New Theory of Rotor Dynamics: Dynamics of Jeffcott Rotor with Moment Unbalance

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1 th FToMM World Congress, Besnçon (Frnce), June8-, 7 New Theory of Rotor Dynmics: Dynmics of Jeffcott Rotor with Moment Unblnce A. Y. Zhivotov * Yuhnoye Stte Design Office Dniepropetrovs, Urin Abstrct The dynmics of the rotor with moment unblnce ws exmined for the first time reltively to rotor with solid shft in resilient supports. These reserches mde possible to describe the physics of rotor rottion process nd define rotor position with respect to rottion xis. The system of equtions obtined mes possible to determine rottionl vribles of rotor, forces nd moments ffecting rotor under different rottion conditions. An importnt prcticl prmeter, which obtined mthemticl expression, is criticl velocity. The criticl velocity depends on the rotor moments of inerti, distnce between supports nd coefficient of elsticity of supports. With tht coefficient of elsticity of supports defines proportion between the restoring elstic force nd rdil displcement of shft section in support. However, ppliction of this formul in determining criticl velocity of rotor fixed on flexible shft, gives result, which differs by hlf s much gin s minimum from n ctul criticl velocity of rotor. This is relted with principlly different suspension/mounting of rotor in mchine. Such specific rotor ttchment on flexible shft requires conducting dditionl reserches of the rotor dynmics for obtining new equtions describing the rotor dynmics, which hs moment unblnce. n this pper the bses re given on cuses of the lc of coincidence of criticl velocities of rotors with their different suspension type in rotor mchine. The principl difference in rottionl fetures of rotor is noted, which is relted with rotor position with respect to rottion xis. The nown eqution of the rotor dynmics with dynmic unblnce is nlyed for the purpose of ccounting distinction reveled. New prmeter ffecting the rotor dynmics is introduced. Trnsformtions of the nown dynmic eqution re performed. New system of equtions is composed. An nlysis of the new system of equtions is conducted, which describe the rotor dynmics. Reserches of the new system of equtions re conducted in reltion to the dis nd cylindricl rotor t subcriticl, trnsfer nd super-criticl rottion conditions. These reserches show tht from the qulittive point of view the rotor dynmics with flexible shft does not differ from the rotor dynmics with solid shft, fixed in resilient supports. However, the equtions obtined cuse essentil qulittive distinctions in determintion of criticl velocities, forces nd moments cting on rotor. The results obtined show tht the rotor dynmics with qusi-sttic or dynmic unblnce moment or dynmic unblnce cnnot be described using the single system of equtions. *E-mil: -hivotov@mil.ru For describing the rotor dynmics with flexible shft, the system of equtions of the rotor dynmics with sttic unblnce is required to be composed s well s the system of equtions of the rotor dynmics with moment unblnce. With tht n interction of these systems of equtions is required to be estblished Keywords: dynmics, rotor, criticl velocity. ntroduction The rotor behvior under effect of sttic unblnce is minly reserched in the vibrtion theory of the rotor dynmics []. The problems of rotor rotting under effect of the so-clled dynmic unblnce re under considertion more seldom []. Rottion of rotor with moment unblnce is mentioned s the importnt problem [3]. However, the theory of rotor dynmics with moment unblnce is bsent. This is relted with tht the vibrtion theory tes into considertion the oscilltions of rotor center of mss with respect to neutrl xis. f for exmple, the Jeffcott rotor hs only moment unblnce, then oscilltions of center of mss re bsent. n this connection the vibrtion rotor dynmics cnnot describe the rotor dynmics with moment unblnce. The inertil theory of the rotor dynmics hs developed specific condition for solving this problem. n the beginning, from positions