/V'1459 -77m - Y,^gi3

NASA Technical Memorandum 82813NASA-TM-82813 19820013730

92N 2-1 (oo`{

Coupled Bending -Bending -TcA of a Mistuned Cascade With

Nonuniform Blades

Krishna Rao V. KazaThe University of ToledoToledo, Ohio

Robert E. Kielb AUG 1 0 1986

Lewis Research Center LANl.LL•V ^pSVA6N;.S„C^_N,ER

Cleveland, Ohio ^t NAMPiUN,

Prepared for theTwenty-third Structures, Structural Dynamics and Materials Conferencecosponsored by the AIAA, ASME, ASCE, and AHSNew Orleans, Louisiana, May 10-12, 1982

k

PJASA

and

COUPLED BENDING—BENDING—TORSION FLUTTER OF AMISIUNED CASCADE WITH NONUNIFORM BLADES

Krishna Rao V. Kaza*

The University of Toledo

Toledo, Ohio 43606 and

National Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

and

Robert E. Kielb**

National Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

w

A set of aeroelastic equations describing the

motion of an arbitrarily mistuned cascade withflexible, pretwisted, nonuniform blades is devel-oped using an extended Hamilton's principle. Thederivation of the equations has its basis in thegeometric nonlinear theory of elasticity in whichthe elongations and shears are negligible comparedto unity. A general expression for foreshortening

of a blade is derived and is explicitly used in theformulation. The blade aerodynamic loading in thesubsonic and supersonic flow regimes is obtainedfrom two—dimensional, unsteady, cascade theories.The aerodynamic, inertial and structural coupling

between the bending (in two planes) and torsionalmotions of the blade is included. The equationsare used to investigate the aeroelastic stabilityand to quantify the effect of frequency mistuningon flutter in turbofans. Results indicate that amoderate amount of intentional mistuning has enough

potential to alleviate flutter problems in en-shrouded, high—aspect—ratio turbofans.

Nnm I.+. .-o

A cross sectional area of the

blade

A0 reference value of A

[A],[A r ] aerodynamic matrices

Aj torsional mode shape

a elastic axis location

a 0 speed of sound

B 1 ,B2 blade sectional constants

b,bR semichord and reference

semichord

C blade chord

E Young's modulus of elasticity

[E],[ E s,r] matrices

e mass and elastic axis offset

(also base for natural

logarithm)

e area centroid and elastic axis

offset

e x ,e yl ez unit vectors along x,y,z axes

ex3' e Y3'

e z 3

unit vectors along x3,y3'z3

axes

shear modulus of elasticity

nondimensional amplitudes of

generalized coordinates

associated with the bending

modes W of the sth

blade measured out of and in

the plane of rotation

nondimensional amplitudes of

generalized coordinates

associated with the bending

modes W in the rth mode

of a tuned cascade

bending moment of inertia

about the major (parallel

to the x—axis) and minor

axis through centroid

reference bending moment of

inertia

unity matrix

V117

torsional stiffness constant

reference value for J

reduced frequency, w0b/Veff

w0b/Meffa0

polar mass radius of gyration

a

bout elastic axis

\k2 k2 + k2m m1 m2/

reference polar radius of

gyration of cross sectional

mass about elastic axis

G

gsi'hsi

gari'hari

Ixx'Izz

Ixxo

[1]

i

J

J0

k

km

k*Adjunct Professor, Mechanical Engineering Depart— m ment, Associate Fellow, AIAA.

**Aerospace Engineer, Structures Branch, Member AIAA.

k k mass radii of gyration about r reference radius of gyration

m l ' m2 0

x and z axesroil running blade coordinate along

k A polar radii of gyration of elastic axis before and

cross-sectional area about after deformation

elastic axis ro,rl position vector of an arbi-

L blade length trary point on the blade

L a lift per unit span, positive before and after deforma-

up (negative z direction) tion, Eqs. (A6) and (A7)

L hhrij " " 'Laarijaerodynamic coefficients, in

Ro,Rl position vector of a point onthe rth mode of cascade

elastic axis before and

1hhr'lharnondimensional lift coeffi-

after deformation, Eqs. (A6)cients due to bending and

and (A7)torsional motions in the rth

Shhsij'" " Saasijstiffness coefficients of the

cascade modesth blade

11 nondimensional moment coeffi-ahr' aar [S],[Ss] stiffness matrices

cients due to bending ands integer specifying blade,

torsional motion in the rths = 0,1,2, N-1; also blade

cascade modespacing

Max axial Mach number, Va/aoTk

kinetic energyMa

aerodynamic moment per unitTc,Tc

2 2blade tension,

T = T c /mo L

span about the elastic axis,[T] transformation matrix

positive nose upt,t0,t1 time, initial time, final time

M r relative Mach number,U strain energy

V a + n r /a o OF radial foreshortening

Meffeffective relative Mach

u,v,w deformations of elastic axis

number, Veff/aoin X., Y Rand Zo

directionsMg ,Mh ,M C, number of generalized coordi-

V a axial velocity

nates with the bendingVeff

effective relative velocity,motions out of and in the

plane of rotation and with Va + a r2

the torsional motion -1• cos [90 - E - tan (V a /or )

