A REAL-SPACE MODAL ANALYSIS METHOD FOR NON- PROPORTIONAL DAMPED STRUCTURES

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1 ECCOMAS Congess 6 VII Euopean Congess on Computatonal Methods n Appled Scences and Engneeng M. Papadakaks, V. Papadopoulos, G. Stefanou, V. Plevs eds. Cete Island, Geece, 5 June 6 A REAL-SPACE MODAL ANALYSIS METHOD FOR NON- PROPORTIONAL DAMPED STRUCTURES E. Stanoev Cha of Wnd Enegy Technology, Faculty of Mechancal Engneeng and Mane Technology, Unvesty of Rostock, Albet-Ensten-St., 859 Rostock, Gemany e-mal: evguen.stanoev@un-ostock.de Keywods: modal decomposton of the equatons of moton, complex egenvalue poblem, popotonal/non-popotonal dampng, synchonous fee vbatons Abstact. The ncluson of dampng n the equatons of moton of FEM-based stuctual models yelds a complex quadatc egenvalue poblem. In ths pape s pesented a vaant of a geneal method [4], [5] fo eal-space modal tansfomaton of damped mult-degee-offeedom-systems MDOFS wth non-modal non-popotonal symmetc dampng matx. The method s based on the conugated complex ght egenvectos of the system, nomalzed elatve to the geneal mass matx. Afte state-space fomulaton of the equatons of moton a eal modal tansfomaton matx s bult by a combnaton of two complex tansfomatons, whch s the man advantage of the pesented method. Analytcally expessons fo the modal tansfomaton bass ae developed be the ad of compute algeba softwae MATLAB. Applyng the suggested method to the specal case of popotonally damped system, an analytcal expesson fo the constant phase lag of the fee vbaton modes has been deved. The conveson of the developed geneal eal tansfomaton matx nto the modal matx of the undamped poblem s analytcally poved by takng nto account the synchonous fee oscllatons n ths specal case. The deved fomulas fo the modal tansfomaton bass contan the eal and the magnay pats of the egenvectos and the assocated egenvalues. A numecal example vbaton of a oto blade of a wnd tubne - demonstates the pefomance of the pesented modal decomposton method fo the geneal case of nonpopotonal damped system. The dampng matx of ths example contans stuctual and aeodynamc dampng. The ntal computaton of the complex egensoluton of the FEM beam model n the pesented example and all subsequent computatons ae done by the ad of the Symbolc Math Toolbox of MATLAB. The suggested pocedue can be appled n stuctual systems contanng dffeent dampng and enegy-loss mechansm n vaous pats of the stuctue and also n stuctue-envonment nteacton poblems, whee a non-modal dampng matx s occung.

2 INTRODUCTION The modal decomposton of the equatons of moton of mult-degee-of-feedom-systems MDOFS s usually appled to systems wthout dampng. The assocated egenvalue poblem has eal egenvectos and eal fee fequences. The ncluson of dampng n the equatons of MDOFS leads to a quadatc egenvalue poblem wth complex conugate pas of egenvalues and egenmodes. The modal decomposton of the equatons has to be pefomed n complex space. Amng to avod the computaton n complex athmetc, a new modal decomposton method, pesented n detals n [] [5], s befly outlned n Sec.. Ths pocedue s based on a eal modal tansfomaton matx, deved fom the complex egenvalue soluton of a MDOFS wth symmetc non-popotonal non-modal dampng matx. In the suggested pocedue the complex egenvectos and egenvalues of the stuctual model should be computed fst. In the pesented example n Sec. 4 vbaton of a oto blade of a wnd geneato - compute algeba softwae was appled to solve the egenvalue poblem. In eal lfe applcatons of the pesented method to hgh dmensonal poblems t must be avalable a elable egenmode solve fo lage complex egenvalue computatons. Thee ae many lteatue efeences fo lage scaled poblems wth vaous soluton stateges, see [] []. The autho has used an mplctly estated Anold/Lanczos method [], [] to solve the complex egenvalue poblem n an applcaton of the method to a flud-stuctuefoundaton nteacton poblem, see n [],[]. Anothe topc of ths pape s to show an analytcal poof of the statement fo the constant phase lag/lead of fee vbatons n the popotonal dampng case see the ntoducton n Sec... The analytcal poof n an ndect manne s based on the pocedue, summazed n Sec.. A fomula fo computng of the constant ato ImX k ReX k has been deved n Sec. 3.. In Sec. 4 the poposed modal analyss method, pesented n Secton, has been appled to a oto blade beam stuctue wth 54 DOF. The numecal example demonstates the pefomance of the method fo the geneal case of non-popotonal dampng. In ths case the dampng matx of the system contans a stffness popotonal Raylegh dampng and aeodynamc non-popotonal dampng pats. In the second vaant of the soluton wth popotonal dampng matx, the fomula fo the constant phase of the esonance modes s vefed numecally.. Fee vbatons of a vscously damped system The equatons of moton of a damped MDOFS ae MV + DV + KV = pt. whee M, D and K ae, espectvely the n x n mass, dampng and stffness matces, and V, V ae the n x dsplacement and velocty vectos and pt s the n x exctaton vecto. In stuctual mechancs poblems we consde the M and K matces to be eal, symmetc and postve defnte, excludng the pesence of gd body modes. The D matx s assumed to be symmetc, non-negatve, she pesents a non-popotonal dampng.

