New Vector Description of Kinetic Pressures on Shaft Bearings of a Rigid Body Nonlinear Dynamics with Coupled Rotations around No Intersecting Axes

Transcription

1 Acta Polytechca Hugaca Vol. No. 7 3 New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes Katca. tevaovć Hedh* Ljljaa Veljovć** *Mathematcal Isttute ANU elgade eba ad aculty of Mechacal Egeeg Uvesty of Nš eba E-mal: khedh@euet.s ** aculty of Mechacal Egeeg Uvesty of Kagujevac eba E-mal: khedh@sbb.s Abstact: New vecto descpto of ketc pessues o shaft beags of a gd body olea dyamcs wth coupled otatos aoud o tesectg axes s fst ma esult peseted ths pape. Mass momet vectos ad vecto otatos coupled fo pole ad oeted axes defed by K. Hedh 99 ae used fo obtag vecto expessos fo ketc pessues o the shaft beags of a gd body dyamcs wth coupled otatos aoud o tesectg axes. A complete aalyss of obtaed vecto expessos fo ketc pessues o shaft beags gve us a sees of the kematcal vectos otatos aoud both dectos detemed by axes of the gd body coupled otatos aoud o tesectg axes. As a example of defed dyamcs we take to cosdeato a heavy gyo-otodsk wth oe degee of feedom ad coupled otatos whe oe compoet of otato s pogammed by costat agula velocty. o ths system wth olea dyamcs sees of gaphcal pesetato tasfomatos ealzatos wth chages of eccetcty ad agle of clato skew posto of heavy gd body-dsk elato to self otato axs ae peseted as well as ealzato wth chagg othogoal dstace betwee axes of coupled otatos. Agula velocty of ketc pessues compoets vecto fom ae expessed by usg agula acceleato ad agula velocty of compoet coupled otatos of gyooto-dsk. Keywods: coupled otato; o tesectg axes; devatoal mass momet vecto; otato; ketc pessues; ketc pessue compoets; olea dyamcs; gyootodsk; eccetcty; agle of clato devato ketc couple; fxed pot; gaphcal pesetatos; thee paamete aalyss 5

2 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes Itoducto No pecsos ad eos the fuctos of gyoscopes caused by eccetcty ad ubalaced gyo oto body as well dstace betwee axes of otatos ae easo to vestgate detemed task as the ttle of ou pape. The classcal book [] by Adoov Vtt ad Hayk cota a classcal ad vey mpotat elemetay dyamcal model of heavy mass patcle elatve moto alog ccle aoud vetcal axs though t s cete. Nolea dyamcs ad sgulates lead to pmtve model of the smple case of the gyo-oto whch epeset a useful dyamcal ad mathematcal model of olea dyamcs. Usg K. Hedh s ee efs. [-] mass momet vectos ad vecto otatos some chaactestcs vecto expessos of lea mometum ad agula mometum ad the devatves fo gd body sgle otato wee obta physcal ad dyamcal vsble popetes of the complex system dyamcs ad the ketc paametes vecto fom fo sgle otato. Thee ae vecto compoets of the shaft beag ketc pessues wth opposte dectos ad same testy that peset devatoal couple effect cotag vecto otatos whose dectos ae same as ketc pessue compoets o coespodg oto shaft beags fo detal see efs. [] ad [5]. The deftos of mass momet vectos coupled to the pole ad the axs [-9] [] ae toduced the foudato of ths vecto method. The ma vecto s def dm of the body mass eta momet at the pot A fo V the axs oeted by the ut vecto ad thee s a coespodg vecto of the gd body mass devatoal momet fo the axs though the pot see efeeces [] ad [5-6]. Ths vecto appoach s vey sutable to obta ew vew to the popetes of dyamcs of pue classcal system dyamcs vestgated by umeous geeatos of the eseaches ad seous scetsts aoud the wold. We poof ths appoach ou publshed efeece []. I Itoducto of ths pape [] a shot evews of the bass of the subjectve selected efeeces about ogal eseach esults of dyamcs ad stablty of gyostats was gve. The s easo that we dd t made ay evews of the papes about gyoscopes. Passg though the cotet of the umeous publshed scetfc pape we ca see that o esults coceg behavo of the ketc pessue dectos ad testy depedg of the olea dyamc egmes. The ou am s to vestgate ketc pessues ad devato ketc couple appeag to the shaft beags of the gd body coupled otatos aoud two o tesectg axes. Two ou efeeces [] fom 8 ad cota shot pesetato of the ketc 5