M hhsij' " ''Maasijinertial coefficients of the

V axial extension of elastic sth blade

e

[M],[M s ] inertial matricesaxis

W work done by aerodynamicm mass per unit length

m reference mass per unit length

loading

N Wi (i = 1,2,...) beam functionsnumber of blades in cascade

P'P' an arbitrary point on the [W]'[W] model function matrices

elastic axis before andXo,Yn,Zn hub-fixed axis system, rotates

after deformationabout the Z.-axis with an

[P] stiffness matrix, Eq. (16)angular velocity a

p quantity defined by Eq. (4)

xyz blade fixed axis system at

[Q] matrix, Eq. (16) arbitrary point on elastic

R hub radius axis

R blade tip radius x3y3z3 blade fixed axis system in the

r integer (0,1,2....N-1) speci- deformed configuration ob-

fying mode of a tuned cas- tained by rotating xyz

cade; also blade coordinate{Xs),{X) column matrices

along elastic axis before{Yr),{Y) column matrices

deformation

2

a angle of twisting deformation,

positive when leading edge

is upward

asi (i = 1,2,...) amplitudes of generalized

coordinates associated with

torsional modes A iof the

sth blade

sariamplitudes of generalized

coordinates associated with

torsional modes A iin the

rth mode of tuned cascade

B rinterblade phase angle in the

rth mode of tuned cascade,

2,rr/N

Y nondimensional eigenvalue,

(w/wo)2

Yyy ,Yyx ,Y yz engineering strain components

a( ) variation of ( )

e yy ,e yx .e yz tensor strain components

^hsl' " ''asl damping ratios of sth blade

n, TT, n a blade running coordinate mea-

sured from hub; n = n/L;

blade elastic axis position;

n a = (a + 1)12

v o reference mass ratio, mo/nPabR

u real part of eigenvalue

pretwist angle

o a ,o m fluid and blade material

density

T nondimensional time, two

v imaginary part of eigenvalue

n rotational speed

w,wo frequency and reference

frequency

W angular velocity vector ex-

pressed in xyz system

wx z'wx zycurvature vectors

3y33

C) derivative a( )/at or

a( ) /aT

( ) derivative a( )/ar or

a( )/an

T Tntrnd,,r+inn

A research program in propulsion system aero-

elasticity is being conducted at the NASA Lewis

Research Center. As a part of this general pro-gram, an effort was made by the authors in Refs. 1and 2, to improve the physical understanding of

turbofan engine aeroelastic characteristics includ-ing blade mistuning (nonidentical blade properties)

effects. Other published work on mistuning wascited in Refs. 1 and 2.

The mathematical formulation considered in

Ref. 1 is a 'typical section' model with twodegrees of freedom, a bending and a pitching aboutthe elastic axis. This model was found to besufficient to elicit physical understanding of mis-tuning effects and to conduct parametric studies.Furthermore, this model was utilized in Ref. 3 toshow that mistuning has enough potential to sig-

nificantly raise the flutter speed of an advancedfan, this potential may have a very practical

significance in eliminating the commonly used mid-span shrouds in advanced turbofan designs. Themain purpose of the midspan shrouds is to increasethe blade natural frequencies and thus to avoid

aeroelastic instabilities. However, the shroudshave an adverse effect on aerodynamic performance.

While the typical section model served the

intended purposes, it is expected to be inadequateto obtain accurate flutter boundaries of moderntechnology turbofans or advanced turboprop bladeswhich have a considerable amount of pretwist. As a

consequence of pretwist, bending out of the planeof rotation couples elastically with bending in theplane of rotation. Consequently, a more realistic

structural model of a blade is required.

The purposes of the research summarized in

this paper are: (1) to develop a more refined andrealistic structural model of a blade than thetypical section model and to combine this modelwith the available unsteady cascade aerodynamicmodels; and (2) to study the effects of mistuning

and other cascade parameters on flutter character-istics of an advanced technology fan stage. To thebest of authors' knowledge, the mentioned effectsutilizing the purposed model have not been studiedin the published literature.

