CHAPTER INSTABIITY IN ROTATING MACHINES I the most of previous chpters, except Chpter 3, we studied how to obti the free d forced resposes of rotor-berig systems i differet modes of vibrtios (e.g., the torsiol, trsverse, d xil vibrtios). Mi ims of these chpters were to obti turl whirl frequecies, mode shpes, criticl speeds d ublce respose. Ublce respose lysis preseted c be exteded to other types of periodic forces with the help of Fourier series especilly for the lier systems, where the priciple of superpositio holds good. Vrious methods especilly suited for lyzig complex rotor systems (prt from geerl methods of vibrtio lysis, like the Newto s secod lw of motio, grge s method, Hmilto s priciple, etc.) hve bee delt i gret detil from its fudmetls (e.g., the trsfer mtrix d fiite elemet methods). Now i ext few chpters, we would explore other kid of pheome i rotor-berig systems clled the istbility, which might cuse the ctstrophic filure of the systems. I certi circumstces, depedig upo the desig, some mchie my be proe to istbility. This mes tht mchie vibrtios set i, eve i the bsece of ublce effects, resultig i high levels of oise d compoet stress d correspodig reduced ftigue life. I lier systems the mgitude of these vibrtios teds towrds ifiity, lthough i prctice shft vibrtios re ofte limited by system o-lierity. I the preset chpter, vrious kids of istbility will be studies. Such mchie istbilities my origite from umber of sources icludig fluid-film berigs, sels, shft stiffess symmetry, iterl frictio betwee mtig compoets, d erodymic forces. A desiger s problem is to ivestigte the possibility of mchie istbility, d to chge the pproprite mchie desig prmeters to esure tht my potetil ustble modes of opertio lie outside the orml opertig regime of the mchie. Aprt from these whe rotors re subjected to gulr ccelertios (uiform or vrible depedig upo the ulimted or limited power of the drive, respectively), trsiet resposes re geerted d study of such trsiet respose is of prcticl importce. The im of the preset chpter is to uderstd vrious kids of istbility with simple sigle mss rotor model d i some cses with cotiuous shft model. I the subsequet chpter, we would explore methods of predictig istbility i lrge rotor-berig systems especilly with fiite elemet methods.. Oil Whirl I rottig mchieries, the ublce is the most commo source of excittio, which comes from rotors. Similrly, oil-film berigs re possibly the most commo source of istbility i such rottig mchieries. Oil-whirl istbility is lso kow s hlf-speed whirl becuse the frequecy of whirl (vibrtios) which re set up is ofte just below hlf the shft rottiol frequecy (typiclly.6-.8
65 times of shft rottiol frequecy). This istbility teds to occur oly i lightly loded oil-film berigs opertig t very smll eccetricity rtio ( e / c, where e r is the eccetricity of the rotor cetre with respect to the berig cetre d c r is the rdil clerce). It is exterlly dgerous coditio becuse berig lod-crryig decreses d results i very high whirl mplitude d cosequetly the destructio of the berigs is possibility. et us cosider fluid-film berig i which jourl is rottig with frequecy, ω, d whirls roud the berig clerce (i.e., e r ) t frequecy r ω s show i Figure.. Becuse the berig is lightly loded (so opertes t oly smll eccetricity e r i the first istce) the vritio i fluid pressure roud the berig circumferece my be cosidered to be egligible so tht the oly fluid flows roud the berig the berig is tht which is iduced by the rottio of the jourl. The lubrict flow rte ito, d out of, the dotted wedge-shped re s show i Figure.() is the give by r Q = ( Rω + )( c + e ) d Q = ( Rω + )( c e ) (.) i r r out r r where ( + Q i ( ( Rω + ) is the verge velocity d ( cr + er ) is the pssge width) d Q out ( Rω ) is the verge velocity d ( c e ) is the pssge width) re flow per uit legth of the berig. r r () Jourl d lubrict i berig durig oil whirl Figure. Oil whirl i fluid film berigs (b) The whirl orbit of the shft cetre Sice jourl is whirlig withi the berig clerce with some frequecy velocity of the jourl ceter will be per uit legth of the berig must be icresig t rte give by ω, the tgetil ω er s show i Figure.(b). So the volume of dotted re, Q = ( ωe )( R) (.) vol r
655 where ( ωe r ) is the tgetil velocity of jourl ceter d ( R) is the shded re of jourl per uit legth of berig. The volume flow rte must be provided by the et lubrict flow ito the dotted re uder cosidertio, so tht we my write Q Q = Q (.3) i out vol O substitutig equtios (.) d (.) ito equtio (.3), we get or Rω( c + e ) Rω( c e ) = Rωe r r r r r ( ) ( ) c + e c e = e e = e (.) r r r r r r r which gives =. 5. So the frequecy of whirl is hlf the rottiol frequecy of jourl. The devitio from the ctul cse (.6 to.8) is due to the ssumptio mde i the lysis regrdig o flow due to pressure vritio cross the circumferece of the berig. Exmple.: et us cosider two ideticl berigs which re symmetriclly supportig light symmetricl rotor t its eds. Through mesuremet the followig dt were foud: the bore of the jourl berig is 3 cm with the rdil clerce of 5 µ m, d the rotor spi speed is 3 rpm. The flow mesuremet were 5 Q i =.87 m 3 /s d Q out 5 =.885 m 3 /s. If the rotor is uder the hlf-speed whirl, obti the eccetricity rtio of the rotor cetre i the jourl. Solutio: We hve the followig dt R = 3 cm, 6 c r = 5 m, π 3 ω = = 3.6 rd/s 6 5 Q i =.87 m 3 /s, Q out 5 =.885 m 3 /s From equtio (.), we hve Q Q = Rωe i out r Hece the eccetricity rtio c be writte s
656 5 (.87.885) e Q Q r i out ε = = = =. 6 c Rωc.3.5 3.6 5 r r Hece, it hs very low eccetricity rtio.. Stbility Alysis usig ierized Stiffess d Dmpig Coefficiets A stble rotor-berig my be defied s oe tht will hve bouded respose for ll possible bouded excittios. To ivestigte the likelihood of oil whirl, ttetio eeds to be give to the berig opertig chrcteristics. For oil-film berigs these my be expressed i terms of the eightlierised stiffess d dmpig coefficiets. () A rotor-berig system with berig forces (b) Free body digrm i y-z ple (c) Free body digrm i z-x ple Figure. A rigid rotor mouted o two ideticl isotropic berigs The reltioship betwee the berig forces d the jourl motio is give by the equtios of motio of the jourl, which i the cse of symmetricl system with rigid rotor (Fig..), cse re k x + k y + c x + c y = mx xx xy xx xy k x + k y + c x + c y = my yx yy yx yy (.5) where x d y re the horizotl d verticl displcemets of the rotor, k s d c s re stiffess d dmpig coefficiets, d m is the hlf of the rotor mss. Here it is ssumed tht rigid rotor is mouted o two ideticl fluid-film berigs d hs purely trsltiol motio. If the rotor is
