Modal Analysis of Periodically Time-varying Linear Rotor Systems using Floquet Theory

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1 7h IFoMM-Confeene on Roo Dynams Venna Ausa Sepembe 2006 Modal Analyss of Peodally me-vayng Lnea Roo Sysems usng Floque heoy Chong-Won Lee Dong-Ju an Seong-Wook ong Cene fo Nose and Vbaon Conol (NOVIC) Depamen of Mehanal Engneeng KAIS Sene own Daejeon Koea Sunaeosys Co Songwon- Nam-myeon Yeong-gun Chungnam Koea Shool of Mehanal Engneeng Kumoh Naonal Insue of ehnology 188 Shnpyung Kum Kyungbuk Koea ABSRAC Geneal oo sysems possess boh saonay and oang asymme popees whose equaon of moon s haaezed by he pesene of peodally me-vayng paamees wh he peod of half he oaon. hs pape employs he Floque heoy o develop he omplex modal analyss mehod fo peodally me-vayng lnea oo sysems. he appoah s based on deomposon of sae anson max leadng o he peodally me-vayng egensoluons. he modal analyss appoah usng he Floque heoy s hen ompaed wh he modulaed oodnae ansfom mehod whh ansfoms he fne ode me-vayng max equaon no an equvalen nfne ode me-nvaan lnea equaon. KEY WORDS geneal oo sysem Floque heoy ll s nfne ode max deonal fequeny esponse funon 1 INRODUCION Aodng o he mehanal popees of he oo and sao pas oo sysems may be lassfed no fou ypes [6 10]: soop (symme) oo sysem - boh oo and sao pas ae axsymme; ansoop oo sysem - he oo pa s axsymme bu he sao pa s no; asymme oo sysem - he sao pa s axsymme bu he oo s no; geneal oo sysem - nehe oo no sao pas ae axsymme. he geneal oo sysem an asymme oo wh ansoop sao hus eveals he oupled effes of ansoop and asymme oo sysems. he asymme (ansoop) oo sysem may look lke a peodally me-vayng lnea sysem when he equaon of moon s wen n he saonay (oang) oodnaes bu an be easly ansfomed no a menvaan lnea sysem by ewng he equaon of moon n he oang (saonay) oodnaes. hus he asymme (ansoop) oo sysem alone essenally edues o a me-nvaan lnea sysem whose modal analyss sheme s well esablshed n he leaue [ ]. On he ohe hand he geneal oo sysem whh s nheenly a peodally me-vayng lnea sysem an no be ansfomed no a me-nvaan sysem by suh a smple oodnae ansfomaon. I leads o a dffuly n modal analyss of he geneal oo sysem due o mahemaal omplexy assoaed wh eamen of peodally me-vayng sysem maes. Many aemps have been made fo dynam analyss of peodally me-vayng lnea sysems by employng he peodally me-vayng egenveos [ ]. Sne hese mehods wee sly based on he me doman analyss he sably analyss was of majo onen. Fo example Snha e al. [14] poposed use of he Lapunov-Floque ansfomaon fo expandng he peod sysem maes n ems of he shfed Chebyshev polynomals of a same peod so ha he ognal dffeenal poblem edues o a se of lnea 1 Pape ID-78