There exist several levels of approximations

in structural theory used in rotary wing aero-elasticity to represent a structural model of ablade. The theory of pretwisted blades presentedin Ref. 4 may be viewed as the first level of

approximation. The structural theories presentedin Refs. 5-9 may be viewed as ones with the second

level of approximation. Since the correspondingunsteady cascade aerodynamic theory consistent with

the second level structural theories is not avail-able, the theory of Ref. 4 is used as a first step.

The governing equations of motion of a

rotating, pretwisted blade are derived by using anextended Hamilton's principle. The derivation ischaracterized by the use of an axial (along the

beam axis) displacement which includes second-degree non inear terms defining the axial fore-shortening lo of the tension axis due to bending,torsion, and the noncoincidence of the elastic axis

and tension axis. The development of a more gen-eral expression for foreshortening than that usedin Ref. 10 and the explicit use of this generalexpression in the axial displacement field arebelieved to be new. The explicit consideration offoreshortening in deriving the equations is appeal-ing in that it parallels the corresponding develop-ment of equations for a nonrotating blade where

retention of second-degree terms in the kineticand potential energy expressions leads to linear

equations (See Ref. 10). A more detailed discus-sion on the role of the general expression forforeshortening in deriving the second-degree non-linear equations of motion and in accounting forthe geometric stiffness effects in beam analyses

3

using finite-element methods will be presented in a

future publication. The disk is assumed to be

rigid. The unsteady, two-dimensional, cascade,aerodynamic loads are calculated by Smith's

theory subsoni44 flow and Adamczyk and

Goldstein's theory 11 in supersonic flow with asubsonic leading edge. The governing equations ofmotion of a mistuned cascade are formulated byassuming that the general motion of a blade is alinear combination of its motions in all possiblemodes of a tuned cascade. The space variable in

the coupled integro-partial differential equationsof motion is eliminated by using a modified

Galerkin's method. The resulting equations arecast as a standard complex eigenvalue problem fromwhich the aeroelastic stability is determined.

A digital computer program is developed to

form and solve the complex eigenvalue problem.This program is written in a modular form such that

it can be applied to advanced turboprops, heli-copter rotors in hover, and wind turbine rotors by

incorporating the appropriate aerodynamic modules.Due to length constraints of the paper, limitedresults are presented only for an advanced turbofanblade. However, additional parametric results willbe presented in a future publication.

II. Theory

The components of a bladed disk system have

complex geometries. The analysis of this system isfurther complicated by blade mistuning. To inves-tigate the aeroelastic stability of a cascade withmistuning, a mathematical model will be developedin this section.

A. Coordinate Systems

Several coordinate systems will be employed in

the derivation of equations of motion; . those whichare common to both the structural and aerodynamicaspects of the derivation for the sth blade areshown in Figs. 1 and 2. The axis system XnYnZQshown in the figures rotates with a constant angu-

lar velocity a about the Xn-axis. The Y2-axiscoincides with the undeformed elastic axis of theblade. The blade principal axes, x and z, of the

cross section at an arbitrary point on the elasticaxis are inclined to the Xn and Zn axes by anangle ^ as shown in Fig. 2. The blade elastic

deformations, u, v, w, and a translate and rotate

the xyz system to the x3y3z3 system as shownin Fig. 1.

B. Tuned Cascade Model

If the blades are tuned, the N-bladed cascade

has N interblade phase angle modes with a con-stant phase angle Or between adjacent blades.Thisinterblade phase angle is restricted byLane 's 13 assumption to the N discrete valuesOr = 2,rr/N were r = 0,1,2,...,N-1. In each of

these modes all blades have the same amplitude.

For a tuned cascade, the modes with differentinterblade phase angles are uncoupled. The bladedeflections are expressed in a traveling wave formin terms of a set of generalized coordinates whichare associated with the nonrotating uncoupled beammodes in pure bending and torsion. The number ofmodes retained in the plane of rotation, in theplane perpendicular to the plane of rotation, and

in torsion are M h , M g , and Ma respectively.The blade deflections expressed in traveling waveform are

w/ bR i

wi w t

u/b R= [W]{X s I e Wo = [E s r ][W](Yr )

e m0

a

(1)

where

W I W 2 ...0 0

[ W ] = 0 0 ... W 1 W2 ...

0 0 ... 0 0 ... A l , A2 ...

T{Xs} (h

sl h s2 " ' g sl g s2 " ' a sl a s2 " 'I

T(Y r} (h

arl h art "' g arl g ar2 — a arl 0 ar2 "')

[ E s r] - e i6 r s [1]

(2)

An interblade phase angle Or in the range

0-180 ` represents a forward traveling wave - a wavetraveling in the direction of rotation.