3 Wth the assumed fee vbaton n the fom t t V Xe, V Xe,. the assocated quadatc egenvalue poblem s λ M + λ D + K X = =,, n.3 In Eq..3 the th egenvalue λ and the coespondng egenmode X appea n complex conugate pas ndex omtted: λ = λ + λ, λ = λ λ.4a X = X + X, X = X X.4b The dynamc equlbum of a vscously damped sngle oscllato s govened by mv t + cv t + kvt = qt esp..5a v t + ηωv t + ω vt = pt.5b whee v s acceleaton, v - velocty, ω = k m η = c mω - fee vbaton fequency, - Leh s dampng ato and pt = qt m. The exponental soluton x e λt, ntoduced nto the homogenous fom of the dffeental equaton.5b, yelds the egenvalue poblem λ + ηω λ + ω =.6 The egenvalue soluton assumng that η, subctcal dampng of Eq..6 s a complex conugate pa: /.7. The constant phase lag poblem D Intepetng the egenvalue pa.4a as the sngle-oscllato-egenvalues.7, we can expess the th fee vbaton of the MDOFS as lnea combnaton of the two complex conugate egenpas.4a,b: V = X e λt = X e ηω±ω η t = = e ηωt [X + X cos ω D t + sn ω D t + X X cos ω D t sn ω D t] = e ηωt X [ cos ω D t X sn ω D t + X cos ω D t + X sn ω D t X cos ω D t + X sn ω D t ]

4 = e ηωt [ X F cos φ cos ω D t X sn ω D t].8 F sn φ The last elaton leads to the eal fom of a damped fee oscllaton fo evey k th DOF: V k = e ηωt [F k cosω D t + φ k ].9 whee φ k = actan X k X k : phase lag/lead fo the k th DOF. Snce the vscous dampng s assumed to be non-popotonal, the fee vbaton soluton.9 epesents non-synchonous damped oscllaton.e. the phase φ k s dffeent fo each DOF. In the case of popotonally damped system we have to deal wth synchonous fee oscllaton.e. the phase φ k s constant the same fo all DOF, fo undamped systems φ k s zeo see [6], [7], p.8. The featues, showed n Eq..9,. ae well known and used n modal analyss, see fo example [6]. In the pesent pape the statement of synchonous fee oscllatons n the popotonal dampng case should be poved analytcally n Sec. 3.. MODAL DECOMPOSITION METHOD INCLUDING THE COMPLEX RIGHT EIGENVECTORS. The sngle mass oscllato The equaton of moton of a damped sngle degee of feedom system SDOFS.5b can be wtten n the fom t t t w t η ω w t p ω v ω v m mq kq p q k q p t p whee the velocty s.a q m k q m.b a w v. t t The exponental soluton q xe, q xe, ntoduced nto the homogenous fom of the dffeental equaton., gves the quadatc egenvalue poblem η ω ω ω m k x m kx.3 The two complex conugate egenvalues, subctcal damped system ae:

