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1 Jounal hoe page : hp:// Sabiliy analysis of oaing beas ubbing on an elasi iula suue Jounal of Sound and Vibaion, Volue 99, Issues 4-5, 6 Febuay 007, Pages N. Lesaffe, J.-J. Sinou and F. Thouveez STABILITY ANALYSIS OF OTATING BEAMS UBBING ON AN ELASTIC CICULA STUCTUE N. Lesaffe*, J-J. Sinou, F. Thouveez Laboaoie de Tibologie e Dynaique des Sysèes UM CNS 553 Eole Cenale de Lyon, 36 avenue Guy de Collongues, 6934 Eully, Fane ABSTACT This pape pesens he sabiliy analysis of a syse oposed of oaing beas on a flexible, iula fixed ing, using he ouh-huwiz ieion. The odel displayed has been fully developed wihin he oaing fae by use of an enegy appoah. The beas onsideed possess wo degees of feedo, a flexual oion as well as a aion/opession oion. In-plane defoaions of he ing will be onsideed. Divegenes and ode ouplings have hus been undesoed wihin he oaing fae and in ode o siplify undesanding of all hese phenoena, he degees of feedo of he beas will fis be eaed sepaaely and hen ogehe. The dynais of adial oaing loads on an elasi ing an eae divegene insabiliies as well as pos-iial ode ouplings. Moeove, he flexual oion of bea ubbing on he ing an also lead o ode ouplings and o he lous veeing phenoenon. The pesene of ubbing sees o ake he syse unsable as soon as he oaional speed of he beas is geae han zeo. Lasly, he influene of an angle beween he beas and he noal o he ing's inne sufae will be sudied wih espe o syse sabiliy, hus highlighing a shif fequeny phenoenon. Keywods: Sabiliy analysis, oaing beas, iula elasi ing, divegene, ode ouplings, ubbing. I. INTODUCTION Pobles aused by loads oving ove elasi suues ou ahe fequenly. The ase of oaing suues has been given speial aenion. A flexible oaing disk exied by loads an expeiene insabiliies, whih fo insane an ou when using a iula saw, as widely sudied by Moe [,], wih opue eoy soage disks, sudied by Iwan and Sahl [3], Iwan and Moelle [4] and Candall [5] aong ohes, o in he field of bake syses, Ouyang e al. [6], Chabee and Jezequel [7]. In all hese exaples, he sudied syse was oposed of a oaing disk whose in-plane vibaions wee onsideed ubbed on is plane by ionay loads o by a fixed disk ubbed by oaing loads. Pos-iial insabiliies have hus been idenified and hei leading paaees deeined. Few sudies howeve ould be found ha fous on a oaing ing ubbed by loads on is inne sufae. Canhi and Pake [8] eenly invesigaed paaei insabiliies due o oaing spings on a iula ing. This kind of syse an be applied, fo insane, in planeay geas o ubo ahiney, in he ase of onas beween he oo blades and he asing [9]. This sudy will hus pesen a sabiliy analysis of a flexible ing ubbed on is inne sufae by oaing

2 beas wih wo degees of feedo: flexual oion and aion/opession. To bee undesand his phenoenon, he wo degees of feedo will fis be sepaaed and hen sudied ogehe. In he fis seion, he ing will be exied by beas feauing only he aion/opession degee of feedo oaing on is inne sufae. Afewads, he beas onsideed will possess us a flexual degee of feedo. Finally, hese wo degees of feedo will be sudied in obinaion wih one anohe. This pape will onlude wih an exainaion of he influene of a onsan angle beween eah bea and he noal o he inne ing sufae. II. THE MODEL The odel onsideed in his sudy onsiss of a flexible ing ubbed by one o seveal beas on is inne sufae, as depied in Fig. a, in he ase of one oaing bea. These Eule-Benoulli beas have wo degees of feedo in he oaing fae, i.e. in he fae aahed o he beas: aion/opession oion u, and flexual oion υ f. An enegy ehod is used o develop he odel; hene, he degees of feedo of he h bea ae expessed by he following iz funions (, ) ( )sin x u x = u, oesponding o he exa aion/opession ode shape of a lap-fee bea, and x υf ( x, ) = υ ( ) os f fo is flexual degee of feedo, wih x being he loal axis along he bea and he ing adius. Conening he ing, is in-plane flexual vibaions ae onsideed, i.e. wo degees of feedo ae onsideed in he oaing fae: adial displaeen us ( φ, ), and angenial displaeen ω( φ, ), wih φ being he angula posiion of he ass ene of a ing's oss-seion in he oaing fae. This lae degee of feedo an be expessed k o using [0]: ω( φ, ) = A ( )os nφ+ B ( )sinnφ, in whih he igid body oion has been eliinaed. n= n n In ode o geneae as siple a odel as possible, only one ode shape, he n h one, will be onsideed fo he ing, hene: ω( φ, ) = An( )os nφ+ Bn( )sinnφ. Moeove, he onsideed ing is assued o be inexensible, hus iplying ha is adial displaeen an be expessed fo is angenial ω ( φ, ) displaeen by: us ( φ, ) =. The bea fee ends ae assued o eain in seady-e ona φ wih he inne sufae of he ing, heefoe, a link elaionship beween he peinen degees of feedo us be wien as follows: us ( φ = φ, ) = u ( x=, )os (, )sin + υf x=, wih being he h angle beween he bea and he noal o he ing's inne sufae. Sine an enegy ehod has been applied o develop he enie odel, he kinei enegy and poenial enegy ae defined fo boh he beas and he ing. ubbing sengh is inodued by defining is wok. The expessions of hese enegies and of his wok ae given in Appendix A, along wih expessions fo he ass aix, siffness aix, iulaoy aix and gyosopi aix assoiaed wih his odel. To bee undesand he phenoenon appeaing wihin his suue howeve, he beas ae fis onsideed o be noal o he ing's inne sufae ( = 0 ). In his ase, he syse an be sepaaed ino siple suues. The fis suh suue onsiss of beas wih us a aion/opession degee of feedo ubbing on he ing. The seond onsiss of beas wih us a flexual oion ubbing on he ing. Then, boh of hese degees of feedo will be obined. In hese ases and fo he sake of sipliiy (o handle odal ass and siffness), he beas will be opaed o adial sping-asses having wo degees of feedo (see Fig. b). The assoiaed odel has been developed in Appendix B. Lasly, he effe of an angle of inlinaion beween he beas and he ing will be analysed.

