Vibrations of a Parametrically and Self-Excited System with Ideal and Non-Ideal Energy Sources

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1 Vibrions of Prmericlly nd Self-Excied Sysem wih J. Wrminski Deprmen of Applied Mechnics, Technicl Universiy of Lublin Ndbysrzyck 36, -68 Lublin. Polnd J. M. Blhzr Deprmen of Sisics nd Applied Mhemics nd Compuions Se Universiy of So Pulo Rio Clro 35-3 Rio Clro, SP. Brzil Vibrions of Prmericlly nd Self-Excied Sysem wih Idel nd Non-Idel Energy Sources Inercions beween prmeric, self-, nd exernlly excied vibrions re nlysed in his pper. The physicl model of he vibring sysem consiss of non-liner spring wih periodiclly chnging siffness of Mhieu ype nd non-liner dmping described by Ryleigh s erm. This sysem is ddiionlly forced by hrmonic force (idel sysem), or by non-idel energy source represened by direc curren moor wih limied power supply. The model of DC moor is considered in wo vrins, s clssicl, in Kononenko s sense model, nd complee elecro-mechnicl sysem. Quniive nd quliive differences of he considered models re compred nd discussed in he pper. Keywords: prmeric vibrions, self-exciion, non-idel sysem, chos Inroducion In engineering prcice we cn disinguish differen ypes of oscillions genered by differen cuses. Among hem, we cn menion he self-excied sysems in which (roughly speking) consn inpu produces periodic oupu. The supply of energy is conrolled by he inernl properies of he sysem. Vibrions cn exis for uonomous sysems wihou exciions depending on ime nd he moion does no depend on iniil condiions, bu on prmeers of he sysem. This kind of vibrion is represened by limi cycles on he phse plne. If he limi cycle is sble, hen he self-exciion is clled sof. On he oher hnd, in opposie siuion, he limi cycle is unsble nd self-exciion is clled hrd or csrophic (becuse soluion cn end o infiniy). Prmericlly excied vibrions belong o he second clss of vibring sysems nd hey re chrcerised by periodiclly chnging in ime prmeers, like siffness or mss momen of ineri. Their chrcerisic feure is h hey re described by homogenous, bu nonuonomous, differenil equions. For some prmeers regions he rivil soluion cn be unsble nd he sysem comes o vibre wih lrge mpliudes. A hird clss, which we cn specify, re vibring sysems excied by n exernl force. Their mhemicl model is described by nonhomogenous differenil equions. All hese vibring sysems were comprehensively nlysed in he curren lierure seprely. Some ppers were lso devoed o he inercions beween wo kinds of vibrion, for exmple: selfnd exernlly excied vibrions (Awrejcewicz nd Mrozowski, 989), self nd prmericlly excied vibrions (Tondl nd Ecker, 999), (Wrminski, b). The influence of he exernl force on prmericlly nd self-excied sysem ws presened in (Szbelski nd Wrminski, 995, b), (Wrminski, ). There were observed inercions beween hree differen vibrion ypes exciing he sysem ogeher he sme ime. If he sysem is forced by funcion independen of he sysem i cs on, hen he model is clled idel. In such cse, he exciion my be formlly expressed on he righ side of differenil equion by funcion of ime. I mens h he force is genered by source wih infinie power (idel). However, if in cerin model he idel source is replced by source wih limied power (nonidel), hen he exciion mus be pu in he form of funcion which depends on he response of he sysem. Therefore, non-idel source cn no be expressed s funcion of ime, bu rher s n equion h reles he source o he sysem of equions h describes he model (Blhzr e l., 997), (Pones e l., ). Hence, non-idel models lwys hve one ddiionl degree of freedom s compred o he idel counerprs. In he clssicl book devoed o non-idel sysems (Kononenko, 969), he energy source is modeled by