2 algeba equaons. oweve he mehod s sll lmed o enhanemen of he sably analyss. Calo and Wesel [2] appled he Floque heoy o develop a modal analyss mehod fo peodally me-vayng onol sysems nodung he peodally me-vayng egenveos deved fom peody of he sae anson max. Alhough he modal analyss mehod s mahemaally sound eques he auae negaon of he sae anson max ove a peod and laks he naual exenson o he fequeny doman analyss. hee ae few nvesgaons on omplee modal analyss of peodally me-vayng sysems vald n boh me and fequeny domans. he majo dffuly s beause he onvenonal modal analyss developed fo lnea me-nvaan sysems anno be dely applable o lnea me-vayng sysems unless hey an be ansfomed no an equvalen me-nvaan sysem [3 14]. Iee [7] developed a mahemaal foundaon fo modal esng of peodally me-vayng oo sysems by expandng he peodally me-vayng modal veos n Foue sees and nodung an nuve bu no goously poven elaon beween modal paamees. In addon alhough he esulng mahemaal eamens ae found o be oe nehe he ompuaonal poedue fo egensoluons no he fequeny doman analyss fo modal esng was desbed. Fo asymmeal oos wh soop saos he peodally me-vayng lnea dffeenal equaon expessed n he saonay oodnaes an be ansfomed o he me-nvaan lnea dffeenal equaon expessed n he oang oodnaes o n he modulaed saonay oodnaes [15]. hen he modal analyss beomes essenally he same as he odnay omplex modal analyss mehod developed fo ansoop oos whh possess asymme popees only n he sao pa [8 12]. On he ohe hand he asymme oo sysem wh ansoop sao anno be ansfomed o a fne ode equaon of moon wh he me-nvaan paamees by oodnae ansfomaon only. In hs pape he omplex modal analyss of peodally me-vayng lnea oo sysems s developed usng he Floque heoy and s ompuaonal effeny n alulaon of egensoluons s dsussed ompaed wh he use of he edued ode ll s max whh esuls fom he modulaed oodnae ansfom appoah. 2 COMPLEX MODAL ANALYSIS OF PERIODICALLY IME-VARYING ROOR SYSEMS 2.1 Equaon of moon n omplex fom Fo a oo sysem wh oang and saonay asymmey he equaon of moon an be onvenenly wen n he omplex saonay oodnaes efeng o Fg.1 as [ ] M (1) 2 () () () { () () ()} j Ω f p + Cf p + Kf p + Mbp + Cb p + Kbp + e { Mp() + Cp() + Kp ()} = g() ee he N 1 omplex esponse and foe veos p () and g() defned by he eal esponse veos y() and z() and he eal exaon veos f () and f () espevely ae y z p() = y() + jz() p() = y() jz() g() = f () + jf () g() = f () jf () (2) y z y z Dsk y Ω ξ Shaf x Beang η z Fgue 1: Geneal oo sysem: smple analyss model whee j means he magnay numbe; N s he dmenson of he omplex oodnae veo; g() nludes he foe and momen; Ω s he oaonal speed; - denoes he omplex onjugae; M C and K denoe he omplex valued N N genealzed mass dampng and sffness maes espevely; and he subsps f and b and efe o he mean value and he devao values fo ansoopy (saonay asymmey) and asymmey 2 Pape ID-78

3 (oang asymmey) espevely. Fo an soop oo C 0 ; fo an ansoop b = Kb = M = C = K = oo M 0; and fo an asymme oo 0. Noe hee ha he peodally me-vayng = C = K = Cb = Kb = ems whh ae peeded by j 2 e Ω n Eq. (1) nheenly appea as boh oang and saonay asymmees exs n he sysem and ha Eq. (1) nludes he exenal and nenal dampng gyosop momen and Cools effe. When ehe oang o saonay asymmey does no exs he equaon of moon beomes o an be ansfomed o a me-nvaan dffeenal equaon. In he followng seon he Floque heoy s appled fo omplex modal analyss of he peodally mevayng lnea oo sysem (1). And s ompuaonal effeny n alulang he unbalane esponse and he dfrfs s dsussed. 2.2 Complex modal soluon by Floque heoy (1) Egenvalue and adjon poblems whee Fom Eq. (1) and s omplex onjugae fom he omplex equaon of moon an be onsued as M() q () + C() q () + K() q() = f() (3) () { q = p () p () } { = } f() g () g () j2ω j2ω j2ω Mf M e M() = C C e b +C f C() = Kf K e b +K K() = j2ω j2ω j2ω Me Mf Cb + Ce Cf K b +Ke Kf Equaon (3) an be ewen n he sae spae fom as whee 0 M() A() = M() 0 B() = M() C() 0 K() A() w () =B() w () +F() (5) (4) q () 0 w () = F() =. (6) q () f () Ulzng he Floque heoy fo hs peodally me-vayng sysem n homogeneous pa of Eq. (5) wh he peod = π / Ω we an expess he 4N 1 omplex sae veo w() n ems of he sae anson max Φ() as [ ] w() = Φ(0) w 0 (7) whee (0) sasfes he dffeenal equaon subje o he nal ondon Φ(00) = I N N Φ 4 4 ( ) Φ (0)= A 1 () B() Φ(0) and he max deomposon elaon gven subje o R(0) = R( ) by [9 2] J 1 Φ (0)= R() e R (0). (9) ee J s he Jodan nomal fom of max whose dagonal enes µ = N ae emed Ponae exponens equvalen o he egenvalues fo me-nvaan sysems. Subsung = no Eq. (9) we oban J 1 Φ ( 0)= R(0) e R (0) o equvalenly [ ] (8) Φ( 0) ω IR(0)= 0. (10) I mples ha ω and ae he egenvalues (haaes mulples) and he oespondng max = e µ R(0) of egenveos espevely of he monodomy max Φ( 0). Subsung Eq. (9) no Eq. (8) we oban R 1 () = A () B () R () R 1 () J o equvalenly () = A () B() µ I () (11) whee () s a olumn veo of R(). Now we an onsu he adjon poblem nodung he adjon sae veo z () o he ognal sysem (5) wh F() = 0 as [2 9] 3 Pape ID-78