3 III. OTATING ADIAL BEAMS UBBING ON A FLEXIBLE ING The sabiliy of an elasi ing ubbed by one o seveal beas an be invesigaed by deeining he soluion λ =a+ ib o he haaeisi equaion de( λ² M+ λ( G+ ) + K ) = 0, whee M, G, and K ae he ass aix, gyosopi aix, iulaoy aix and siffness aix of he syse, espeively. The syse beoes unsable if one o oe of he eigenvalue eal pas a ae posiive. Thoughou his seion, he beas ae assued o be adial o he ing's inne sufae ( = 0 ).. BEAMS WITH JUST A TACTION/COMPESSION DEGEE OF FEEDOM In his seion, he beas onsideed onain only a aion/opession degee of feedo. They an hus be epesened by adial oaing sping-asses ubbing on he elasi ing, as ploed on Figue in he paiula ase of us one oaing load. Due o he link elaionship beween he adial degees of feedo of he odel, he syse has wo degees of feedo and he assoiaed aies an be dedued fo he oplee syse developed in Appendix B. In he ase of us one oaing load, he dynai behaviou of he syse an hus be desibed by he following aix equaion: ( + ) µ + ( ) 0 ( + ) + M n n n A n B n h M n n h Kn ( n ) Mn ( n + ) µ + ( n ) ( k ) n An + B n 0 Kn ( n ) Mn ( n + ) + ( k ) n ( ) 0 Mn n + A n + Mn ( n + ) 0 B n h µ + + = n ( n ) ( NU) EI wih M = ρs and K = 3 I is obvious ha ubbing akes he ass and siffness aies asyei, whih is known o be haaeisi of a poenially-unsable syse. Soe poenial iial speeds of he syse ay be deeined analyially using he ouh-huwiz ieion. The haaeisi polynoial of his 4 3 aix equaion aually has he following fo: () () Ps = As + Bs + Cs + Ds+ E wih: ( ) ( ) A= M n + + M n + n h B= + ( n ) µ Mn ( n + ) ( ) ( ) [ ] ( ) C = K n ( n ) M n + + n + M n n + M + M n + n k ( ) ( ) h D= n k + µ M n n + ( ) ( ( ) ) E = M ( ) ( ) n n Kn n n M n Kn n n k Aoding o he ouh-huwiz ieion, he polynoial Ps () = As + Bs + Cs + Ds+ E has all is oos wih eal pas negaive if A, B, BC AD, ( ) BC AD D B E and E have he sae sign. Eah sign B BC AD hange of one of hese es iplies ha one of he oos of he haaeisi polynoial osses he veial axis, aking is eal pa posiive, hene he syse beoes unsable. I is obvious ha A and B ae always posiive. egading he e BC AD B : () 3

4 4 - If ( )( ) M ( n ) ω ( )( ) M n + + n > n, i is posiive if: + ( + ) ( )( ) + n n M n > 4 = 4 4 M n + + n n M n + + n n n ( ) n + M n + povided ha ω >, ohewise = 0. 4 M n If ( )( ) n ( ) n + M n + ω < M ( n + ) ( ) M n + + n < n, i is negaive if egading he e ( ), ohewise > 4 povided ha : = 0. BC AD D B E : BC AD n ( ) n + M n + - If ω > : he nueao is posiive if M ( n + ) ( ) ( ) 4 ( ) ( ) ( ) 4 ( ) 5 = < < 5 = ( n + ) ( n + ) ω n n n 4n n ω n 0 ; ohewise, if ω n + + n n + n n + ω n ω n + + n n n n + ω n 4 povided ha ( ) ( ) ( ) ω ( n ) n ( n ) 4n ( n ) ω n 0 < < 5 = ( n + ) < The denoinao is posiive 4 fo >4 if M ( n + )( + n ) > n ; ohewise, i is always negaive. n ( ) n + M n + n - If > ω > : he nueao is posiive if M ( n + ) ( n + ) ω ( n + ) + n ( n ) + 4n ( n + ) ω n ω ( n + ) + n ( n ) 4n ( n + ) ω n 5 = < < 5 = n + n + ( ) and ω ( ) ( ) ( ) ω ω ( ) ( ) 4 ( ) 0 < < 5 = ( n + ) < n n n 4n n n 0 n + + n n n n + ω n ; ohewise, if The denoinao is negaive fo 4 4 M n + + n < n ; ohewise, i is always posiive. n - If >ω he nueao is always negaive. The sign of he denoinao is he sae ( n + ) as in he above ase. ω The las e E is negaive beween and wih = + and K = M ( n ) n + > if ( )( ). ( ) M n M ( n + ) + + ( + ), oesponding o he ing's fis iial speed. These las wo oaional speeds deeine he oaional speed ange ove whih he syse is unsable even wihou ubbing. I will be shown below ha his insabiliy onsiss of a divegene in he ing's fowad ode shape. This phenoenon is lose o ha shown by Canhi and Pake [8] o by Iwan and Sahl [3], and Iwan and Moelle [4] in he ase of a disk insead of a ing, wih he influenes of he load paaees also being quie siila. 4

5 k In all hese expessions, ω = is he squaed angula fequeny of he adial sping-ass. I an hus be seen ha his kind of syse wih ubbing is alos always unsable. As a ae of fa, i only lies wihin speifi oaional speed anges, i.e. only beween 5 and 5 an he above oeffiiens all be posiive in he ase of a lighweigh syse in opaison wih he ing's 4 n (i.e. n < M( n + )( + n )) and a siffness, suh ha: ω >. ( n + ) The effes of boh a ass ubbing on he ing and of he siffness ay be sepaaed. Figues 3a and 3 display he sabiliy analysis of a adial siffness (wihou ass) ubbing agains he ing wih µ = 0.0 and µ = 0., espeively. Figues 3b and 3d show assoiaed zoos of Figs. 3a and 3, espeively. As explained peviously, a divegene insabiliy in he fowad ode shape of he ing beween and an be obseved. Moeove, as expeed, one he oaional speed is geae han 0 PM, he syse, and espeially he bakwad ode shape of he ing, is unsable beause of ubbing. This ubbing effe is well-known and has been highlighed, fo insane, in he ase of a odal epesenaion of a ubine engine exied by ubbing foes []. I hus appeas ha as he ubbing oeffiien ineases, insabiliy ises even fase. The ase of us one ass ubbing on a ing will now be onsideed; Figue 4 pesens he assoiaed sabiliy analysis. Hee again, he syse is unsable as soon as he oaional speed diffes fo zeo, wih he unsable ode now howeve being he ing's fowad ode shape. A divegene insabiliy in he fowad ode shape noneheless eains afe eahing he iial syse speed, beween and, and ode oupling beween he fowad and bakwad ode shapes of he ing. I should be noed ha his ode-oupling appeas even wihou ubbing and is due o load displaeen on he elasi ing. This phenoenon has been epoed in he ase of a disk insead of a ing (see [3,4]). Figues 4 and 4d show he sabiliy analysis fo he sae syse as in Figues 4a and 4b, bu wih a highe ubbing oeffiien. The effe of he ubbing oeffiien is he sae as befoe. Boh a siffness and ass will now be onsideed. Figues 5a and 5 (wih he assoiaed zoos in Figs. 5b and 5d, espeively) display sabiliy analysis fo a adial sping-ass ubbing on he ing's wo-node diaee ode shape. In boh ases, he ass is = 00kg, ye Figues 5a and 5b inlude 6 =.0 N. 5, wheeas =.0 N. in Figues 5 and 5d. A oss beween he eal pa uves k k of he ing's fowad and bakwad ode shapes an be obseved, whih sees o be oe in opaison wih he las esuls aken sepaaely. A low oaional speeds, he siffness aually desabilises he bakwad ode shape of he ing, bu a highe oaional speeds he ass, wih a negaive siffening effe popoional o oaional speed ( ), desabilises he ing's fowad ode shape. This oss only ous if ω > ωn, whee ω is he angula fequeny of he sping-ass h and ω n he angula fequeny of he ing's n nodal diaee ode shape. Moeove, as indiaed on hese las figues, he oss ous a a oaional speed beween 5 and 5. As ealie disussed, he syse ay be sable beween 5 and 5, as shown in Figues 5b and 5d. Figue 6 pesens a sabiliy analysis fo he sae syse as in Figue 5a, bu wih a highe ubbing oeffiien, one again ephasising is effe. I an be poined ou ha he ubbing oeffiien exes no effe on he eakable iial oaional speeds,, and. As he nube of nodal diaees of he ing's ode shape ineases o infiniy, he speed ange, 5 5 ollapses o, whih iself ends o infiniy. The ase of seveal adial oaing sping-asses ubbing on he ing will now be invesigaed. Ceain onfiguaions appea o avoid he divegene insabiliy of he fowad ode shape beween and, as shown in Figue 7 in he ase of sping-asses wih ω = 00 ad / s and µ =

