Research on Modal Parameters Identification of Parallel Manipulator with Flexible Multi-Body System

Transcription

1 Reseach Joual of Applied Scieces, Egieeig ad echology 5(): , 3 ISS: ; e-iss: Maxwell Scietific Ogaizatio, 3 Submitted: Septembe 6, Accepted: Octobe 3, Published: Mach 5, 3 Reseach o Modal Paametes Idetificatio of Paallel Maipulato with Flexible Multi-Body System Chuxia Zhu, Jima Luo, Da Wag ad Big Cai School of affic ad Mechaical Egieeig, Sheyag Jiazhu Uivesity, Sheyag 68, Chia Abstact: I this study, a ew method based o simulatio is poposed. Ad the aalysis method based o flexible multi-body system of paallel maipulato is povided i the same time. Fistly, modal aalysis piciple of paallel maipulato was aalyzed i theoy ad the paametes of dyamic chaacteistic wee idetified by theoetical aalysis. he vibatio model of flexible multi-body fo paallel maipulato was built i vitual pototype softwae ad fomed vibatio system of igid ad flexible couplig fo simulatio aalysis ad fom the simulatio esults got the value of paametes fo vibatio chaacteistic of paallel maipulato. Ad the dyamic chaacteistic paametes wee idetified accodig to the simulatio esults. he esults showed that the simulatio method ad esult dates ae validated. So the itegatio simulatio method is feasible, which ca povide efeece fo dyamic optimal desig. Keywods: Dyamic chaacteistic, flexible multi-body system, paallel maipulato, paametes idetificatio IRODUCIO he vibatio ad defomatio of paallel maipulato will be havig impotat effect o quality of machiig (Xu ad Jiao, 998). he paallel maipulato with high pecisio, low coaseess ad high automatio ae eeded as the equiemet of mode maufactue. It is a mai eseach wok to educe ad void vibatio fo esuig the maipulato will ot geeate self-oscillatio whe it woks i the age of ated powe (Sakamoto, et al., ). May study shows that good dyamic chaacteistic of paallel maipulato will be moe eeded ad moe impotat with gadually impovemet of machiig popety (Jiwook et al., ), so paametes idetificatio aalysis of dyamic chaacteistic is moe impotat. Vibatio aalysis have bee a idispesable eseach cotet fo paallel maipulato desig, maufactue ad use ad is paid moe attetio as a impotat aspect of dyamic chaacteistic (Yu et al., 4). he expeiece ad epeated measued ae mai meas of taditioal stuctue desig while cosideig dyamic facto, it geatly slowdow the desig speed ad the desig quality also had to be optimizatio (Kis et al., 4). It is ievitable ted to aalyze vibatio chaacteistic of paallel maipulato ad eplace static desig by dyamic desig. Vitual pototype simulatio techique is oe of mai ways fo stuctue dyamic desig; it ca solve some mai poblems fo stuctue dyamic desig (Zhu et al., ). I this study this method is bee cosideed fo aalyzig dyamic chaacteistic of a kid of 3-P paallel maipulato ad the esults ca be used fo optimizig stuctue, the the desig pefomace of paallel maipulato will be impoved ad desig cycle will be shoteed. I this study, a ew method based o simulatio is poposed. Ad the aalysis method based o flexible multi-body system of paallel maipulato is povided i the same time. Fistly, modal aalysis piciple of paallel maipulato was aalyzed i theoy ad the paametes of dyamic chaacteistic wee idetified by theoetical aalysis. he vibatio model of flexible multi-body fo paallel maipulato was built i vitual pototype softwae ad fomed vibatio system of igid ad flexible couplig fo simulatio aalysis ad fom the simulatio esults got the value of paametes fo vibatio chaacteistic of paallel maipulato. Ad the dyamic chaacteistic paametes wee idetified accodig to the simulatio esults. he esults showed that the simulatio method ad esult dates ae validated. So the itegatio simulatio method is feasible, which ca povide efeece fo dyamic optimal desig. MODAL AALYSIS PRICIPLE FOR PARALLEL MAIPULAOR Basic theoy: Modal aalysis is a mode method ad mea fo aalyzig dyamics fo most egieeig system, such as mechaical system, civil egieeig Coespodig Autho: Chuxia Zhu, School of affic ad Mechaical Egieeig, Sheyag Jiazhu Uivesity, Sheyag 68, Chia 974

