THE BOUNDED ROTATIONAL AND TRANSLATION MOTION OF A BODY (A MATERIAL POINT AS PHYSICAL POINT) ON A CIRCLE

Transcription

1 1 THE BOUDED ROTATIOAL AD TRASLATIO MOTIO OF A BODY (A MATERIAL POIT AS PHYSICAL POIT) O A CIRCLE A.Yu. Zhivotov, Yu.G. Zhivotov Yuzhnoy Stat Dsign Offic This articl is ddicatd to th analysis of a forcd motion of a rotor having on dgr of frdom. Both rotational and translational motions ar considrd. Th rotor motion is rprsntd as a matrial point rotation around th fixd point assuming that a matrial point posss both a mass and a momnt of inrtia. Rotation and translation of th matrial point ar tratd as motion of th systm matrial point fixd point in an inrtial rfrnc systm. An attntion is drawn to th fact that in ordr to nabl rotation and translation of a matrial point with a constant vlocity, it is ncssary to apply an additional torqu for th purpos of ovrcoming of th inrtial momnt bing cratd by a momnt of inrtia of th matrial point, which is producd du to a displacmnt of th matrial point with rspct to th fixd point. Th spcial attntion is paid to th cas of a simultanous rotation and linar displacmnt of th matrial point by thr virtual displacmnts with rspct to th fixd point. Th first law of wton says that if thr is no ffct of any othr body to a body bing undr considration, this body will maintain th quiscnt stat or kp a uniform rctilinar inrtial motion. Such a motion is rfrrd to as a fr motion. Th fr-moving body consists of th matrial points bing connctd to ach othr crating th singl whol without intracting as indpndnt bodis do. Such an assumption is usd in th classical mchanics. Thrfor, on can say that th matrial points of th fr-moving body ar in th wightlssnss stat. Th first law of wton says nothing about fr rotation of a body possssing th principal momnt of inrtia about th principal axis of inrtia, although such motion xists. Th rason is as follows: wton has considrd th plantary motion as a boundd motion of th matrial point possssing only th mass, whil a siz and a momnt of inrtia could b nglctd. In fact, wton considrd th boundd translation of th matrial point along a curvilinar trajctory. Such a conclusion is vidntly provn through th xampls of th matrial point motion. On of th xampls providd in th physics is a boundd motion of th body undr ffct of paralll gravitational forcs of th Earth. A thrown body prforms a complx motion consisting of inrtial translation with a constant linar vlocity in th dfind dirction, and of th uniformly acclratd fr fall. Bcaus of ths two motions, th body movs along a curvilinar trajctory. Gravitational forcs can b rfrrd to as mass forcs du to th fact that thy ar applid in a continuous mannr to ach matrial point of th body. In this cas, th matrial points as indpndnt bodis ar not intracting btwn thmslvs and, thrfor, thy ar in th wightlssnss stat. Th mor complx xampl of th boundd motion is th plantary motion around th Sun. Th motion of a plant as a matrial point is prformd in th fild of th gravitational forcs convrging to th cntr of th Sun. Du to such a fatur of th fild of th gravitational forcs, th fr motion of a plant is prformd whil changing th dirction of th vlocity. Th gravitational fild of th Earth affcts in th sam mannr th moving satllits. It is considrd that th motion of plants and satllits is prformd along closd curvilinar trajctoris. Dspit a changd dirction of th plant vlocity, th principl of body s matrial point wightlssnss is consrvd also for that typ of a motion. In cas of boundd translation, th body possssing th principal momnt of inrtia can prform a uniform rotation about principal axis of rotation as in th cas whn no forcs ar affcting th body. Such a fact dmonstrats that th fild of gravitational forcs dos not affct th body s rotation, bcaus th gravitational forcs ar th mass forcs not producing a torqu. Th fr motion of th body possssing th principal momnt of inrtia is not lft without du attntion. Th physics considr a rotation of th Earth, and it is shown that in th cours of rotation th fild of cntrifugal acclrations is producd. It affcts th body s matrial points and crats divrging cntrifugal forcs affcting ach matrial point as gravitational forcs do. Thr is anothr diffrnc in addition to th path of th forc application: th valu of th cntrifugal forc dpnds on a radius of th body s rotation about th principal axis of th inrtia. Th body would b dstroyd into sparat matrial points if th forcs of anothr natur will not xist. Ths forcs prvnt from such dstruction. In cas th matrial points bing nabld to mov insid th body ar prsntd, thn such matrial points ar intracting btwn thmslvs and thir intraction rsults in dvlopmnt of th pondrability stat.

