage is infinie he sevo syse has a banwih of he syse ha oes no have any backlash. hough his wok i has becoe possible o eeine he axiu peissible agniue of oal backlash o saisfy he esie banwih fo a sevo syse wih a gea euce. Inex es Ani-esonance Backlash Banwih I. IRODUCIO I EIGE syses such as sall auonoous vehicles unanne aiplanes an guie issiles equie api esponsiveness an ousaning aapabiliy o envionens. Such equieens nee auoaic sevo evices wih fas esponses. Up o he pesen ie sevo syses wih gea euces have been wiely use in fiels incluing auonoous vehicles an guie issiles whee hee ae liiaions on insallen space an weigh. he size an weigh of insalle sevo syses on flying objecs such as guie issiles ae vey ipoan as inceases in he weigh of he loae sevo syse shoens he isance ha he objec can fly on a consan aoun of fuel. heefoe when he weigh of he sevo syse inceases he aoun of fuel shoul also be incease o ainain he sae flying isance. Howeve since he incease aoun of fuel also inceases he weigh of a guie issile he weigh incease of he sevo syse has a significan effec on he pefoance of a guie issile. he velociy conol banwih an esponsiveness of sevo syses ae gealy liie by he ani-esonance an esonance Manuscip eceive Januay 5 3. his wok was suppoe in pa by G Innoek Copany.. J. H. Baek is wih G Innoek Co. 48- Mabuk-i Guseong-eup Yongin-ciy Kyonggi-o Koea (el: ; fax: ; e-ail: jhbaekb@lginnoek.co. Y. K. Kwak an S. H. Ki ae wih he Depaen of Mechanical Engineeing Koea Avance Insiue of Science an echnology 373- Guseong-ong Yuseong-gu Daejeon 35-7 Koea (e-ail: kyk@kais.ac.k kish@kais.ac.k. fequency appeaing in he oo angula velociy oupu o he oo oque inpu [] []. he ani-esonance fequency is efine as he sevo syse s velociy conol banwih so i is necessay o incease he ani-esonance fequency of he syse fo he sevo syse o have an even fase acking an esponsiveness []. Fo his pupose we nees he eseach ha ais owas esiaing he banwih of a sevo syse wih a gea euce a he esign sage an expaning he banwih a he esign evise sage. In elae eseach Rue epoe ha he siffness of iving linkage such as a shaf gealy affece he banwih of he syse [3]. Dhaouai e al. suie ha he backlash affece he ani-esonance an esonance fequencies of he syse [4]. Jang an Oh veifie ha he incease of backlash agniue ae he banwih euce in he expeien [5]. hese woks showe ha he banwih eucion of he syse ue o backlash is sall. In oe o expan he banwih wihou inceasing he weigh of he syse Bigley e al. use he opial conol echnique o sae equalizaion conol echnique o iniize o eliinae he effec of he ani-esonance fequency on he sevo syse [] [] [6] [7]. Howeve he opial conol echnique equies pecise oeling an he sae equalizaion conol echnique nees vey lage oo oupu when he oen of ineia of loa is lage aking i ifficul o anufacue he syse. In his pape i is shown ha he backlash has a significan effec on he banwih of he syse accoing o he apliue of inpu volage of he oo. I is also foun ha a echanical way o expan he banwih wihou inceasing he weigh of he syse is o euce he agniue of backlash. heefoe he effec of backlash agniue will be analyze in he change of oo s inpu volage. A. Moeling II. HEOREICA APPROACH Fig. is a scheaic iaga of a sevo syse wih a gea euce insalle on a guie issile he segen gea on he fixe shaf oes no oae an he enie poion of he shae aea in Fig. pinion he oaing shaf gea pinion he oo an he beaing oae aoun he OO ' axis wih he oaion of he oo. I is assue ha he beaings in each oaing shaf suppo each shaf wihou cleaance by peloa.

