Aeroelastic Vibrations and Stability in Cyclic Symmetric

International Journal of Rotating Machinery2000, Vol. 6, No. 6, pp. 445-452Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2000 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in Malaysia.

Aeroelastic Vibrations and Stability inCyclic Symmetric Domains

BERNARD LALANNEa’* and MAURICE TOURATIERb

aTurbomca, 64511 Bordes, France," bLMeS, UPRES A 8007, ENSAM,151 Bd de l’H6pital, 75013 Paris, France

(Received 13 June 1997," Revised 10 March 1999; In finalform 12 June 1999)

This paper deals with rotating cyclic symmetric structures, immersed in light fluid flow.Firstly, general and usual cyclic symmetric properties are recovered from the Floquettheorem for differential equations, having space periodic coefficients in conjunction with adiscrete space Fourier series development. The approach for aeroelastic problem havingcyclic symmetry is then formulated based on the twin mode approach. In addition to modalone, rotating and stationary wave bases are introduced to derive the equilibrium equations fora non-dissipative system subjected to aerodynamic loading. Rotating wave basis is the natu-ral one, and it also permits consistently to prescribe the aerodynamic pressure on the bound-ary between the fluid and the structure. The aerodynamic load is then derived from aharmonic analysis of the fluid flow extending to turbomachinery as in the case of aeroplanewing. In this way, aerodynamic forces may be obtained as general as required, depending onsuccessive time derivatives of degrees of freedom in addition to themselves. Finally, somespecial cases are given and stability is studied for a cyclic periodic blade assembly, even whenmistuning between sectors can occur.

Keywords: Cyclic symmetry, Forced vibrations, Aeroelasticity, Stability, Mistuning

INTRODUCTION

Vibration mastery is an important issue forcompressors and turbines of turbomachines. Thesestructures in general have circumferential periodicsymmetry. It is well known that the studies ofvibration mode of cyclic symmetric structurescan be reduced by analysing one repeated sector,see Balasubramanian et al. (1993), Lalanne and

Corresponding author. Tel.: 05 59 12 5000. Fax: 05 59 53 15 12.

445

Touratier (1996), M6zire (1994), Thomas (1974),Wagner and Griffin (1994). In a turbomachinery,turbines are, among others, loaded by the fluid flowand the corresponding vibration problem ofturbinemay be solved using cyclic symmetric properties ifthe fluid flow satisfies these ones.

Hereafter, an analysis of fluid-structure vibra-tion analysis for cyclic symmetric structures is pre-sented. The cyclic symmetric decomposition for

446 B. LALANNE AND M. TOURATIER

linear problems is recovered by using the Floquettheorem. The principle of virtual work for thewhole structure deals with the partial differentialequations, having space periodic coefficients. Then,using Fourier series development of all the fields,the sector formulation for cyclic symmetric prob-lems can be easily deduced from either continuousor discrete approaches, see Lalanne and Touratier(1996). To analyse aeroelastic responses, manyways are possible, see for example Wagner andGriffin (1994). Here, stationary and rotating wavebases are introduced in conjunction with the twinmode approach, Ewins (1988), by taking intoaccount aeroelastic forces from a harmonic anal-ysis of the fluid flow. Generalized forces associatedto aerodynamic loading are then expanded as func-tions of generalized co-ordinates and their succes-sive time derivatives. Examples are given whereingeneralized co-ordinates and their first time deri-vatives are kept, and some particular cases arestudied based on a weak aeroelastic coupling.Finally, stability analysis is presented for anaeroelastic problem associated with a bladeassembly wherein a small mistuning (see forexample Wei and Pierre, 1988) between blades hasoccurred. The effect of mistuning on stability isanalysed based on a perturbation technique.