3 Acta Polytechca Hugaca Vol. No. 7 3 pessue to the gyooto self otato beags ad otatos as well as pesetato of the olea dyamcs of the heavy gyo-oto but ot completed. Ths s easo that we take to a lage cossdeato ad vestgato thee paametes aalyss of the vecto expessos of shaft beag ketc pesues ad the cmpoets based o ou pevous esults o applcatos vecto method ad publshed ou efeeces []. Ths pape cota ew ezults based o the pevous ou esults. gazatos of ths pape based o the vecto method applcatos wth use of the mass momet vectos ad vecto otatos fo descbg vecto expessos of ketc pessues of the shaft beags of the gd body coupled otatos aoud two o tesectg axes ad coespodg ketc devato couple appeag by opposte ketc pessues to shaft beags ad shaft beag eactos. yamcs of a gyo-oto wth oe degee of feedom ad coupled otatos whe oe compoet of otato s pogammed by costat agula velocty s cosdeed as a example. o ths system of olea dyamcs the sees of gaphcal pesetato of the ketc pessues of the shaft beag of a gd body self otato ae peseted. z v v u u u v N dm u u v y u v x u u gue A gd body coupled otatos aoud two o tesectg axes. ystem s wth two degees of moblty ad oe degees of feedom whee ad ae heoomc ad geealzed coodates. Vecto otatos ad ae peseted.ssetal coectos 53

4 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes Model of a gd ody otato Aoud Two Axes wthout Itesecto Let us to cosde gd body coupled otatos aoud two o tesectg axes peseted gue. fst axs s oeted by ut vecto wth fxed posto ad secod axs s oeted by ut vecto whch s otatg aoud fxed axs wth agula velocty. Axes of otato ae o tesectg axes. gd body s postoed o the movg otatg axs oeted by ut vecto ad otate aoud self otatg axs wth agula velocty ad aoud fxed axs oeted by ut vecto wth agula velocty. All geometcal paametes ae peseted gue. Whe ay of thee ma cetal axes of gd body mass eta momets s ot decto of self otato axs the we ca see that gd body s scew postoed to the body self otato axs. The agles of gd body cetal ma eta axes clatos acodg self-lf otato axs ae. These agles ae agles of scew posto of gd body to the body self otato axs. Whe cete of the mass of gd body s ot o body self otato axs of gd body otato we ca say that gd body s scew postoed. Eccetcty of body posto s omal dstace betwee body mass cete ad axs of self otato ad t s defed by e. Hee s vecto posto of mass cete wth og pot ad posto vecto of mass cete wth fxed og pot s. 3 Vecto Equatos of yamc Equlbum of gd ody oupled otatos aoud Two No Itesectg Axes y usg theoems of lea mometum ad agula mometum wth espect to tme we ca wte two equatos of dyamc equlbum of the cosdeed gd body coupled otatos about two o tesectg axes peseted gue the followg equatos fo detal see ef. [7] ad Appedx: dk M dt P P G G AN N Am AN N Am 54

5 Acta Polytechca Hugaca Vol. No P Am A N AN A P N G G M M dt d L whee P... 3 ae actve foces ad G s weght of gyo oto A ad ae eactve foces of fxed axs shaft beag eactos ad A ad ae foces of self otato shaft beag eactos. om pevous obtaed vecto equatos t s ot dffcult to obta ketc pessues to both shaft beags A ad as well as A ad o both shafts beags as well as two dffeetal equatos alog ad of the gd body coupled otatos about two o tesectg axes ad to obta tme solutos of ukow geealzed coodate ad o f we kow these coodate to fd ukow exteal actve foces. o the case that axes ae pepedcula some tems pevous vecto expessos ad vecto equatos ae equal to zeo but these equatos ae olea alog agle coodates ad ad coupled by geealzed coodates ad ad the devatves ad also by foces of shaft beags eactos. Two vecto equates ad ae vald fo gd body coupled otatos aoud o tesectg axes as well fo the case tesectg axes as ts specal case. Also these equatos ae vald fo the system dyamcs wth two degees of moblty ad fo thee dffeet cases. 4 Vecto otatos of gd ody oupled otatos aoud Two No Itesectg Axes We ca see that pevous vecto equatos ad tems fo devatve of lea mometum ad agula mometum cota two sets of the vecto otatos: 3 4