of the inertil dynmics, the Jeffcott rotor rottion with sttic unblnce ws considered [4]. Then rottion of rotor with qusisttic unblnce in resilient supports ws considered [5], s well s the dynmics of console rotor ws considered [6]. For the first time the dynmics of rotor with moment unblnce ws considered using s n exmple rotor rottion, solid shft of which is fixed in resilient supports [7]. These reserches mde possible to describe the physics of rotor rottionl process nd define rotor position with respect to rottion xis. The system of equtions obtined mde possible to define the rottionl prmeters of rotor, forces nd moments cting rotor t different rottionl conditions. However, the ttempts on using the obtined reserch results to define the rottionl chrcteristics of the Jeffcott rotor with moment unblnce reveled the discrepncies. n prticulr, criticl velocity of the Jeffcott rotor ws higher thn rted velocity. n this connection our ttention ws pid to

2 th FToMM World Congress, Besnçon (Frnce), June8-, 7 the structurl fetures of the Jeffcott rotor mounted in solid supports from rotor fixed to solid shft in resilient supports, s well s on the discrepncies in displcement of shft ttchment point with rotor to rottion ngle of rotor cross-section with respect to rottion xis. These specific fetures were ssumed s bsis of reserches given below.. Reserch object nd coordinte system Let us consider the rottion of the Jeffcott rotor with mss m under ero-g conditions nd vcuum. A rotor is mde s body of revolution nd hs n xil moment of inerti nd equtoril moment of inerti. Assume tht <. Rotor is fixed in the middle of flexible shft. The shft hs the elstic properties. Under effect of lods the shft bends, nd when unloded, it is returned to n originl position. The shft is mounted in two solid supports. Supports re plced t distnce l from rotor center of mss. Moment unblnce we set by n inclintion ngle ϕ of min centrl xis of inerti to geometricl rotor xis. The rottion of rotor we consider in the rotting coordinte system OXYZ. An origin О of the coordinte system OXYZ coincides with center of mss of rotor (See Fig. ). Axis Z coincides with rottion xis. At the beginning moment of the rottion n xis X is in plne coincides with plne of ction of moment unblnce. Axis Y is not shown in Fig.. To reserch rotor motion we choose two mesurement plnes - nd -. We consider tht distnce from center of mss of rotor to the mesurement plnes is lso equl to l. We shll study the rotor motions in the mesurement plnes - nd - ccording to trcs of rottion xis (points A nd A ), ccording to trcs of geometricl xis of rotor (points B nd B ), ccording to trcs of min centrl xis of inerti of rotor (points nd C ). C Fig.. System of cylindricl rotor supports When the rottion begins, geometricl rotor xis devites from rottion xis by n ngle. An ngle γ is formed between rottion xis nd min centrl rotor xis of inerti. Angles of the rotor devition from rottion xis we te s low. We lso consider tht distnce between points A nd ( A nd ) equls to d, distnce between points B B B C B C A nd C ( nd ( nd ) equls to b nd distnce between points A nd C ) equls to σ. A,, C nd ositions of the trcs of xes (points B points A, B, C ) in the mesurement plnes - nd - re the sme. Therefore in the next we concentrte on the results observed in plne -. From the beginning of the rottion, rotor turns bout geometricl xis by n ngle β in such mnner s it is shown in Fig.. Fig.. Schemtic lyout of forces nd moments cting on rotor The forces nd moments presented on the Fig. described below. Equtions for their determintion re provided below lso.. Coefficient of shft ngulr rigidity The shft hs definite elsticity nd under the ction of forces nd moments it cn bend. A shft bending cn te plce under the ction of force pplied for exmple to center of mss of rotor. n the rotor dynmics with sttic unblnce the proportionlity is estblished between this force cting on rotor nd displcement vlue of rotor shft t the point of force ppliction from rottion xis. Usully the point of force ppliction is center of mss of rotor. A coefficient of proportionlity is coefficient of the rotor rigidity. A coefficient of the rotor rigidity depends on the detils of shft fstening in resilient supports nd rotor position with respect to these supports. n the rotor dynmics with moment unblnce, when rotor is fixed on flexible shft, center of mss of rotor coincides with geometric center nd there is no displcement of center with respect to rottion xis. This cse excludes the possibility on using the coefficient