5 ω /.4 whee,, D.5 The two coespondng complex conugate egenvectos φ /, at fst nomalzed elatve to the mass matx,, k ω k T k k k mx x x.6 ae combned nto a modal matx:.7 Due to nomalzaton Eq..6 the othogonalty elatonshps can be deved: ω ω T T T m.8 ω ω ω ω T T T k.9 The nvese of the complex modal matx, can be expessed analytcally usng compute algeba softwae: Q P Q P. whee.a Q P.b. The damped mult-degee-of-feedom-system The equatons of moton. of damped MDOFS n DOF wll be wtten n the statespace fom: P Q K Q M p V W K K D V W K M G G t ; W V.a P Q K Q M G G,.b

6 whee M G and K G ae, espectvely the n x n symmetc genealzed mass and the genealzed stffness matces. The symmetc dampng matx D s non-negatve and epesents a non-popotonal dampng. The exponental soluton., substtuted nto the homogenous fom of Eq.., leads to the n-dmensonal egenvalue poblem Χ MG KG.3 Χ The soluton of Eq..3 s gven by n complex conugate egenpas.4, now wtten n the fom: Χ Χ ; ;, n.4 Χ Χ Each th egenvecto-pa mass matx M : G X, Χ s nomalzed ndex omtted elatve to the geneal T X Χ Χ, A B MG.5a A B Χ Χ T Χ Χ Χ, A B MG.5b A B Χ Χ Subect to the nomalzaton.5 follow the othogonalty elatonshps expessed n tems of the th egenvecto-pa ndex omtted: T T M K M G D K K K G.6.7 The n x n complex squae modal matx, denoted by G, s made up of the n egenvecto-pas, see Eqn..5: n n n n G.8 n n The othogonalty popetes see Eq..6,.7, ae used to pefom a modal decomposton of the equatons of moton.: T M T D K T pt G A G G G A G.9 K K E dag whee

7 W V A G G a b a n b n T. s a coodnate tansfomaton, and a b Intoducng eal modal coodnates a, b ae new complex vaables. x, y fo each th egenpa,.e.: x,. y the dffeental equatons.9 can be tansfomed n pas nto the eal fom of SDOFSequaton ndex omtted, egadng Eqs..8,.9 and usng.5: T ω x y x y T ω ω x y x y T T p t T p t g h t t. the matx, Fo each egenvalue pa SDOFS can be computed by Eqs..4,.5,.,...3 The eal modal tansfomaton bass of the coespondng Usng both tansfomatons.9 and., the equatons of moton. wll be uncoupled nto n eal SDOFS block equatons as follows: [ Y T [ M ω K ] Y ω n ] x y + Y T D K [ ] Y K x n η [ y n] ω ω X [ ω η n ω n ω n ω n ] x y x n [ y n ] X = Y T [ p ] g h g n [ h n ].3 The new n x n tansfomaton bass Υ s defned by combnaton of two complex tansfomatons.9,.: x W y G G Ψ X Υ X V.4 n Υ xn Ψ yn X It can be shown that the Υ -matx n Eq..4 and all load -vectos [gt ht] T, see Eq.., ae puely eal. Afte component multplcaton of the analytcally expessed tems

8 of G and of Ψ all magnay pats cancel each othe, see detals n [4]. Ths s befly sketched below by developng the two columns of Υ, belongng to the th egenvecto-pa: G Ψ V y V x W y W x Y Y Y Y.5 Wth egad to Eq..5,.,. the multplcaton Q P Q P.6 leads to puely eal components of the two columns of the tansfomaton bass Υ :.7a-d In the same manne we develop the load -vecto n Eq.. [ gt ht ] = η [ + P Q P + Q ] [λ + λ T + T λ λ T T ] pt.8 gt = ω η { η + η T + η η T } pt.9a ht = ω η { T T } pt.9b Each th SDOFS block equaton n.3 can be easly solved, elmnatng the modal coodnate x to obtan the usual fom of the SDOFS equaton of moton ndex omtted: t h y x.3a t h t h t g y y y.3b V y V x W y W x Y Y Y Y D D D D