he sionry chrcerisic of he DC moor. To obin dynmicl model, which is close o he relisic sysem, i is necessry o consider lso n ddiionl equion clled he elecricl equion of he moor (Belo e l., ). The min purpose of his pper is o nlyse he influence of he exernl force, genered by he non-idel energy source, on prmericlly nd self-excied sysem. Also, o presen resuls obined for wo non-idel models: he firs model simplified in clssicl Kononenko sense (Wrminski, c), (Blhzr e l., 997), in which he qusi-sic chrcerisic of he energy source is ken ino ccoun, nd he second model complee elecromechnicl one (Wrminski nd Blhzr, ). New dynmic phenomen nd differences beween regulr nd choic moion, in ech level of simplificions, will be emphsized in he pper. Dynmicl Model of he Sysem Le us consider prmeric nd self-excied model, which includes direc curren (DC) moor wih limied power supply, opering on srucure (Fig. ). The exciion of he sysem is limied by he chrcerisic of he energy source (non-idel energy source). Then, he coupling of he vibring oscillor nd he DC moor kes plce. As he vibrion of he mechnicl sysem depends on he moion of he DC moor, lso he moion of energy source depends on vibrions of he sysem. Hence, i is imporn o nlyse wh hppens o he moor, s he response of he sysem chnges. Pper cceped Ocober, 3. Technicl Edior: Ail P. Silv Freire. Figure. Mechnicl model of he vibring sysem, nd he elecricl schemic represenion of DC moor. J. of he Brz. Soc. of Mech. Sci. & Eng. Copyrigh 3 by ABCM Ocober-December 3, Vol. XXV, No. 4 / 43

2 J. Wrminski nd J. M. Blhzr Le us ssume h he mhemicl model consiss of DC moor which is suppored by non-liner spring wih periodiclly chnging siffness of Mhieu ype, nd h dmping of he sysem is described by non-liner Ryleigh s funcion. These wo erms gurnee coexisence of wo ypes of vibrions in he dynmicl sysem: prmeric nd self-exciion. The direc curren moor, wih roing mss, is n exernl energy source forcing his sysem. The elecricl scheme of he DC moor represenion is presened in Fig.. The equions governing he moion of he DC moor re ypiclly wrien in he form (Pelczynski nd Krynke, 984): d f J () () () = Mm Mz H () d di () U() = RI () + L + E() () d where: ime funcions U () nd I () re he volge nd he curren in he rmure, R nd L is resisnce nd inducnce of he rmure, E() is he inernlly genered volge, M z () is n exernl orque pplied o he moor drive shf, H() is fricionl orque nd M m () denoes he orque genered by he moor. The orque M m () nd inernl genered volge E() cn be expressed s M () = c Φ I() (3) m M E() = c Φ w() (4) E where: c M,c E re mechnicl nd elecricl consns nd Φ is he mgneic flux. Le us ssume h exernl exciing curren I m nd volge U m re consn nd hen he mgneic flux Φ is lso consn in he considered model. Tking ino ccoun Eqs. ()-(4) nd mechnicl model published in (Wrminski e l., ), we cn wrie he differenil equions of he complee elecro-mechnicl sysem presened in Fig. s follows: di () R ceφ U () = I () f + (5) d L L L Jf = cmφi () H( f ) + mrx cosf mgr cosf (6) ( ) ( )( ) mx + f x, x + k k cosf x + k x = m rf sin f m rf cosf 3 where prime denoes derivive wih respec o dimensionl ime. The funcion f ( x, x ) = ( c + cx ˆ ) x is clled Ryleigh s funcion nd i describes non-liner dmping of he sysem. Inroducing dimensionless ime = w, where w = km is he nurl frequency of he sysem nd m = m + m, we cn wrie he sysem of differenil equions (5)-(7) in dimensionless form ( ) ( ) o (7) I = pi p f + U (8) f = p I H q X f (9) ( ) ( ) 3 cos ( b ) ( mcosf)( g ) ( f sinf fcosf) X + + X X + + X X = q () R ceφ cmφ mr where: p =, p =, p3 = Izn, q =, Lw LI zn J w mxs mrx s U ( ) q =, U ( ) = H ( ), H ( ) = c, =, J Lw I J w mw n cˆw k x b = xs, g = kx s, m =, X =, I I =, nd x s mens m k xs I r sic displcemen of he sysem, I r is red curren in he rmure nd dos indice differeniions wih respec o dimensionless ime. In he curren lierure, very ofen he model of DC moor is simplified (Kononenko, 969), (Blhzr e l., 997), by king ino ccoun h I =, nd he momen genered by he moor cn be expressed by M m = p p p. () 3 U p p w Then, srigh line pproximes he chrcerisic of he DC moor model. The mhemicl model of he vibring sysem described by Eqs. (8)-() cn be considered in hree vrins: Idel sysem, if here is no coupling beween moion of he roor nd vibring sysem ( ) ( )( ) X + + bx X + mcosw + gx X = qw sin w () where f = w. On he righ side of equion (), funcion of ime is presen. non-idel sysem in Kononenko s sense cos (3) f = Γ( ) qx f ( b ) ( mcosf)( g ) ( f sinf fcosf) X + + X X + + X X = q where Γ ( f) = Mm ( f) H( f) (4) is he difference beween he orque genered by he moor nd he resisnce orque. Funcion Γ ( f ) = u uf is pproximed by srigh line, where u is conrol prmeer nd i cn be chnged ccording o he volge, u is consn prmeer, chrcerisic for he model of he moor. Full non-idel elecro-mechnicl sysem described by Eqs. (8)-(). All bove hree models describe he sme problem in differen simplificion levels. However, hey my led o quliive nd quniive differences in heir behviours. The comprison of he resuls obined for idel nd non-idel problems nd for differen of DC moor models re presened in he nex secion. Anlysis of he Vibring Sysem in Differen Simplificion Levels The numericl simulion of he vibring sysem ws crried ou for prmeers of he DC moor ken from (Pe³czynski nd Krynke, 984) nd mechnicl prmeers, which correspond o he pper (Wrminski e l., ). Numericl dimensionless d for he model presened in Fig. re following: 44 / Vol. XXV, No. 4, Ocober-December 3 ABCM

3 Vibrions of Prmericlly nd Self-Excied Sysem wih p =.3, p = 3., p3 =.5, =., b =.5, g =., m =., q =., q =.3 (5) In he idel model described by (), he exernl force is expressed on he righ side of he differenil equion by pure funcion of ime. Anlysis for his idel problem ws crried ou in deils in (Szbelski nd Wrminski, 995, b). Addiionl soluions, hving shpe of n inernl loop, pper in he synchronision region. However only he upper pr of his loop is sble q q q q q q = Figure 3. Ampliude curves versus prmeer q The shpe of he resonnce curve depends on he vlue of he mpliude of he exernl exciion. The inernl loop is visible only for smll level of he exernl exciion. For lrge vlue of he q prmeer, he loop disppers (Fig.3) Figure. Ampliude curves round he min prmeric resonnce for he idel sysem, wihou n exernl force α =., β =.5, γ =., µ =., q =, nd for he sysem forced by idel hrmonic funcion α =., β =.5, γ =., µ =., q =.. If he exernl force is no presen ( q = ) hen he resonnce curve hs he shpe presened in Fig.. Inercion beween prmeric nd self-excied sysem leds o he synchronision phenomenon ner he min prmeric resonnce. In his region, prmeric vibrions domine. They pull in he frequency of selfexcied vibrions nd he sysem vibres wih single frequency nd wih consn mpliude (solid line in Fig. ). Ouside he synchronision regions, influence of he self-exciion is bigger nd wo frequencies in he response of he sysem occurs. Sysem vibres qusi-periodiclly nd is moion is visible s qusiperiodic limi cycle on he phse plne (see deils Szbelski nd Wrminski, (995,b) ). Vericl lines in Fig., which denoe mximl nd miniml vlues of he moduled mpliude, mrk his moion. The exernl force cuses very imporn quliive chnges. Behviour of he idel sysem for q =. is presened in Fig.. Ω u u Figure 4. Ampliude of vibring oscillor nd ngulr velociy of he moor versus conrol prmeer u. J. of he Brz. Soc. of Mech. Sci. & Eng. Copyrigh 3 by ABCM Ocober-December 3, Vol. XXV, No. 4 / 45