6 Sine his divegene insabiliy is pesen wihou ubbing, he ouh-huwiz ieion applied o he haaeisi polynoial of he syse wih µ = 0 an yield a suffiien ondiion fo he disappeaane of divegene. This ondiion ay be wien as: k sin ( nφ ) os ( ) = k nφ k sin( nφ )os( ) 0 nφ = sin ( nφ ) os ( ) = nφ sin( nφ )os( ) 0 nφ = (3) These ondiions ae obviously saisfied in he ase shown in Figue 7 sine all he sping-asses have he sae paaee values and ae loaed a φ = 60, φ = 0 and φ 3 = 80, whih is no ue in he ases shown on he ohe figues. I an be noed ha even in he ase wih no divegene of he ing's fowad ode shape, he syse is sill unsable one he oaional speed diffes fo 0 PM.. BEAMS WITH JUST ONE FLEXUAL DEGEE OF FEEDOM Hee again, his syse is quie siila o a ubbing oaing sping-ass angen o he ing, as depied in Figue 8, in he ase of one sping-ass. The aix equaion fo he dynai behaviou of suh syses is now: ( + nube of loads) ( + nube of loads). Fo a sabiliy analysis poin of view, he diffeenes beween he bea odel wih us one flexual degee of feedo and he angen sping-ass odel se fo he spin-sofening es. Those assoiaed wih he bea odel do no ake ino aoun he enie flexual odal ass, bu ahe ρ b I b. Anohe 8 diffeene also onens aix, whih is neihe syei no skew-syei (see Appendies A and B). The phenoenon ouing fo he angen sping-asses ubbing on he ing should howeve be he sae as fo beas wih us a flexual degee of feedo ubbing on a ing. The sabiliy analysis of suh syses an hus be pefoed using he angen sping-ass odel, whih allows onsideing load odal paaees. In he ase of only one angen sping-ass ubbing on he asing, he haaeisi polynoial of is aix equaion is: ( ) ( ( )) ( µ ) ( ρ ( )) ( ) P() s = s + s + k s M n + + Kn ( n ) Mn n + + s Mn n + By alulaing us he oos of his polynoial, whih oespond wih he oos of is fis ebe s + s( µ ) + k, sping-ass sabiliy an be sudied. The disiinan of his fis ebe is: ( ) ( k = µ + ) 4k. If <, hen < 0 and he oos of his polynoial ae: ( µ + ) µ + i µ i s = and s =, k hus e( s) < 0 and e( s) < 0 and he sping-ass is sable. Now, if >, hen > 0 ( µ + ) µ + µ and s = and s =, hus e( s ) < 0. Conening he eal pa of s, i is negaive if < k = ω and posiive if > ω, oesponding o a syse divegene. All hese esuls 6

7 ae valid fo he bea, bu he eakable oaional speeds ae 4k µρbsb + 4 ρbib 8 k insead of and k k insead of. Figue 9 pesens he sabiliy analysis of ( µ + ) ρbib 8 6 one angen sping-ass ubbing on he ing's wo-node diaee ode shape, wih k =.0 N. and = 00kg, and wih: a) µ = 0.0 and b) µ = 0.. As expeed, he sping-ass is sable unil > ω, a whih poin i expeienes divegene insabiliy. The effe of he ubbing oeffiien is he sae as befoe. In he ase of seveal angen sping-asses ubbing agains he ing, no addiional phenoenon ous. I an also be obseved ha boh of he ing's ode shapes appea o be pefely sable. Sine he effes of eah degee of feedo fo a bea ubbing on an elasi ing have been sudied sepaaely, he beas an now be onsideed o possess boh degees of feedo. 3. BEAMS WITH BOTH A TACTION/COMPESSION DEGEE OF FEEDOM AND A FLEXUAL DEGEE OF FEEDOM The bea's wo degees of feedo will now be onsideed. One again, his syse, as deailed in Appendix A, is siila o a sping-ass wih wo degees of feedo (see Appendix B), as displayed in Figue. The diffeenes beween hese wo odels, in addiion o all hose desibed above, se fo he gyosopi es pesen sine he sping-asses have wo degees of feedo ha ae no expessed in he sae anne. These gyosopi es ae likely o eae new ode ouplings in he syse. Neveheless, as seen in he lae ase, he phenoenon appeaing fo hese wo syses should be he sae. The sping-ass syse will be sudied in ode o easily handle odal paaees and afewads will be opaed wih he bea odel. Figue 0 shows he sabiliy analysis of he wo-node diaee ode shape of he ing ubbed by a 6 sping-ass wih = 00kg, k = k =.0 N. and µ = 0.. All phenoena sudied ealie esuling fo a angen sping-ass o a adial sping-ass ubbing on he ing ae one again pesen. The effe of he ubbing oeffiien (no epesened hee) is sill he sae: an inease in he slope of he uves' eal pa. A lous veeing phenoenon is also in effe beween he ing's bakwad ode shape and he sping-ass, followed by ode oupling beween he ing's fowad ode shape and he sping-ass. This ode oupling ay esul fo he gyosopi es. Moeove, he speed ange onened by his ode oupling is vey sensiive o he angenial siffness k of he sping-ass, as shown in Figue. The geae he angenial siffness, he wide he ange in ode oupling speed. Lasly, he effes of seveal oaing sping-asses an also be sudied. Figue shows he sabiliy analysis fo he wo-node diaee ode shape of he ing ubbed by wo idenial spingasses wih = 00kg, k = k =.0 N. and µ = 0.; Figue a oesponds o wo loads 6 sepaaed by 60 fo eah ohe, wheeas Figue b oesponds o wo loads sepaaed by 80. In boh ases, i appeas ha only one sping-ass exhanges is ode shape wih he bakwad ode shape of he ing (lous veeing) and hen expeienes ode oupling wih his ing's fowad ode shape. Moeove, in he fis ase (i.e. sping-asses sepaaed by 60 ), he eigenfequenies of boh sping-asses slighly inease afe hei heoeial divegene, as shown in Figue a; his does no ou when he wo sping-asses ae diaeially opposed (see Fig. b). This phenoenon will be analysed fuhe below. Figue 3 exhibis he sabiliy analysis fo he wonode diaee ode shape of he ing ubbed by hee idenial oaing sping-asses wih 6 = 00kg, k = k =.0 N. and µ = 0., bu eihe sepaaed by 60 fo eah ohe (Fig. 3a), o wo sepaaed by 60 and he hid a 80 fo one of he ohe wo (Fig. 3b). In he fis ase, 7