657 mometrily displced from its equilibrium positio by some rdom iput, free vibrtios of the rotor i the horizotl d verticl directios will tke the form x = X e λt d y = Ye λt (.6) where λ is prmeter d i geerl it is complex qutity with the rel prt represet the dmpig d the imgiry prt s the whirl turl frequecy; d X d Y re the vibrtio mplitudes i the horizotl d verticl directios, respectively. Equtio (.6) gives x = X λe λt ; = y X λe λt ; ; x = X λ e λt (.7) y = X λ e λt O substitutig equtios (.6) d (.7) ito equtio (.5), d dividig whole equtio by we get t e λ, k X k Y c X c Y m X xx + xy + xx λ + xy λ = ( λ ) yx + yy + yx λ + yy λ = ( λ ) k X k Y c X c Y m Y (.8) which c be simplified s X ( mλ k c λ) = Y ( k + c λ) xx xx xy xy X k + c = Y m k c ( yx yxλ) ( λ yy yyλ) (.9) Equtio (.9) c be combied s X c λ + k mλ + λc + k = = Y mλ c λ k c λ k xy xy yy yy + xx + xx yx + yx (.) which gives the frequecy equtio s
658 ( m ) λ + m( c + c ) λ + ( mk + mk + c c c c ) λ 3 xx yy xy yy xx yy xy yx + ( k c + c k k c c k ) λ + ( k k k k ) = yx xx yy xx xy yx xy yx xx yy xy yx (.) The most direct pproch for ivestigtig the stbility of lier rotor-berig system is to determie the roots of the bove chrcteristic polyomil (i.e., the frequecy equtio). Equtio (.) hs four roots of λ. I geerl the roots of λ will both rel d imgiry prts, idictig tht the trsiet motio of the jourl will tke form of hrmoic wve hvig decyig mplitude whe the system is stble (i.e., whe the rel prts of ll roots of λ re egtive). The imgiry prts of the roots of λ idicte the frequecy of the resultig vibrtios. Physiclly to test for the system stbility oe must exmie the motio of the jourl which follows mometry displcemet from its stedy ruig positio. Does the jourl retur to stble equilibrium or ot? If the jourl were to retur to stble equilibrium positio the this would be chrcterized by vlues of displcemet x d y which decreses with time, tht is by egtive vlues of Re(). If the jourl motio is ubouded with time tht mes positive vlue of Re(). The circumstces uder which the rel prts of ll roots of re egtive re give by the Routh-Hurwitz stbility criteri. This criterio ws developed idepedetly by Hurwitz (895) i Germy d Routh (89) i Uited Sttes. Two pproches re preseted i which the first oe determies the stbility of the system with the help of Routh tble (Sih, 995)) d the secod oe fids by set of domits. Method : et the chrcteristic polyomil be give by 3 ( ) + λ = λ + λ + λ + λ + + λ + λ + λ + λ + + (.) + 3 The the Routh tble (Tble.) is costructed bsed o the coefficiets of the polyomil. The Routh-Hurwitz criterio sttes tht the umber of roots with positive rel prts is equl to the umber of chges i sig i the first colum of the Routh tble. Hece, for the system to be stble, o sig chges should tke plce i the first colum of the tble.
659 Tble. Routh tble for fidig the stbility of lier rotor-berig system λ + + 3 λ λ ( + ) b = λ ( b b ) c = b b c = = ( ) 3 + ( b b ) b b c = = ( ) 5 + 6 ( b b ) 6 3 b λ h Method : For the stbility of lier system the followig coditios re to be met (i) ll coefficiets of the chrcteristic equtio must hve the sme sig, d (ii) ech of the followig determits must be positive R =, R =, 3 3 3 5 R =, 3 R = 3 5 7 6 3 5, R = (.3) 3 5 7 9 6 8 3 5 7 3 6 3 5 5 6 where system chrcteristics equtio tkes the form (.) 3 λ + + + λ + + λ + 3λ + λ + λ + = Substitutio of pproprite vlues ito equtio (.7) the llows the desiger to determie whether mchie is likely to be stble or ustble. It does ot idicte how stble (or ustble) mchie my be. It will be oted from bove tht system stbility depeds upo the stiffess d dmpig of berigs. Hece, for equtio (.), we hve the followig stbility coditios
66 m( c + c ), ( mk + mk + c c c c ), xx yy xy yy xx yy xy yx ( k c + k c k c k c ), ( k k k k ) yx xx xx yy xy yx yx xy xx yy xy yx (.5) d R = 3 = ( k c + k c k c k c )( mk + mk + c c c c ) m( k k k k )( c + c ) yx xx xx yy xy yx yx xy xy yy xx yy xy yx xx yy xy yx xx yy R = 3 3 3 ( kyxcxx kxxcyy kxycyx k yxcxy )( mkxy mkyy cxxcyy cxycyx ) m( cxx cyy ) ( kyxcxx + kxxcyy kxycyx kyxcxy ) ( m ) ( kxxkyy kxyk yx ) m( cxx + cyy ) = + + + + (.6) Exmple. For rigid rotor of mss kg is supported o two ideticl fluid film berigs with the followig properties: k xx = MN/m, k yy = 5 MN/m, k xy = -.5 MN/m, k yx = 5 MN/m, c xx = kn-s/m, c xy = 5 kn-s/m, c yx = kn-s/m d c yy = kn-s/m. Fid the stbility of the rotor. Solutio: Hlf of the mss of the rotor m = 5kg. O substitutig the give rotor d berig properties i equtio (.), we get 3 3 6 5λ + 5 ( + ) λ + ( 5.5 + 5 5 + 5 ) λ 9 + (5 + +.5 5 5) λ + ( 5 +.5 5) = () O simplifictio of equtio (), we get 6 3 5λ 3 λ 5.9 λ 9.6 λ 3.38 + + + + = (b) Equtio hs the followig form + + + + = (c) 3 λ 3λ λ λ Method : The Routh tble for the bove chrcteristic equtio is give i Tble.. Sice the degree of polyomil is ( + = ), hece = 3. It c be observed tht there is o sig chge i the first colum of the Routh tble d hece the system is i stble coditio. Tble. Routh tble for fidig the stbility of lier rotor-berig system ( = 3)
66 λ = 5 = 5.9 = 3.38 3 λ 3 = 3 6 = 9.6 = λ ( 3 ) b3 = 3 = 5.9 λ ( b3 3b ) c3 = b3 = 9.3 λ ( c3b b3c ) d3 = c3 = 3.38 b ( ) = = 3 3 c = 3.38 ( b b ) 3 3 = = b3 b ( ) 3 3 = = 3 c ( b b ) 3 3 3 = = b3 d = d = Method : Hece, the first coditio of stbility is stisfied, i.e., ll coefficiets of the chrcteristic equtio must hve the sme sig. The secod coditio is tht the followig determit must be positive: R 9.6 = =, R 9.6 3.38 3 = = = 5.58 6 3 3. 5.9, 3 3 5 3 9.6 3.38. R = = 3. 5.9 9.6 =.67 6 3. 5. 3. 6 It c be see tht ll determits re positive, hece, the rotor-berig system is stble. Exmple.3 For rigid rotor of mss kg is supported o two ideticl fluid film berigs with the followig properties: k xx =. MN/m, k yy =.5 MN/m, k xy =. MN/m, k yx = - MN/m, c xx = kn-s/m, c xy = 5 kn-s/m, c yx = 5 kn-s/m d c yy = kn-s/m. Fid the stbility of the rotor. Solutio: Hlf of the mss of the rotor m = 5kg. O substitutig the give rotor d berig properties i equtio (.), we get 6 3 3 5λ λ.753 λ.5 λ.3 + + + = () Equtio hs the followig form