4 J. Wrminski nd J. M. Blhzr If we ssume h he sysem is forced by DC moor wih limied power, hen i is necessry o consider is dynmics in he model nd o solve he sysem of differenil equions (3)-(4). The orque genered by DC moor is limied nd, ccording o clssicl Kononenko heory, is ssumed s srigh line. Trnsiion hrough he resonnce region is possible if he prmeer u, conneced wih volge supplied o he moor, is incresed. The vibrion mpliude of he oscillor nd he ngulr velociy of he roor re presened in Fig Ω Figure 5. Ampliude versus ngulr velociy of he moor. Ouside he synchronision region he oscillor vibres qusiperiodiclly nd he moor ngulr velociy chnges hve qusiperiodic chrcer s well (vericl lines in Fig.4). Inside he synchronision re he moion of he sysem is periodic (solid line in Fig.4). During rnsiion hrough his resonnce, locl decresing of he mpliude nd ngulr velociy kes plce. This phenomenon ffecs he resonnce curve, mpliude versus exciion frequency (Fig.5). On he resonnce curve, n inernl loop is visible, however is lies on he lef brnch of he curve nd is compleely sble. Anlyicl soluions nd sbiliy nlysis hve been crried ou in (Wrminski e l., ). Time hisories of he vibrion of he oscillor nd he vriion of he ngulr velociy of DC moor re presened in Fig.6. In Fig.6, he sysem vibres wih he consn mpliude while, in Fig.6, (c), (d) he moion is qusi-periodic. The mos deque model o he relisic problem is described by elecro-mechnicl equions (8)-(). To simule he mhemicl model of his sysem, MATLAB, Simulink TM nd Dynmics pckge (Nusse nd York, 998) were pplied. Differenil equions were solved by fifh order Runge-Ku mehod wih uomic sep lengh nd inegrion error conrol. Behviour of he sysem ws observed during slow volge incresing, nd hen vibrion mpliude of he oscillor nd ngulr velociy of he moor were ploed X (c) (d) Figure 6. Times hisories for non-idel sysem nd chosen conrol prmeers u ; u =.3, u =., (c) u =.4, (d) u = / Vol. XXV, No. 4, Ocober-December 3 ABCM

5 Vibrions of Prmericlly nd Self-Excied Sysem wih U X X U Figure.7. Displcemen of he oscillor nd ngulr velociy of he shf versus conrol prmeer U nd versus dimensionless ime ner he min prmeric resonnce, simplified nd complee model of DC moor. The dimensionless volge pplied cross he rmure U ( ) (, 5) is he conrol prmeer. In Fig. 7 we cn compre resuls obined for wo differen models of non-idel problem. The model simplified in Kononenko sense is limied by influence of he DC moor chrcerisic described by he Eq. (). The sionry chrcerisic of he moor, for ssumed d (5), is expressed by he funcion Γ ( f ) = f. Trnsiion hrough resonnce for his model is presened in Fig. 7. The resuls for he complee elecro-mechnicl model described by he full sysem of differenil equions (8)-() re presened in Fig.7. Trnsiion hrough he min prmeric resonnce kes plce for volge ner U ( 3, 4) nd for ime ( 6, 8). Compring Fig. 7 nd, we cn find h he moions of he oscillor X hve similr chrcer. Ner U = 3.5, in boh figures, locl decresing of vibrion mpliude of he oscillor is visible. Trnsformions of Figs.7 nd o he dependence ( Ω ) i.e. he mpliude versus ngulr velociy curve, led o he loop presened in Fig.5. The imporn difference occurs in DC moor dynmics. The chnges of ngulr velociy re much more complex for he complee elecromechnicl model (Fig.7 ) hn for he simplified non-idel model (Fig.7 ). Figure 8. Lypunov exponens digrm versus conrol prmeer U, simplified nd complee DC moor model, regulr moion (µ=.). J. of he Brz. Soc. of Mech. Sci. & Eng. Copyrigh 3 by ABCM Ocober-December 3, Vol. XXV, No. 4 / 47 Figure 9. Lypunov exponens digrm versus conrol prmeer U, simplified nd complee DC moor model, choic regions (µ=.).