9 The dynamc esponse y t can be obtaned by step-by-step ntegaton, appled to Eqs..3b,a. The fnal tme esponse of the ognal n DOFs s calculated by supeposton of the modal coodnates x, y n accodance to Eq..4. The mao advantage of the suggested method s the developed new modal tansfomaton matx Υ, see.4, n eal space fo damped MDOFS wth symmetc non-dagonalsable.e. non-modal dampng matx. The method has also the usual modal supeposton advantage - an uncompleted tansfomaton employng only a few modes k<<n n the Y -bass leads wth suffcent numecal accuacy - afte the fnal back coodnate tansfomaton - to the dynamc esponse of all n DOF. 3 THE PROPORTIONAL DAMPED SYSTEM 3. Modal tansfomaton of the equatons of moton A smple method to constuct a dampng matx D p, pesentng a popotonal dampng, s the Raylegh dampng assumpton: D p = α M + β K 3.a whee α, β : unknown weghtng paamete, see Eq.3.7,3.8a,b The modal dampng matx s a patcula case of a moe geneal popotonal dampng assumpton, see [7] p.5, n the fom: n D p = k= a k M M K k 3.b The matx 3.b tuns fo n = to D p = a M + a K, whch s the Raylegh appoach 3.a. The egenvalue poblem λ M + K X =, 3. coespondng to the equatons of moton of MDOFS wthout dampng MV + KV = pt, 3.3 has the soluton: λ = ω wth the fee fequency ω X, =,, n eal egenvectos The modal matx U, belongng to 3. U = [U U U n ] 3.4 compses n eal, mass nomalzed egenvectos X U = =,,, n. 3.5 T X M X

10 The mass nomalsaton 3.5 leads to the othogonalty elatonshps U T MU = [ ] 3.6a U T KU = [ ω ω ω n ] = Ω 3.6b The Raylegh dampng matx D p can be dagonalzed applyng 3.6a,b to 3.a: U T D p U = [ α + βω α + βω n ] = [ η ω η n ω n ] 3.7 The geneal fom of the D p matx 3.b can also be tansfomed n dagonal fom by use of the eal modal matx U, as shown n [7] p. 5. The tems n the man dagonal of the ght sde of 3.7 ae set to be equal to the modal dampng tem ηω of the equaton of moton fo SDOFS.. The two unknown paamete α and β can be calculated by solvng a system of two equatons α + βω = η ω, =,, usng the fst two lowest fee fequences ω and by abtay choose of two appopate dampng atos η : α = ω ω ω η ω η β = ω ω ω η ω η ω ω 3.8a,b Typcally fo the vscous dampng s evdently the fequency elated dampng paametes. Below the paamete α and β ae set to be known. In the geneal case 3.b the unknown coeffcents a k ae to be detemned by solvng a system of n lnea equaton, see [7] fo detals. We consde now the popotonally damped system. wth D = D p. Assumng the soluton., the assocated quadatc egenvalue poblem.3 gves λ M + λ α M + β K D p + K = 3.9 The mass nomalzed egenvectos n Eq. 3.9 ae geneally complex conugate, see Eq But the egenvalue poblem 3.9 possess also classcal eal egenmodes, dentcal to the egenmodes U, belongng to the egenvalue poblem wthout dampng, see Eq Usng the U egenmodes nstead of, the egenvalue poblem 3.9 can be tansfomed, wth egad to 3.6, 3.7, to

11 {λ M + λ α + βω M D p M + ω K } U = 3.a The coespondng complex egenvalue λ s then computed fom λ + λ α + βω + ω = λ, = α + βω λ, = η ω λ η ω ± ω η λ ± 4 α + βω ω = 3.b By compang 3.b to.7 s evdently, that the fee fequency ω, computed accodng to.4,.5 ω = λ + λ = {ηω + ω η } 3.c by use of the th conugate complex egenvalues λ of the popotonal damped system, s dentc to the fee fequency ω of the coespondng system wthout dampng. Wth the egenvalue λ, Eq.3.b, the elatonshp 3.a poves that U s a egenvecto of the popotonal damped system 3.9. In the consdeed case the equatons of moton. can be tansfomed n modal space Eq. 3., wth egad to 3.6, 3.7: U T MU y + U T D p U y + U T KU y = U T pt, 3. In 3. the modal supeposton of the ognal DOF s supposed by use of the classcal modal matx U of the undamped poblem, see Eq.3.4, : y y V = [U U U n ] [ ] = U y 3.3 y n In ode to tansfom the state space fom of the equatons of moton. we constuct a n xn tansfomaton matx Y U by the mass nomalzed egenvectos U 3.4, 3.5 n the fom Y U = [ U U U U U n ] 3.4 U n Eq.. can be tansfomed nto n uncoupled eal SDOFS block equatons by the ad of Y U, wth egad to Eqs. 3.6, 3.7:

12 Y T U [ M ω [ K ] Y U ω n ] x y + Y T U [ D p K ] Y U K x n η [ y n] ω ω [ ω η n ω n ω n ω n ] x y x n [ y n ] = Y T U [ p ] g g n [ ] 3.5 whee the modal velocty s x = y 3.6 Equaton 3.5 s anothe fom of the modal decomposton 3.. Note the dffeence of Eq. 3.6 fom Eq..3a n the geneal case of non-popotonal dampng, see futhe Eq We compae now the tansfomed equatons 3.5 fo the case of popotonal dampng wth the geneal fom.3 whee D = D p. Fo both of the compaed SDOFS block equatons to be dentc, t s evdently that each load tem h fom.3 must be equal to zeo, see.9b: ω h = { η T T } p = 3.7 All tems n Eq. 3.7 exclusve of p belong to the consdeed th egenmode. Thus, wth egad to.a,.5 T T = k = η + η = = k η η η η = const. 3.8 fo all k th DOF of the th egenmode pa ± wth coespondng egenvalue λ ± λ. Equaton 3.8 poves the statement of a constant phase lag/lead, see Eq..,.e. n the case of popotonally damped system each fee vbaton s a synchonous moton of all DOF. 3. The tansfomaton matx Y The modal equatons 3.5 demonstate, that fo the nvestgated case of popotonal dampng the modal tansfomaton matx Y U, Eq. 3.4, must be dentcal to the matx Y, Eq..3,.4, deved fo the case D = D p. By compang the two columns of Y, see Eq..7b,c, to the coespondng zeo-columns of Y U, t follows Y W y = η ω ω = Y V x = η + = 3.9a 3.9b

13 .5.5 m 8 x.5 m E. Stanoev + = 3. The elatonshp 3. leads agan to k k = = const. Eq.3.8 fo all k th DOF of the consdeed egenmode. Thus the modal tansfomaton matx Y, see Eq..5 and 3.4, has n ths case the fom Y W x Y = [ V Y y ] = [ U U ] 3. 4 NUMERICAL EXAMPLE 4. Stuctual system, stffness and geomety data x 4 3 z y Fg. Roto blade beam model subected to wnd loads The stffness data of the blade coss sectons have been calculated n [4]. The genec aeodynamc blade geomety has been deved fom eal blade data. Below ae gven fo nstance the stffness data, efeed to the ogn of the coodnate system of the coss secton, at the dstance of. m fom the blade oot see Fg. : Cente of mass F.4, -.9 [m] dstbuted mass [kg/m] EA = [N] axal stffness EA y = [Nm] EA z = [Nm] EA yy = [Nm ] EA zz = 644. [Nm ] EA yz = 846. [Nm ] GI T = [Nm ] tosonal stffness

14 The fnte element soluton s based on the numecal ntegaton of the system of dffeental equatons fo the Benoull-beam. The efeence axs of the beam model concdes wth the cente of the ccula-secton at the oot t s the eal otatonal axs of the oto blade. Theeby the dffeental equatons and all coss secton stffness data ae efeed to ths axs, accountng fo the eccentc mass applcaton. z y Fg. Roto blade sectons at. m thn wall coss secton model 4. Wnd loads The wnd loads ae calculated accodng to the fomula fo the aeodynamc lft foce pe unt length of an aeofol, see [3] p.59: L = ρ c W C L 4. whee: W : a velocty elatve to the aeofol ρ : a densty =.5 [kg/m 3 ] c : chod of the aeofol : lft coeffcent C L = π α = π π C L 8 the flow angle α s assumed to be 6. [deg] 6. =.658, The a velocty W s the vecto sum of the otatonal speed Ω wth assumed 6 pm and the wnd speed u, ncdent on the aeofol n accodance wth the Betz-theoy: W = Ω + 3 u whee Ω = 6 π n [ad/s] 4. The wnd speed functon s assumed to be 3 ut = snπft, whee f = [Hz] 4.3 The esultng wnd thust loads pe unt length along the x-axs of the oto blade ae gven below. In the stuctual model the wnd thust loads ae actng as summazed nodal foces.