2 In th whol, th uniform rotation of th body as a motion has a principal diffrnc with rspct to an inrtial rctilinar uniform motion of th body. Th boundd motion of a body along a curvilinar trajctory undr th action of both th activ forc bing gnratd by anothr body and th gravitational forcs bing gnratd by th Earth rprsnts a nw qualitativ class of th motion. Undr th action of a forc, th body prforms an acclratd motion in th dirction of th forc application. In this cas th inrtial proprtis of th body opposing its acclratd motion ar manifsting thmslvs. Th mass of th body bing includd into th dfinition of th activ forc accounts for th inrtial proprtis of th body in an inrtial rfrnc systm. Thrfor, th inrtia forcs of th body ar takn into considration only in a non-inrtial rfrnc systm. Th distinctiv fatur of such translational motion of th body is an intraction of th matrial points of th body btwn thmslvs. Thrfor th matrial points of th body ar in pondrability stat. A mor complx motion of th body undr action of an activ forc in cas whn th path of th forc application dos not go through th body's cntr of gravity might occur. In such a cas, in addition to th rctilinar motion of th body occurs a complmntary acclratd rotation of th body possssing th principal momnt of inrtia about th principal axis of inrtia. Such a motion can b rfrrd to as a boundd rotation undr th action of th torqu about th cntr of gravity rsultd from th action of an activ forc. Hnc, th boundd rotation of th body is prformd about th principal axis of inrtia. It is a uniformly acclratd motion, and it could occur only in prsnc of th torqu-producing forcs. Howvr thr is also th third typ of rotation of th body possssing th momnt of inrtia. Such typ can b rfrrd to as th boundd rotation of th body. Such a typ of th body's rotation has bn producd in an artificial way as a rsult of imposing th mchanical constraints onto th body. Such constraints forc th body to prform an inrtial rotation about prviously chosn axis or a point undr th action of th torqu-producing forcs. Du to that fact, a nw typ of th systm has bn stablishd. It could b rfrrd to as th systm with th mchanical constraints or as th mchanical systm. Th distinctiv fatur of th systm rotation is th rotation with th sam vlocity of all its points bing situatd at th diffrnt radial distancs around th sam fixd axis or point. In this cas, any variation of th matrial points' rlativ position is finit and rsults from th dformation of th mchanical constraints. It is obvious that such systms in addition to th principal momnt of inrtia can possss a complmntary on, bcaus a dtrmind axis of rotation may not coincid with th principal axis of inrtia. If in th cours of rotation a systm allows changs of th matrial points' position as a function of th spd of rotation, thn such a systm bcam a systm with variabl momnts of inrtia. Th common fatur of all bodis and systms prforming a rotation is th fild of th cntrifugal acclrations divrging from th axis of rotation. Such acclrations impl ach matrial point to rturn to th uniform rctilinar motion. It is vidnt that in som cass a mchanical systm can b rprsntd as a matrial point possssing th mass and th principal momnt of inrtia, which is connctd to a fixd axis in som particular way. Th mchanical systm has bn occurrd as a rsult of th tchnological progrss and thy ar th products of nginring activitis. Usually th dsign forss th coincidnc of th body's cntr of gravity with th rotation axis. It rsults in th fact that th rsultant of all cntrifugal forcs of th rotor s matrial points is qual to zro. In practic, it is not always possibl to locat th body's cntr of gravity on th rotation axis. As a rsult, th rsultant of cntrifugal forcs mrgs. It is applid at th rotor's cntr of gravity and is dirctd along th rotation radius. Usually th dsign approachs fors suprposition of th rotor's principal axis of inrtia with th rotation axis. Such a systm posssss th principal momnt of inrtia uniformly distributd around th rotation axis in th sam mannr as for th body prforming a fr rotation about th principal axis of inrtia. Du to this fact th systm is nabld to prform a uniform rotation during unlimitd priod of tim if thr ar no applid torqu-producing activ forcs. Th rotating systm having th rotation axis coinciding with th principal axis of inrtia prforms a uniformly acclratd rotary motion undr th action of th torqu-producing activ forc whn no rsistanc forcs ar prsnt. Howvr, in practic, it is impossibl to achiv a full coincidnc btwn th principal axis of inrtia of th systm and th axis