2 Res. J. Appl. Sci. Eg. echol., 5(): , 3 stuctue ad bidge ad so o Zhag et al. (). Modal aalysis is a kid of coodiate tasfomatio accodig its essece ad its pupose is to descibe the espose vecto which was descibed i oigial physical coodiate i modal coodiate system, so that the complex actual stuctues system ca be simplified to be a modal model fo aalyzig ad pedictig modal espose (Li et al., ). Fo abitay feedom liea mechaical system, its vibatio diffeetial equatio ca be showed as follows: Mx () t Cx () t Kx() t f () t () M is mass matix. C is stiffess matix, K is dampig matix, f(t) is exteal excited vecto, x(t) is displacemet espose vecto. Eq. () is tasfeed by Fouie tasfom: Ad afte coodiate tasfomatio, the equatio ca be expessed: [ x] qi{ ϕi} i i (7) I Eq. (7), {φ i }is th ode vibatio mode which is detemied accodig to [ M]{ x} [ K]{ x}.big Eq. (4) ito () the followig equatio ca be expessed: [ M] q i{ ϕi} [ C] q i{ ϕi} [ K] qi{ ϕi} { f( t)} i i i (8) Eq. (8) left multiplicity {φ } accodig to the othogoality of modal shape vecto about mass matix [M] ad stiffess matix [K]. he the followig equatio ca be got: [ ] X( ) F( ) ω M ωc K ω ω ω A vaiable X(ω) Laplace tasfom fo x(t) F(ω) Laplace tasfom fo f(t) he espose of system is : () { ϕ} [ C]{ ϕi} αϕ { } [ M]{ ϕi} βϕ { } [ K]{ ϕi}, i αm βk c, i m Modal mass k Modal stiffess c Modal dampig coefficiet (9) X( ω) [ ω M ωc K] F( ω) H( ω) F( ω) (3) he Eq. (9) ca be simplified as follows: H(ω)is displacemet tasfe fuctio matix. he system espose i modal piciple ca be witte as follows: X ( ω) { ψfψ } { }{ } ω m cω k m Modal mass c Modal dampig k Modal stiffess {ψ } System mode vecto (4) Fo the equatio of elatioship is moe visual betwee espose ad exteal foce, Eq. (4) ca be expessed i matix fom as follows: { } { } X ( ω) ψψ { F} ω m cω k (5) mq cq kq f t { ϕ } { ( )} () Set {f(t)} {F}e ωt,the q i Q e ωt,put it ito Eq. (): ωt ωt ( ω m ωc k) Qe { ϕ}{ F} e Q F ω { ϕ }{ } m ωc k () () If the iitial coditios ot be cosideed, the displacemet espose of system ca be got fom Eq. (7): X X ωt ωt { x} e qi{ ϕi} Qi{ ϕi} e i i X (3) he tasfe fuctio fom which is showed the elatioship betwee exteal foce o poit ad espose o poit i ca be expessed as follows: X ψψ ψψ / m Hi F m c k m i i i ω ω ω ζ ωω ω (6) 975 X X { ϕ }{ F} { X} Qi{ ϕi} i i ω m ωc k X { ϕ }{ ϕ } i ω m ωc k { F} (4)