In Eq. (2) the standard nonrotating ortho-

normal modes for a beam with fixed-free boundaryconditions are given by

W j (n) = cosh(p j n) - cos(pjn)

(cos p j + cosh pj)

sin p j + sinh pj[sin(p j n) - sin(p j n)] (3)

A j (n) = %1-2- sin[(2j - 1) 7/2 n]

where the value of pj is obtained from

cos pj cash pj + 1 = 0

(4)

In the case of a tuned cascade it is adequate

to analyze the motion of a cascade in each of theinterblade phase angle modes separately. Hence,the total number of degrees of freedom of a tuned

cascade is (Mh + Mg * Ma ) for each value of 9r.

C. Mistuned Cascade Model

In the case of an arbitrarily mistuned cas-

cade, the blades can have different amplitudes, andthe phase angle between adjacent blades can vary.However, the general motion of a blade in a mis-

tuned cascade can be expressed as a linear combina-tion of the motions in all possible interbladephase angle modes of the corresponding tuned cas-cade. Consequently, the blade deflections can bewritten as

4

w/bR i w

Wu/b R= [W]{X s } e

d

a

N-1

Y, [Es r][W]{Y r ) e Wd (5)

r=0

For a case with N mistuned blades, Eq. (5)

can be generalized as

i W t i 14[W]{X} e d = [ E ][ W ]( Y ) e o (6)

where

[W][W]

[W]

[W]

{X0

} {Y0 }

{X 1 } {Y1}

{X} = ^ > {Y} =

{XN-11^l {YN-1}

[E O 0] [EO 1 1 ...

[E] _ [T1,03 [EI 1 ] ... (7)

[EN-1,N-1]

The total number of degrees of freedom for

this general case is N times (Mh + M g + Mc').

D. Aerod y namic Model

The unsteady, two-dimensional, cascade, aero-

dynamic loads were calculated by using Smith'stheory in s bsonic flow, and Adamczyk andGoldstein's^2 theory in supersonic flow with asubsonic leading edge. In these theories, the air-foil thickness, camber and steady state angle ofattack are neglected, and the flow is assumed to beisentropic and irrotational. At any radial station

the relative Mach number is a function of theinflow conditions and the rotor speed. Most cur-rent fan designs have supersonic flow at the tipand subsonic flow at the root. As a result, some

region of the blade span encounters transonicflow. Since the above unsteady theories are notvalid in the transonic region, the subsonic the-ory with Mefff = 0.9 for stations in the range0.9 < MO ff < 1 and the supersonic theory with

Meff = 1.1 for stations in the range 1.0 < Meff < 1.1were used. The motion-dependent aerodynamic liftand moment were expressed in Ref. 1 in terms of

nondimensional coefficients. The expressions forlift and moment per unit span as a function of the

same coefficients can. be written as

N-1

L32 (b )2

a = -npab3I hhr b

r=0

M Mg

cos C Wj(n)harj + sin Wj(n)9arj

j=1 j=1

L)3

Ma

_ i(w T + Srs/

+ l har (b Aj(n)aarj e (8a)

R

j=1

N-1

43 (b 13

M a = ,rpabR.

Lahr bR

r=0

M Mg

cos f

X,

Wi (n)h arj + sin E Wj(n)9arj

j=1 j=1

4 Ma

_ m T + Br S)

+ l aar (bR ) Aj(n)aarj e o (8b)

j=1

The coefficients l hhr , lhar, la r arecalculated for specified values of Meff , ^, s/c,^, and a. In the aerodynamic theories used

herein, the steady state angle of attack is neg-

lected. In applying these theories to the present

case, the blade steady state angle of attack isalso set to zero and the effective relative veloc-

ity is assumed to be the component of the bladerelative velocity along the blade chord as indi-cated in Fig. 2.

E. Structural Model

The structural model of each blade consists of

a straight, slender, twisted, nonuniform elasticbeam with a symmetric cross section. The elasticaxis, the inertia axis, and the tension axis aretaken to be noncoincident. The effect of warpingis not explicitly considered. However, a partial

effect of warping enters into the equations ofmotion as it does in Ref. 4. The blade is assumed

to be rigid in the direction along the elasticaxis. Consequently, the axial equation of motionis eliminated. The structural model has its basis

5

/• t 1J (dTk - 6U + 6W)dt = 0 (9)t0

[(E I z z cos 24 + EIxx sin 2E)u"

(14a)

in the geometric nonlinear theory of elasticity in

which elongations and shears are negligible com-pared to unity and the squares of the derivatives

of the extensional deformation of the elastic axisare negligible compared to the squares of the bend-ing slopes. This level of the geometric nonlineartheory of elasticity is required to derive a set of

linear coupled bending-torsion equations of motion.