66 + + + + = (b) 3 λ 3λ λ λ Method : The Routh tble for the bove chrcteristic equtio is give i Tble.3. Sice the degree of polyomil is ( + = ), hece = 3. It c be observed tht there is sig chge i the first colum of the Routh tble d hece the system is i ustble coditio. Tble.3 Routh tble for fidig the stbility of lier rotor-berig system ( = 3) λ = 5 =.753 =.3 3 3 λ 3 = 6 = -.5 = λ ( 3 ) b3 = 3 = -.75 λ ( b3 3b ) c3 = b3 = -.5 λ ( c3b b3c ) d3 = c3 = -. 8 b ( ) = = 3 3 c = -. 8 ( b b ) 3 3 = = b3 b ( ) 3 3 = = 3 c ( b b ) 3 3 3 = = b3 d = d = Method : Hece, the first coditio of stbility is stisfied, i.e., ll coefficiets of the chrcteristic equtio () must hve the sme sig, which is ot sme. The secod coditio is tht the followig determit must be positive: R.5 = =, R =. 3 =, R = = 8.8 3 3 5 3 7 It c be see tht ll determits re lso ot positive, hece, the rotor-berig system is ustble..3 Stbility Alysis with Fluid-Film No-ierity I previous method the oil-whirl predictio bsed o eight lierlised fluid-film coefficiets (d lso mss of the rotor). ierised fluid-film coefficiets re vlid for smll displcemets of the jourl roud its sttic equilibrium positio. But fluid-whirl implies lrge mplitude of vibrtios. So fluid-whirl istbility lysis with fluid-film o-lierity is more relevt. Fluid-film forces re
663 determied by solvig the Reyolds equtio s discussed i Chpter 3. The fluid-film forces due to mometrily displcemet of the jourl roud its sttic equilibrium positio is lso obtied usig the Reyolds equtio with time depedet terms retied i the equtio. Figure.3 The free-body digrm of jourl durig whirlig i berig Tkig compoets of fluid forces d other forces (Figure.3) such s jourl weight, ublce d ierti forces i the directio of rdil d tgetil, we get d with d f t + me + W = m e e r ( ) ω cosψ cos φ ( r rφ ) f t + me W = m e + e (.7) t ( ) ω siψ si φ ( rφ rφ) π = f ( t) R p( θ, z)cosθ dθdz r π = θ θ θ (.8) f ( t) R p(, z)si d dz t where, f r d f t re the rdil d tgetil fluid forces to be obtied from the solutio of Reyolds equtio for the pressure vritio, p(, z), of fluid-film over the circumferece of the berig (refer Chpter 3), R is the rdius of the rotor, d z re the circumferetil d xil coordites of berig surfce, W = mg is the weight of the rotor per berig, e r is the eccetricity of the shft cetre with the berig cetre, φ, is the ltitude gle of the shft cetre with respect to the berig cetre, me is the rotor ublce d m is the effective mss of the rotor t ech berig. Terms m( e φ + e φ) re the rdil d tgetil ccelertios of the shft cetre (i.e., r r m e ( r erφ ) d e r is the rdil
66 ccelertio of the cetre of the rotor due to lier motio, r, is the cetripetl rdil ccelertio e φ due to rottiol motio, e rφ, is the tgetil ccelertio due to rottio, d e r φ is the Coriolis compoet of ccelertio due to lier d gulr motios). The ukow i equtios (.7) re e r d φ, d its derivtives. The differetil equtios my be itegrted usig Euler or Rge-Kutt method to obtied e r d φ for some mometry disturbces. Thus it is possible to determie the jourl positio, described by e r d φ t vrious time istces followig the iitil disturbce of the system. For stble coditio with ublce i the system the pth of the jourl (jourl orbit) for mometry disturbce will settle dow to ellipticl shpe (or orbit with fixed shpe) fter sufficiet itertios (Figure.). () Stble (b) Ustble Fig.. Jourl ceter pth due to perturbtio For o ublce i the system it should coverge to poit. For ustble system whirl orbit will ot be o ellipse (or orbit with fixed shpe) but the whirl mplitude will icrese with time (Figure.b), idictig subsequet destructio of the berig d the the rotor system itself. For multi-dof rotor-berig systems the stbility lysis will lso be similr to wht discussed here, i.e., (i) Routh Herwitz criteri, (ii) eige vlue lysis, d (iii) plottig of the orbit for mometry disturbces. However, lst two methods re more prcticl to implemet.. Resot Whip Whe the shft rottes t bout twice the speed ssocited with the first criticl speed of the system, the oil whirl tkes plce t the hlf the rottiol speed d hece equl to the first criticl speed of the system. This coditio is clled oil-whip. I these circumstces very severe ustble vibrtios ideed re itroduced d the situtio is the most udesirble. The effect of vibrtio ssocited with oil whirl combies with system criticl speed to produce most excessive vibrtio. It is the cross-
665 coupled stiffess (k xy d k yx ) of the fluid-film berig which destbilizes the rotor-berig system. Although the dmpig i fluid-film berigs is high, it is ot sufficiet to suppress the oil whip t high rotor speeds. I this situtio berigs will be i ustble opertig regime. Muszysk (986) explied the differece betwee the oil whip d oil whirl. The oil whirl is stble d its frequecy is lwys roud hlf the rotor speed, wheres, the oil whip is ustble d it hs fixed frequecy of twice the first criticl speed of the system. Oil whirl d oil whirl re olier vibrtio pheome d cot be described s such by lier vibrtio lysis. As show i Figure.5 i zoe A: No oil whirl is preset d oly sigifict vibrtio is ssocited with the ublce t shft rottio. I zoe B: Oil whirl is preset t ω d criticl speed effect. I zoe C: Whe the oil-whirl vibrtio correspods to system resot frequecy d oe hvig extremely high mplitude. I zoe D: Oil-whirl subsides d oly out of blce respose will be preset. Figure.5 Jourl vibrtio frequecy spectr showig the oil-whirl d the oil-whip Exmple 3.3 A rigid rotor of mss kg is supported by two ideticl fluid film berigs with the followig properties: k xx = MN/m, c xx = kn-s/m. Obti the frequecy of oil whip. Solutio: The udmped turl frequecy of the rotor-berig system, with mss of m d effective support stiffess of k xx, is
666 ω f 6 kxx = = = rd/s m Hece, the dmped turl frequecy is give s ω ω ζ d f = f =. = 989.98 rd/s with 3 c ζ = = =. mω f Hece, the oil whip will tke plce whe the rotor is t resoce d the frequecy of the oil whip will be 995 rd/s (i.e., t the hlf of the resoce frequecy)..5 Iterl Frictio Egieerig mterils show some resistce to their deformtio which is fuctio of their rte of deformtio. Such mteril property my be represeted by dmpig force whe modelig the mteril behviour. The dmpig effect lso produced by the frictio forces betwee mtig compoets of shft whe the shft deflects d the compoets move reltive to ech other. These forces re prticulrly sigifict where iterferece fit compoets re preset (Fig..6). Figure.6 A shft-berig system with iterferece fit Whe shft elogtes t the loctio of the iterfce AA (Figure.6b), the frictio force opposes the shft deformtio, these forces provide hysteretic dmpig effect. Similr effect will be there t iterfce BB. This kid of dmpig c be modeled s force proportiol to the rte of shft deformtio, s compred to the viscous dmpig which is proportiol to bsolute velocity of the rotor.