3 3 of rotation. Th momnt of inrtia of such a systm is gratr than th principal momnt of inrtia. On could say that th systm posssss a complmntary momnt of inrtia bcaus of not complt coincidnc btwn th principal axis of inrtia and th rotation axis. An important fatur of a complmntary momnt of inrtia is th fact that it introducs asymmtry into th systm s momnt of inrtia. Thrfor a complmntary momnt of inrtia of th systm could b considrd as a disturbing factor that countracts th systm rotation. It is obvious that in ordr to ovrcom th ngativ ffct of a complmntary momnt of inrtia th torqu should b applid to th systm. Exactly bcaus of this typ of th rotating systm, disagrmnts in physics and nginring mchanics hav bn occurrd. On th on sid, it is considrd that th mchanical systm "solid body fixd axis posssss on dgr of frdom, and a forcd acclratd rotation of th body about fixd axis (in accordanc with th nrgy consrvation law) is prformd undr th action of forcs bing abl to produc a torqu. In cas such forcs ar not prsnt, a mchanical systm prforms a uniform rotation about a rotation axis if th systm posssss only th principal momnt of inrtia. Howvr, whil analyzing a systm matrial point-fixd axis it is considrd that th matrial point bing includd into th systm posssss two dgrs of frdom, and a rotation of th matrial point about fixd axis is prformd undr th action of a cntriptal forc, which is not abl to produc a torqu. In this cas, it is not considrd a fact that such a systm posssss a crtain momnt of inrtia, which includs th principal momnt of a matrial point. Th law of th kintic nrgy consrvation lavs a room for a ral cntrifugal forc, whil rotation of th matrial point about fixd axis rquirs considring a cntrifugal forc as an imaginary on. Th discrpancy btwn th modl of th matrial point rotation about fixd axis and th law of th kintic nrgy consrvation can b xplaind by a simpl carryovr of th plantary rotation physics provisions to th mchanical systm rotation. Considring abov-writtn commnts, w can analyz a forcd rotation of th matrial point C about fixd point undr th action of th activ forc in th horizontal plan (s Fig. 1). This matrial point posssss th mass m F a and th momnt of inrtia I z. F a C ω O Fig. 1 F u W assum that th matrial point C is put off to th distanc and is mchanically connctd with th fixd point O. Th mchanical connction is idal. Such a systm posssss on dgr of frdom. In cas of sufficintly big distanc btwn th matrial point and th fixd point, th motion of th matrial point with th rotation spd ω can b considrd as th on bing clos to th rctilinar motion. Thrfor, at first glanc, th law of th uniformly variabl curvilinar motion of th matrial point would look lik th law of th uniformly variabl rctilinar motion of th matrial point. Th only diffrnc is th following: in cas of curvilinar motion of th matrial point th linar vlocity and tangntial acclration ar takn into account []