F. Eouations of Motion

The equations of motion will be derived by

using the extended Hamilton's principle in the form

[(EI zz

sin 2C + EIxx cos t) w"

- (EIzz - EI xx )u" sin f cos - EB 2^'a' sin E

- Tc eAa cos E

J

- (Tcw') + mw + mea cos C

2- n [(mrea cos + mw + mea cos ^] _ -L a cos c

The expressions for strain energy, kineticenergy, and virtual work of aerodynamic forces canbe written as

URTff [EYyy + G f

Yy z + f 2 x)]dr dx dz

R A \

(10)

ffldrl drl

Tk = ^ m dt dt dr dx dz (11)

fR A

R aW (-La sin ^ du - L a cos & 5w + M a aa)dr

RH

(12)

- (EI zZ - EI xx )w" sin ^ cos E - EB2t'a' cos C

- T c e A a sin ^J

- (Tc u') + mu + mea sin f

- 0 2 (mrea sin 4) = -L a sin ^ (141b)

JGJa' + EB1^12a' + Tck2Aa'

+ EB 2{'(u" cos E - w" sin 01 - T c e Aw" cos E

- T c eAu" sin + mlkma + eu sin G + ew cos CI

+ msi 2 re(w' cos C + u' sin ^) - rM2

where

dr l arldt = at + w x rl

W = Q(e x cos ^ + e sin ^)

r = (r - U F )ey + (u cos - w sin ^)ex

x

+ (u sin + w cos O e z + [T] T rz

^k 2 - Jk 2a acos 2t + we cos 4J = M (14c)

L\ 2

The sectional properties in these equations are

defined in Appendix B.

By substituting Eqs. (8a) and (8b) into Eqs.

(14a), (14b), and (14c), nondimensionalizing theresulting equations, applying Galerkin's method,and extending the resultant equations to all the

blades by using Eq. (6), the equations of motion ofan arbitrarily mistuned cascade can be simplified as

[Pj{Y1 = Y [Qj {Y1 (15)

(13)where

The expressions for the strain components,

position vector, foreshortening, and transformationmatrix are developed in Appendix A. Substituting

Eqs. (A9b), (A9c), and (A14) into Eq. (10), Eq.(13) into Eq. (11), Eqs. (8a) and (8b) into Eq.(12), the resulting equations into Eq. (9), takingthe indicated variations, integrating over the

cross section of the blade wherever necessary,integrating by parts over time, neglecting Coriolisand rotary inertia terms, and retaining only first-degree terms in u, w, and a yields

[Q] = [E] 1 [ M ][E] + [A]

[ P ] = [E] 1[S][E]

Y = (W/Wo)2

6

[MO ]5. For the mode of interest check whether the

imaginary part of the eigenvalue is within

[M1]tabltolerance of the reference

anacceI ffrequency. go back to step (3)[M] = with a new reference frequency.

6. For the mode of interest, check whether thereal part of the eigenvalue is within an

[M acceptableacceptable tolerance of unity. If not,modify the rotational speed and repeat theprocess from step (2). If positive, reduce

[AO ] speed; if negative, increase speed.7. To find the flutter boundary for another

[A1 ] value of the axial Mach number go back tostep (1).

It should be noted that the above method

[AN-1 ]utilizes a rotor speed iteration within the axialMach number iteration loop. It would appear thatit would be more efficient to place the Mach number

[S O ]

iteration within the rotor speed iteration loop,

since the mass and stiffness matrices have to be

[S 1 ]

updated each time the rotor speed is changed.

However, it was found that the time required to

[S] _ (16) assemble these matrices is small relative to the

time required to assemble the aerodynamic matrixand to extract the eigenvalues and eigenvectors.

[SN13 In addition, the flutter boundary was more

"distinct" using the method described above.

As indicated in Eq. (B2), structural damping

is added by multiplying the direct stiffnesscoefficients by (1 + 2i; hsl) , ( 1 + 2i;hs2),(1 + 2is as2 ) •••

III. Results and Discussion

A. Solution

The aeroelastic stability boundaries are ob-

tained by solving the standard complex eigenvalue

problem represented by Eq. (15). The relationbetween the frequency w and y is

i w = i iv (17)wo

Flutter occurs when u > 0.

B. Computer Program and Verification

A computer program (ASTROMIC) was written to

assemble and solve the generalized eigenvalue prob-lem given in Eq. (15). The program can be used topredict the in vacuum natural frequencies of a

nonuniform rotating beam by setting the aerodynamicmatrix [A] to zero. In this case the problem is

reduced to a standard eigensolution. For theflutter problem, the aerodynamic matrix is a func-

tion of the eigenvalues. Hence, an iterative solu-tion is required. The iterative technique used

herein to calculate flutter boundaries (once theproblem is completely described) is briefly summa-rized as follows:

Select an axial Mach number.