667 Figure.7 The rotor motio with respect to the fixed d rottig frme of refereces From Figure.7, we hve CF = x = DO, CD = y = FO CG = ξ = EO ; CE = η = GO ; Notig bove expressios, we hve d et d x = DO = HO DH = OE cosωt CE siωt = ξ cosωt η siωt (.9) y = CD = ID + CI = EH + CI = OE siωt + CE cosωt = ξ siωt + η cosωt (.) s = x + jy (.) ζ = ξ + jη (.) O substitutig equtios (.9) d (.) ito equtio (.), we get s = x + j y = ( ξ cosωt η si ωt) + j( ξ siωt + ηcos ωt) = ξ (cosωt + jsi ωt) + η( siωt + jcos ωt) = ξe + j ηe = ( ξ + j η) e jω t jω t jω t Notig, equtio (.), we get s j t = ζ e ω (.3)
668 Equtio (.3) is the trsformtio betwee sttiory d rottig coordite systems, where ω is the spi speed of the shft. O tkig the first d secod differetitios of equtio (.3) gives d s = ζ e + ζ (j ω) e = ( ζ + j ωζ ) e j ω t j ω t j ω t (.) s = ζ e + ζ (j ω) e + ζ ( j ω) e + ζ (j ω) e = ( ζ + j ωζ ω ζ ) e j ω t j ω t j ω t j ω t j ω t (.5) Equtios of motio of the Jeffcott rotor c be writte s mx + c x + kx = d my + c y + ky = (.6) V V where c V is the viscous dmpig. O combiig equtio (.6), otig equtio (.), we get ms + c s + ks = (.7) V O substitutig equtios (.3)-(.5) ito equtio (.7), we get m c k (.8) ( ζ + j ωζ ω ζ ) + V ( ζ + j ωζ ) + ζ = Seprtig the rel d imgiry prts of equtio (.8), we get d m c k (.9) ( ξ ωη ω ξ ) + V ( ξ ωη) + ξ = m c k (.3) ( η + ωξ ω η) + V ( η + ωξ ) + η = Fig..8 The hysteretic dmpig force i rotor i the rottig coordite system
669 Now with the hysteretic dmpig, c H, lso (Fig..8), hysteretic dmpig forces ct log ξ d η directios with vlues of c Hξ d modified s chη, respectively. Hece, equtios (.9) d (.3) c be d m c c k (.3) ( ξ ωη ω ξ ) + Hξ + V ( ξ ωη) + ξ = m c c k (.3) ( η + ωξ ω η) + Hη + V ( η + ωξ ) + η = Agi combiig equtios (.3) d (.3), we get m c c k or ( ζ + j ωζ ω ζ ) + Hζ + V ( ζ + j ωζ ) + ζ = m ζ + mωζ + c + c ζ + k mω + c ω ζ = (.33) (j H V ) ( j V ) where ζ is the complex displcemet d for the sychroous whirl c be defied s ζ j t = ζ e λ (.3) which gives ζ = jζ λ d j t ζ = ζ λ e λ (.35) j t e λ where ζ is the complex whirl mplitude i rottig coordite systems, λ is defied s the reltive whirl frequecy (or the whirl frequecy i the rottig coordite system ξ-η) of the rotor ( λ = λ ω ) d is the whirl frequecy i the sttiory coordite system x-y. It should be oted tht for λ =, we hve the sychroous whirl coditio ( λ = ω ) d for such cse there will ot be y hysteretic dmpig, sice the shft whirls s rigid body i the bed cofigurtio. O substitutig equtios (.3) d (.35) i equtio (.33), it gives { } { ( )} ( ) mλ + ωm + j c + c λ + k mω + jc ω = (.36) H V V A geerl form of equtio (.36), qudrtic polyomil with complex coefficiets, is
67 ( + j b ) λ + ( + j b ) λ + ( + j b ) = (.37) For which the Routh-Hurwitz stbility criteri re (which is of differet form s described previously i Sectio.) b b > b b b d > b b b (.38) s the coditios for the imgiry prt of λ to be egtive (tht is, for the mplitude of ζ to decrese with time). O comprig equtios (.36) d (.37), we get = m, b =, = ωm, b = c + c, H V k mω b =, V = c ω (.39) From the first coditio of equtio (.38), we hve m mω > c + c H V m( c + c ) > ( c c ) H V + > (.) H V From the secod coditio of equtio (.38), we hve m m k m ω ω c + c c ω m ωm k mω H V V which c be expded s c + c c ω H V V > (.) c + c c ω H V V m m ωm k mω > H V V c + c c ω or m( ch + cv ) ( m)( m) > c c c ω ωm k mω cv ω c H + c V c Vω H + V V
67 or or or or ( ){ ( )( )} m c + c c m c + c k m m c > H V V ω H V ω Vω ( )( ) ch cv ω cv ω ch cv chcv ωf ω cv ω + + + + > with ( ch cv ) ωf ( ch cv ) + + ω > ω = k / m f c V + c ω < ω H f cv + ch (.) I geerl, c / c, we hve the followig coditio V H c ω < + c V H ω f (.3) To summrise coditios of the stbility of the rotor system with the hysteretic dmpig from equtios (.) d (.3), it c be writte s with c H f c V + cv > d ω < + ω f (.) ch ω = k / m (.5) The secod coditio idictes tht system is lwys stble, eve i the presece of hysteretic dmpig, below the criticl speed ω f. For the preset cse, both c d V c re ssumed to be H positive. I presece of viscous dmpig, however, hs the effect of risig the speed t which the system becomes ustble, so tht if sufficiet viscous dmpig is desiged ito the system the the istbility threshold speed (i.e., the speed below which the rotor hs lwys stble opertio) due to hysteretic dmpig c be rised beyod the orml opertig speed rge of the mchie, however, the shift would be mrgil.