3 Fom above equatio, its deomiato is ielevat to excited poit ad espose poit i ad oly elevat to fequet ad dampig. So the tasfe fuctio ad its deomiato ae all same i spite of ay measued poit i mechaical system. he umeato of Eq. (4) detemied by modal mass, vibatio modal value of excited poit ad espose poit, Above equatio shows that the tasfe fuctio icludes all paametes of dyamics, modal paametes ae eeded to be idetified afte gettig tasfe fuctio, amely, measued value theoetical fomula of tasfe fuctio ae caied out cuve fittig to get modal paametes such as atue fequecy, dampig atio ad mode shape. he paametes of vibatio chaacteistics ca be idetified as follows: atual Fequecy: he atual fequecy ca be idetified fom the amplitude fequecy cuves, because the max amplitude is happed o atual fequecy. Dampig Ratio: Fom above aalysis, the tasfe fuctio of system accodig to Eq. (6) is showed as followig: Fo gettig the fequecy which coespodig the maximum espose, the udamped atual fequecy is: he dampig atio is: he damped atual fequecy is: Due to the dampig of geeal egieeig stuctue is small (ξ.), so the damped atual fequecy ca be cosideed as followed: Fom above we ca got that: the peak value of cuve of amplitude fequecy espose chaacteistics is the atual fequecy of system. Fo decidig the dampig atio of system, it is ca be got by the half powe fequecy. O the poit of half powe, the maximum value of fequecy espose fuctio ca be showed as follows: Res. J. Appl. Sci. Eg. echol., 5(): , 3 { ϕ } { ϕ}{ ϕ} i i X ( ω) H ( ω) F( ω) ω m ωc k ω m ωc k k ω π f m c ξ mω d ω ω ξ ωd ω H ( ω) m ( ω ω ) ( ξω ω) mξω ξ he two appoximate solutios of above equatios ae: ω ω ξ ω ( ξ) a he, the maximum value of tasfe fuctio is: H max 4 kξ( ξ), ωb ω ξ ω( ξ) he, the maximum value of tasfe fuctio is: H max 4 kξ( ξ) he the dampig atio is: ωb ωa ξ ω he stiffess of system is: k Aξ So, i the vibatio testig expeimet, the atual fequecy ca be tested, fom the cuves of amplitude fequecy chaacteistic, the poit of half powe is foud out ad accodig above computig fomulas, the paametes of vibatio chaacteistic ca be got. SIMULAIO AALYSIS OF VIBRAIO CHARACERISIC FOR PARALLEL MAIPULAOR It is a effective appoach fo stuctue optimizatio ad test of dyamic chaacteistic to combie closely theoetical aalysis, expeimet ad simulatio, ADAMS softwae adopted vitual pototype techique ca solves effectively above poblems. he ADAMS/Vibatio module is used fo test foced modal shapes ad aalyzig fequecy domai (Zhu et al., 9), it ca povide the fuctio multi-iput ad multioutput ad ca descibe them i vibatio fom, so this module is used fo makig aalysis of vibatio simulatio. Ad the model of paallel maipulato with multi-body system fo paallel maipulato is showed i Fig.. O the desig stage, it is ca pe-evaluate dyamic pefomace of paallel maipulato to utilize vitual pototype to simulate, so that paallel maipulato poduct ca be aalyzed fo fidig stuctue weakess, evisio ad optimizatio, the put fowad impovemet suggestios fo pefectig the pototype of paallel maipulato. his kid of vitual pototype 976

4 Res. J. Appl. Sci. Eg. echol., 5(): , 3 Fig. : he model of paallel maipulato with multi-body system 6.E-4 X diectio 9 Y diectio 4.5E-4 3.E E-4 5 (a) (b) 5 Z diectio 4 Space 5.E-4 3 (c) (d) Fig. : Fequecy espose of motio platfom simulatio techique poceeds fom multi-body dyamics aalysis ad utilizes compute simulatio ad opeatio to get the esults, its model eeds to be built ad impoved accodig to the measuemet esults fo assuig it cofom to eality. 977 hee ae two millig types whe simulatig wok situatio of paallel maipulato, oe is paallel maipulato is wokig with o-load, ude this coditio, the liks speed ad dive foce ae be measued whe tool is movig with cetai speed ad

5 Res. J. Appl. Sci. Eg. echol., 5(): , 3 5.E E-4.5E-4.5E-4 X diectio Y diectio (a) (b) Space 87 Z diectio 5 (c) (d) Fig. 3: Fequecy espose of taslatio platfom acceleatio; the othe is tool feed with uifom speed ad bea millig foce. Fo millig tool with diffeet mateials pocess diffeet wokpiece, whe wokig paametes ae detemied, the millig foce ca be calculated by coespodig equatio (Zhag et al., 9), take millig tool with W8C4V ad wokpiece with 45# fo example, the mai millig foce F t ca be calculated by the followig equatio: F Ca f d a z t F e z p C F Coefficiet of millig foce, it is 88 α e Millig width (mm) α p Millig depth (mm) f z he feed of evey gea tooth (mm/z) d he oute diamete of millig tool (mm) z ooth umbe of millig cutte Accodig to the stuctue paametes of paallel maipulato, the ed millig tool ad wokpiece with steel 45# ae chose, the coespodig pocessig paametes ae: simulatio is caied out. he iitial phase is, the vibatio aalysis is caied o with apid scaig style. ADAMS softwae ca output the displacemet espose i X, Y, Z diectio ad space, the esults ae showed i Fig. to 3 ad the fequecy age is fom to Hz. I Fig. it shows the fequecy espose of motio platfom, the maximum displacemet esposes i X diectio occued o 8Hz, displacemet espose is 48 mm; the maximum displacemet esposes i Y diectio occued o 3 Hz, displacemet espose is 9 mm; the maximum displacemet esposes i Z diectio occued o 3Hz, displacemet espose is 5 mm; the maximum displacemet esposes i space occued o 3Hz, displacemet espose is 35 mm: ω 5.Hz; H 37mm ω.8 Hz, ω 7.Hz a b Stiffess ad dampig ae as followig: k 7.69E5( / m); ξ x 76; C F 88, α e mm, α p mm, f z. mm/z,d 6 mm, z 6,So F t 6 I Fig. 3 it shows the fequecy espose of taslatio platfom, the maximum displacemet he foce is applied o tool tip as exteal esposes i X diectio occued o Hz, but the value excited foce accodig to above calculatio whe of displacemet espose is vey small ad ca be 978