BASIC FORMULATION OF CYCLICSYMMETRIC PROBLEMS

Let f be a circumferentially periodic structurearound the z-axis in cylindrical co-ordinates

(r, 0, z) and 2s, one repeated sector allowing to gen-erate ft by c-rotation around the z-axis. Typically,ft can be decomposed into N sectors fts. Let usconsider small linearly varying displacements forthe structure ft. The latter being circumferentiallyperiodic, by applying the principle ofvirtual work itis clear that linear second-order partial differentialequations are derived, with periodic coefficients.Thus, the Floquet theory is available, see Ince(1956), Stocker (1992) to take into account ofthe cyclic periodicity. Let u(r, O,z, t) be then the

three-dimensional displacement field in cylindricalco-ordinates, and the time. Recalling the closedcondition associated to the circumferential peri-odicity as u(r, 0 + 27r, z, t) u(r, O, z, t), then theFloquet theory allows to write (Touratier, 1986)

u(r, 0 + c, z, t) exp(jqc)u(r, O, z, t),

j2=--1; q=0,1,...,N- 1,

where q is the Floquet wave number.Since deduced motion equations are linear and

second order, with periodic coefficients, it is con-venient to search the u-unknown from a Fourierseries development with respect to the circumfer-ential 0-variable (Balasubramanian et al., 1993):

u(r, O, z, t) Z uk(r, z, ,)exp(jkO)

n=0 k=-oc

blkN+n(l", Z, t)expj(kN +/- n)O.

(2)

This equation may be reduced to

+u(r, O,z, t) E(Hn(r, O,z, t) -+- H_n(r, O,z, /)),

n--0

N N-if N odd. (3)N-- if N even, 2--

2

Introducing the s-index attached to any sectorfrom Eqs. (3) and (1) we may still write

u+(r, 0 + sc, z, t) exp(+/-jns a)u:(r, O, z, t). (4)

Replacing Eqs. (3) and (4) into the principle of vir-tual work, expected equations of motion are thendeduced. Finally, we could demonstrate that thesolution of the given boundary value problem forcircumferentially periodic structures is reduced tosolve N uncoupled problems P(n) (identically P(-n)associated to the phase difference index-n) on thereference sector fro, for each Fourier series com-

ponent un (identically U-n) of the displacement u

AEROELASTIC VIBRATIONS 447

which satisfies cyclic periodic properties. Theseconclusions are also achieved when using a discreteapproach (finite elements for example) to solve theboundary value problem.

THE AEROELASTIC PROBLEM

Our interest is now for a rotating circumferentiallysymmetric structure in a fluid flow. Firstly, weconsider a cyclic symmetric structure excited by afluid flow which does not respect cyclic symmetricconditions. Then, excitation by the fluid flow isinduced by the vibration of the structure, and thisis called the aeroelastic effect. From the aboveparagraph, it is clear that results for the boundaryvalue problem are still available if cyclic symmetricconditions are satisfied, as consistently assumedby Lane (1956). To be complete, it is necessary toinclude rotatory effects as in Childs (1993) and togive the aerodynamic loading, to be included intothe boundary conditions (pressure).However, gyroscopic effects will be hereafter

neglected. From those latter, coupling betweenfluid and structure will be prescribed. Aerodynamicloading (pressure on external surfaces of turbine)may be characterized from experimental, analyticalor purely numerical ways. Wind tunnel is needed forexperimentally achieving the bench tests on rotoror cascade to deduce pressure on it, while analyti-cal computations using a cascade model (Jacquet-Richardet and Henry, 1994) give approximateexpressions for the total (steady and unsteady)pressure acting on blades of a turbine. The mostcomplete and accurate approach deals with thenumerical computations of the fluid flow (Marshalland Imregun, 1996) within steady and unsteadystate phases. Anyway, the aerodynamic solution ofthe problem is given by reference to rotating wavebasis.