67 Exmple.3 For Jeffcott rotor, with mss disc of kg, d shft of dimeter of. m d legth of.6 m. It is foud tht the rtio of the coefficiets of viscous d hysteretic dmpig to be.. For the shft tke E =. N/m. Fid the speed of the istbility threshold. Solutio: The stiffess of the shft is give s k 8EI 8..99 l.6 = = =.9 N/m 3 3 The turl frequecy of the rotor system is give s ω f k.9 = = = 7.3 rd/s m Hece the speed of istbility threshold from equtio (.) is : 7.3. = 8.3 rd/s. It c be observed tht for the preset cse, from equtio (.), we get 8. rd/s, which is very close to the bove pproximte vlue. Exmple. For Jeffcott rotor, with mss disc of kg, d shft of dimeter of. m d legth of.6 m. It is foud tht the rtio of the coefficiets of viscous d the hysteretic dmpig to be.. The viscous dmpig rtio i the system is.. For the shft tke E =. N/m. Plot the respose i time domi for some iitil coditio t followig speeds (i) ω =.ω f (ii) ω =.9ω f (iii) ω =.6ω f d (iv) ω =.ω f, where ωf is the udmped turl frequecy. Solutio: From equtio (.33), the equtio of motio of the rotor i rottig coordite system with the viscous d hysteretic dmpigs, we hve m c c k () ( ζ + j ωζ ω ζ ) + Hζ + V ( ζ + j ωζ ) + ζ = The trsformtio from the rottig to sttiory coordite systems is give s - j t ζ = se ω (b) so tht = ( s jωs) e - jωt d ζ = ( jω ω ) ζ (c) - j s s s e ωt
673 O subsistig equtios (b) d (c) i equtio (), we get ( ω ) ms + c s j s + c s + ks = (d) H V O seprtig rel d imgiry prts of equtio (d), we get d ( ) ω mx + c + c x + kx + c y = (e) H V H ( ) ω my + c + c y + ky c x = (f) H V H It should be oted tht equtios (e) d (f) re lier coupled ordiry differetil equtios. For obtiig the respose both equtios hve to be itegrted simulteously by y direct umericlly itegrtio techique. Typicl vibrtio resposes for some iitil coditios hve bee geerted t followig speeds (i) ω =.ω f (ii) ω =.9ω f (iii) ω =.6ω f d (iv) ω =.ω f d re show i Figs..9-., respectively. Figures coti free resposes i time domi d its orbit plots (i.e., x-y plot). It c be see tht for the first three cses the system is stble (Figs..9-.) d for fourth cse it is ustble (Fig..). It should be oted tht for lrge oscilltios lier theory would cese to be vlid d respose the would be govered by olier behviour of the system to prevet very lrge oscilltios before filure. Displcemet (mm) Time (s) Fig..9 () The rotor free vibrtio respose i time domi for ω =.ω f
67 Fig..9 (b) The rotor orbit respose for short time itervl for ω =. ω f (S is the strtig poit) Displcemet i x-directio (mm) Fig.9 (c) The rotor orbit respose for log time itervl for ω =. ω f Displcemet (mm) Displcemet i y-directio (mm) Displcemet i y-directio (mm) Displcemet i x-directio (mm) Time (s) Fig.. () The rotor free vibrtio respose i time domi for ω =.9ω f
675 Displcemet i x-directio (mm) Fig.. (b) The rotor orbit respose for short time itervl for ω =.9 ω f (S is the strtig poit) Displcemet i x-directio (mm) Fig. (c) The rotor orbit respose for log time itervl for ω =.9 ω f Displcemet (mm) Displcemet i y-directio (mm) Displcemet i y-directio (mm) Time (s) Fig.. () The rotor free vibrtio respose i time domi for ω =.6ω f
676 Displcemet i x-directio (mm) Fig.. (b) The rotor orbit respose for short time itervl for ω =.6 ω f (S is the strtig poit) Displcemet i x-directio (mm) Fig. (c) The rotor orbit respose for log time itervl for ω =.6 ω f Displcemet (mm) Displcemet i y-directio (mm) Displcemet i y-directio (mm) Time (s) Fig.. () The rotor free vibrtio respose i time domi for ω =.ω f
677 Displcemet i y-directio (mm) Displcemet i x-directio (mm) Fig. (b) The rotor orbit respose for log time itervl for ω =.ω f.6 Effect of Rotor Polr Asymmetry I my mchies the lterl stiffess of the rotor is differet i two orthogol directios. For electricl motor or geertor, the rotor (Figure.3) my hve slots cotiig electricl widigs o some, but ot ll, prts of its surfce. Figure.3 A geertor rotor For such rotors the stiffess bout x-xis will be less s compred to y-xis (i.e. the shft deforms more whe x-xis is horizotl s gist whe y-xis is horizotl, due to grvity lod). I some cse to mke the stiffess i two directios closer, stiffess-compestig slots i pole fce re mde. But i severl cses it cot be ssured the sme stiffess i ll trsverse directios of the shft.
678 Fig.. Asymmetric rotor with the sttiory d rottig coordite systems Equtios of motio c be developed o the similr lies s pervious sectio with the cosidertio of stiffess i ξ d η directio s k ξ d k η ((i.e., with respect to rottig frme of referece ξ η s show i Fig..). or kξξ = m( ξ ωη ω ξ ) d ηη = η + ωξ ω η k m( ) + ( ) = d m m ( k η ) mξ mωη k ξ ω ξ η + ωξ + ω η = (.6) The resultig motio will be periodic d will tke the form ξ( t) t = ξe λ t d η( t) ηe λ = (.7) where ξ d η re complex mplitudes i rottig coordite systems d λ is the reltive whirl frequecy i rottig coordite system. O substitutig equtio (.7) ito equtio (.6), it gives d { } t ξ λ ωη λ ω e λ ξ ω ξ + ( ) = (.8) { } t η λ ωξ λ ω e λ η ω η + + ( ) = (.9) with ω = k m d ξ η ω = k m (.5) η η
679 Equtios (.8) d (.9) c be combied i mtrix form s [ A]{ X } = {} (.5) with ( λ + ωξ ω ) ( ωλ ) [ A] = ωλ ( λ + ωη ω ) d { X} ξ = η The o-trivil solutio of equtio (.5) is give by λ + ( ω + ω + ω ) λ + ( ω ω )( ω ω ) = (.5) ξ η ξ η The bove equtio c be solved to obti the roots of for vrious ruig speeds ω. Roots will be i geerl complex. The rel prt of λ is if egtive the the system is stble d if positive the system is ustble. As discussed previously tht for egtive vlue of the rel prt of λ, ξ (s well s η ) decreses expoetilly, d is give s ( ) αt ξ ( t) = e Acos βt + Bsi βt with λ = α ± iβ (.53) Altertively stbility my be ivestigted usig the Routh-Hurwitz criteri for polyomil of th degree i λ, i.e., equtio (.3) (or qudrtic i λ ). With either pproch it is foud tht there is potetil regio of istbility defied by (ll the coefficiets of the chrcteristic polyomil must be of the sme sig) ( ωξ ω )( ωη ω ) < for istbility which gives ω ω ω ξ < < η for ωξ ωη < (.5) I prctice, there my be sufficiet dmpig i the system to ihibit ustble vibrtio. The previous lysis is bsed upo the ssumptio tht the shft vibrtio frequecy ω correspods to the mchie ruig speed (ublce). This is stisfctory ssumptio sice i most cses the predomit vibrtio frequecy compoet is tht ssocited with mchie ublce. However, i the cse of rotors with stiffess polr symmetry, which re mouted, horizotlly there is compoet of vibrtio frequecy t twice mchie ruig speed.