3 th FToMM World Congress, Besnçon (Frnce), June8-, 7 of а shft rigidity mentioned. n spite of the lc of the displcement of geometric rotor center from rottion xis, shft bending tes plce under the ction of moment pplied to rotor for exmple with respect to its geometric center. Therefore in order to estblish the interction between cting moment nd n ngulr displcement of а rotor cross-section pssed through its geometric center, we introduce the concept of n ngulr rigidity of shft, which is chrcteried by coefficient of n ngulr rigidity of shft. A rotor cross-section pssed through its geometric center, we nme s centrl cross-section. A coefficient of n ngulr rigidity of shft depends on the fixturing detils of shft in resilient supports nd on position of rotor with respect to these supports. With ccount of coefficient of n ngulr rigidity of shft, we cn write n eqution defining the interction of n cting moment nd n ngulr displcement of centrl cross-section: = =. () d M sin The coefficient of n ngulr rigidity of shft is possible to express using the coefficient of the generlly ccepted rigidity of shft, relted with displcement of centrl cross-section of rotor from rottion xis by mens of the coefficient of proportionlity. =. () n prticulr, for console rotor the coefficient of proportionlity is defined s follows [5]: V. Forces cting on rotor 3 =. (3) l During the rottion, the following forces nd moments ct on rotor (See Fig. ): moment N γ, occurring s result of devition of min centrl xis of inerti from rottion xis, which is possible to write in the form: γ ω γ γ ω γ ω N = ( ) sin cos =. (4) = ( ) sin = ( ) γ Moment N γ (4) cn be represented s two forces nd rotor: γ γ, cting on n rm l bout center of mss of ( ) sin cos γ γ l ( ) ω sin γ ( ) ω ω γ γ = = = ; (5) γ = = l l moment, creting by the shft elsticity in Moment N y connection with devition of geometric xis of rotor from rottion xis by n ngle : N y Ny sin = =. (6) cn be represented s two forces y nd y, cting on n rm l bout center of mss of rotor: y sin = y = = ; (7) l l rottionl moment, which cn be represented s two forces: N кр кр кр = кр = N N кр l sin = l ; (8) moment of inerti Nu = Nu + Nu, cting bout rottion xis in n opposite direction of rottion: N u = ( ) ω sin cos = ( ) ω sin = ( ) ω = ( ) ω sin cos Nи = Nи ( ) ω sin ( ) ω = = V. Dynmic equtions of rotor = =, (9). () Ting into ccount tht t constnt speed of rottion, the system rotor flexible shft supports is t unchngeble position in the rotting coordinte system, we cn me up the equtions of moment with respect to the points A, B, C. n ming up the system of equtions we shll use the geometric dependencies: = =, () d l sin l b = l sinϕ = l ϕ, () σ = γ = γ, (3) l sin l

4 th FToMM World Congress, Besnçon (Frnce), June8-, 7 B Е = db sin β l sin sinϕsin β lϕsin β σ = sinγ =, (4) γ σ = d + b + dbcosβ, (5) γ = + ϕ+ ϕ β, (6) sin sin sin sin sin cos smll ngulr devitions of rotor from rottion xis. As is nown, criticl speed of rotor is chieved if cos β =. Consequently, we hve the following: ω = r. (9) = + + cos, (7) γ ϕ ϕ β C N = b sin β = l sinϕsin β = l ϕsin β, (8) BN = bcosβ = l sinϕcosβ = lϕcosβ. (9) Ming up the equtions of moments with respect to the points A, A, B, B, C, C we obtin the following: d N =, () p u BE γ N u =, () B N + N C N =. () p u y After substitution of vlues, we obtin the following: N p l sin ( ) ω sin =, (3) l sin ( ) ω sinγ l sinsinϕ sin β l sinγ, (4) ( ) ω sin = N p l sinϕcos β ( ) ω sin + l sin l sinsinϕsinβ = l. (5) There is criticl speed for the rotors where >. Such