8 he suffiien ondiion fo eliinaing he divegene insabiliy beween and is saisfied, hene Figue 3a shows no divegene beween hese oaional speeds, wheeas his divegene is sill pesen in Figue 3b. One again, in boh ases, only one sping-ass exhanges is ode shape wih he ing's bakwad ode shape and hen expeienes ode oupling wih his ing's fowad ode shape. Moeove, afe he heoeial divegene in oaional speed fo he hee sping-asses, wo of he also see o have slighly ineased in eigenfequeny. The sping-ass eigenfequenies an be adused hough hei asses and siffness. A sabiliy analysis fo he wonode diaee ode shape of he ing ubbed by wo sping-asses wih wo diffeen eigenfequenies sepaaed by 60 fo eah ohe has been ploed on Figue 4. In his ase, us one sping-ass exhanges is ode shape wih he ing's bakwad shape, ye boh sping-asses expeiene ode oupling wih he ing's fowad ode shape. The syse, like in all ohe exaples, is unsable one he oaional speed diffes fo zeo. Conening he inease in eigenfequenies of he divegen beas (see Figs. a and 3), hee is aually a ansiion beween he divegene and flue of wo, and only wo, beas beoing oupled hough he ing. The ode shape being onsideed fo he ing is indeed vey ipoan fo his oupling beween wo beas and he ing ha povides seady-e ona. Alhough all siulaions pesened in his pape peain o he ing's wo-node diaee ode shape, Figue 5 shows he angula egions whee a bea loaed a 60 in he oaing fae an ouple wih anohe bea fo: a) he ing's wo-node diaee ode shape, and b) is hee-node diaee ode shape. I hus appeas ha fou egions exis fo he ing's wo-node diaee ode shape, wheeas six exis wih he ing's hee-node diaee ode shape. I an oeove be seen ha as ing defoaion ineases, he oupling egions beoe naowe. Fo he wo-node diaee ode shape fo exaple, eah oupling egion is 36 wide, while fo he hee-node diaee ode shape, eah one is 6 wide. I an be noed ha he whole angula posiion ange, ove whih wo blades an ouple, is geae in he ase of he ing's wo-node diaee ode shape (44 ) han in he ase of is heenode diaee ode shape (96 ). Figue 6 shows a zoo of Figue a nea he oupling egion. I an lealy be seen on his iage ha afe bea divegene, he assoiaed eal pas ouple wih one anohe, heeby leading o an unsable dynai onfiguaion. A ase of hanging insabiliy has been unoveed by Gaul and Wagne [], wheeby a oaing syse expeiened insabiliy divegen fo ode-oupling. Moeove, he oaional speed a whih oupling appeas vaies ove he oupling angula egion, as indiaed in Table fo he ase of he wo-node diaee ode 6 shape of he ing ubbed ( µ = 0.) by wo sping-asses wih = 00kg and k = k =.0 N., he fis one being a 60 in he oaing fae and he ohe beween 87 and 3. I us be poined ou ha his phenoenon also ous wih beas feauing diffeen odal paaees. I has been said pio ha phenoena ouing in he ase of a sping-ass wih wo degees of feedo ubbing on he flexible ing ae he sae as hose ouing in he ase of a bea also wih wo degees of feedo ubbing on his ing. This an be onfied fo one oaing load ubbing on he ing's wo-node diaee ode shape. Figue 7 displays a sabiliy analysis fo his ing's ode shape ubbed eihe by a sping-ass (Fig. 7a) o by a bea (Fig. 7b), boh having he sae odal paaees: ω = 5 ad / s and ω = 00 ad / s (fo he sping-ass: = 4.8kg, k k 6 = N. and 7 =.75.0 N. and k 6 k =.48.0 N. and fo he bea: =.0 N. ). As expeed, he sae phenoena ou in boh ases: = kg, = 00kg, lous veeing is in effe beween he sping-ass and he ing's bakwad ode shape, along wih a divegene in is fowad ode shape beween and, ode oupling beween his fowad ode shape and he sping-ass, a divegene in his sping-ass and ode oupling beween he ing's fowad and bakwad ode shapes. The only diffeene beween hese wo syses is he offse of hese phenoena due o odal paaee diffeenes. The syse naually beoes unsable one he oaional speed diffes fo 0 PM. 8

9 IV. OTATING BEAMS UBBING ON A FLEXIBLE ING WITH AN ANGLE OF INCLINATION The effe of an angle of inlinaion of a bea ubbing on a oaing disk has been sudied by, aong ohes, Chabee and Jezequel [7], ye no sudies have been found in he lieaue on he influene of suh an angle beween beas oaing wih ubbing on he inne sufae of an elasi ing. The pevious sudy by Chabee and Jezequel [7] deonsaed ha his kind of angle an odify he paaei doains whee he syse is unsable: he invesigaed syse was a oaing disk exied by a bea wih boh aion/opession oion and flexual oion. I has been shown ha he sae kind of insabiliies as hose inluded in he pesen sudy, i.e. divegene afe he iial speed and ode oupling, ould be obained and odified by he angle beween he bea and he disk. The ain esuls fo his sudy wee in fa ha as he bea beae oe heavily inlined, he ing's divegene speed ange naowed and he ing's ode oupling was oe heavily delayed. I has also been epoed ha ode ouplings ould aise even befoe he iial speed if he fequeny of he bea's flexual degee of feedo was below he disk's fequeny o lose o i. The influene of he angle of inlinaion on he pesen syse, in he ase of us one bea ubbing on he ing, ould hus be sudied when he fequeny of he bea's flexual oion lies below o above o lose o ha of he ing. I appeas howeve ha he ehaniss ouing due o his angle of inlinaion ae he sae in all hee of hese ases; heefoe, only he ase whee he bea's flexual oion fequenies lie below ing fequenies will be deailed heein. The ing's wo-node diaee ode shape has been se a 30 Hz and he flexual oion of he bea a 0 Hz. Figue 8 pesens he sabiliy analysis fo a bea ubbing on he ing's wo-node diaee ode shape, wih: a) = 0, b) = 5, ) = 0 and d) = 89. These values of have been hosen beause of he high evoluion in syse fequenies fo low values of. Figue 9 exhibis he assoiaed appopiae zoos of Figue 8. Fis of all, i ay be obseved on Figue 8 ha as ineases, he oaional speed a whih he syse expeienes ode oupling beween he ing's fowad and bakwad ode shapes ises, as does he oaional speed a whih he bea's flexual degee of feedo diveges. Moeove, ode oupling beween he ing's fowad ode shape and his flexual degee of feedo of he bea appeas befoe divegen insabiliy in he lae, when ineases, as shown on Figue 9. On his sae figue, i appeas ha he oaional speed ange ove whih he ing's fowad ode shape diveges deeases as ineases and hen finally disappeas. Fo low values of, a lous veeing phenoenon beween he ing's fowad ode shape and he bea's flexual oion ay be obseved, espeially on Figue 9a. Duing his phenoenon, he syse is bound o be unsable; his insabiliy (ode oupling ype) an hus ake effe befoe he ing's iial speed. As ineases, he flexual degee of feedo fequeny ineases and hene he lous veeing phenoenon wih he fowad ode shape disappeas. As fo ode oupling beween he ing's fowad ode shape and he bea's flexual oion enioned above, Figue 9 shows ha as ineases, he assoiaed oaional speed ange begins lae and has geae values. This insabiliy ous afe he ing's iial speed. All hese phenoena ay be seen oninuously as a funion of, as indiaed on Figues 0 and. The evoluion in he oaional speed a whih boh ode shapes of he ing ouple an be onioed on Figue 0. Figue shows he evoluion of he pos-iial ode ouplings beween he ing's fowad ode shape and he bea's flexual oion, as well as he divegene of his lae degee of feedo. The oaional speed ange assoiaed wih his ode oupling ay be ineased abou 500 PM, fo a onfiguaion a = 0 o one a = 89. The ehanis involved in he inlinaion of a bea ubbing on an elasi ing hus piaily onsiss of an inease in he bea's flexual fequeny. If, when he bea is adial o he ing, is flexual fequeny is below ha of he ing beause of he evoluion wih oaional speed, eihe lous veeing o ode oupling an ou beween he ing's fowad ode shape and he bea's flexual oion even befoe he ing's iial speed. In his ase, boh eigenfequenies (of he bea and he ing) ae vey lose o eah ohe (see Fig. 9a), and he syse is bound o be unsable. As ineases, bea fequenies inease unil eahing he fequeny of is aion/opession degee of feedo. One he bea fequeny has isen above ing fequenies (fo > 5 ), lous 9