4 4 at S = S0 + V0t+, (1) whr S is th pathway travrsd by th matrial point, S 0 - is th pathway travrsd by th matrial point by th tim of initiation of th uniformly acclratd motion - tangntial vlocity of th matrial point prior to initiation of th uniformly acclratd motion, V 0 t - tim of th matrial point motion, a - tangntial acclration of th rotating matrial point. Th xprssion (1) can b also rprsntd in th following form: ξ t S = S0 + ωt+, () whr ω - is th angular vlocity of th matrial point, ξ - angular acclration of th matrial point. Howvr, th matrial point maks part of th systm, and a rotation of th matrial point is prformd about an axis, which is paralll to th principal axis of inrtia of th matrial point at th distanc. In this cas, th systm's momnt of inrtia consists of principal momnt of inrtia of th matrial point and th complmntary momnt of inrtia of th matrial point about th fixd point 0 z I = I + m. (3) Gnral momnt of inrtia of th systm is an asymmtrical momnt of inrtia. Thrfor, so calld inrtial momnt is producd whil matrial point is rotatd about th fixd point. Such inrtial momnt countracts th systm's rotation [3] и M = mω. (4) Th inrtial momnt of th systm acts around th fixd point opposing a dirction of th systm's rotation. Thn, it can b rprsntd as th forc, which is acting onto th matrial point in th dirction opposit to th dirction of th forc F a F и application и F = mω. (5) In this cas, for th tim point t of th rotor's spinning, th variabl rsultant forc whos action nabls th matrial point motion can b drivd R = F F = ma* mω = ma, (6) a whr a * - is th acclration dtrmining th valu of th dfind forc, a - an instantanous acclration at th vlocity valu ω at th givn tim point. * It is vidnt that th valu of rotation in turn dpnds on duration of th forc F и и * dpnds on th systm spd of rotation, whil th spd of F a application.

5 5 In this cas a gnral quation of th rotary motion of th systm may b drivd in th form shown blow basing on th law of th kintic nrgy consrvation and taking into account th known gnral quation of th rotary motion []: dω I0 = ma * mω. (7) dt Th quation shows that undr th action of th forc having a constant valu, th systm's rotation is passs with variabl acclration. In ordr to nsur th systm's rotation with a constant valu of acclration, which is charactrizd with th quation (1), it is ncssary to apply a variabl forc, whos valu is varid in accordanc with a crtain law. Sparating th variabls and rsolving th diffrntial quation, w can driv a dpndnc btwn th duration of th forc F and th matrial point rotation spd ω a I0 a* + ω t = + ln. (8) m a* a* ω dω Th maximum valu of th systm rotation spd ω max is achivd at = 0. dt In this cas, from th quation (7) w can driv: max ma* mω = 0, (9) a * ω max =, (10) a* max = ω, (11) ξ* max = ω, (1) whr ξ * - is an initial angular acclration at th bginning of th matrial point motion. Stting various valus to ω th tim of ncssary to rach th rquird lvl of th matrial point rotation spd can b valuatd using th quation (8). Th xprssion (9) shows th condition for th uniform rotation of th systm too. It follows from th condition that th uniform rotation of th systm can b achivd only in th cas whn th activ forc F a is applid, or in cas th torqu is applid to th systm in ordr to countract its inrtial momnt. Th quation (9) shows that in cas th valu of both th forc Fa and th distanc ar constant, th matrial point initially prforms a circular motion with acclration, whil aftr a tim it starts a motion about th fixd point with a finit rotation spd ω. Th tangntial vlocity of th matrial point, which w will rfr to as V will b upon a crtain tim th finit on too. In th procss of th matrial point circular motion, th xisting mchanical ti forcs th matrial point to chang th rotation spd dirction whil kping on th distanc btwn th matrial point and th fixd point. Lt us assum that during a small priod t th dirction of th tangntial vlocity has bn changd by a small angl ϕ. It is vidnt that th tangntial vlocity will obtain an incrmnt max