Select a rotation speed and assemble thestiffness and mass matrices.Calculate the variation of the reducedfrequency based on an assumed referencefrequency wo and relative Mach number

with span and construct the aerodynamic

matrix. The initial value for the refer-ence frequency is the same as the naturalfrequency of the mode of interest.

Solve the eigenvalue problem.

The correctness of the program was checkea by

constructing a hypothetical blade model with con-stant structural and aerodynamic properties alongthe span. The blade frequencies obtained by usingthe program were checked against known solutions.The eigensolutions with aerodynamics included werechecked against the results obtained by using thetypical section model program (MISER) developed inRefs. 1 and 2.

For a cascade with nonuniform blades, there

are no published theoretical results on flutter.However the results from ASTROMIC are compared withthose from MISER by choosing the properties for atypical section at different spanwise stations.

These results will be discussed later.

C. Aeroelastic Stability of an Advanced Fan

An advanced unshrouded fan stage (aspectratio = 3.3), representative of a next generationfan, was chosen for analysis. A similar stage wasanalyzed in Ref. 3 by using the typical sectionmodel. The results in Ref. 3 indicated that thefan design without shrouds did not meet the flutter

requirements, but that it may be feasible to usemistuning as a p assive flutter control. The moti-vation for choosing this fan stage for analysisherein is to further investigate the use of mis-tuning as a passive flutter control.

The properties of the fan blade are listed in

Table I and the reference properties are listed inTable II. The design point is at a rotor speed of4267 rev/min. and an axial Mach number of 0.55.At these conditions the tip relative Mach numberis 1.45.

The analyses are performed using two modes

each in the plane of rotation, in the plane per-

pendicular to the plane of rotation, and in tor-

sion. Since there are 28 blades in this stage andhence 28 interblade phase angle modes, the totalnumber of degrees of freedom for an arbitrarilymistuned cascade is 168. However, for alternate

mistuning, in which every other blade is identical,

certain symmetry properties are exploited in theanalysis. Because of symmetry, the 0. modecouples with the (Sr + n) mode only. The analy-

sis for this case consists of 14 separate eigen-

solutions of 12 degrees of freedom each rather tharone eigensolution of 168 degrees of freedom for an

arbitrarily mistuned cascade.

The blade was analyzed for vibration to gen-erate a Campbell diagram which is shown in Fig. 3.

Only the first and second bending frequencies andthe first torsion frequency are shown. Since themass ratio for this blade is high, the aerodynamicforces result in flutter frequencies which are notsignificantly different from those in a vacuum.

In Refs. 1-3, a typical section model, which

was originally used for fixed wing flutter analy-sis, was adapted for rotating blades. In the case

of a fixed wing, the relative velocity of eachstrip along the span is constant; the structuralproperties of the typical section are obtained bytheir respective values at the 3/4 blade spanstation. In the case of a rotating blade, therelative velocity of each strip along the blade

varies. Then, a question arises as to which span-wise station should be used as a reference to cal-

culate the structural and aerodynamic properties ofthe typical section. To answer this question, the

eigenvalues of the tuned cascade at the designcondition from both the present beam and typicalsection analyses using various spanwise locationsas reference are compared in Fig. 4. As expected,the cascade is unstable over a range of interbladephase angles corresponding to forward traveling

waves. It should be noted that, whereas there are168 eigenvalues, only the predominantly torsional

eigenvalues are shown. The other modes were foundto be stable at the design condition. Also, it can

be seen that the typical section model correspond-ing to approximately the 7/10 span reference sta-tion gives the best correlation for the unstablemodes with the nonuniform blade model. Thisfinding is in close agreement with the 3/4 spanreference station usually used for fixed wing aero-

elastic calculations. This observation is veryuseful in performing preliminary aeroelasticanalyses.

Previous publications 1 - 3 using the typical

section model have shown that torsional frequencymistuning could have a significant stabilizing

effect on the cascade. The effect of mistuning isfurther investigated herein with the nonuniform

blade model. The method used to vary the frequencyfrom blade-to-blade was simply to vary the tor-sional stiffness, J. For alternate mistuning, thetorsional stiffness of the odd numbered blades wasincreased by 10 percent over that of a tuned bladeat each spanwise location. Likewise, the torsionalstiffnesses of the even numbered blades was de-creased by 10 percent from that of a tuned blade.The result was a total torsional frequency varia-tion of approximately 7 percent. The eigenvaluesfor this mistuned rotor at the design conditionsare shown in Fig. 5, along with the correspondingtuned values. These predominately torsional eigen-values have split into high and low frequencyfamilies corresponding to modes with major partici-pation of the odd and even blades, respectively.As can be seen, this type of mistuning has stabi-lized the cascade to such an extent that it isstable at the design point.