2 - Poenioee Fixe Shaf (shaf achoee Beaing b O P O M k Moo Pinion Gea oa Roaing Shaf (shaf Pinion Gea Base p J J Fig.. he sucue an oel of sevo syse he sucue of sevo syse he oel of sevo syse he aping effec of each coponen can be oele by Yang an Sun s eho [8]. Howeve I is assue ha he aping effec can be neglece because he consiee syse consiss of siff coponens wih sho lengh an high siffness. he effec of viscous ficion is also assue o be negligible because i can be consiee using esonan Q faco if necessay [] [9]. Since i is in a sevo syse wih a wo-sage gea euce if pinion is fixe he oal backlash b of he syse easue a he oaing axis OO ' of he loa can be expesse as b = b + b ( whee b is he angula backlash agniue expesse in egees obaine by easuing he backlash beween pinion an gea ; b is he angula backlash aoun expesse in egees obaine by easuing he backlash beween pinion an gea ; an is he evoluion gea aio beween pinion an gea. he oal backlash was easue using poenioee shown in Fig.. Because he backlash b eceases wih he gea aio he backlash b has a oinan effec on he syse [5] []. heefoe i is assue ha he oal backlash b easue a he final loa sage exiss a he locaion of backlash b. Because he oens of ineia of gea he oaing shaf an pinion ae vey sall copae o he agniue of he oens of ineia of he oo an loa hese oens of ineia wee assue o be negligible an only he osion siffness was consiee. he poion enclose by a ouble o line is he oaing pa of Fig. an he oel of his is shown in Fig.. he equivalen osion siffness in he Fig. can be epesene as whee k ( s k kks k = ( k + k is he equivalen osion siffness of gea an he oaing shaf ( / a ; is he osion siffness ue o k ooh siffness beween pinion an gea ( / a ; an k is he osion siffness of he oaing shaf ( / a. s he osion siffness is given as follows []: k k E pz pez = (3 4( E Z + E Z p whee is he pich cicle iaee of gea ( ; p an ae he ouli of elasiciy of pinion an gea E especively ( / ; an ae he elasiciy Z p efoaion facos of pinion an gea especively; an is he ooh face of gea (. An he elasiciy efoaion faco is also given as follows []: yi Z = i (4 whee siffness y i k s s Z y i E p is he ewis fo faco of gea i. Also he osion of he oaing shaf use in ( can be calculae as 4 π Gs s ks = (5 3 s whee G is he shea oulus of elasiciy of he oaing shaf ( / is he iaee of he oaing shaf ( an s s s is he lengh of he oaing shaf beween gea an pinion (. In he sae way he equivalen osion siffness beween gea an he fixe shaf can also be wien as kg ks k = (6 k + k whee ( g is he equivalen osion siffness of gea an he fixe shaf ( / a is he osion siffness ue o ooh siffness beween pinion an gea an k is he osion siffness of he fixe shaf. A his ie he osion siffnesses an can be obaine by equaions like (3 an (5. kg k S B. Equaions of Moion k g In his suy a peanen agneic fiel ype DC oo wih a achoee was use. he elecical equaions fo his oo ae given as follows []: ia V = a + Ria + k & b (7a s

3 - 3 = kia (7b whee is he angula ansission eo V ( = p (a is he angle of oaion of he = k & s (7c whee V is he oo inpu volage (V loa (a an δ is / of he agniue of oal backlash a is he inucance of he oo s aaue ( H i is he cuen of easue a he final loa sage when pinion is fixe he oo s aaue ( A R is he esisance of he oo s aaue (Ω k b is he back ef consan ( V s a is he angle of oaion of he oo pinion ( a is he oo oque ( k is he oo oque consan ( A V is he oupu volage of he achoee (V an is he achoee sensiiviy V s a. he equaion of k s ( oion fo he oo can be wien as J & = f sign( & (8 whee ( kg J is he oen of ineia of he oo oo is he oque exee on boh ens of he g equivalen osion sping ( is he gea aio beween pinion an gea is he oo s saic ficion oque ( f an sign ( is he sign of he value insie (. he elaion beween he angle of oaion of pinion an he angle of oaion of gea can be expesse as = (9 whee is he angle of oaion of gea ( a. he oque can be epesene as = k p ( whee p is he angle of oaion of pinion ( a. hen he elaion beween he oque an he oque of he loa can be epesene as = ( whee is he oque of he loa ( is he evoluion gea aio beween pinion an gea = + an is he aio of he pich cicle iaee ( beween pinion an gea ( = a g p /. Fo he elaion beween ( an ( he angle of oaion of pinion can be obaine as follows: p = k ( Due o he backlash beween pinion an gea he oque of he loa can be expesse as follows: k( δ = k( + δ p > δ < δ < δ (3 b o ( δ = π b 36 (a. Hee he ea-zone oel was use fo he backlash oel [] [3]. he equaion of oion fo he loa can be epesene as J & = sign( & (4 f whee J is he oen of ineia of loa ( kg an is he loa s saic ficion oque (. f C. ansfe Funcion Analysis When he backlash δ he ficion oques an of he equaions of oion eive in secion II.B ae assue o be he sevo syse wih a gea euce becoes a linea syse. In his case he ansfe funcion of oo angula velociy & o he oo oque can be obaine as f ( J s + keq [ J J s + k ( J + J ] f & ( s = (5 ( s s whee is he oal gea aio ( = an k is he oal equivalen osion siffness of he syse. hen he oal equivalen osion siffness can be expesse as kk keq = (6 k + k eq he fequency coesponing o he zeo an pole of (5 is calle ani-esonance fequency an esonance fequency especively an hey can be wien as an f f AR R keq = (7a π J keq = (7b π J J /( J + J whee f is he ani-esonance fequency ( Hz an f is AR he esonance fequency (Hz. Fo (7a an (7b i can be seen ha he ani-esonance an esonance fequencies of a sevo syse wih a gea euce ae eeine by he gea euce s sucual siffness gea aio an he oens of ineia of he oo an loa when backlash an ficion oques wee assue o be. D. he Effec of Backlash Incease he oel s escibing funcion gain can be obaine by applying he escibing funcion eho on he backlash oel as shown in Fig.. he noalize escibing funcion gain is given as follows [](his is also pesene in Fig..: eq R

4 - 4 δ δ δ ( A = sin + (8 π A A A whee (A is he noalize escibing funcion gain an A is he apliue of he inpu sinusoial signal. he backlash has a gain beween an an he incease in backlash agniue causes a ecease in he equivalen osion siffness of he syse. he effecive equivalen osion siffness euce by he backlash can be expesse as k eff = ( A k (9 whee (. eff δ is he effecive equivalen osion siffness δ Gain δ / A Fig.. he oel an escibing funcion gain of backlash Backlash oel using ea-zone oel he escibing funcion gain of ea-zone oel In he en he backlash incease of he gea euce insie he sevo syse euces he effecive equivalen osion siffness. heefoe he ani-esonance an esonance fequencies of he syse ae euce. III. SIMUAIOS A. he Effec of he apliue of Moo Inpu Volage he siulaions wee pefoe by Malab Siulink. he sinusoial volage was supplie o he oo fo Hz o 3 Hz. he oo s angula velociy esponse wih espec o each excie fequency was saple wih he ie ineval of.5 s an he syse s gain an phase wee foun by he fequency analysis of he obaine oo s angula velociy aa. he fequency esponse chaaceisics of he sevo syse obaine in his way ae shown in Fig. 3 an Fig. 3. In he case of Fig. 3 he aoun of ecease of he ani-esonance an esonance poins when he agniue of oal backlash is.8 ae salle han he case in Fig. 3 an he aoun of ecease of he ani-esonance an esonance poins ue o aiional backlash incease is salle han in Fig. 3. Because he escibing funcion gain of he backlash oel is a funcion of no only he agniue of backlash bu also he apliue of he inpu sinusoial signal he angula ansission eo is incease accoing o he incease in oo velociy ue o he incease of he oo inpu volage. In conclusion he gain of he escibing funcion is incease in spie of he sae aoun of backlash. heefoe fo he sae aoun of backlash Fig. 3 which has a lage oo inpu volage han Fig. 3 has highe ani-esonance an esonance fequencies. he eason why hee is a significan eucion of he ani-esonance an esonance fequencies when he agniue of he backlash is.8 is because when he backlash is he fequencies ae no affece by he apliue of oo inpu volage howeve when he backlash is no hey ae affece by he oo s inpu volage. heefoe when analyzing he effec of backlash on fequency esponse chaaceisic he axiu inpu volage of he oo is vey ipoan. When he aoun of oal backlash is.8 he change in ani-esonance an esonance fequencies ue o he incease in he oo s inpu volage is pesene in Fig. 4 an when he oo inpu volage is incease infiniely i can be seen ha he syse has a banwih of he syse ha oes no have any backlash. Howeve since hee is a lii o he axiu inpu volage of he oos he oo s axiu volage use in his suy was liie o 7.9 V pk. B (V / V Phase (Degee b = o b =.8 o b =.6 o b =.4 o Fequency (Hz B (V / V Phase (Degee b = o b =.8 o b =.6 o b =.4 o Fequency (Hz Fig. 3. he Boe iaga V /V of gea eucion sevo syse V =. V pk V = 7.9 V pk Fequency (Hz he inpu volage of oo (V pk Ani-esonance Fig. 4. he change of ani-esonance & esonance fequencies accoing o he inpu volage of oo (b =.8