15 c wnd thust Ft [m] [m] [N/m] The wnd thust functons Ft ae actng on the oto blade as shown n Fg. 3 fo sec. Ft =.4 [5.333 snπft ] Ft Fg. 3 Wnd thust functon at.5 m 4.3 Relatonshps and data fo the dampng appoach Statng pont of the computaton ae the equatons of moton M W D K W Pt K V K V 4.4 whee P t s the nodal foce vecto, epesentng the wnd thust accodng to Sec. 4.. The system equatons 4.4 wll be solved applyng the poposed modal analyss method n Sec. fo two cases: non-popotonal and popotonal dampng. The lowest fou fee-vbaton fequences and assocated peods fo the undamped system ae calculated to

16 f =.643 [s ] f = 4.6 [s ] f 3 = 7.94 [s ] f 4 = 6.65 [s ] T =.378 [s] T =.6 [s] T 3 =.6 [s] T 4 =.6 [s] 4.5 Assumng stffness popotonal dampng n accodance wth Eq.3., the dampng system matx s D p = β K 4.6 Wth an assumed dampng ato η =.8, see [3] p.49, fo the fst natual peod T, we obtan wth egad to Eq.3.8a,b β = η = η T =.964[s] 4.7 ω π The non-popotonal symmetc dampng matx D np s buld addng to the D p -matx a new matx D a, whch epesents the aeodynamc dampng. The fomulaton s based on a smple expesson fo the aeodynamc dampng coeffcent pe unt length c, gven n [3], p. 47: c = ρ Ω c dc L dα [ kg s ], whee dc L = π 4.8 m dα Wth Eq. 4., 4., the coespondng dampng coeffcents along the x-axs of the oto blade ae calculated to c c [m] [m] [kg/s.m] The coeffcents c, whch epesent the aeodynamc dampng, ae actve fo vbaton n z-decton of the coss-secton coodnate system, see Fg.. The assocate symmetc dampng matx fo the Benoull-beam element s deved by analogy wth the method used to deve the fnte element mass matx, see [5]. Fnally the symmetc system dampng matx, D np, s assembled n a fnte-element manne, ncludng stuctual popotonal and aeodynamc dampng: D np = D p + D a Non-popotonal damped system We use hee the matx D np Eq.4.9. The vecto of the fst ten complex conugate egenvalue pas of the matx A M K G, see Eq..3, s G

17 4. The numbe of modes consdeed n the modal tansfomaton s lmted to the fst fou egenvecto pas ths ae n ascendng ode the #3,, 4, 5 of the vecto n 4.. The stuctual system has n Fg. has 54 DOF. The coespondng 8x8 modal matx G wth mass nomalzed egenvectos Eq..8, s computed to only the fst ten ows ae pnted 4. The matx Ψ s now calculated n the case of fou nvolved egenmodes accodng to Eq..4: Ψ = φ φ φ 3 φ 4 ] = [

18 4. Fnally the 8x8 eal tansfomaton matx Y s computed accodng to.4 hee only the fst ten ows: Y = 4.3 Afte the modal tansfomaton n egad to.3 the tme-dependent load vecto hee fo the tme 5 sec s calculated to be, see also Fg. 3, g t h t = Y T [ p ] = g 3 t [ h 3 t] 4.4 The esultant fou uncoupled SDOFS block equatons fom type of Eq..3, pepaed n the fom.3a,b, ae solved by step-by-step ntegaton: x η ω ω x g ω y ω y h + =, n = 4 x n η n ω n ω n x n g n [ ω n ] [ y n] [ ω n ] [ y n ] [ h n ] whee [ω ] = X X 4.5 [η ] = 4.6a,b

19 The effect of the mpled addtonal aeodynamc dampng esults evdently n the lage dampng ato η =.3369 fo the fst fee vbaton. The vbaton-esponse has been detemned n the tme s, the tme step length fo the appled Newmak ntegaton method s.5 s. The tme esponse of the modal coodnates fgue 4 fo the tme 5 sec: y t,,...4, ae shown n the followng Fg. 4 Tme esponse of the modal coodnates y t fo the case non-popotonal dampng By a back tansfomaton accodng to Eq..4 the total esponse Fgs. 5a-c: V t s obtaned - see Fg.5a Total vbaton t m u at the oto blade tp y-decton at node #