By using the iterative procedure defined above

the flutter boundaries for both the tuned and mis-tuned cascades were calculated. The results areshown in Fig. 6. For,the tuned cascade it is seen

that the design point lies well inside in the un-

stable region. As expected, the axial Mach numberat flutter monotonically decreases with increasingrotor speed. For the tuned cascade it should alsobe noted that along the flutter boundary the tiprelative Mach number is approximately 1.15.Consequently, a region near the blade tip mayexperience nonlinearities associated with transonicflow which are not accounted for by the unsteadyaerodynamic theories used herein. The relative tipMach number at the design conditions is 1.45.

The inclusion of an alternate frequency mis-

tuning of approximately 7 percent has significantlyincreased the cascade stability by moving theflutter boundary to the right. Furthermore, thedesign point is stable for this level of mistuning.The mode defining the boundary has maximum partici-pation of the even-numbered (or low frequency)blades. The tip relative Mach number along the

boundary of the mistuned cascade is 1.53. Con-sequently, the transonic region is considerably

inboard of the tip and the transonic effects shouldbe minimal.

Figure 6 also illustrates the effect of struc-

tural damping on flutter of a tuned cascade. Astructural damping ratio of 0.002 is included ineach of the torsional modes. As can be seen, thedamping has a significantly stabilizing effect on

flutter speed of a tuned cascade.

IV. Conclusions

The major conclusions from this investigation

are summarized as follows:

1. A general expression for foreshortening ofa blade was derived and was explicitly used in

deriving the first-degree linear equations ofmotion of a rotating blade.

2. An aeroelastic model and an associated

computer program for a mistuned cascade with non-uniform blades were developed.

3. For the blade analyzed herein a typical

section model corresponding to the 7110 span refer-ence station gives the best correlation with thenonuniform blade model.

4. An advanced, unshrouded, high-aspect ratiofan was modelled and analyzed with and without mis-tuning. The results show that a moderate amount ofmistuning has enough potential to alleviate flutter

problems in unshrouded turbofans.

Appendix A

Strain Displacement Relations

A schematic representation of the deformed and

undeformed elastic axis is shown in Fig. 1. Since

the pretwist angle f varies with r, with ini-tial curvature of the elastic axis is

d^wxyz = ey Ur = ey ,' (Al)

The elastic deformations translate and rotate , + 1 2 + ,2)

the triad xyz to x3y3z3. Let the expressions E (uyy v 2w

for the curvature vector of the deformed elasticaxis and for the transformation matrix between the — z[(u" + aw") sin + (w" — au") cos C]x3y3z3. and xyz systems be

W = e + W e + e (A2)- x [(u" + aw") cos - (w" - au") sin ^]

x 3y3 z 3x3 x3y3 y3z3 z3

2 + 2+ ? —r x (a' 2 — 2a V ) (A9a)

e x 3 ex

ey= [T] ey(A3) Eyx = 2 -a' - (u'2 + w'2)

3

ez3ez

+ (u' sin + w' cos (u' cos - w' sin ^)

Following the procedure presented in Ref. 8, one (A9b)can develop the second-degree expressions for the

curvature components and for the transformation x (u,2 + w,2)matrix in terms of u, v, w, and a as Eyz = - 2 [-a - 2

W = (u" + aw") sin ^ + (w" - au") cos E3

(u'2 w'2

3 T_ )

,+ (u' sin t + w' cos t) (u' cos - w' sin ^)

mz 3 = -(u" + aw") cos C + (w" - au") sin C (A4)

+ (u' sin ^ + w' cos (u' cos w' sin ^)

(A9c)

Invoking the small strain assumption, the

engineering strains are related to the componentsof the strain tensor according to:

Yyy = Eyy ; Yyx = 2Eyx ; Yyz = 2 Eyz (A10)

21 - 2 (u' cos - w' sin 2 -

2u' cos - w' sin -a

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

[T] _ -(u' cos E - w' sin ^) 1 - u' 2 - w' 2a(u' cos ^ - w' sin c)

(A5)- a(u' sin g + w' cos ^) 2 2 - (u' sin g + w' cos ^)

- - - - - - - - - - - - - - - - - ' - - - - - - - - - - -' - - - - - - - - - - - - - - - - -

a - (u' cos - w' sin ^) u' sin + w' cos c , au

1 (sin

c + w' cos

0 2

• (u' sin E + w' cos f) 1 - 2 - 2

From Fig. 1, the position vector of an arbi-trary point on the blade before deformation is

r = R0 (r) + Xex + ze z (A6)

and that of the same point after deformation is

r 1 = (r 1)+ xex + ze z(A7)

3 3

Then, the Green's strain tensor based on aLagrangian description is given by

dx

dr 1 dr 1 - dro • dr o = 2[dx dr dzj[Ei^] or (A8)

dz

Substituting Eqs. (Ab) and (A7) into Eq. (A8)

and retaining terms up to second-degree in u, v,w, and a, one obtains the following expressionsfor the required components of strain:

It should be pointed out that in arriving at

the expressions given in Eq. (A9) several higher-order terms have been discarded based either onconsiderations related to small deformations or an

approximations due to slenderness of the blade.