Here, to model aeroelastic forces, a harmonicanalysis of the fluid flow is made, prescribing at thefluid-structure, the boundary motion as in Poiron

(1995) for aeroplane wing. The following expres-sions for displacements in rotating wave basis are

considered assuming harmonic motions:

tl ttl/JUn )ll exp(jc t);bln (hin Xn

u (c,’:-ju. llxL.II exp(-ja t)

tz and " are twin mode shapes associated towhere un "nin the stationary wave basisthe eigenfrequencies con

and are function of the cylindrical co-ordinates,with the index of the twin mode, and wheredenotes the modulus of the generalized co-ordi-

and xn in the rotating wave basis. Thennates xn

the displacement vector in Eq. (3) can be written as

xnu +jxn un ). (6)

’[ and jx" are the generalized co-In this Eq. (6), x ntl and " ttlordinates associated to the mode shapes un u

Let us consider firstly the structural responseunder an arbitrary excitation force f(t) acting at a

point P and time on the cyclic symmetric fluid-structure boundary I. This excitation force f(t)being developable in Fourier series (see Eq. (8)),then equilibrium equations for a non-dissipativesystem in stationary wave basis may be written as

.ti 2_ ti / ,,tixn + (COn) xn un .f(t) ds.J l’

.tti 2 tti fXn -[- (Odn) Xn Jr -JUn f(t) as

Vi 1,...,

Vn 0,...,2,

(7)

where the centred dot denotes the scalar productand the overdot denotes d/dt. In Eq. (7), is a

positive integer, and from Eq. (3) n- 0,..., +N.Here, the excitation forcefwill be generated from

the fluid flow for which no assumption is made,except the small disturbances. Denoting the exter-nal unitary normal on I’ at the point P by m, thepressure p at the point P and time can be intro-duced and we havef--pm. Excitation forces in tur-bomachinery, generally come from flow distortionsgenerated by blades andvanes and distortion ofinlet.They can be observed in rotatory co-ordinate sys-tem with respect to the cyclic symmetric structure.


Then, a Fourier series development of the exci-tation forcefmay be introduced with respect to thecircumferential co-ordinate 0- t associated to a

rotatory co-ordinate system at the angular speed ,and the flow distortion is

f(t) Z fq(r, z)expjq(O -t). (8)

Using the basis changing between stationary androtating wave bases, generalized co-ordinates withrespect to the rotating wave basis can be expressedas follows:

ti tti ti ttiXn @ Xn Xn Xn (9)Xn 2 X-n "Thus from Eq. (7), equilibrium equations of thecyclic periodic structure in rotating wave basis for anon-dissipative system are deduced:

(lo)

It can be shown (Wildheim, 1979; Lalanne, 1989)that resonance condition requires

q- (kN + n)-COn (11)

for all positive integers k,n, taking into accountthe inequality n <_ N/2. Since the system consideredis non-dissipative, the resonance condition isreached when excitation frequency becomes equalto any eigenfrequency, and the generalized forces inEq. (10) then being non-zero.

Let us suppose that the aeroelastic load isgenerated from the displacement u at the boundaryand that now this load satisfies cyclic symmetric

conditions. In the rotating wave basis, we can write

ttlx tl ttl\

n

ZZ(u/+ u-)" (12)n

Within the linearity assumption kept here, a har-monic excitation at circular frequency coo involvesidentical frequency harmonic load. Thus, in the casewherein the general excitation load f(t) is an aero-elastic one, we can write for a harmonic motion

fn (jco) -(r, z, O,jco)Xn (jco), (13)

where

Xn(j ) IJXnll exp(jcot),

XZ-n(J ) IIx ll exp(--jcot)

fn(^t r,z, O,jco) -Ptn (r, z, O,jco)m.

and (14)

The generalized forcef/i(jco) generated by the har-monic displacement un (jco) for the ith wave of gen-eralized co-ordinate xn(jco) is given by

f, [J;co\ hnli (jco)Trpb2co2x/ (Jco) (15)

with

jf (r, O,jco)m [.tln .^,,iz?Ju ds.

2-p z, (16)

In these last expressions, p is the mass density of thefluid, b the mid-chord or a characteristic size, andthen bco/Vis the Strouhal number while Vis velocityfar away in the fluid flow. h(jco) is the complexamplification factor at the frequency co.