6 J. Wrminski nd J. M. Blhzr If we compre Lypunov exponens digrms obined for he sme prmeers (Fig. 8), we see h for he considered inervl of he conrol prmeer, he moion of he sysem is regulr; Lypunov exponens re negive or equl o zero. Neverheless, we cn noice h round he inervls ner U =. nd U (3, 4), behviour of he sysem in he digrm Fig. 8 is more complex. I confirms resuls presened in Fig. 7. For he idel sysem (Eq. ()) nd for wide rnge of prmeers used, i ws no possible o find choic moion of he model. The idel sysem vibres regulrly, periodiclly or qusiperiodiclly (Szbelski nd Wrminski, 995,b). For non-idel model, bsed on he pper (Wrminski, b), we cn expec h he increse of he prmeric exciion cn led o choic moion. Therefore, le us ssume for boh considered non-idel models, h he prmeric exciion is µ=.. Lypunov exponens digrm for h cse is ploed in Fig. 9. We see h in few inervls of he conrol prmeer U, mximl Lypunov exponen hs posiive sign. However, he endency in rnsiion o choic moion is differen. Choic moion for he full elecro-mechnicl model (Fig. 9) ppers in wider region, in opposiion o he simplified model (Kononenko pproch) for which choic moion ppers only for few smll regions. Figure. Poincré digrms for equivlen DC moor models, regulr rcor for simplified model nd srnge choic rcor for complee model, µ=., U=.. Poincré digrms for equivlen prmeers nd differen nonidel models re presened in Fig.. The moion of he simplified model is regulr nd i is represened by closed orbi in Fig., wih Lypunov exponens l =, l =, 7, l3 =.6, l4 = 5.739, whils he moion of he complee model for he sme prmeers, is choic. In Fig. he choic rcor wih one Lypunov s exponen posiive is presened ( l =.77, l =, l3 =.59, l 4 =.348, l 5 =.56 ). This resul shows h he difference in dynmic behviour for differen models is significn, priculrly for regions where choic moion is possible. Remrks nd Conclusions Anlysis crried ou in his pper emphsizes differences in modeling of idel nd non-idel sysems for chosen clss of self-, prmeric nd exernlly excied vibrions. Behviours of he idel nd non-idel sysem re differen. The exernl force genered by he idel moor inroduces ddiionl soluions in he synchronision region. These soluions re observed s n inernl loop inside he resonnce curve wih only he upper pr sble. However if he prmeric nd self-excied sysem is forced by non-idel energy source, he loop moves o he lef brnch of he curve nd becomes sble. The obined resuls le us lso conclude h wo differen pproches o he modeling of non-idel sysems cn led o imporn differences. The clssicl model proposed by Kononenko (969) is bsed on he pure mechnicl model of he DC moor nd kes ino ccoun he sionry chrcerisic of he energy source. Th pproch considers only mechnicl inercions beween he oscilling sysem nd he energy source, which is limied by n ssumed srigh line. To be close o he relisic sysem, he model should ke ino ccoun lso influence of he dynmics of he oscilling mechnicl elemens on elecricl properies of he DC moor. Therefore, wo lernive models were nlysed in his pper: he simplified clssicl model nd he complee elecromechnicl model. Numericl simulions show h for regulr moion, nd for slow rnsiion hrough he resonnce region, behviour of wo