68 Figure.5 A rotor with symmetry due to grvity lod () less sg (b) more sg The secod momet of re bout x-x i cse () will be greter th for cse (b) i Figure.5. For this reso there will be greter sg of the rotor due to grvity for the ltter rotor positio s compred with the former. Sice the mjor d mior xes of the rotor sectio chge oriettio twice per revolutio, there will be strog rotor vibrtio frequecy compoet t twice the mchie ruig speed. For this reso there will be ustble mchie opertig frequecy rge, for the horizotlly mouted rotor, defied by ω < v < ω or ω < ω < ω (.55) ξ η ξ η This c be rrived usig the previous lysis with excittio frequecy s i the previous cse it ws v = ω due to ublce oly. Exmple. A ellipticl shft with the legth of m, d the mjor d mior xes of. m d.9 m, respectively. The shft crries disc of mss kg t the mid-sp. For the steel shft tke E =. N/m. Fid the zoes of the istbility i the rotor system due to symmetry of the shft cross-sectio. Solutio: The stiffess of the shft i two pricipl directios re give s with d 8EI 8. 7.95 l. 8 ξ 5 ξ = = = 8. N/m 3 3 k π b π.5.5 I ξ 8 = = = 7.95 m
68 with 8EI 8. 8.836 l.6 8 η 5 η = = = 8.9 N/m 3 3 k πb π.5.5 I η 8 = = = 8.836 m Now, we hve d ω ξ 5 kξ 8. = = = 633.5 rd/s m ω η 5 kη 8.9 = = = 667.6 rd/s m Hece, the rotor will be ustble i speed rge 633.5 rd/s to 667.6 rd/s d i speed rge of 36.67 to 333.73 rd/s..7 A Asymmetric Rotor with Uiformly Distributed Mss et us cosider rotor s show i Figure.6 with uiform distributio of mss. It could be ssumed s lrge umber of thi discs uiformly distributed log the shft with ifiite umber of discs s the limitig cse (Todl, 965). Figure.6 A rotor with distributed mss property. et E be the modulus of elsticity of the shft mteril, I d I re the pricipl re momets of ierti of the shft sectio, mk is the polr mss momet of ierti per uit legth of the shft elemet (e.g., thi disc) with respect to xis of rottio, mk is the dimetrl mss momet of ierti per uit legth of the shft elemet (e.g., thi disc) with respect to dimetrl xis, m is the mss per uit legth of shft, k is the rdius of gyrtio of the shft elemet, ω is the gulr velocity of the shft, ϕ x d ϕ y re the gulr displcemets bout the x d y xes, respectively; d these re chose positive i ccordce with right hd xis covectio s show i Figures.7. et us first
68 derive equtios of motio for I = I, the symmetry of the shft cross-sectio would be itroduced subsequetly. Figure.7 A rotor elemet Figure.8 Positive directios of slopes d gulr displcemets (left) z-x ple (right) y-z ple Equtios of Motio: The gulr displcemet of the shft elemet (deoted by ϕx d ϕ y ) re give by (see Figure.8) y( z, t) ϕx = z d x( z, t) φy = z (.56) where x d y re lier displcemets. It should be oted tht the positive directio of gulr displcemet, ϕ x, is opposite to the positive slope, dy/dz, directio; d the positive directio of gulr displcemet, ϕ y, is sme to the positive slope, dx/dz, directio. So tht the gulr velocity d ccelertio c be writte s
683 d y ϕx = z t d 3 y ϕx = z t d x ϕy = z t (.57) 3 x ϕ y = z t (.58) () A free-body digrm of the shft elemet i z-x ple (b) A free-body digrm of the shft elemet i y-z ple Figure.9 Gyroscopic d ierti momets o the shft elemet () z-x d (b) y-z ples Momet due to ierti forces of elemets (tkig momets i the y-z d z-x ples, d respective directios positive i directios of ϕ x d ϕ y ) re give s (see Figure.9) d M M I ϕ = (.59) zx g y d y
68 M + M I ϕ = (.6) zy gx d x with gyroscopic momets (ierti momet) to the rotors re give s M = I ωϕ d g y P x M = I ωϕ (.6) gx P y where M zx d M zx re momets which re pplied to the rotor (disc) by the shft. It should be oted tht i ple z-x the chge i gulr mometum over the time itervl cosidered is I Pωϕ x d so the gyroscopic couple which must be pplied to produce this chge, equl to the rte of chge of gulr mometum, is give by I Pωϕ x. I Figure.9() the gyroscopic couple pplied to the shft through the rotor, I Pωϕ x, must ct bout y-xis correspodig to the oriettio of the vector ( I pω) i y-z ple (i.e., i the egtive y-xis directio). Hece, the rotor will get rective couple, M = I ωϕ. Similrly, i the ple y-z the chge i gulr mometum over the time itervl g y P x cosidered is IPωϕ y d so the gyroscopic couple which must be pplied to produce this chge equl to the rte of chge of gulr mometum, is give by IPωϕ y. I Figure.9() the gyroscopic couple pplied to the rotor, IPωϕ y, must ct bout x-xis correspodig to the oriettio of the vector ( I pω) i z-x ple (i.e., i the positive x-xis directio). Hece, the rotor will get rective couple, M = I ωϕ. Equtios (.59) d (.6), c be writte s gx P y M = I ϕ + I ωϕ (.6) zx d y P x d M = I ϕ I ωϕ (.63) zy d x P y For the shft elemet (i.e., thi disc) we hve, Id = mk dz d I p Id mk dz = = ; d otig equtios (.57) d (.58), from equtios (.59) d (.6), we get d 3 3 x y x y M zx = ( mk dz) ( mk dz) ω = mk dz + ω z t z t z t z t 3 3 y x y x M yz = ( mk dz) + ( mk dz) ω = mk dz ω z t z t z t z t (.6) (.65)
685 Figure. A positive sig covetio for the bedig momet d the sher force d its effect o bem For the bedig momets of the elstic forces hve which gives M y M z x = EI z y 3 x = EI 3 z M y d M x i z-x d y-z ples, respectively, we y d M x = EI (.66) z d M z x 3 y = EI 3 z (.67) The sig i bove equtios hs bee chose such tht it is cosistet with the choice of coordite xes i Figure.8 d the defiitio of positive bedig momet s tht which produces curvture cocve upwrds (Figure.). For coordite xes s show i Figure.8, we see tht whe the curvture is cocve upwrds, the slope dx/dz lgebriclly icreses cotiuously (from left to right eve fter the zero slope) with z d hece d x / dz is positive. ikewise, whe the curvture is cocve dowwrds (egtive bedig momet), the slope dx/dz is lgebriclly decresig with z d hece d x / dz is egtive. Thus Similrly, the slope dy/dz lgebriclly icresig with z d hece d x / dz hs lwys sme i sig to bedig momet. d y / dz is positive.