10 veeing onens is bakwad ode shape and, as seen on Figue 9d, he speifi eigenfequenies ae no longe lose o one anohe. This veeing does no ause syse insabiliy. This ehanis is he sae as in he ase whee he bea's flexual fequeny is highe han ha of he ing. As ineases, he bea's flexual fequeny ineases; howeve, sine i always eains above he ing's, lous veeing ay ou even befoe he ing's iial speed ye an only onen is bakwad ode shape. In his ase, boh fequenies ae no vey lose o one anohe and his veeing does no ake he syse unsable. While he bea's flexual fequeny deeases wih an inease in oaional speed, ode oupling wih he ing's fowad ode shape hen ous. The influene of he angle of inlinaion of he bea ubbing on an elasi ing is heefoe lose o ha of a bea ubbing on a disk (see [7]). This angle as upon he sae iial phenoena. As he inlinaion angle ineases, he oaional speed ange ove whih he ing's fowad ode shape diveges an in fa be odified (edued), as an he oaional speed fo boh ode shapes of he ing ouple (pu away). The syse an also be ade unsable befoe he ing's iial speed by eans of ode oupling beween he bea's flexual oion and he ing's fowad ode shape povided he flexual fequeny lies below he ing's fequeny. The bea's angle of inlinaion aually odifies he values of he noal and angenial sengh beween boh suues in ona, heeby odifying phenoena like divegene o he ing's ode oupling. This angle also odifies he flexual fequeny of he bea in ona wih he ing, aking ode ouplings possible o no povided ies have fequenies lose o eah ohe. All siulaions have been ondued fo a ing's wo-node diaee ode shape, ye he sae phenoena ae pesen fo ohe ode shapes as well. Moeove, only one ode shape fo he ing and beas has been onsideed heein; he phenoena ageed in his sudy howeve ae quie siila o hose ha ay aise when onsideing seveal ode shapes fo eah ie of he odel, as illusaed by Iwan and Sahl [3] and Iwan and Moelle [4]. V. CONCLUSION The sabiliy of oaing beas ubbing on an elasi ing has been sudied in his aile. An enegy odel of flexible beas possessing wo degees of feedo in seady-e ona wih an elasi ing possessing us one in-plane ode shape has been developed wihin he oaing fae. This odel, devoid of ie-dependen es, has been sudied fo a sabiliy poin of view. I appeas ha ubbing always akes he syse unsable one he bea's oaional speed is nonzeo. I has also been shown ha a adial siffness ubbing on he ing ends o ake is bakwad ode shape unsable, wheeas a onenaed ass ubbing on a ing akes he fowad ode shape unsable. The aion/opession degee of feedo of a bea ubbing on a ing, in addiion o he unsable phenoena ouing even wihou ubbing (divegene of he ing's fowad ode shape nea is iial speed and pos-iial ode oupling beween fowad and bakwad ode shapes), hus sas by aking is bakwad ode shape unsable and hen is fowad ode shape. The eakable oaional speeds of hese phenoena have been deeined analyially. As he ubbing oeffiien ises, he gadien of he eigenvalue eal pas also ises. The beas' flexual degee of feedo yields ode ouplings and lous veeing wih he ing. The influene of seveal beas ubbing on a ing has been exained and soe ases of oupling beween beas highlighed. Lasly, an angle of inlinaion beween he beas and he ing has been onsideed. I has also been deonsaed ha he ain esul of his paaee was he inease in he bea's flexual fequeny wih inlinaion, hus leading o veeing and ode ouplings. 0

11 APPENDIX A Expessions of he kinei enegy and poenial enegy, as well as he wok of ubbing sengh assoiaed wih he odel of oaing beas loaed a φ wihin he oaing fae ubbing on he flexible inexensible ing wih inlinaion angle. The expession of he kinei enegy of he syse is given by: us w T = ρs us ( φ, ) ( φ, ) w( φ, ) ( φ, ) dφ + φ φ + ρb S { (,) (,) ( (,)) (,) ( (,)( (,)) (,) (,) b u x υ f x x u x υ f x υ f x x u x u xυ f x ) } dx 0 υ f ( x, ) + ρb I b + x 0 The expession of he poenial enegy of he syse is given by : dx ² υ s f (, ) (, ) 3 s b b b b φ x 0 x 0 E I u u ( x, ) ( x, ) γ = φ u φ dφ E S dx E I dx ² When inluding iz funions fo he degees of feedo in he above expessions and in onsideing he elaionship beween he ing's adial degee of feedo and boh he bea's degees of feedo: u ( φ = φ, ) = u ( x=, )os + υ ( x=, )sin, hese enegies and poenials an be wien s f by: us w T = ρs us ( φ, ) ( φ, ) w( φ, ) ( φ, ) dφ + φ φ u (, ) an (, ) s φ u s φ υf + ρb S (, ) b u s φ υf + os os os 3 4 ( 4 + 8) us( φ, ) υ f + ρb S an b υf υ f + + os u (, ) (, ) an s φ 8 u s φ + ρb S (, ) b us φ υf os os os ρb S an an b υf υf ρb I b υf + υ f + 8 The expession of he poenial enegy of he syse is now given by: EI ² us us φ (, ) (, ) 3 us d Eb S b φ (, ) γ = φ + φ φ+ ² 8 os