considered models is similr. Neverheless, if he dynmics of he sysems becomes complex, hen he difference in he response of hose wo models is more significn. Trnsiion from regulr moion o chos is possible for boh models. However, he endency in going o chos for he elecro-mechnicl model is bigger. The generl conclusion is h simplified model enues dynmics of he relisic sysem. For exmple, if he choic region is found for he simplified model, we cn expec h for he complee elecro-mechnicl model choic moion will pper in much wider re. The opposie siuion ws no observed. References Awrejcewicz, J. nd Mrozowski, J., 989, Bifurcion nd Chos of Priculr Vn der Pol-Duffing's Oscillor, Journl of Sound nd Vibrion, 3 (), pp Blhzr, J.M., Rene, M.L. nd Dvi, V.M., 997, Some Remrks On he Behviour of Non-Idel Dynmicl Sysem, in Nonliner Dynmics, Chos, Conrol nd Their Applicions o Engineering Sciences, eds. Blhzr, J.M., Mook, D.T. & Rosrio, J.M. (Americn Acdemy of Mechnics nd Associco Brsileir de Ciencis Mecnics) Vol., pp Belo, D., Weber, H.I., Blhzr J.M. nd Mook, D.T.,, Choic Vibrions of Nonidel Elecro-Mechnicl Sysem, In. Journl of Solids nd Srucures, 38, pp Kononenko, V.O., 969, Vibring Sysems wih Limied Power Supply, Illife. Nusse, H.E. nd Yorke, J.A., 998, Dynmics: Numericl Explorions, Springer Verlg. 48 / Vol. XXV, No. 4, Ocober-December 3 ABCM

7 Vibrions of Prmericlly nd Self-Excied Sysem wih Pe³czyñski, W., nd Krynke, M., 984, Mehod of Se Vribles in Anlysis of Power Trnsmission Sysem Dynmics, WNT, Wrsw, Polnd, (in Polish). Pones, B.R., Oliveir, V.A. nd Blhzr, J.M.,, On Fricion- Driven Vibrions in Mss-Block-Moor Sysem wih Limied Power Supply, Journl of Sound nd Vibrion, 34(4), pp , doi:.6/jsvi..88. Szbelski, K. nd Wrminski, J., 995, The prmeric self excied non-liner sysem vibrions nlysis wih he ineril exciion, In. Journl of Non-Liner Mechnics Vol.3, No, pp Szbelski, K. nd Wrminski, J., 995 b, The self-excied sysem vibrions wih he prmeric nd exernl exciions, Journl of Sound nd Vibrion, 87 (4) pp Tondl, A. nd Ecker, H., 999, Cncelling of Self-Excied Vibrions by Mens of Prmeric Exciion, Proceedings of he 999 ASME Design Engineering Technicl Conferences, Sepember -5, Ls Vegs, Nevd, USA, DETC99/VIB-87, pp. -9. Wrminski, J., Blhzr, J.M. nd Brsil, R.M.L.R.F.,, Vibrions of Non-Idel Prmericlly nd Self-Excied Model, Journl of Sound nd Vibrion, 45 (), pp , doi:.6/jsvi Wrminski, J.,, Influence of he Exernl Force on he Prmeric nd Self-Excied Sysem, Journl of Technicl Physics, 4, pp Wrminski, J., b, Synchronision Effecs nd Chos in vn der Pol-Mhieu Oscillor, Journl of Theoreicl nd Applied Mechnics, 4, 39, pp. -4. Wrminski, J., c, Regulr nd Choic Vibrions of Prmericlly nd Self-Excied Sysems wih Idel nd Non-Idel Energy Sources, Technicl Universiy of Lublin Publisher, Lublin, Polnd, (in Polish). Wrminski, J. nd Blhzr, J.M.,, Vibrions of The Prmericlly nd Self-Excied Sysem Wih Non-Idel Energy Source, 6h Inernionl Conference, Dynmicl Sysems Theory nd Applicions, eds. Awrejcewicz, J., Grbski, J., & Nowkowski, J., ódÿ, Polnd, pp J. of he Brz. Soc. of Mech. Sci. & Eng. Copyrigh 3 by ABCM Ocober-December 3, Vol. XXV, No. 4 / 49

8 J. Wrminski nd J. M. Blhzr 4 / Vol. XXV, No. 4, Ocober-December 3 ABCM