20 Fg.5b Total vbaton t m u 3 at the oto blade tp z-decton at node # Fg.5c Total otaton t ad at the oto blade tp y-axs at node # Fg.5d Total tosonal otaton t ad at the oto blade tp x-axs at node #

21 The vbaton esponses, computed by dect step-by-step ntegaton of the equatons 4.4, ae pactcally dentcal to those n Fg. 5a-d. 4.5 Popotonal damped system In ths case we use the deved symmetc dampng matx D p Eq.4.6, 4.7. The fst ten lowest complex conugate egenvalue pas, esultng fom Eq..3, ae now: 4.7 The coespondng 8x8 G modal matx Eq..8, compses the fst fou mass nomalzed complex conugate egenvecto pas. In ode to vefy the deved elatonshp k k = η = const., see 3.4, we compute ths ato fo all components of the n- η volved ± =, 4 egenvectos fo nstance the fst ten ows only: = η η, =, The coespondng dampng atos η, see Eq. 4.b, ae computed n accodance wth Eq..5. The next step s the computaton of the matx Ψ, Eq..4. The 8x8 eal tansfomaton matx Y, computed n egad wth Eq..4,.7, has now the fom of 3.: Y = 4.9 In 4.9 ae pnted agan only the fst ten ows of Y.

22 The tme-dependent load vecto n the geneal modal tansfomed equatons.3 s now hee fo the tme 5 sec - see also Eq. 3.5 and 4.: g t h t = g 4 t [ h 4 t] 4. Eq. 4. mples x = y, see Eq.3.5, 3.6, contay to the geneal case Eq..3a. In the esultant fou uncoupled SDOFS block equatons, see 4.5, the fee fequences and the modal dampng atos ae esp. [ω ] = [η ] = 4.a,b Afte step-by-step ntegaton of the fou modal equatons 4.5, the tme sees of the modal coodnates x t, y t,,...4, ae obtaned Fg. 6: Fg. 6 Tme esponse of the modal coodnates y t fo the case popotonal dampng

23 The total esponses Fgs. 7a-d: E. Stanoev V t ae computed by a back tansfomaton accodng to Eq..4 see Fg 7a - Total vbaton t m u at the oto blade tp y-decton at node # Fg 7b - Total vbaton u 3 t m at the oto blade tp z-decton at node # Fg 7c - Total toson t ad at the oto blade tp x-decton at node #

24 Fg 7d - Total otaton t ad at the oto blade tp y-axs at node # 5 CONCLUSIONS A geneal modal decomposton method of MDOFS wth non-popotonal dampng s befly pesented n Sec.. The pocedue s based on the complex egenvalue soluton of a stuctual model wth symmetc non-popotonal dampng matx. By use of the ght complex conugate egenvecto pas, nomalzed elatve to the geneal mass matx, a new eal tansfomaton matx Y, see Eq..4,.7, s developed analytcally to pefom a modal decomposton of the equatons of moton n eal athmetc. The complex conugate egenpas egenvalues and the coespondng egenvectos ae to be computed fst, at least fo the lowest few modal shapes. The equatons of moton ae tansfomed nto uncoupled SDOFS block equatons. Employng only a few k egenvecto pas n the Y -bass k<<n s leadng typcal fo a modal tansfomaton pocedue wth suffcent numecal accuacy to the total tme esponse of all n DOF. The modal equatons ae numecally ntegated and fnally tansfomed back to the ognal DOF. In moe detals the method has been descbed n [3], and n [4] has been developed a smla method, based on the ght and left egenvecto pas. The applcaton of the suggested method to the specal case of popotonal damped system s consdeed n detals n Sec. 3. Employng a Raylegh dampng matx, t has been shown that the modal tansfomaton fom Sec. mples a ato k k = const. fo all k th DOF of each consdeed egenmode ±,.e. the constant phase statement. Ths poves n an ndect manne that the fee vbatons n the popotonal dampng case ae synchonous. A smple fomula fo computng of the constant ato has been also deved, expessng t though the assocated modal dampng ato η. In Secton 4 a numecal example vbaton of a oto blade wth 54 DOF - demonstates the pefomance of the pesented modal method fo the two cases nonpopotonal and popotonal Raylegh dampng. In the fst vaant the dampng matx of the system contans a stffness-popotonal pat and a smple appoxmated aeodynamc dampng pat. In the second vaant the fomula fo the constant phase of the esonance modes s vefed numecally. Real lfe applcatons of the poposed modal analyss method and possble numecal complcatons ae dscussed moe wdely n [4], [5]. The pesent pape studes some