rotors we consider s cylindricl ones. Dis rotors do not hve criticl speed, since <. V. Dynmic fetures of cylindricl rotor The gyro-dynmics of cylindricl rotor is described by n eqution form of which coincides completely with the generl dynmic eqution (6). A. Sub-criticl rottion mode We nme the rnge of velocities s sub-criticl rottion mode, for which we cn ssume with sufficient ccurcy level, tht cos β =. n this cse n eqution (8) tes the form: ( ) ω ( + ϕ) =. (3) Let us ssume, tht rotor is blnced nd ϕ =. n this cse, from n eqution (3), we hve: ω = ch. (3) An eqution (3) defines frequency of nturl rotor oscilltions ω ch, which coincides with criticl rotor velocity. As consequence of this feture, when n unblnced rotor reches criticl rotor velocity, the resonnce phenomen occur. Let us note tht n eqution (3), if ϕ =, then it is possible to write it eeping vrible, in the form: After simple trnsformtions, we hve the following: p ω ω N = ( ) sin = ( ), (6) sin sin β = sinϕ = ϕ, (7) ( ) ω ( + ϕcos β ) =. (8) ( ) ω =. (3) Such form of n eqution llows ming conclusion on the stbility of this system t rndom devition of rotor by some ngle. f t ω = const, the inequlity shll be observed t subcriticl velocities: ( ) > ω. (33) The dynmics of rotor we consider with ccount of

5 th FToMM World Congress, Besnçon (Frnce), June8-, 7 nd then stble rotor rottion bout rottion xis is possible if ω < Defining from n eqution (3), we hve: B. Trnsfer rotor mode. (34) ( ) ωϕ =. (35) ( ) ω Eqution (8) cn be written in the following form, if cos β : ( ) ( ) ω + ϕ =. (36) t follows from n eqution (36): = 4 ( ) ωϕ. (37) 4 ( ) ω ( ) ω ( ) + We obtin = ϕ from n eqution (37), if ω = ωr. f ω, then from n eqution (37) we hve: ϕ =. (38) The results obtined demonstrte tht the resonnce phenomenon provides rotor trnsfer to supercriticl rottion mode. The resonnce phenomenon cuses n incresing. A vlue becomes higher tht vlue ϕ nd the regulrity described by n eqution (7), is violted. This ensured tht the conditions occur when rotor trnsfers to supercriticl rottion mode nd n eqution (36) is not fulfilled with incresing velocity. Let us note tht fulfilling n eqution (36) with n unlimited incresing velocity would cuse the rotor rottion in the mode, which is not dvntgeous from the energy point of view. t follows from equtions (7) nd (37) tht in the trnsfer rotor rottion mode: = ( ) sin β = = ϕ 4 ( ) ω 4 ( ) ω ( ) ω +. (39) n the trnsfer mode condition (33) of the stble rotor rottion is observed. However, with incresing velocity, the stbility mrgin decreses. Beginning from some velocity, rndom devition of rotor from rottion xis cuses n unstble rottion mode, which is chrcteried by the resonnce phenomenon. C. Supercriticl Rotor Mode The resonnce phenomenon cuses violtion of regulrities described by n eqution (36). However, if rotor psses through criticl velocity, we observe the rottion stbilied. n stbilition of the rottion, the retrogrde precession of rotor tes plce, which exists until n ngle β becomes equl to 8. f we ssume tht cos β =, then n eqution (8) tes the form: ( ) ω ( ϕ) =. (4) This eqution exists if > nd ϕ >. Defining from n eqution (4), we hve the following: ω ϕ ω ( ) = ( ). (4) Let us ssume tht ω. n this cse we obtin = ϕ from n eqution (4). The results obtined indicte tht rotor self-centering expressed s n pproximtion of min centrl xis of inerti nd rotor rottion xis, is provided. For the supercriticl rottion mode min centrl xis of inerti is plced between geometric xis nd rottion xis. With incresing velocity, vlue γ decreses. A condition (33) t supercriticl velocities tes the form: ( ) ω γ < γ. (4) A stbility fctor of the rotting rotor increses depending on incresing velocity. V. Dynmic eqution of dis rotor As dis rotor we nme rotor for which >. For dis rotor moment unblnce devites min centrl xis of inerti from geometric xis of rotor by some ngle in the direction where min moment of unblnces cts (See Fig. 3). As result, rottion xis is plced between geometric xis nd min centrl xis of inerti of rotor. n this cse - < ϕ.