The generalized loads in Eq. (10) can now beexplicited using a time harmonic analysis basedon decomposition associated with the generalized

and their successive derivativescoordinates x


..t ...z Some additional state variables mayXn, Xn, Xn,

be added to include the delay effects and acousticresonances (see Poiron, 1995; Crawley, 1988). Froman identification procedure between definition ofthe aerodynamic loading (see Eq. (7) for example)and the above introduced time harmonic analysis,the generalized forces in Eq. (10) can be written as

un .f(t) ds

anXn +bnxn -at-cnxn +...

ttl i" ttl ..ttl+ anxn +onxn + cnxn +...),(17)

tti

-LJU, .f(t) ds

anXn -+-bn-n + CnXn --"’"tl tl tl+ ax + bx + cnxn +...),

an, hn, bn,/, c,, g,..., being complex numbers andwhere indexes i, have been omitted. These coeffi-cients are depending on the aeroelastic propertiesof the fluid structure given by Eqs. (13)-(16). InEq. (10), aeroelastic quantities modify the dynamicsof the system, but from the orthogonality proper-ties, Eq. (10) associated to the phase differenceindex n are formally unchanged and can be sepa-rately solved. This is not true in the modal basis.

Using Eq. (4), the aeroelastic problem (10)-(17)could be defined on the reference sector fo and inthe rotating wave basis.

SPECIAL CASES

associated to in-phase and quadrature terms inthe harmonic motion. Indexes i,l are omittedbefore, and coefficients an, b, hn, b, are now realnumbers.

Rigid Disk

In this case, u u) for all n and p, and for all pointin the reference sector fo. From Eq. (4)

H r, 00 - ----N-’ Z, Ui(r, 00, Z, t)exp(27rjns/N).

(18)

The linearity assumption, in addition to the rigiddisk one, allows to obtain pressure on the referencesector during motion as the superimposition ofpresslres on each other sector. Thus, the harmonicmotion involves

N

in(r,z, O) Z exp(27rjns/N)

i.e. 0 E 00,00 +--

on 0

(19)

Blade Interaction

From experiments and calculations, it is admittedthat any blade is sensitive to the interactionsproduced from the motion of two adjacent bladeson either side of the blade considered (Szechenyi,1987). So, aeroelastic loading knowledge for fivevalues of the phase difference index n allows tocompute load for any phase difference index.

Weak Aeroelastic Coupling

The fluid is light, and then it is assumed that theaeroelastic forces are small by comparison with thatof mechanical ones. Aeroelastic loads are only sig-nificant near resonances, then, they will be given by

tl tl ttl 7 ttlanXn q- bnx -+- anXn -+- OnXn

and

ttl .ttl tl .tlanXn + OnXn + anXn + bnxn,

Efficient Strategy

The above particular cases furnish a way to reducecomputational cost from the Fourier analysisusing the rotating basis. From the Eq. (4), where

u u; on f0, for all n and p, the blade interactionassumption allows to keep only five values for thephase difference index. Then, mode decompositionshould use the rotating wave basis. Finally, if theaeroelastic coupling is weak, stability analysis willbe made near each resonance.

45O B. LALANNE AND M. TOURATIER

TWIN MODE APPROACH WEAKAEROELASTIC COUPLING- STABILITY

If roots of the twin modes are far, those latter willbe almost uncoupled, when mode veering doesnot occur (Morand and Ohayon, 1992). FromEq. (19) and the above weak aeroelastic couplingassumption, the following equilibrium equationsare found, in the rotating basis:

"i .i /i+ + + + o,in -[- (Oicn /icn)i_n -[- (oi__kn /i_kn)Xin O,

(20)with

N-1

2coi.(i siOcn ,,-n -+- 2Ctn COS nsc,

s:0

N-12 - si

ozkn (COn) + % cos ns,s:0

N-1

cn % sin ns,&n s=0

N-1si sin nS

s=0

(21)

According to the above weak aeroelastic couplingassumption, we have the relations

i2 ii-a, oz, (COn) bn Cen 2COniC’n,~i /n b~in /cin"--an

In Eq. (21), coefficients have the following phys-ical sense"

/3i are viscous and pseudo-viscousOcn, cn

aeroelastic terms,

OZkn/n

are in-phase force terms due to motionof the sector fs at the mode i,are viscous damping coefficients,are in quadrature force terms generatedfrom the motion of the sector fs atthe mode i,are stiffness and pseudo-stiffnessaeroelastic terms.