25 known featues of popotonally damped systems the synchonous fee vbatons fom a vewpont of a new poposed modal analyss method. REFERENCES [] E. Stanoev, H. Came. u Boden-Bauwek-Inteakton auf de Bass de modalen Analyse. BKI 3/3, Unvestät de Bundesweh München, München 3, p [] H. Came, E. Stanoev. u modalen Analyse be gedämpften Mehmassensystemen.. Desdne Baustatk-Semna: Neue Bauwesen Tends n Statk und Dynamk, TU Desden, Lehstuhl fü Statk, Desden, 6, p. 8-9 [3] H. Came, E. Stanoev. En Vefahen zu modalen Analyse gedämpfte Systeme de Stuktumechank. Rostocke Bechte aus dem Insttut fü Baungeneuwesen, Heft 9, Unvestät Rostock, Insttut fü Baungeneuwesen, 8, ISSN , p [4] E. Stanoev. A modfed modal analyss method fo damped mult-degee-of-feedom- systems n stuctual mechancs. etschft fü angewandte Mathematk und Mechank AMM, 3, 3 3, [5] E. Stanoev. A modal analyss method fo stuctual models wth non-modal dampng. MS Multbody system dynamcs and modal educton n the fame of th Wold Congess on Computatonal Mechancs WCCM XI, 5th Euopean Confeence on Computatonal Mechancs ECCM V, -5 Jul 4, Bacelona, ISBN: , Tomo IV, p [6] Petes, H. J.; Tso, P.; Goosen, J.F.L.; van Keulen, F. Modfyng esonance modes of dsspatve stuctues usng magntude and phase nfomaton, th Wold Congess on Computatonal Mechancs WCCM XI, 5th Euopean Confeence on Computatonal Mechancs ECCM V, -5 Jul 4, Bacelona, ISBN: , Tomo II, p [7] Géadn, M; Rxen, D. Mechancal vbatons theoy and applcatons to stuctual dynamcs, John Wley & Sons Ltd, 997 [8] Meskous, K. Baudynamk Modelle, Methoden, Paxsbespele, Enst & Sohn, 999 [9] Chopa, A.K. Dynamcs of Stuctues. Theoy and Applcatons to Eathquake Engneeng, Peason Pentce Hall, New Jesey, 7 [] M.-C. Km, L.-W. Lee. Egenpoblems fo lage stuctues wth non-popotonal dampng. Eathquake Engng. Stuct. Dyn, 8, 57-7, 999 [] D. C. Soensen. Implctly Restated Anold/Lanczos Methods fo Lage Scale Egenvalue Calculatons. Dep. Comp. Appl. Math., Rce Unvesty, Houston, 995 [] R. Lehoucq, D. C. Soensen. Implctly Restated Lanczos Method. Secton 4.5,7.6 :. Ba, J. Demmel, J. Dongaa, A. Ruhe and H. van de Vost, edtos, Templates fo the Soluton of Algebac Egenvalue Poblems: A Pactcal Gude, SIAM, Phladelpha,, 67-8, [3] T. Buton, T.; Jenkns, N.; Shape D.; Bossany, E. Wnd Enegy Handbook, John Wley & Sons,, chapte 5.7, 5.8

26 [4] Nan L. Beechnung de Queschnttsstefgketen des Rotoblatts ene WEA duch en FE-Vefahen fü dünnwandge mehzellge Pofle mt Ensatz von MATLAB, Studenabet, Unvesty of Rostock, Cha of Wnd Enegy Technology, 5 [5] E. Stanoev. Ene altenatve FE-Fomuleung de knetschen Effekte bem äumlch belasteten Stab. Rostocke Bechte aus dem Insttut fü Baungeneuwesen, Heft 7, Unvestät Rostock, Insttut fü Baungeneuwesen, 7, ISSN , p.43-6