6 6 V = V ϕ if ϕ 0. In such a cas, th variation of th tangntial vlocity in small intrval of tim dfins in th limit an acclration of th matrial point undr th action of th raction forc F R bing applid from th fixd point by mans of th mchanical connction, which is sufficint for changing th matrial point vlocity dirction Vd ϕ a= = V ω = ω. (13) dt Th raction forc F R of th fixd point (s Fig. ) occurs as a rspons to attmpts of th matrial point to prform its inrtial rctilinar. C Fc f ω O F R Fig. It is obvious that th matrial point applis to th fixd point th sam forc trying to impart th sam acclration to th fixd point. By this is mant that th inrtia of th matrial point producs initially a cntrifugal forc, and as th rspons to its action (in accordanc with th third law of wton), th raction forc of th fixd point is occurrd cf F = mω = F. (14) R Th xprssion (13) also shows that th occurring acclration is qual to th tangnt acclration in trms of magnitud at th bginning of th matrial point motion undr th action of th activ forc. Usually th mchanics dal with a motion of mor complx systms undr th action of svral forcs of diffrnt natur including also th wight. Thrfor th raction forc dpnds on all forcs bing applid to th matrial point. Du to this fact, th concpt of th cntrifugal forc is xtnsivly usd in th rotor dynamics. This forc could b asily considrd in th analysis togthr with othr forcs bing applid to th body. Thrfor, a particular fatur of th forcd rotation of th matrial point in th slctd coordinat systm (as distinct from a rctilinar motion of th matrial point undr th action of a forc) is an origination of two additional forcs: th cntrifugal forc and th support raction forc. Thr wr som attmpts in th rotor dynamics thory bing basd on th oscillation thory to introduc an ffct of Coriolis forcs onto th rotor spinning. It is pointd-out that considration of Coriolis forcs incrass th calculation accuracy [1]. In this connction, an analysis of th rotary motion faturs of th systm "matrial point fixd point" plays a ky rol in th cas whn th matrial point is abl to prform not only a rotation, but also a translation with som linar vlocity rlativ to th fixd point (.g. in th radial dirction). In th bginning, w shall considr uniform circular motion of th matrial point about th fixd point O with a radius (s Fig. 3).

7 V лин 7 V C O ω Fig. 3 W assum that th matrial point prforms a rotary motion along a circl with a constant tangntial vlocity V and in addition it movs with a linar vlocity V лин along th sam circl. W assum that th tangntial and th linar vlocity of th matrial point hav th sam dirction. An absolut vlocity of th matrial point can b dfind as a лин V = V + V. (15) An acclration acting on th matrial point during its rotation and changing th vlocity dirction as is wll known, can b dfind as V a a =. (16) Thrfor an absolut cntrifugal acclration of th matrial point will b qual to a a ( V ) ( Vлин) V Vлин = + + or to ( Vлин) aa = ω + + ωvлин. (17) An analysis of th rlationship (17) shows that acclration of th matrial point is composd of th following parts: th acclration, which could hav a matrial point in cas it would prform only rotary motion; th acclration, which could hav a matrial point in cas it would prform only a circular motion with a linar vlocity; th complmntary acclration, which is rfrrd to as th Coriolis acclration. Of thr abov mntiond acclrations th two formr ons hav th sam dirctions. Th dirction of th Coriolis acclration is orthogonal to thm []. That mans that for th rotation of such a systm it is ncssary to incras an additional torqu. It is gratr than th torqu M rquird for rotation of a fixd matrial point кр ( Vлин) M кр = m( ω + + ωvлин ). (18) If th linar vlocity V лин of th matrial point is oppositly dirctd, thn th absolut cntrifugal acclration can b dfind as

8 8 ( Vлин) aa = ω + ωvлин. (19) If th matrial point movs with som small angl α with rspct to a mridian circl, thn th absolut cntrifugal acclration can b dfind in th following way (dpnding on th dirction of a motion) ( лин) aa = [ ω + ± ωvлин]cosα. (0) V In cas whn th matrial point motion is dirctd at a grat angl α, it is ncssary to tak into account a displacmnt of th matrial point along th radius whil dfining th absolut cntrifugal acclration. Lt us assuming that th matrial point movs with a constant linar vlocity along th rotation radius from th point to th point M. Such a vlocity will b rfrrd to as (s Fig. 4). V лин O ω V лин C M F и M V лин M F и Fig. 4 Th forc at th point M и F acts onto th matrial point at th point M bing producd by an inrtial momnt, whil th forc Fи acts, which is also producd by an inrtial momnt. W shall mak an assumption that 0. Th tangntial vlocity of th matrial point at th point is qual to M V = ω. (1) Th linar vlocity of th matrial point can b dfind in th following way: лин M V = V V. () Th tangntial vlocity of th matrial point at th point M bing dfind from th rlationship (0) is qual to M лин V = V V. (3) Basing on th quation (3), w can dfin an absolut cntrifugal acclration at any point M