4 fom whh we an defne he adjon max L () suh ha L A B L L 1 () = () () () + () whee l () s a olumn veo of L (). We an ewe Eq. (11) usng he deny elaon z ()= A 1 () B() z () (12) J o equvalenly 1 { } l () = A () B() µ () I l 1 R() R ()= I as (13) R () = R () A () B () + JR 1 (). (14) Fom de ompason of Eq. (13) wh Eq. (14) we an oban he bohonomaly ondon as [2] whee j L () R() = I o equvalenly 4N 4N l () () ; j = 1 o 4N (15) j = δ j δ s he Koneke dela he supesp means he anspose and () and l () ae he -h olumn veo of R () and L() espevely. (2) Suue of he egenveos and he adjon veos Subsung he elaon { } homogeneous pa of Eq. (5) we oban he elaon gven by w () = q () q () = () η() wh q() u () η() and Eq. (11) no he { + µ } = () = u () u () u (). (16) Lkewse subsung he elaon z() = l () ζ () wh l () =A () l () and Eq. (13) no he adjon equaon (12) we oban he elaon gven by 1 { µ } v () () () + () () M M v l =. (17) v () ee he modal and he adjon veos ae omposed espevely of () { u () ˆ () } = u u { ˆ } v () = v () v () (18) and fo he omplex equaon of moon as n Eq.(1) holds n geneal uˆ( ) u () vˆ () v (). (19) Noe ha he egenveo ( ) (and hus u ( )) and he adjon veo l( ) (and hus v( )) ae peodally mevayng veos wh he peod = π / Ω. Fo he me-nvaan sysem.e. () = l () = A l A() = A and B() = B Eqs. (11) and (13) and Eqs. (16) and (17) edue o he fom of µ A = B µ A l = B l (20) and { µ } = u u = { µ v } v espevely whh ae onssen wh he pevous esuls n [11]. l u = { u u ˆ } { ˆ } v = v v (21) 4 Pape ID-78

5 (3) Modal equaons and egensoluons he omplex sae and adjon veos w() and z() an be expanded n ems of he egenveos and he adjon veos espevely fo he oo sysem (5) as 4N N w () = { () η() } = '{ () η() } 4N { } N = ζ = { = 1 =BF =N z () l () () ' l () ζ () } (22) = 1 = BF =N whee he pme noaon n he summaon mples exluson of = 0 η () and ζ () ae he pnpal oodnaes of he ognal and adjon sysems espevely and he supesps B and F efe o he bakwad and fowad modes espevely followng he well-esablshed onvenon fo mode lassfaon n oodynams [11]. Subsung Eq. (22) no Eq. (5) usng he elaon (11) pemulplyng by l k and usng s he bohonomaly ondon (15) we an oban he 4N ses of omplex modal equaons of moon as η () = µ η () + v () f() = µ η () + v () g()+ vˆ () g() ; = ±1 ±2 ±N ; = B F. (23) Reallng he Floque heoy ha fom he one peod soluon he ene me esponse of he egensoluons an be expessed peodally wh he base of ha peod we an expand he egenveo u () and he adjon veo () n Eq. (18) by Foue sees as [7] v j2mω j2mω ˆ ˆ = ( m) e u = ( m) e m= m= u () u () u v j 2 mω j () ˆ ˆ 2 mω = v ( m) e v () = v ( m) e (24) whee u ae he omplex Foue oeffen veos assoaed wh he omplex ( ) ˆ m u ( m) v ˆ ( m) and v( m) j2m hamon funon of e Ω. Fom Eqs. (23) and (24) we an oban he foed esponse of he geneal oo sysem (5) as N m= m= N ( 2 )( ) { } { ˆ 0 ( ) ( ) ; ( ) ( ) ; } + j mω η m m n n τ m mn n p ( ) = ' u ( ) ( ) = ' e µ τ u v g ( ) + u v g ( τ ) d τ (25) =BF = N =BF =-N m= n= whee g = g. 2 ; () () e j n Ω n (4) De numeal soluon mehod In hs de modal analyss appoah fo he peodally me-vayng paamee sysem (1) he egenvalues and he oespondng peodally me-vayng egenveos an be analyally obaned fom Eqs. (8) o (15) a leas n heoy. oweve he losed fom soluons ae lmed only o a few smple ases beause of mahemaal omplexy. Fo mos of paal applaons numeal appoah s nsead aken as follows. Fs he monodomy max Φ( 0) s obaned by numeal negaon of Eq. (8) wh espe o me fo gven A() B() and nal ondon Φ(00) = I. Seond he haaes mulples ω 4N 4N and he oespondng max of egenveos R(0) of Φ ( 0) ae alulaed fom Eq.(10). hen he Jodan nomal µ = log ω / and R() an be solved by numeal fom of max J s fomed wh s dagonal enes ( ) negaon of Eq. (11) wh he nal ondon R(0). he same poedue apples o he adjon max L() usng R() and he bohonomaly ondon (15). Noe ha one he peodally me-vayng modal (adjon) veos ae obaned alulaon of he Foue oeffen modal (adjon) veos whh ae onsan veos n Eq. (24) beomes saghfowad. Alhough he above poedue looks lke a novel analyal appoah one of s al dawbaks s he numeal nsably sne suffes fom seous aumulaed eo wh exensve numeal negaon poesses [5]. Fo example he omplex Foue oeffen modal (adjon) veos ae vey vulneable o he numeal eos wh R() and L(). 5 Pape ID-78