5 - 5 B. he Effec of Backlash Magniue his secion looks ino changes in he ani-esonance an esonance fequencies of sevo syses wih a gea euce which ae cause by he change in he gea euce s backlash agniue when he axiu inpu volages of he oo ae. V pk an 7.9 V pk. Boh Fig. 5 an Fig. 5 show ha he syse s ani-esonance an esonance fequencies ecease as backlash inceases. he ecease ae of he ani-esonance an esonance fequencies seen in Fig. 5 which has salle oo inpu volage is uch geae han he ecease ae seen in Fig. 5 an also when he agniue of oal backlash is equal he ani-esonance an esonance fequencies shown in Fig. 5 ae uch salle han hose shown in Fig. 5. When he oo of he sevo syse is eeine an consequenly he axiu inpu volage is liie i can be seen ha he aequae eho o expan he syse s banwih wihou inceasing he syse s weigh is o euce he agniue of he syse s backlash. hee ae wo ways o euce he agniue of backlash in he syse. One is o euce a isance beween he shaf of a pinion an ha of a gea he ohe is o use a pecision anufacuing achine [4]. Fequency (Hz Fequency (Hz oal Backlash (Degee Ani-esonance oal Backlash (Degee Ani-esonance Fig. 5. Ani-esonance & esonance accoing o backlash vaiaion V = 7.9 V pk V =. V pk C. Deeining he Moo s Inpu Volage an Backlash Magniue his secion eals wih he change in ani-esonance fequency cause by he agniue of backlash an he inpu volage of he oo. he esuls ae illusae in Fig. 6. In he case ha he sevo syse has ani-esonance fequency ha is banwih ha he esigne inens he inpu volage of he oo an he agniue of backlash can be eeine by Fig. 6. Fo exaple when he sevo syse wih a gea euce has a banwih of ove Hz an a oo wih a axiu volage of V is use he axiu allowable backlash agniue on he syse woul be.3. Howeve if a oo wih a axiu inpu volage of 3. V is use o euce he coss of pocessing an anufacuing he allowable backlash agniue woul be.8. Ani-esonan Fequency (Hz oal Backlash (Degee. V 6. V 3. V 7.9 V. V Fig. 6. Ani-esonance fequency accoing o he backlash an he inpu volage of oo Fo his i can be seen ha he axiu allowable backlash agniue in he syse epens on he axiu inpu volage of he oo an he aequae backlash agniue an he equie oo volage can be eeine fo Fig. 6. Howeve i is expece ha he value of ani-esonance fequency shown in Fig. 6 woul be euce in an expeien because i is obaine une he assupion ha he effecs of aping an viscous ficion ae negligible. heefoe in oe o fin he ani-esonan fequency oe accuaely he effecs shoul be consiee by calculaing he fequency eucion aio ue o he esonan Q faco. he eaile conens fo his consieaion ae in [] an [9]. IV. COCUSIO In his suy he change in he syse s banwih ue o he change in he apliue of he inpu volage of he oo an he agniue of backlash has been invesigae. I has been foun ha when backlash exiss he effec of he oo s inpu volage on banwih is vey significan. I has also been shown ha when analyzing he banwih of he sevo syse wih a gea euce one shoul consie he axiu inpu volage of he oo as he backlash eucion aouns equie o expan he banwih of he sevo syse iffes epening on he ha axiu inpu volage. Also i has becoe possible hough his suy o eeine he aoun of axiu allowable backlash o saisfy he banwih equieen of he syse if he oo has aleay been eeine.

6 - 6 ACKOWEDGME We woul like o hank G Innoek Co. fo suppoing his suy. REFERECES [] W. J. Bigley Wieban base oion isolaion conol via he sae equalizaion echnique Opical Engineeing vol. 3 no. pp [] W. J. Bigley an V. J. Rizzo Wieban linea quaaic conol of a gyo-sabilize eleco-opical sigh syse IEEE Conol Syses Magazine pp [3] A. K. Rue Pecision sabilizaion syses IEEE ans. Aeospace an Eleconic Syses vol. AES- pp [4] R. Dhaouai K. Kubo an M. obise Analysis an copensaion of spee ive syses wih osional loas IEEE Inenaional Wokshop on Avance Moion Conol Yokohaa 993 pp [5] S. W. Jang an J. H. Oh A suy on he ynaics of gea syse wih backlash Poceeings of he KSPE sping annual eeing (in Koean pp [6] W. J. Bigley an S. P. sao Opial oion sabilizaion conol of an eleco-opical sigh syse Poceeings SPIE Acquisiion acking an Poining vol. 989 pp.6-. [7] W. J. Bigley an F. Schupan Wieban base oion isolaion conol fo a obile plafo Poceeings of he Aeican Conol Confeence vol. 987 pp [8] D. C. H. Yang an Z. S. Sun A oay oel fo spu gea ynaics Jounal of Mechaniss ansissions an Auoaion in Design vol pp [9]. Meiovich Pinciples an echniques of vibaions USA Penice-Hall 997 [] B. A. Chubb Moen analyical esign of insuen sevoechaniss USA Aison-Wesley Publishing Copany 967 pp [] M. Cliffo Moen eleconic oos USA Penice-Hall 99. [] J. E. Sloine an W. i Applie nonlinea conol USA Penice-hall 99 pp [3] M. oin J. Galic an P. O. Guan ew oels fo backlash an gea play Inenaional Jounal of Aapive Conol an Signal Pocessing vol. pp [4] C. J. Richas Mechanical engineeing in aa an counicaions UK Van osan Reinhol Copany 969 pp