6 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes 5 st two vecto otatos ad ae othogoal to the decto of the fst fxed axs ad thd vecto otato s othogoal to the self otato axs. ut fst vecto otato s coupled fo pole o the fxed axs ad secod ad thd vecto otatos ad ae coupled fo the pole o self otato axs ad fo coespodg decto oeted by dectos of compoet agula veloctes of coupled otatos. Itestes of two fst otatos ae equal ad ae expessed by agula velocty ad agula acceleato of the fst compoet otato ad testy of thd vecto otatos s expessed by agula velocty ad agula acceleato of the secod compoet otato ad they ae the 4 4 followg foms: ad. Lets toduce otato ad deote dffeece betwee coespodg compoet agles of otato ad of the gd body compoet otatos ad coespodg absolute agles of otato of pue kematcs vecto otatos about axes oeted by ut vectos ad. These agula veloctes of elatve kematcs vectos otatos ad whch otate about coespodg axs elato to the compoet agula veloctes of the gd body compoet otatos ae: 4 ad I gue Vecto otatos ad ae peseted. 4 We ca see that pevous vecto expesso fo devatve of agula mometum ae toduced vecto otatos the followg vecto fom: u

7 Acta Polytechca Hugaca Vol. No. 7 3 v * u v u v a u e u v A A Vecto otatos a* b* c* gue a* ad b* elatos to coespodg mass momet vectos ad the coespodg devatoal compoets ad ad as well as to coespodg devatoal plaes. c* Model of heavy gyo oto wth two compoet coupled otatos aoud othogoal axes wthout tesectos The fst s othogoal to the fxed axs oeted by ut vecto ad secod s othogoal to the self otato axs oeted by ut vecto. Itesty of fst otato s equal to testy of pevous defed otato ad testy of secod otato s equal to testy of pevous defed otato defed by expessos 7. I gue vecto otatos gue a* ad gue.b* elatos to coespodg mass momet vectos ad ad the coespodg devatoal compoets ad as well as to coespodg devatoal plaes ae peseted. Vecto otatos ad ae pue kematcal vectos fst peseted efeece [89] as a fucto o agula velocty ad agula acceleato a fom u w. otatos fom fst set ae otated aoud though pole ad axs decto of fst compoet otato agula velocty ad deped of agula velocty ad agula acceleato. Thee ae two vectos of such type ad all tees have equal testy. otatos fom secod set ae otated aoud axs decto of secod compoet otato ad deped of agula velocty ad agula acceleato. Thee ae two vectos of such type ad they have equal testy. 57

8 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes Lets toduce otato ad deote dffeece betwee coespodg compoet agles of otato ad of the gd body compoet otatos ad coespodg absolute agles of pue kematcs vecto otatos about axes oeted by ut vectos ad though pole. These agula velocty of elatve kematcs vectos otatos ad whch otate about axes coespodg dectos elato to the compoet agula veloctes of the gd body compoet otatos though pole ae expessed as ad. 5 Vecto Expessos of Ketc Pessues Ketc eactos o haft eags of gd ody oupled otatos aoud Two No Itesectg Axes Ketc pessues beag eactos wth out pats eactos duced by exteal foces o fxed shaft beags fo the case that sphecal beag s at the pole ad cyldcal ths fxed axs defed by vecto posto ae the followg fom: 8 AN N N 9 dev M It s ot dffcult by use system decomposto to obta ketc pessues o body self otato shaft beags fo the case that sphecal beag s at the pole. y aalyss vecto equatos ad ad coespodg expessos 8 ad 9 fo ketc pessues o the both shafts beags we ca coclude that the system to the both shaft beags appea the pa of beags two opposte compoets of ketc pessues wth devato couple. I fxed shaft beags A ad appea the followg opposte compoets: dev Nad N vecto dev elato: N but dffeet pots of appeaace beags A N ad dev wth dstace M ad buld oe couple dev N N AN 58