12 an + E S + E I E S u os 4 b an (, ) b b b 3 υf b b s φ υf The expession fo ubbing wok is given by: h h us ( φ, ) Wex = Tb (, ) (, )( os sin an ) (, )an ( ) ω φ υf x = + us φ δ φ φ φ h wih, in he die enipeal fae, Tb ( ) = µ Nb sign Vslip being he ubbing sengh of he bea on he ing. In his expession, V slip is he slip speed of he bea on he ing and Nb he adial load of he h bea on he o. Fo insane, in he ase of onas beween blades of a oaing ahine and he asing, i an be expessed by he adial load due o he unbalaned ass NU plus a dynai load due o he dynais of he h bea. In ode o inlude his ubbing sengh ino he aix equaion of syse dynai behaviou and pefo a sabiliy analysis, his ubbing sengh an be expessed by: ρ b S b Tb ( (, ) (, )) (, ) (, )an ( ) = µ NU + us φ us φ + Eb S b u s φ ρb S b u s φ δ φ φ µ ρb S ( os sin an ) (, ) (, )sin ( ) b + υf x= ρb S b ρ b S b ρb I b υ f x= δ φ φ µ 4 ρb S (, )sin ( ) b Eb S b ρ b S b E b I b + ρ 3 b S b υf x= δ φ φ 8 3. This iplies ha ubbing sengh always follows he sae dieion, aking his odel valid if he adial load due o unbalaned ass is fa geae han he dynai load due o dynais of he h bea, whih is aepable, and if V slip always eains he sae sign. This lae ondiion is ue fo a suffiien oaional speed. In all ases, he ain pupose of his odel and of his sudy is o dee he appeaane of insabiliies and no o alulae poenial lii yles his fa ino he sudy. The expession of T an be obained by he Hailon piniple using Lagangian uliplies. I should be noed ha he iz funions fo he beas' equal uniy a hei end ubbing agains he ing; bea paaees appeaing in his ubbing sengh ae hene aually odal paaees of he beas a hei end ubbing agains he ing. Only one ode shape of he o has been onsideed a his ie, i.e.: ω( φ, ) = An( )os nφ+ Bn( )sinnφ and us( φ, ) = nan( )sin nφ+ nbn( )osnφ. The aix equaion of syse dynai behaviou is: MX + G + X + KX = F T wih: X = { An Bn υ f υ } f ( ) M M 0 0 M M 0 0 an an µ ( os ) ( ) + sinan nsin( nφ ) µ os + sinan nos( nφ) M os os M = an an µ ( os sin an ) sin( ) + n nφ µ os sin an nos( nφ ) M os + os ( ) ( + )( + ) sin ( nφ ) h M = M ( n + ) + n + µ ( ) os( ) sin( ) an sin( ) n nφ n nφ n nφ os + b

13 sin( nφ )os( nφ) h M = n µ ( ) os( ) sin( ) an os( ) + n nφ n nφ n nφ os sin( nφ )os( nφ) h M = n + µ ( ) sin( ) os( )an sin( ) n nφ n nφ n nφ os + + os ( nφ ) h M = M ( n + ) + n µ ( ) sin( ) os( ) an os( ) + n nφ + n nφ n nφ os M = + an µ os + sin an ( )sin ( ) 33 ( ) M µ ( + )( + ) = + an os sin an ( )sin + sin( ) sin( nφ ) nφ 0 Mn ( n + ) ρb S b n ρ b S b n os os os( ) os( nφ ) nφ Mn ( n + ) 0 ρbs b n ρ b S b n os os sin( nφ) os( nφ) ρbs b n ρ b S b n 0 0 G = os os sin( nφ) os( n ) φ ρb S 0 0 b n ρ b S b n os os 3 ( + ) 3 ( + ) 3 3 µρbs ( os ) b + sin an = ( ) ( ) µρb S ( os sin an ) b h = µρb S sin( ) an ( ) os( ) sin( ) an b n nφ n nφ n nφ + h = µρb S os( ) an ( ) os( ) sin( ) an b n nφ n nφ n nφ + h = µρb S sin( ) an ( ) sin( ) os( ) an b n nφ n nφ n nφ + + h = µρb S os( )an ( ) sin( ) os( )an b n nφ n nφ n nφ + + h 3 = µρbs ( os ) ( ) b + sin an n os( nφ ) nsin( nφ ) an + h ( + ) = µρb S ( os sin an ) ( ) os( ) sin( ) an b + n nφ n nφ + h 3 = µρbs ( os ) ( ) b + sin an n sin( nφ ) nos( nφ ) an + + 3

14 h ( + ) = µρb S ( os sin an ) ( ) sin( ) os( ) an b + n nφ n nφ = µρbs sin( ( ) b n nφ ) an os + sin an 3 = µρbs os( ( ) b n nφ ) an os + sin an ( ) = µρb S sin( ) an ( os sin an ) b n nφ + + ( ) = µρb S os( ) an ( os sin an ) b n nφ + + K K K K K K K K an an ( k ) sin( ( ) ( ) ( ) n nφ ) µ os + sinan k os( n nφ ) µ os + sinan K os os K = an an ( k ) sin( ) n nφ µ ( os + sin an ) ( k ) nos( nφ ) µ ( os + sin an ) K os os 3 ( + ) 3 ( + ) ( + )( + ) sin ( nφ ) h K = Kn ( n ) Mn ( n + ) + ( k ) ( ) sin( ) ( ) os( ) sin( ) an n + µ k n nφ n nφ n nφ os + sin( nφ )os( nφ) h K = ( k ) ( ) os( ) ( ) os( ) sin( ) an n µ k n nφ n nφ n nφ os + sin( nφ )os( nφ) h K = ( k ) ( ) sin( ) ( ) sin( ) os( ) an n + µ k n nφ n nφ n nφ os + + os ( nφ ) h K = Kn ( n ) Mn ( n + ) + ( k ) ( ) os( ) ( ) sin( ) os( ) an n µ k n nφ n nφ n nφ os + + an h K3 = ( k ) sin( ( ) ( ) n nφ ) + µ + n os( nφ ) nsin( nφ ) an k sin k ρ bi b os 8 an h K( + ) = ( k ) sin( ) ( ) os( ) sin( ) an ( ) sin n nφ + µ n nφ n nφ k k ρ b I b os + 8 an h K3 = ( k ) os( ( ) ( ) n nφ ) + µ + n sin( nφ ) + nos( nφ ) an k sin k ρ b I b os 8 an h K( + ) = ( k ) os( ) ( ) sin( ) os( ) an ( ) sin n nφ + µ n nφ n nφ k k ρ b I b os K33 = k an ( ) ( ) + k an + ρ os b I b µ + sin an k sin k ρ bi b 8 8 K( + )( + ) = k an an ( os sin an ) ( ) sin + k + ρ b I b µ + k k ρ b I b 8 8 4