6 th FToMM World Congress, Besnçon (Frnce), June8-, 7 n obtining n eqution (8) of the rotor dynmics it ws expected tht nd re rbitrry vlues. Therefore n eqution (8) shll describe the dynmics of ny rotor including one for which >. Fig. 3. Dis Rotor-Supports System f ϕ >, then single eqution, which llows tht >, is n eqution, which is given in the form: ( ( ) ω ϕ ) =. (43) From this conclusion follows tht becuse of specific combintion of moments of inerti nd, dis rotor rottion meets the super criticl conditions. Therefore, criticl velocity cn not be chieved t n unlimiting velocity incresing. t is followed from n eqution (43): ( ) + ωϕ =. (44) ( ) ω Let us ssume tht ω. n this cse from n eqution (44) we obtin = ϕ. Consequently, rotor self-centering tes plce, when n inclintion ngle of geometric xis to rottion xis increses. When velocity incresing is unlimited ϕ, nd γ. The self-centering phenomenon cuses n incresing the mchine vibrtions. The rottion stbility condition hs the following form: rotor dynmics with moment unblnce. For the first time the generl system of equtions of the cylindricl Jeffcott rotor dynmics with moment unblnce ws obtined; this system defines specific feture of rotor behvior, forces nd moments cting on rotor. The system of equtions describes the rotor dynmics within the whole rnge of velocities under ll rottionl conditions. The system of equtions of the Jeffcott rotor dynmics with moment unblnce is obtined which defines specific rotor behvior, forces nd moments cting on rotor. The system of equtions describes the rotor dynmics within the whole velocity rnge. t ws shown tht dis rotor motion t ny velocities meets the supercriticl rotor rotting conditions. t ws shown tht the self-centering phenomenon cuses decresing the mchine vibrtions with cylindricl rotor nd incresing the mchine vibrtions with dis rotor. The reserch results cn be used in designing the rotor mchines, in development of the dignostic systems nd systems for blncing the rotors. They re ey element to study the rotor dynmics under effect of the sttic nd moment unblnce. References [] F.M. Dimentberg, K.T. Shtlov, A.A. Gusrov. Oscilltions of Mchines. М.: Mshinostroenye, p. [] A.S. Kelon, Y.. Tsimnsy, V.. Yovlev. Dynmics of Rotor in Resilient Supports. М.: Nu. Min Editing Stff of hysicl- Mthemticl Literture, p. [3] R. Gsch, H. fuetner. Rotordynmi Eine Einfuehrung, Springer Verlg Berlin-Heidelberg-New Yor, 975, SBN X [4] A.Y. Zhivotov. New Theory of Rotor Dynmics: Dynmics of Dic Rotor with Sttic Unblnce // FToMM Sixth nterntionl Conference on Rotor Dynmics. - Sidney, Austrli. - roceedings - Volume p. 57. [5] A.Y. Zhivotov, Y.G Zhivotov, Y.G. Brlu New Theory of Rotor Dynmics: Dynmics of Ubrell-Type Rotor with Resilient Support // The -nd nterntionl Symposium on Stbility Control of Rotting Mchinery. SCORMA -. - Gdns, olnd p [6] A.Y. Zhivotov, Y.G. Zhivotov. New Theory of Rotor Dynmics: Dynmics of Outbord Rotor with Qusi-Sttic Unblnce// Mshinostroenye nd Electrotechnology. Mshinintelet, Sofi, Bulgri p. -4. [7] A.Y Zhivotov. Fetures of Rotor Dynmics with Moment Unblnce // nstrument-ming Technology.. NT, Urine, Khrov. ( ) ω γ < γ. (45) A condition (45) is observed in rottion of dis rotor. A rotor rottion is stble one. According to condition (43), stbility mrgin increses depending on incresing velocity. V. Conclusions The need for introduction of the ngulr rigidity of shft ide ws for the first time substntited to study the

INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION. The three general approaches to the solution of kinetics problems are: INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The