Example As an example, we consider now abladed-disk incorporating a rigid disk surrounded

by 20 blades allowing the cyclic symmetry assump-tion available. Blade vibrations may be bending,torsion or any other mode type. Hereafter are givendata corresponding to an abstract problem usefulonly to analyse trends for the dynamic behaviourof the fluid-structure system.

Aeroelastic interaction concerns only one sectoradjacent to the reference sector. Thus, requiredterms are restricted in Eq. (21) to s values N 1,0, 1.Non-zero coefficients are given by

c- 5.72 s-1 c 10s-1, ozN-1 2 S-170_0.01 s-Z, ,l- 75 000 s-2, ,yl- 25 000 s-2,COni- 6283 rad s-1 N-20, - 0.001,

where the indexes n and are omitted.Root calculation from the characteristic equa-

tion of the differential system (20), having expo-nential solutions, allows to determine stable andunstable zone in the complex plane, see Fig. 1.Modes corresponding to phase difference indexesn=8 and 9 are unstable. The Routh-Hurwitzcriterion also gives the same following stabilitycondition:

(O cin)2 2--(cn) > 0 and C cn > 0. (22)

SMALL MISTUNING BETWEEN SECTORSFOR AN AEROELASTIC PROBLEM

The twin mode approach is still supposed to beavailable in modal basis. So, corresponding equi-librium equations for an aeroelastic problem arededuced from Eq. (20). By adding the basis changedefined in Eq. (9), they are finally given in thestationary wave basis as

.ti .ti ,’i .tti ti ,’i ttiXn -- OcnXn -- /JcnXn -- OZknXn @/O/cnXn O,.tti .tti ,’i .ti tti /i xti O.Xn -Jr- OZcnX @/9cnXn -- OknXn -- kn (23)

Assuming a small mistuning between sectors of a

cyclic periodic structure, it is still possible to re-

strict stability analysis to twin mode formulation as

previously suggested in Wei and Pierre (1988). Here


3o RAD/S

6300

X2 X7 |X3

X4X8 6;P90._[.."

-25 -20 .15 .10 -5 0 5

REAL PART OF EIGENVALUES

FIGURE Stability of the tuned system.

x X2 x3

XO

Xl

X4

X3

-25 -2o .,t5

X5

X4

63o RAD/S

6300

6X 7 6290 "1"

X8 X6280

0

X 5X 6

6270 .--t0 -5 0 5

REAL PART OF EIGENVALIJES

FIGURE 2 Stability of the mistuned system.

coupling comes from aerodynamics. Thus, consid-ering a small perturbation Ah of the coefficient

the coefficient (cn + Ah/n)is substituted to cknOZkn

in one of the Eq. (23). Then, the correspondingeigenvalue equation is usually written. Finally, theRouth-Hurwitz criterion is used and the stabilitycondition is derived as follows:

It has been proved that stability margin increaseswhen the mistuning occurs. Next, a mistuning to theabove example showed in previous section isapplied as follows:

Ah/i_6273rads- and con+- 6293rads-COn

(ccAh)2 + 2[Cc2 -/32J [2/32 (2OZk Jr- Ah)] > 0 and Cc > 0,+oz

where indexes and n are omitted.

(24)

Then, the system becomes stable as shown in Fig. 2.Physically, if we introduce some small lack of

symmetry into the assembly, the twin modes no

longer have the same frequency. It follows that the


unsteady forces caused by either of the modes nolonger have the ideal frequency for exciting theother.