9 Acta Polytechca Hugaca Vol. No. 7 3 kow ude the ame devato couple ad detfed lke ou vestgated system dyamcs fo whch we obta the followg vecto expesso: M dev M Also t s possble to coclude fo two opposte compoets of ketc pessues to the self otato shaft beags dev N ad vecto elato: dev N but dffeet pots of appeaace beags A N ad wth dstace dev ad buld oe couple M dev N kow AN ude the ame devato couple ad also detfed lke ou vestgated system dyamcs. 6 yamc of gd ody oupled otatos aoud Two thogoal No Itesectg Axes ad wth e egee of eedom We ae gog to take to cosdeato specal case of the cosdeed heavy gd body wth coupled otatos about two axes wthout tesecto wth oe degee of feedom ad the gavtato feld. o ths case geealzed coodate s depedet ad coodate s pogammed. I that case we say that coodate s heoomc coodate ad system s wth kematcal exctato pogammed by foced suppot otato by costat agula velocty. Whe the agula velocty of shaft suppot axs s costat cost we have that heoomc coodate s lea fucto of tme t ad agula acceleato aoud fxed axs s equal to zeo. pecal case s whe the suppot shaft axs s vetcal ad the gyo-oto shaft axs s hozotal ad all tme hozotal plae ad whe axes ae o tesectg at omal dstace a. o we ae gog to cosde that example peseted gue c*. The omal dstace betwee axes s a. The agle of self otato aoud moveable self otato axs oeted by the ut vecto s ad the agula velocty s. The agle of otato aoud the shaft suppot axs oeted by the ut vecto s ad the agula velocty s costat. The agula velocty of oto s. The agle s geealzed coodates case whe we vestgate system wth oe degees of feedom but system have two degees of moblty. Also wthout loose of geealty we take that gd body 59

10 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes s a dsk eccetcally postoed o the self otato shaft axs wth eccetcty e ad that agle of skew cled posto betwee oe of ma axes of dsk ad self otato axs s as t s vsble gue c*. o that example dffeetal equato of the heavy gyo oto-dsk self otato of evewed model gue fo the case coupled otatos about two othogoal o tesectg axes by usg afte multplyg scala by ad takg to accout othogoal betwee axes of coupled otatos we ca obta the followg fom: cos s cos whee u v mges u v meas u v e 4 Hee t s cosdeed a eccetc dsc eccetcty s e wth mass m ad adus whch s cled to the axs of ts ow self otato by the agle see gue 4 so that pevous costats dffeetal equato become the followg foms: s s e 4 g s e s ea s 3 e s elatve olea dyamcs of the heavy gyo-oto-dsk aoud self otato shaft axs s possble to peset by meas of phase potat method. oms of phase tajectoes ad the tasfomatos by chages of tal codtos ad fo dffeet cases of dsk eccetcty ad agle of ts skew posto as well as fo dffeet values of othogoal dstace betwee axes of compoet otatos may peset chaacte of olea oscllatos. o that easo t s ecessay to fd fst tegal of the dffeetal equato. Afte tegato of the dffeetal equato 6 the o-lea equato of the phase tajectoes of the heavy gyo oto dsk dyamcs wth the tal codtos t t t we obta the followg fom: cos cos s cos cos s 4 The aalyzed system s cosevatve ad equato 4 s the eegy tegal. 6

11 Acta Polytechca Hugaca Vol. No. 7 3 I cosdeed case fo the heavy gyo-oto-dsk olea dyamcs the gavtatoal feld wth oe degee of feedom ad wth costat agula velocty about fxed axs we have thee sets of vecto otatos. Thee of these vecto otatos ad fom fst set ae wth same costat testy cos tat ad otate wth costat agula velocty ad equal to the agula velocty of gd body compoet pecesso otato about fxed axs but two of these thee vecto otatos ad ae coected to the pole o the self otato axs ad ae othogoal to the axs paallel decto as decto of the fxed axs. All these thee vecto otatos ad ae dffeet dectos see gues ad 3. Two of these vecto otatos ad fom secod set ae wth same testy 4 equal to ad coecte to the pole ad othogoal to the self otato axs oeted by ut vecto ad otate about ths axs wth elatve agula velocty defed by expesso 6 espect to the self otato agula velocty see gues a* b* ad c*. y use expessos 3-5 ad 7 we ca lst followg sees of vecto otatos of the gyo-oto dsk wth coupled otato aoud othogoal o tesectg axes ad wth cost : v v u ~ s s v ~ cosu ~ ~ cos s s 4 u 5 ~ ~ u s cos v cos cos s cos 4 v u u whch ~ s agle betwee elatve vecto posto of gd body mass cete ad self otato axs oeted by ut vecto. e of the vectos otatos fom the thd set s wth testy ad decto: 6