6 Res. J. Appl. Sci. Eg. echol., 5(): , 3 eglected; the maximum displacemet esposes i Y diectio occued o 3 Hz, displacemet espose is 7 mm; the maximum displacemet esposes i Z diectio occued o 3Hz, displacemet espose is 7 mm; the maximum displacemet esposes i space occued o 3Hz, displacemet espose is 75 mm: ω 5.4Hz; H 74mm ω 3. Hz, ω 7.6Hz a b Stiffess ad dampig ae as followig: k.64e6( / m); ξ 75; Fom above simulatio esults, the followig coclusio ca be go: the esoace fequecy of 3-P paallel maipulato is 3Hz i the age fom Hz to Hz, so this fequecy should be avoid whe paallel maipulato is wokig. COCLUSIO I this study, we wat to seek fo paametes idetificatio aalysis method fo paallel maipulato. Fom this aalysis pocess ad simulatio, we edeavo to seek some commo modelig methods of idetificatio fo multi-body system of paallel maipulato. Ad i the couse of this exploatio, we ae able to see that simulatio method with vitual pototype techology is ievitable ted. he simulatio is caied out ad get the esoace fequecy is o 3Hz ad displacemet espose is 35 mm; It shows that the simulatio esults ae high pecisio accodig expeimet esults; it ca be povide efeece ad basis fo paallel maipulato stuctue desig ad dyamic chaacteistic aalysis. he simulatio techique of paametes idetificatio fo paallel maipulato has some advatages, such as lowe cost ad shote peiod compaed with expeimet, so it has efeece value fo elated measuig. But the dyamics modelig of igidflexible couplig system is based o the assumptio that the system s compoets ae pefect. Howeve, fictio, mateial o-uifomities ad maufactuig ad assembly eos i actual systems have vey impotat effects o simulatio esults. ACKOWLEDGME his eseach was suppoted by Poect suppoted by the atioal atual Sciece Foudatio of Chia (Gat o. 5558) ad Suppoted by Doctoal Scietific Reseach Foudatio of Liaoig Povice (Gat o.79). REFERECES Jiwook, K., F.C. Pak ad K. Mumsag,. Geometic vibatio aalysis of obotic mechaisms. Poceedigs of IEEE Iteatioal Cofeece o Robotics ad Automatio, Sa Facisco, CA, USA. Kis, K., E.U. Imme ad S. William, 4. Locally lieaized dyamic aalysis of paallel maipulatos ad applicatio of iput shapig to educe vibatios. J. Mech. Desig, 6: Li, C., Y. Hou, C. Du ad Y. Dig,. A aalysis of the electic spidle s dyamic chaacteistics of high speed gide. J. Adv. Mauf. Syst., (): Sakamoto, H. ad. Matsuda ad S. Shimizu,. Effect of clamped tool holdes o dyamic chaacteistics of spidle system of machiig cete. It. J. Autom. ech., 6(): Xu, B. ad Q. Jiao, 998. Mechaical Vibatio ad Modal Aalysis Foudatio. Machie Pess, Beiig, Chia. Yu, M., J. Zhao, L. Zhag ad Y. Wag, 4. Study o dyamic chaacteistics of vitual-axis hybid polishig machie tool by flexible multi-body dyamics. P. I. Mech. Eg. B. J. Eg., 8(9): Zhag, Y., X. Li, C. Liu ad G. Cai, 9. Dyamic chaacteistic aalysis ad validatio of spidle assembly of high speed machie tool. Key Eg. Mat., 47-48: Zhag, K., H. Yu, Z. Liu ad X. Lai,. Dyamic chaacteistic aalysis of BM tuelig i mixedface coditios. Simul. Model. Pact. h., 8(7): 9-3. Zhu, L., C. Zhu,. Yu ad W. Wag,. Study o simulatio ad expeimet of dyamic chaacteistics of the liea ollig guide i tumillig cete. J. Adv. Sci. Lett., 4(6-7): Zhu, C., L. Zhu, W. Jiaog, S. Cog ad W. Washa, 9. Aalysis of dyamic chaacteistic ad stuctue optimizatio fo hybid machie tool. It. J. Model. Idetificatio Cotol, 7():