15 sin( nφ ) h 4ρ b S ( ) os( ) sin( ) an 4 b + N U n + µ + n nφ n nφ ρb S b + N U os os( nφ ) h 4ρ b S ( ) sin( b + N U n + µ + n nφ) + nos( nφ)an 4ρb S b + N U os F = 4ρb S an [ ] b + N U µ os+ sinan 4ρb S b + N U 4ρb S an os sin an 4 b + N U µ + ρb S b + N U In hese expessions, = ρ S and b b 3 4 = ρ b S b + ρb I b 8 aion/opession and flexue, espeively. k = E b S b and k 8 siffness of aion/opession and flexue, espeively. ae he odal ass of 4 = E I ae he odal 3 b b 3 5

16 APPENDIX B Expessions of he kinei enegy and poenial enegy, as well as of he wok of he ubbing sengh assoiaed wih he siplified odel of oaing sping-asses feauing wo degees of feedo ubbing on he flexible inexensible ing. The expession of syse kinei enegy is given by: us w T = ρs us ( φ, ) ( φ, ) w( φ, ) ( φ, ) dφ us ( φ, ) us ( φ, ) δ φ φ φ φ + x ( φ, ) + x δ( φ φ ) + u ( φ, ) x( φ, ) + x ( φ, ) ( u ( φ, ) ) δ ( φ φ ) s s The expession of syse poenial enegy is given by: E I ² us γ = (, ) u (, ) (, ) (, ) 3 s d k u s k x φ + φ φ + φ δ φ φ + φ δ φ φ φ² ( ) ( ) ( ) ( ) The expession of he ubbing wok an now be given by: h h us ( φ, ) Wex = T (, ) (, ) ( ) ω φ x φ δ φ φ φ wih: T ( (,) (,) (,) ) (,) ( ) = µ NU + us φ us φ + + x φ + k u s φ δ φ φ being he ubbing sengh of ass on he o. The sae eaks as hose offeed in Appendix A an be fowaded hee onening validiy ondiions of he ubbing odel. Hee again, only one ode shape of he o has been onsideed a a ie. The aix equaion of syse dynai behaviou is as follows: MX + ( G + ) X + KX = F. T X = A B x x wih: { n n } M M 0 0 M M 0 0 µ n sin( nφ ) µ n os( nφ) 0 0 M = µ n sin( nφ ) µ n os( nφ) h M = M ( n + ) + n sin ( nφ ) + µ + ( n ) n sin( nφ )os( nφ ) h M = n sin( nφ )os( nφ ) µ + ( n ) n os ( nφ) h M = n sin( nφ )os( nφ ) + µ + ( n ) n sin ( nφ) h M = M ( n + ) + n os ( nφ ) µ + ( n ) n sin( nφ )os( nφ ) 6

17 ( ) 0 Mn n + nsin( nφ ) nsin( nφ ) Mn ( n + ) 0 nos( nφ) nos( nφ) nsin( nφ) nos( nφ) 0 0 G = nsin( nφ) nos( nφ) 0 0 h h 0 0 µ os( nφ ) + ( n ) µ os( nφ) + ( n ) h h µ sin( nφ ) + ( n ) µ sin( nφ) + ( n ) = µ µ K K 0 0 K K 0 0 µ ( k ) ( ) nsin( nφ ) µ k nos( nφ) k 0 0 K = µ ( k ) sin( ) ( ) os( ) n nφ µ k n nφ k h ( ) ( ) ( ) K K n n M n n k n n n k n n n = ( ) + + ( ) sin ( φ ) + µ + sin( φ )os( φ ) h K = ( k ) sin( )os( ) os ( ) n nφ nφ µ + n k n nφ h K = ( k ) sin( )os( ) ( ) ( ) sin ( ) n nφ nφ + µ + n k n nφ h K = Kn ( n ) Mn ( n + ) + ( k ) os ( ) ( ) ( ) sin( )os( ) n nφ µ + n k n nφ nφ h ( + NU) nsin( nφ ) + µ + ( n ) ( + NU) os( nφ) h ( + NU) nos( nφ ) + µ + ( n ) ( + NU) sin( nφ) F = µ ( + NU ) µ ( + NU) ( ) ( ) Unde suh ondiions, diffeenes beween his syse and he bea odel se fo spin-sofening es sine hose assoiaed wih he bea odel do no ake ino aoun he enie flexual odal ass, bu insead ρ b I b. Anohe diffeene onening boh he aix and gyosopi 8 es has also been idenified. 7

18 EFEENCES [] C. D. Moe, Sabiliy of Ciula Plaes Subeed o Moving Loads, Jounal of Fanklin Insiue, 90 (970), [] C. D. Moe, Moving Load Sabiliy of a Ciula Plae on a Floaing Cenal Colla, Jounal of he Aousial Soiey of Aeia, 6(977), [3] W. D. Iwan and K. J. Sahl, The esponse of an Elasi Disk Wih a Moving Mass Syse, Jounal of Applied Meahnis, 40 (973), [4] W. D. Iwan and T. L. Moelle, The Sabiliy of a Spinning Elasi Disk Wih a Tansvese Load Syse, Jounal of Applied Mehanis, 43(976), [5] S. H. Candall, oodynais, in W. Klieann and N. S. Naahhivaya, eds., Nonlinea Dynais and Sohasi Mehanis, CC Pess, Boa aon, 995, pp [6] H. Ouyang, J. E. Moeshead, M. P. Caell and M. I. Fiswell, Fiion Indued Paaei esonanes In Diss/ effes of a Negaive Fiion-Veloiy elaionship, Jounal of Sound and Vibaion, 09(998), [7] P. Chabee and L. Jezequel, Sabiliy of a Bea ubbed agains a oaing Dis, Euopean Jounal of Mehanis, A/Solids, (99), [8] S. V. Canhi and. G. Pake, Paaei Insabiliy of a Ciula ing Subeed o Moving Spings, Jounal of Sound and Vibaion, 93(006), [9] S. K. Sinha, Dynai Chaaeisis of a Flexible Bladed-oo wih Coulob Daping Due o Tipub, Jounal of Sound and Vibaion, 73(004), [0] Love, A. E. H., A Teaise on The Maheaial Theoy of Elasiiy, New Yok Dove Publiaions, 944. [] V. Gallado and C. Lawene, Tubine Engine Sabiliy/Insabiliy Wih ub Foes, NASA/TM [] L. Gaul and N. Wagne, Eigenpah Dynais of Non-onsevaive Mehanial Syses suh as Dis Bakes, IMAC XXII, Deabon, Mihigan, 004. Angula posiion of seond blade ( PM ) X ( ) X PM Table Coupling oaional speeds beween wo sping-asses wih = 00kg and k k.0 N. 6 = =, he fis one being a 60 in he oaing fae and he ohe one, in he fis oupling angula egion (87-3 ) 8

19 ω ω u s υ f u x k x k u s (a) (b) Figue a) Model of Eule-Benoulli bea ubbing on an elasi ing, b) odel of ing ubbed by one oaing load having wo degees of feedo u ω s k Figue Model of adial sping-ass ubbing agains a ing 9

20 = 64PM = 7PM Bakwad Fowad (a) (b) Bakwad = 7PM Fowad = 64PM () (d) 6 Figue 3 Sabiliy analysis fo a adial siffness of k =.0 N. ubbing on he wo nodal diaee ode shape of he ing wih a) µ = 0.0, b) being he assoiaed zoo and ) µ = 0., d) being he assoiaed zoo 0