6 (5) Infne ode deonal fequeny esponse max (dfrm) Foue ansfomng Eq. (25) we oban P u v u vˆ N N ( m) ( mn) ( m) ( mn) ( jω) = ' ( ω) ' ˆ G;n j + G; -n( jω) n= m= = BF = N j( ω 2 mω) µ m= = BF =N j( ω 2 mω) µ n= { ˆ g ( jω) ;n( jω) ( jω) ; ;-n( jω) ;-n p G g-n p G } = + (26a) whee P ( jω) G ( jω) and G ˆ ( jω) ae he Foue ansfoms of p () g () and g () espevely and holds { ω } Gˆ ( jω) = ˆ { j ( ω + 2 nω ) } G; n( jω) = G j( 2 nω) ; n G. (26b) Alhough hee ae sll an nfne numbe of dfrms n Eq. (26) we nodue fou dfrms ha ae mpoan n haaezng he sysem asymmey and ansoopy as N um ( ) v m ( ) ( jω) = g ( ) ' ;0p jω = m= = B F =N j( ω 2 mω) µ N um ( ) v ( m) ˆ ( jω) = ( jω) = ' ;0 (27) g p m= = B F =N j( ω 2 mω) µ N um ( ) v ( m+ 1) ( jω) = ( jω) = ' m= = B F =N j( ω 2 mω) µ g; 1p N um ( ) v m ( 1) ( jω) = ( jω) = ' m= = B F =N j( ω 2 mω) µ g; 1p ee ( jω) s efeed o as he nomal dfrm ha epesens he sysem symmey ˆ ( jω) s efeed o as he evese dfrm ha epesens he effe of sysem ansoopy and ( jω) and ( jω) ae he modulaed dfrms ha epesen he effe of sysem asymmey and he oupled effe of sysem ansoopy and asymmey espevely. 2.3 Complex modal soluon by oodnae ansfomaon I has been poven ha noduon of omplex modulaed oodnaes suessfully ansfoms he fne ode me-vayng max equaon no an equvalen nfne ode me-nvaan lnea equaon leadng o an nfne se of onsan egensoluons [6]. hen he modal analyss of he me-nvaan lnea sysem beomes saghfowad defnng he ll s nfne ode max. I an be easly shown ha he modulaed oodnae ansfom appoah leads o he expessons fo he dfrms denal o Eqs.(26) and (27). oweve he numeal poedues fo he dfrm esmaes ae dffeen fom eah ohe. In paula he unaon shemes of he nfne sees expanson (o equvalenly he nfne summaon) wh espe o he ndex m ae dffeenen. Fo example unaon of he nfne summaon s done wh he omplex Foue sees expanson of he peodally me-vayng egenveos Eq.(24) fo he Floque appoah wheeas he ode eduon s done wh he ll s nfne ode max fo he oodnae ansfom appoah [6]. 3 NUMERICAL EXAMPLE hs seon demonsaes and ompaes he modal analyss mehods wh a smple ye geneal oo sysem model whh onsss of a gd oo wh asymme mass momens of nea a massless shaf wh asymme shaf sffnesses and wo ohoop beangs as shown n Fg. 1. he dealed despons of he oo model ae eaed n [6 12]. he egensoluons of he analyss oo model wee alulaed by usng he Floque appoah wh he hee em appoxmaon (he ndex m was aken o be -1 0 and 1 n Eq. (24)) fo he omplex Foue sees expanson of he me-vayng egenveos. he alulaed egenvalues wee n good ageemen wh hose fom he edued ll s max of ode 24 whh was found o yeld faly auae esuls [6]. Fgue 2 ompaes he ypal unbalane esponses of he oo alulaed by fou dffeen mehods. Noe ha boh he edued ll s max of ode 24 and he Floque s mehod wh hee em appoxmaon yelded non-dsngushable esuls fom he exa numeal one. Boh mehods ae found o be supeo n alulaon of unbalane esponse o he onvenonal hamon balane mehod [4] as shown n Fg Pape ID-78