9 9 M ( Vлин) aa = ω + + ωvлин. (4) If th linar vlocity of th matrial point is dirctd towards th fixd point O, thn an absolut cntrifugal acclration at any point can b dfind in accordanc with th following rlationship M ( Vлин) aa = ω + ωvлин. (5) M Th driving torqu M кр rquird for a uniform rotation of a movabl matrial point can b drivd from th following rlationship: M ( Vлин) M кр = m( ω + + ωvлин). (6) If a uniformly rotating matrial point prforms at th sam tim a motion along both th radius and th circular trajctory with constant linar vlocitis, thn, obviously, an absolut acclration of th matrial point will b dfind in trms of absolut cntrifugal acclrations of th matrial point in considrd dirctions. A motion of th matrial point can also b prformd along a gnratrix, which is prpndicular to th rotation plan. This is th third virtual displacmnt of th matrial point, which dos not mak any influnc on th valu of a cntrifugal acclration bcaus a tangntial vlocity of th matrial point dos not chang its magnitud in cas of such displacmnt. For most rotors th gomtry and mass charactristics ar constant valus. Howvr, it is typical of th rotors of sparation cntrifugs to chang thir masss. Thrfor, allowing for th absolut cntrifugal acclration and with it an ffct of th Coriolis forcs to th rotor dynamics ar applicabl only to th limitd class of variabl-mass rotors. In addition to th mchanical systms prforming a rotation about som dfind axis, thr ar mchanical systms nabling translational motion of a body around dfind axis along a circl. At ral conditions, a mchanical systm usually includs bodis, which prform thir motion, and bodis prforming a translational motion along a circl. Ths typs of body s motion hav som similaritis; howvr thy hav som qualitativ diffrncs and distinct physical ssnc. As distinct from a rotary motion of th mchanical systm, in cas of a translational motion of a systm along a circl, th principal momnt of inrtia of th systm can b disrgardd. For what concrns th rotor dynamics, a translational motion of a body along a circl has bn rronously considrd as a rotary motion. Howvr a translational motion of particular lmnts of th rotating mchanical systm plays also a significant rol in th rotor dynamics. Th quation for th matrial point translational motion along a circl can rprsntd as dω = a* ω. (7) dt Th matrial point rotation spd and duration of a forc application ar boundd a* + ω t = + ln. (8) a* a* ω Th studis has bn prformd show that th systm "matrial point fixd point, in th sam way as th systm rotor supports is a systm with asymmtrical momnt of inrtia, which dos not dpnd on th rotor s rotation spd. Corrspondingly, an inrtial momnt mrgs in th

10 10 systm. It dpnds on vlocity and countracts th rotation of th systm. Obtaind quations and rlations nabl valuating of th rotary motion spcialtis, momnts of inrtia of th systm, inrtial momnts of th systm, and a running torqu of a driv for ach particular cas. Th studis ar prformd following principal laws of physics and nginring mchanics. Thy ar also vrifid by th xprimnt. Th comparativ analysis of th obtaind data with known xprimntal data [4] confirms th appropriatnss of th study has bn prformd. Th rsults obtaind from th study mak part of th furthr activitis aimd at an stablishmnt of a nw modl of rotation of th rotors basd on a flxibl shaft. REFERECES 1. Ishida Y, Murakati S, Inou T. onstationary oscillations of a nonlinar rotor systm with intrnal rsonanc during acclration through critical spds. // IFToMM Sixth Intrnational Confrnc on Rotor Dynamics, Procdings Volum, Sidny, Australia p.. Тарг С.М. Краткий курс теоретической механики. М.: Наука, с. 3. Животов А. Ю.: Особенности Вращения Дискового Ротора со Статической Неуравновешенностью. J. Visnik of th East Ukrainian ational Univrsity (36): 18-5, Щепетильников В.А. Основы балансировочной техники // Уравновешивание жестких роторов и механизмов. М.: Машиностроение Том с. A.Yu. Zhivotov, Yu.G. Zhivotov