CONCLUDING REMARKS

Using the twin mode approach, the linearizedapproximation given for aeroelastic problemsencountered in turbomachinery, having light fluidin which turbine is immersed, has allowed torecover resonance conditions and to derive stabilityconditions, when cyclic symmetric properties aresatisfied.From general conditions associated to cyclic

symmetric structures with aeroelastic loading basedon weak aeroelastic coupling, N independantstability problems defined in the reference sector

f0 are deduced in rotating wave basis, according tothe N values of the phase difference index asso-ciated to the N sectors of the structure considered.A stability study is proposed when a small

mistuning occurred between sectors of an aero-elastic cyclic symmetric structure. The solution isbased on a perturbation technique. It has beenestablished that an unstable, perfectly, periodicstructure can become stable with a small mistuningbetween sectors of this structure.

References

Balasubramanian, P., Jagadeesh, J.G., Suhas, H.K.,Srivastava, A.K. and Ramamurti, V. (1993) On the use ofthe Ritz-Wilson method for the dynamic response analysisof cyclic symmetric structures, Journal of Sound andVibration, 164(2), 193-206.

Childs, D. (1993) Turbomachinery Rotordynamics, John Wileyand Sons, New York, Chichester, Brisbane, Toronto,Singapore.

Crawley, E.F. (1988) Aeroelastic formulation for tuned andmistuned rotors, AGARD A-G 298, Vol. 2.

Ewins, D.J. (1988) Structural dynamic characteristics of bladedassemblies, AGARD, A-G 298, Vol. 2.

Ince, E.L. (1956) Ordinary Differential Equations, Dover Pub-lications, Inc., New York.

Jacquet-Richardet, G. and Henry, R.A. (1994) Modal aero-elastic finite element analysis method for advanced turbo-machinery stages, International Journal for NumericalMethods in Engineering, 37, 4205-4217.

Lalanne, B. (1989) Vibrations of cyclic symmetric structures.Modes Resonances Damping Stability Control (inFrench), 9th French Congress of Mechanics, Metz.

Lalanne, B. and Touratier, M. (1996) Friction damping ofblade vibrations of gas turbines (in French), DRET Reportof the Grant, No. 92/471.

Lane, F. (1956) System mode shapes in the flutter ofcompressor blade rows, Journal of the Aeronautical Sciences,23, 65-66.

Marshall, J.G. and Imregun, M. (1996) Review of aeroelasticitymethods with emphasis on turbomachinery applications,Journal of Fluid and Structures, 10(3), 237-267.

M6zire, L. (1994) Vibrations of cyclic symmetric structures.Application to turbomachines (in French), AGARD, CP 537,Vol. 1.

Morand, H.J.P. and Ohayon, R. (1992) Fluid Structure Inter-actions (in French), Masson, Paris, Milan, Barcelone, Bonn.

Poiron, F. (1995) Time model in aeroservoelasticity systems.Application to time delay effects (in French), La RechercheAOrospatiale, No. 2, 103-114.

Stocker, J.J. (1992) Non-Linear Vibrations in Mechanical andElectrical Systems, John Wiley and Sons, Inc., New York.

Szechenyi, E. (1987) Understanding fan blade flutter throughlinear cascade aeroelastic testing, AGARD A-G 298, Vol. 1.

Thomas, D.L. (1974) Standing waves in rotationally periodicstructures, Journal of Sound and Vibration, 37, 288-290.

Touratier, M. (1986) Floquet waves in a body with a slenderperiodic structure, Wave Motion, 8, 485-495.

Wagner, L.F. and Griffin, J.H. (1994) Forced harmonicresponse of grouped blade system: Part discrete theory,International Gas Turbine and Aeroengine Congress andExposition, ASME, 94-GT-203, The Hague.

Wei, S.T. and Pierre, C. (1988) Localization phenomena inmistuned assemblies with cyclic symmetry, Trans. ASME,Journal of Vibration, Acoustics, Stress and Reliability inDesign, 110(4), 61-79.

Wildheim, S.J. (1979) Excitations of rotationally periodicstructures, Trans. ASME, Journal of Applied Mechanics, 46,878-882.

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Aeroelastic Vibrations and Stability in Cyclic Symmetric