686 () z-x ple (b) y-z ple Figure. Free body digrm of the shft elemet Applyig coditios for equilibrium of momets i ple z-x (Figure.), we get M y M y + M y + dz+ M zx + mdzx ( dz ) Sxdz = (.68) z or M y dz + M zx +.5 mxdz S x dz = z (.69) d similrly i y-z ple (Figure.b), we get or M x M x M x + dz + M yz mdzy( dz ) + S ydz = z (.7) M x dz + M yz.5 mydz + S y dz = (.7) z O substitutig equtios (.59) (.66) i equtios (.68) d (.7), eglectig terms cotiig dz, we get 3 3 x x y EI mk S 3 + ω x = z z t z t (.7)
687 d 3 3 y y x EI mk S 3 ω y = z z t z t (.73) O differetitig equtios (.7) d (.73) with respect to z, we get d 3 x x y S EI mk x + ω = z z t z t z 3 y y x S y EI mk ω = z z t z t z (.7) (.75) From the coditio for the equilibrium of forces ctig o the elemet we hve S z x Sx Sx + dz = mdz x t or Sx z x = m t (.76) Similrly, we hve S y y S y S y + = m z t or S y z y = m t (.77) O substitutig equtios (.76) d (.77) ito equtios (.7) d (.75), we get d 3 x x y x EI mk m + ω + = z z t z t t 3 y y x y EI mk m ω + = z z t z t t (.78) (.79) Deotig s = x + jy, equtios (.78) d (.79) c be combied s ( y jx) 3 ( x + j y) ( x + j y) ( x + j y) EI mk ω m + + = z z t z t t (.8)
688 which c be writte s 3 s s s s EI mk j m ω + = z z t z t t (.8) Figure. The sttiory d rottig coordite systems et the complex displcemet i the rottig coordite system ( ξ η) s show i Figure. be defied s ζ ( z, t) = ξ ( z, t) + j η( z, t) (.8) The trsformtio from the sttiory coordite system (x-y) to the rottig coordite system ( ξ η) is give s s( z, t) = ζ ( z, t) e jωt (.83) O differetitig equtio (.83) with respect to time, t, d spce, z; we get j ζ ωζ ; s = ( ζ + jωζ + jωζ + j ω ζ ) = ( ζ + jωζ ω ζ ) e ωt (.8) j s = ( + j ) e ωt d s j t = ζ e ω ; j ζ ωζ ; s = ( ζ + jωζ ω ζ ) e ωt (.85) j s = ( + j ) e ωt Trsformig equtio (.8) ito the rottig co-ordite system ( ξ η) by substitutig equtios (.83) d (.8), we get
689 {( ζ ωζ ω ζ ) ω ζ ωζ } ( ζ ωζ ω ζ ) ζ EI mk + j j ( + j ) + m + j = (.86) z which c be simplified s ζ ζ ζ ζ ζ EI mk ω m jω ωζ + + + = z z t z t t (.87) Seprtig the rel d imgiry prts d itroducig the uequl momets of ierti I d I, log ξ d η directios, respectively, we get d EI ' + + = ( ) k m ξ ξ ω ξ ξ ωη ω ξ (.88) EI ' k + + + = ( ) m η η ω η η ωξ ω η (.89) Boudry Coditios & Frequecy Equtio: For simply supported shft correspodig boudry coditios re ξ (, t) = ξ (, t) = η(, t) = η(, t) = for displcemets (.9) d ξ (, t) = ξ (, t) = η (, t) = η (, t) = for bedig momets (.9) Equtios of motio (equtios (.88) d (.89)) d boudry coditios (equtios (.9) d (.9)) the stisfy the followig solutio ξ π z λ = Asi cos t d Bsi si π z η = λ t (.9) where A d B re costts, = (,, 3, ), is the legth of the shft, λ is the turl frequecy i the sttiory coordite system (x, y, z) d λ = λ ω is the turl frequecy i the rottig coordite system ( ξ, η, z). π z ξ λ λ π z = A si si t, η = Bλ si cosλ t
69 π z ξ = Aλ si cosλt, π z η = Bλ si si λt π π z ξ = A si cosλt, π π z η = B si si λt πλ = A π z ξ si cosλt, η πλ = B π z si si λt π π z ξ = A si cosλt, π π z η = B si si λt (.93) O substitutig ssumed solutios from equtio (.9) d its derivtives from equtio (.93) i equtios of motio (equtios (.88) d (.89)), we get EI π πλ π π z A k A k ω A + ( Aλ ) ωbλ ω A si cosλt = m d EI π πλ π π z B k B k ω B + ( Bλ ) + ω ( Aλ ) ω B si si λt = m which simplifies to d EI π π π z k ( λ ω ) ( λ + ω ) A + ( ωλ ) Bsi cosλt = m EI π π π z ( ωλ ) ( ) ( A + k λ ω λ + ω ) Bsi si λt = m (.9) (.95) For o-trivil solutio, the determit of coefficiets of lier homogeeous equtios (.9) d (.95) must be zero
69 EI π π k ( λ ω ) ( λ + ω ) ωλ m EI π π ωλ k λ ω λ + ω m ( ) ( ) = (.96) which gives the frequecy equtio s EI π EI k π ( ) ( ) π k π λ ω λ + ω ( λ ω ) ( λ + ω ) ( ωλ ) = m m or E I I π EI π EI π EI π π m m m m 8 8 6 6 6 6 k ( λ ω ) ( λ 8 6 + ω ) k 6 ( λ ω ) + k ( λ ω ) π EI π π + k ( λ ω ) ( λ + ω ) + k ( λ ω ) + ( λ + ω ) ( ωλ ) = m or π π π k k k + + + λ EI π EI π EI π π EI π + k k k + m m m m 6 6 6 6 ω ω ω 6 6 λ E I I π EI π EI π EI π π + + + + m m m m 8 8 6 6 6 6 k ω ω k ω k ω 8 6 6 + k or π EI π π ω ω ω ω m k + = k π π + k + λ E( I + I ) π E( I + I ) π π + k m m 6 6 k ω ω 6 λ 8 8 6 6 E II π E( I + I) π E( I + I) π π π + + k ω ω + k ω k ω ω 8 6 + + = m m m
69 or π π E( I + I) π π π λ ω λ ω k k k k + + + + + m Deotig ( ) π π + m I m π 8 8 E I + I k E I ω = 8 (.97) kπ + = d kπ = b (.98) Whirl Nturl Frequecies & Criticl Speeds: Equtio (.97) gives the reltive turl frequecy s k π E( I + I ) π ( λ ) = + ω + m 8 8 π E( I I) π E( I I ) π E II π ω b ω b ω 8 k + + ± + + + m m m (.99) To simplify equtio (.99), let us tke terms withi the squre brcket seprtely d expd s or 8 8 π π E( I I ) π E ( I I) π ω ω 8 k k + + + + + + m m 8 8 E( I + I ) π E II π ( b ) ω bω + 8 m m (.) k π k π k π E( I + I ) π E( I + I ) π + + + + + m m 8 8 8 8 b ω b ω 8 8 E ( I + I) E II π + 8 m m (.)