12 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes u. Ths vecto otato s coecte to the pole ad othogoal to the axs oeted by ut vecto ad elatve otate about ths axs. Itesty of ths vecto otato expessed by geealzed coodate agle of self otato of heavy dsk takg to accout fst tegal 9 of the dffeetal equato 6 obta the followg fom: cos cos s cos cos 6 s Itesty of two of these vecto otatos ad fom secod set depeds o agula velocty ad agula acceleato. o the cosdeed system of the heavy gyo-oto-dsk dyamcs fo obtag expessos of testes of vecto otatos ad fom secod set the fucto of the geealzed coodate agle of self otato of heavy dsk self otato we take to accout a fst tegal 4 of olea dffeetal equato ad by usg these esult ad pevous expessos 5 of vecto otato we ca wte: *testes of the vectos otatos ad coected fo the pole ad otate aoud self otato axs the followg fom: cos s cos cos cos s cos cos s 7 *vecto otatos othogoal to the self otato axes ae the followg vecto foms: cos s cos cos cos s cos cos s cos s cos cos cos s cos cos s 8 6

13 Acta Polytechca Hugaca Vol. No Ketc Pessues to haft eags of gd ody oupled otatos aoud Two thogoal No Itesectg Axes ad wth e egee of eedom y use pevous deved vecto equatos ad ad appoach to obtag vecto expessos 8 ad 9 fo ketc pessues A ad to fxed shaft beags of gd body coupled otatos aoud two o tesectg othogoal axes ad fo system wth oe degee of feedom t s easy to obta vecto expessos fo ketc pessues A ad cludg compoet eactos of the gd body weght to self otato shaft beags A ad of gd body coupled otatos aoud two othogoal o tesectg axes ad fo cosdeed patcula example the followg fom: G A 9 * * G G * * G G AT whee s matx of teso of mass eta momets fo pole. Pevous expessos cota membe whch coespod to the beag eactos of the oto pope weght. Afte takg to accout mass eta momet vecto fo cled dsk ad dsk posto wth eccetcty of mass body cete we ca wte scala fom compoets of ketc pessues A ad cludg compoet eactos of the gd body weght to o beags A ad of the self otato axs the followg fom: s cos cos s s cos s e mg e m e Au cos cos s cos e m v v v u

14 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes Av u e s s e s s s mg cos m e cos e u v v coscos v m cos s 3 e m e s cos e s s cos mg cos s e u v v cos s v m cos cos 4 v e s s e s s s mg cos m e cos e u v v coscos v m cos s 5 Pevous obtaed expessos -5 of the compoets of ketc pessues A ad cludg compoet eactos of the gd body weght to beags A ad of the self otato axs scala fom s possble peset the followg vecto fom: k k k uu vv mes v u v v cos su cosv a m a es cos e cos cosu sv u v su cosv k k k A Auu Avv mes v u v v cos su cosv a m a es cos e cos cosu sv u v su cosv Pevous scala expessos ae sutable fo aalyss o the bass decompostos to the sepaate compoets wth specfc popetes of testy dectos fluece of some mass ad geometcal popetes ad stuctue paametes as well as agula veloctes ad othe ketc paametes of cosdeed specal example

15 Acta Polytechca Hugaca Vol. No. 7 3 y toducg the followg ut vectos w w ad w 3 w u s v cos w u s v cos w3 u cos v s pessues Ag ad g eactos of the gd body weght to beags A ad of the self otato axs s possble to expess the followg vecto fom: e e Ag Aug u Agvv mg cos su mg cos cosv e e g guu gvv mg cos su mg cos cos om last foms of the pessues Ag ad g v 8 eactos of the gd body weght to beags A ad of the self otato axs we ca see that s possble to sepaate compoet wth same testy ad opposte dectos ad also compoet wth same agula velocty oe o othe dectos. fo aalyss ketc pessues cludg compoet eactos of the gd body weght to beags A ad of the self otato axs used expessos 8 I gue 3 some of the toduced ut vectos u v w w ad w 3 schematcally ae peseted wth coespodg agula velocty ad dectos of otatos. v v w w 3 w u w u 3 w w w u w v v v v w w u u u 3 w 3 3 w w w 3 w w 3 w 3 gue 3 chematcally pesetato of the ut vectos u v w w ad 3 kematcal elatos wth coespodg agula velocty ad dectos of otatos w ad the geometcal ad 65