7 Z (mm) Y (mm) Fgue 2: Unbalane esponse a 4000 pm ( δ = = 0.3): Numeal negaon; ll s max of ode 24; Floque s mehod; amon balane mehod Fgues 3 and 4 ompae he waefall plos fo he magnude of he fou mpoan dfrfs (gven n Eq.(27)) obaned by he Floque mehod usng he hee em appoxmaon and he edued ll s max of ode 24 espevely as he oaonal speed s vaed up o pm fo he analyss oo model wh δ = = 0.3. he numbe of assumed modes used fo alulaon of dfrfs s kep unhanged fo he lae mehod bu vaes fo he fome mehod dependng upon he ype of dfrfs due o he nheen naue of appoxmaon. I ofen leads o elavely lage dsepanes n he logahmally saled dfrf esmaes obaned by wo mehods. he dfrfs shown n Fgs. 3(a) and 4(a) ae almos denal bu ohe ypes of dfrfs show some dsepanes wh he ode of magnude less by 5 o 6 han he domnan peak values n he nomal dfrfs whh oesponds pehaps o he measuemen nose level n pae. he oveall ends of he dfrfs paulaly he esonan peaks ae no muh dffeen fo boh mehods. Some dsepanes ae howeve noable a he an-esonan egons and nea he unsable egons. In geneal he numbe of assumed modes used n he Floque mehod s less han o equal a bes o he oodnae ansfom mehod. hus an be onluded ha he oodnae ansfom mehod s supeo n esmaon of dfrfs han he Floque mehod. Noe ha he oodnae ansfom mehod whh s essenally a fequeny doman appoah sueeds n appoxmang he dfrfs wh a lmed numbe of assumed modes (Rz veos) wheeas he Floque mehod whh s essenally a me doman appoah fals n usng an effeve se of base hamons equed o bee esmae he dfrfs n he fequeny doman. heoeally speakng as he numbe of assumed modes neases ndefnely boh mehods wll evenually lead o he denal esuls. Alhough hee exs some dsepanes n he logahm magnudes of dfrfs he ypal esponse alulaons n he me doman by boh mehods yeld lle dffeene as demonsaed pevously n Fg.2. 4 CONCLUSION he omplex modal analyss by he Floque heoy s developed fo peodally me-vayng lnea oo sysems and ompaed wh he oodnae ansfom appoah. he Floque mehod povdes lea physal undesandng of he egenvalues and he oespondng egenveos. I s found ha he Floque mehod wh a few appoxmaon ems esmaes he egenvalues and he egenveos of lowe ode wh fa auay leadng o sasfaoy esponse alulaons n he me doman. oweve s no effen n alulang he egenveos of hghe ode and hus he fequeny doman haaess nludng he deonal fequeny esponse funons ompaed wh he oodnae ansfom appoah. 7 Pape ID-78

8 (a) (b) () (d) Fgue 3: Waefall plos of dfrfs by he Floque mehod: δ = = 0.3 (a) (b) g0 p( jω) ĝ;0 p( j ω) () ĝ; 1p ( j ω) (d) g; 1p( jω) (a) (b) () (d) Fgue 4: Waefall plos of dfrfs fom he ll s max of ode 24: δ = = 0.3 (a) (b) g0 p( jω) ĝ;0 p( j ω) () ĝ; 1p ( j ω) (d) g; 1p( jω) 8 Pape ID-78

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