693 From equtio (.98), we c hve d k π k π k π b = + = b 8 8 8 k π k π k π k π k π = + = = + 8 (.) (.3) O substitutig equtios (.) d (.3) ito expressio (.), we get k π k π k π k π + + + 8 8 8 8 8 8 8 8 ω 8 8 k π E( I + I ) π E( I + I ) π k π E ( I I ) π + + + ω + 8 m m m (.) or k π E( I + I) E ( I I) π ω + ω + mk m k (.5) O substitutig equtio (.5) ito equtio (.99), we get k π E( I + I) π ( λ ),,3, = ± + ω + m k π E( I + I) E ( I I) π ω ω ± + + mk m k / (.6) where = (,, 3, ). The bsolute frequecy is the λ ( λ + ω (.7) = ),,3, Cse I: λ so tht λ = ω i.e. the whirl frequecy is thus equl to the gulr velocity, which = gives the sychroous forwrd precessio or whirl. Equtio (.97) gives (for λ = )
69 E( I + I ) π ω E I I π ω + = (.8) 8 8 8 m b m b ω 8 8 E( I + I) π E( I + I) π E II π = ± 8 m b bm m b {( ) } π E( I + I ) π E E π = ± + + = + ± I I II I I I I ( I I) mb bm mb E π = + ± mb l [ I I ( I I )] which gives ω π EI π EI / m * = = b m k π / (.9) d π EI ω = (.) ** b m where =,, 3, * ** For ω d ω to be lwys rel, we hve k π < or < (.) kπ Equtio (.) will give the umber of criticl speeds, so fiite umber of sychroous forwrd whirl criticl speeds re possible. Cse II: For λ = ω ( hece, it gives λ = ω tht mes the sychroous bckwrd whirl or the ti-sychroous precessio. Thus, for λ = ω from equtio (.97), we get
695 k π E ( I + I ) π E ( I + I ) π b b m m (6 ω ) + ( ) ω + ω + ω ω which c be rerrged to 8 8 E II π + = 8 m (.) 8 8 k E( I I) E( I I) E II 6 8 π + b π + π + + ω + b π ω + = 8 m m m (.3) From equtio (.36) we c get criticl speeds (, ) for ti-sychroous whirl, d criticl speeds of ti-sychroous whirl exit for ll vlue of. So there re ifiite umber of criticl speeds exits for the preset cse. Cse III: For momet of ierti forces d gyroscopic effect re egligible i.e. k =. The chrcteristic equtio (.97) will reduce to 8 8 E( I + I) π E( I + I) π E II π λ ω + λ + ω ω + = 8 m l m l m l which c be rerrged to E ( I + I ) π EI π EI π λ ω + λ + ω ω = m l m l m l (.) For sychroous whirl λ =, we get criticl speeds, s ω π EI = = d l m * ω k = ω π l EI m ** = ω k = = (.5) I this cse we obti ifiite umber of criticl speeds d ifiite umber of itervls of istbility, ω ω with =,,. Equtio (.) c be solved directly to get turl (, ) frequecy of the systems s (To be expded)
696 E( I + I ) π π E( I + I ) E ( I I ) π λ ω ω ω m m m,,3, = ± + ± + = ω ± ω + ± ω ω + ω + ω ω ω + ω ( ) ( ) (.6) where ω d ω re give s i equtio (.5). Equtio (.6) will give ifiite umber of grphs of λ plotted s fuctio of ω ( =,, 3, ) d ifiite umber of itervls of istbility. Stbility Alysis of Asymmetric Shft with Gyroscopic Effects: Notig equtio (.98), equtio (.97) c be writte s 8 8 E( I + I) π E( I + I) π E II π λ + bω + λ + b ω b ω + = 8 m m m (.7) which c be rerrged s EI π EI π EI π EI π λ + bω + + λ + b ω b ω = m m m m (.8) From equtios (.9) d (.), we get * EI π bω = d m ** EI π bω = (.9) m Notig equtio (.9), equtio (.8) gives {( ) } ( )( ) λ + b ω + ω + ω λ + ω ω ω ω = (.) * ** * ** For the cse of sigle disc with mssless shft or the cse of polr symmetry the chrcteristic equtio (equtio (.5)) is λ + ( ω + ω + ω ) λ + ( ω ω )( ω ω ) = (.) ξ η ξ η d by the Routh-Hurwitz criteri the regio of istbility is defied by < or ωξ < ω < ωη (.) ( ωξ ω )( ωη ω )
697 Sice we hve k ξ < k d o comprig chrcteristic equtio (.) d (.), o the similr η lie we c obti the coditio of istbility s ( ω )( ) ω * ** ω ω < (.3) * ** The bove coditio of istbility is fulfilled oly iside the itervl ( ω, ) (it is ssumed here ** * tht ω > ω, i.e., I > I). Thus we hve fiite umber of itervls of istbility. It my hppe tht two itervls overlp, i.e. tht they merge ito sigle itervl of istbility. The coditio for this to hppe is, tht ω ω (.) ** * > ω + sice is fiite so fiite umber of itervl of istbility. Exmple.: Cosider rotor system with the followig rotor prmeters: m =.98 kg, k = 3 cm, I = 5 cm, I = cm, = cm d E =. N/m. Obti the istbility plots up to third mode d tbulte ll the frequecy rge of istbility. Obti the criticl speeds for (i) sychroous whirl with gyroscopic effect (ii) ti-sychroous whirl with gyroscopic effect, d (iii) without gyroscopic effect. Solutio: For the cse whe k, < / ( π k) = / ( π 3) =.6, so tht =. The gyroscopic effect thus cuses the itervls of istbility to be reduced to fiite umber (i.e., for the preset * ** cse). Hece, we hve itervls of istbility { ( ω, ω ), =,,, } s give i the secod d third colums i Tble. for the sychroous whirl. No istbility itervls overlp (or merge) for the sychroous whirl. Figures.3 d. show vritio of the forwrd whirl turl frequecy with the spi speed for the first mode d up to third modes, respectively. The shded frequecy itervl represet the istbility zoes, i Fig.. it c be see tht these zoes re distict d o overlp is foud i the istbility zoes up to third modes. Tble. lso lists istbility zoes for ti-sychroous whirl i fourth d fifth colums. It c be see tht for the ti-sychroous whirl the overlp of istbility itervl exist for differet eighbourig modes. Tble. lso lists first istbility zoes for the cse of o gyroscopic effect, i.e. k = d the overlp of istbility itervl exist for differet eighbourig modes.
698 3 5 Whirl frequecy, rd/s 5 5 = * =. = ** = 3.5 - - 3 - - 3 Spi frequecy, rd/s Figure.3 Vritio of whirl frequecy with the spi speed for first mode (Plot of equtio (.97) for = )