16 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes k ompoets k k 3 ad k 4 of pue ketc pessue of the self otato axs ae the followg foms: k to beag k k v cos u v a k m a es cos e cos 4 mes v k 3 k om pevous expessos fo compoets k ketc pessue k k 3 ad k 4 of pue to beag of the self otato axs we ca coclude that fluece of dsk posto eccetcty s stoge to the compoets k pue ketc pessue ad that testy of compoet k k 3 of k cease ad testy of the compoet decease wth ceasg of dsk eccetcty. k Itesty of the pue ketc pessue cease wth ceasg of dsk eccetcty..85 A.83 A.775 A A.738 A A A A3 a* b* 5 5 gue 4 Itesty tasfomato of ketc pessue compoet k A to self otato shaft sphecal beag A of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom decto of the self otato shaft axs fo dffeet dsk eccetcty 8 Gaphcal Pesetato of Ketc Pessues to elf otato haft eags of gd ody oupled otatos y use pevous lsted expessos as well as othe o lsted he ad Mathad as a softwae tool a umecal expemet was followed fo aalyss popetes of the ketc pessues ad the coespodg compoets to the both shaft beags. elected gaphcal pesetato s doe the gues 4-. All gaphcal 66

17 Acta Polytechca Hugaca Vol. No. 7 3 pesetato ae obtaed by aalytcal expessos deved pevous chaptes of ths pape. I gue 4a* ad b gaphcal pesetato of testy tasfomato of ketc k pessue compoet A to self otato shaft sphecal beag A of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom decto of the self otato shaft axs ad fucto of self otato elatve agle fo dffeet dsk eccetcty s peseted. I gue 5 gaphcal pesetato of testy tasfomato of ketc pessue k compoet A to self otato shaft sphecal beag A N of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fucto of self otato elatve agle fo dffeet dsk eccetcty s peseted. Au Au Au 5 5 Av Av Av 5 5 a* 5 5 b* 5 5 gue 5 k a* ad b* Itesty of ketc pessue compoet to self otato shaft sphecal beag A A N of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk eccetcty k I gue 6 a* c* ad d* the testy of ketc pessue compoet of to self N otato cyldcal beag of gd body coupled otatos aoud two othogoal o tesectg axes decto of ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk agle skew posto s peseted. I gue 8 b* Itesty of the vecto otato fucto of the value of dsk agle skew postos s peseted. I gue 7 gaphcal pesetato of testy tasfomato of ketc pessue compoet k to self otato shaft cyldcal beag N of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fucto of self otato elatve agle fo dffeet dsk eccetcty s peseted. 67

18 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes a* / 9 /6 / 4 /3 b* / 9 / 6 / 4 / / 3 / 9 / 4 / 6 c*. 5 5 d* gue 6 k a* c* ad d* Itesty of ketc pessue compoet of to self otato cyldcal N beag of gd body coupled otatos aoud two othogoal o tesectg axes decto of ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk agle skew posto..b* Itesty of the vecto otato fucto of the value of dsk agle skew posto Itesty of ketc pessue compoet 5 5 gue 7 k to self otato shaft cyldcal beag N of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk eccetcty I gue 8 gaphcal pesetato of testy tasfomato of ketc pessue k compoet to self otato shaft cyldcal beag N of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fucto of self otato elatve agle fo dffeet dsk eccetcty s peseted. I gue 9 testes of ketc pessue devato couple to self otato shaft beags of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk eccetcty ae peseted. 68