21 = 64PM = 6PM Fowad Bakwad (a) (b) = 64PM = 6PM Fowad Bakwad Fowad Bakwad () (d) Figue 4 Sabiliy analysis fo a ass of 00kg ubbing agains he wo nodal diaee ode shape of he ing wih a) µ = 0.0, b) being he assoiaed zoo and ) µ = 0., d) being he assoiaed zoo

22 = 4PM = 64PM Bakwad 783PM Fowad 5 = 90PM 5 = 553PM (a) (b) = 64PM = 6PM Fowad 5 = 3PM 5 = 49PM 666PM Bakwad () (d) Figue 5 Sabiliy analysis of he wo nodal diaee ode shape of he ing exied by a ubbing ( µ = 0.0) adial sping-ass wih a) ω = 00 ad / s, b) being he assoiaed zoo and ) ω = 3.6 ad / s, d) being he assoiaed zoo wheeas ω = 34.4 ad / s n = 4PM = 64PM Bakwad Fowad 5 = 90PM 5 = 553PM (a) (b)

23 Figue 6 a) Sabiliy analysis of he wo nodal diaee ode shape of he ing exied by a ubbing adial sping-ass wih ω = 00 ad / s and µ = 0., b) being he assoiaed zoo = 4PM = 48PM = 64PM = 97PM 783PM 646PM (a) (b) No divegene = 83PM = 04PM 597PM 597PM () (d) Figue 7 Sabiliy analysis of he wo nodal diaee ode shape of he ing exied by a) one adial sping-ass b) wo adial sping-asses sepaaed fo 60 fo eah ohe, ) hee adial spingasses sepaaed fo 60 fo eah ohe, d) hee adial sping-asses wo being a 60 fo eah ohe and he hid one being a 80 fo one of he lae wo ω x u s k Figue 8 Model of ubbing oaing sping-ass angen o he ing 3

24 Sping-ass Sping-ass 60 k Sping-ass 60 = 955PM k = 955PM (a) (b) Figue 9 Sabiliy analysis fo a angen sping-ass ubbing agains he wo nodal diaee ode shape of he ing wih a) µ = 0.0, and b) µ = 0. = 4PM = 64PM 60 k = 955PM Mode oupling B-F Mode oupling S-F F B Mode oupling Lous veeing S-F S B B S F B S F B S Divegene of S Mode oupling B-F F F (a) (b) Figue 0 a) Sabiliy analysis of he wo nodal diaee ode shape of he ing ubbed by a sping-ass 6 having = 00kg, k = k =.0 N. and µ = 0., b) being he assoiaed zoo S = Sping-ass, F = Fowad, B = Bakwad = 56PM 759PM = 4PM 68PM = 64PM = 64PM 60 k = 955PM 733PM 60 k = 300PM (a) (b) Figue Sabiliy analysis of he wo nodal diaee ode shape of he ing ubbed by a sping-ass of 7 = 00kg, a) k =.0 N. 6, k =.0 N. 6 b) k =.0 N. 7, k =.0 N. and µ = 0. 4

25 60 k Mode oupling B-F 60 k Mode oupling B-F Lous veeing Mode oupling S-F Lous veeing Mode oupling S-F S S B S S F B F Divegene of S and S Inease S B SS S (a) (b) Figue Sabiliy analysis of he wo nodal diaee ode shape of he ing ubbed by wo spingasses having = 00kg, k = k =.0 N. and µ = 0., a) sepaaed fo 60 fo eah ohe b) 6 sepaaed fo 80 fo eah ohe S = Sping-ass, F = Fowad, B = Bakwad F B F Divegene of S and S No divegene 60 k Mode oupling B-F 60 k Mode oupling B-F S S S3 B F Lous veeing S S3 S Mode oupling S-F B Inease of wo eigen F fequenies S S S3 Mode oupling Lous veeing S-F S B S S3 S S3 B S (a) (b) Figue 3 Sabiliy analysis of he wo nodal diaee ode shape of he ing ubbed by hee spingasses having = 00kg, k = k =.0 N. and µ = 0., a) sepaaed fo 60 fo eah ohe b) 6 wo being a 60 fo eah ohe and he hid one a 80 fo one of he lae wo S = Sping-ass, F = Fowad, B = Bakwad F Inease of wo eigen F fequenies S S S3 5

26 60 k Mode oupling B-F 60 k S F Mode oupling S-F B S Lous veeing S- B F S Mode oupling S-F B Divegene of S F Divegene of S Figue 4 Sabiliy analysis of he wo nodal diaee ode shape of he ing ubbed by wo spingasses sepaaed fo 60 fo eah ohe having = 00kg, k =.0 N. 7 6, k =.0 N., = 00kg, k 7 =.0 N., k 6 =.0 N. and µ = 0. S = Sping-ass, F = Fowad, B = Bakwad Figue 5 Coupling egions fo a) he wo nodal diaee ode shape of he ing, b) he hee nodal diaee ode shape of he ing 6

27 Figue 6 Coupling beween wo sping-asses ubbing ( µ = 0.) on he wo nodal diaee ode shape of he ing, one being a 60 in he oaing fae and he ohe, a 0, wih = 00kg and k = k =.0 N. 6 = 490PM 60 k = 955PM Mode oupling B-F = 550PM 60 k = 040PM ρbib 8 Mode oupling B-F = 64PM = 64PM PM 84PM (a) (b) Figue 7 Sabiliy analysis fo he wo nodal diaee ode shape of he ing ubbed by a) one spingass having wo degees of feedo b) one bea having wo degees of feedo S = Sping-ass, b= flexue oion of he bea F = Fowad, B = Bakwad 7

28 09PM 57PM 76PM 7740PM 453PM (a) 48PM (b) 808PM () (d) Figue 8 Sabiliy analysis of he wo nodal diaee ode shape of he ing (30Hz) ubbed by one bea (0Hz) a a) = 0, b) = 5, ) = 0 and d) = 89 8

29 007PM 900PM 900PM 050PM 937PM 98PM Bakwad Bakwad Fowad Bea Bea Fowad 09PM Fowad Bea Bea Fowad 57PM 900PM (a) 93PM (b) 906PM 388PM Fowad Bea Bakwad Fowad Bea 453PM Bea Bakwad Fowad Bakwad Bea Fowad 40PM 7PM () (d) Figue 9 Zoos assoiaed wih Fig. 8 evoluion of he fowad ode shape divegene of he ing and ode ouplings as a funion of, fo a) = 0, b) = 5, ) = 0 and d) = 89 9

30 Figue 0 Capbell diaga of he wo nodal diaee ode shape of he ing (30Hz) ubbed by one bea (0Hz), as a funion of - evoluion of Figue Capbell diaga of he wo nodal diaee ode shape of he ing (30Hz) ubbed by one bea (0Hz), as a funion of - posiion of he ode ouplings beween he fowad ode shape and he flexue oion of he bea as well as he evoluion of he divegene of his lae degee of feedo 30