19 Acta Polytechca Hugaca Vol. No u u u 5 5 v v v gue 8 k Itesty of ketc pessue compoet to self otato shaft cyldcal beag N of gd body coupled otatos aoud two othogoal axes wthout tesecto ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk eccetcty. a* M M M3 M b* N N N3 N gue 9 Itesty of ketc pessue devato couple to self otato shaft beags of gd body coupled otatos aoud two othogoal o tesectg axes ad fo system wth oe degee of feedom othogoal decto to the self otato shaft axs fo dffeet value of dsk eccetcty. ocludg emaks omplexty of the sgle gd body moto wth coupled otatos about o tesectg axes by vecto method based o the mass momet vectos ad vecto otatos coupled fo pole o selfotato axs ad compoet agula velocty axes s peseted by sample vecto expessos them usually scala foms pofessoal books ths aea. New appoach ad ew composto of ths vecto method ope ew way fo applcatos to the mult-body system dyamcs wth coupled mult-otatos about otesectg axes. New vecto expessos fo lea mometum ad agula mometum ad the devatves of the sgle gd body complex moto by coupled otatos about otesectg axes expessed by ew toduced mass momets vectos ad the vey elegat fom ope ew possblty fo geealzatos these expessos fo descbg mult gd body system complex moto by coupled mult-otatos about hghe umbes of otesectg axes lage peset may eal mechacal egeeg systems ad obotc system dyamcs wth coupled mult-otatos. Ackowledgemet Pats of ths eseach wee suppoted by the Msty of ceces ad Techology of epublc of eba though Mathematcal Isttute ANU elgade Gat N74 yamcs of hybd systems wth complex stuctues. Mechacs of mateals ad aculty of Mechacal Egeeg Uvesty of Nš ad aculty of Mechacal Egeeg Uvesty of Kagujevac

20 Katca. tevaovċ Hedh et al. New Vecto escpto of Ketc Pessues o haft eags of a gd ody Nolea yamcs wth oupled otatos aoud No Itesectg Axes efeeces [] Adoov A. A. Vtt A. A. Hayk. E. 98 Teoya kolebay Nauka Moskva p. 568 [] Hedh tevaovć K. 99 ome Itepetatos of the gd odes Ketc Paametes XVIIIth ITAM HAIA Apstacts pp [3] Hedh tevaovć K. 993 ame Vectoal Itepetatos of the Ketc Paametes of old Mateal Les ZAMM. Agew.Math. Mech T53-T56 [4] Hedh tevaovć K. 993 The Mass Momet Vectos at -dmesoal oodate ystem Teso apa Vol pp [5] Hedh tevaovć K. Vecto Method of the Heavy oto Ketc Paamete Aalyss ad Nolea yamcs Uvesty of Nš Moogaph p. 5 Eglsh YU IN [6] Hedh tevaovć K. 998 Vectos of the ody Mass Momets Moogaph pape Topcs fom Mathematcs ad Mechacs Mathematcal sttute ANU elgade Zbok adova pp Publshed 999 Eglsh Zetalblatt evew [7] Hedh tevaovć K. 998 evatves of the Mass Momets Vectos wth Applcatos Ivted Lectue Poceedgs 5 th Natoal ogess o Mechacs Ioaa 998 pp [8] Hedh tevaovć K. 994 Itepetato of the Moto of a Heavy ody aoud a tatoay Axs the eld wth Tubulet ampg ad Ketc Pessues o eag by Meas of the Mass Momet Vecto fo the Pole ad the Axs acta Uvestats ees Mechacs Automatc otol ad obotcs Vol. No pp [9] Hedh tevaovć K. 998 Vecto Method of the Ketc Paametes Aalyss of the oto wth May Axes ad Nolea yamcs Paallel Geeal Lectue Thd Iteatoal ofeece o Nolea Mechacs INM III August hagha ha pp [] Hedh tevaovć K. 4 otbuto to the oupled oto Nolea yamcs Advaces olea ceces Moogaph elgade Academy of Nolea ceces IN X U :538 UKUP. T. 6 [] Hedh tevaovć K. Itepetato of the Moto Equatos of a Vaable Mass bject otatg aoud a Atatoay Axs by Meas of the Mass Momet Vecto fo the Pole ad the Axs Pocedgs of the 4 th Geek Natoal ogess o Mechacs Vol. Mechacs of olds emoctus Uvesty of Tace Xath 995 pp [] Hedh tevaovc K. ad Veljovc Lj. Vecto otatos of gd ody yamcs wth oupled otatos aoud Axes wthout Itesecto Hdaw Publshg opoato Mathematcal Poblems Egeeg Volume Atcle I pages do:.55//3569 7