It is convenient to eliminate the axial equa-

tion of motion. This is done by explicitly con-sidering the foreshortening due to bending andtorsion. The expression for foreshortening forthe present case is derived by making use of theequilibrium condition that the integral of thelongitudinal stress over the cross section must beequal to the total tension. Thus,

T E ff yyy dx dz

A

= EAIv + 2 (u' 2 + w ' 2 ) - eA[(u" + caw") cos

(_2

2(w" - au") sin ^1 + kA a -a'^') (All)

9

from which

V' = v - OF(Al2)

where

Tc

ve=EA

OF =f

r

j 2 (u 2 + w' 2 ) - eA [(u" + aw") cos

H l

( 2

- (w" - au") sin ^] + kA 1 a2 - a'^'^ d (A13)

An alternate expression for T c results from

the kinetic energy and is given in Appendix B.

Substituting Eqs. (Al2) and , (A13) into Eq. (Aga)and invoking the assumption that the blade is rigid(EA + -) along its elastic axis, one obtains

Yyy = -z[(u" + aw") sin 4 + (w" - au") cos ^]

- (x - eA )[(u" + aw") cos ^ - (w" - au") sin ^]

+ 1/z2 + x 2 - kA) (a' 2 - 2a'^') (A14)

The \other components of strain given by Eqs.(A9b) and (A9c) remain unchanged.

Appendix B

Sectional Properties and

Definition of Matrices

Tc fRT

ma 2 r Or

r

I xx = !f z 2 dx dz

I zZ =ffx

2 dx dz - e 2A

B1 =ff Cx2 + z2 - k 2 I dx dz

B2 = / J (x - e A ) ^.c 2 + z 2 - k 22 ) dx dz

J=ff (x 2 + z2) dxdz

A = `f dx dz

AeA =ffx dx dz

AkA2 = ff (x 2 + z 2 ) dx dz

m =Jf pm dx dz

me = ffpm

x dx dz

mkm

= 3 f

pm z2 dx dz1

mkm = //pmx2 dx dz2

k 2 = k 2 + k2m

mlm2

Note the expression for J is valid only for a

circular section. For a noncircular section awarping correction must be included.

(B1)

10

Mhhsll Mhhsl2 " ' M hgsll M hgsl2 " ' M hasll Mhas12

Mhhs21 Mhhs22 " ' M hgs21 M hgs22 " ' M has21 Mhas22

Mghsll Mghs12 " ' Mggsll Mggs12 " ' Mgasll Mgas12 "

[ M S ] =

Mghs21 Mghs22 " ' Mggs21 Mggs22 " ' Mgas21 Mgas22 "

Mghsll Mahs12 " ' Mggsll Mags12 " ' Maas11 Maas12

Mahs21 Mghs22 " ' Mags21 Mags22 " ' Maas21 Maas22

S hhSll (1 + 2i ^hsl ) Shhs12 " ' S hgsll S hgsl2 " ' Shasll S has12 " '

S hhs21 Shhs22(1 + 2ichs2)

[S S ] _ (B2)

Saas21 Saas22 1 2i;a52

Lhhrll L hhrl2 " ' L hgrll L hgrl2 " ' L harll L har12 "

L hhr21 L hhr22 " ' L hgr21 L hgr22 " ' L har21 L har22 " '

L ghrll Lghr12 " ' Lggrll Lggr12 " ' Lgarll Lgar12 "

[ A r]

Lghr21 Lghr22 " ' Lggr21 Lggr22 " ' L gar21 Lggr22 " (B3)

L ghrll Lahr12 " ' Lagrll agr12 " ' Lgarll Laar12

Lahr21 Lo,

hr22 " ' Lagr21 Lc, gr22 " ' Laar21 Laar22

11

ShhsijEOxxo2

0)

n2

m L 4 w2 S lsij + ( w S7sij - S3sij]

Lhari J u1 C

bR)3

Thar A'W• cosj

do0 0

Shgsij

_ _ El xxo S

2sijm0L4wo

1 2

Lghrij = u 0 ( bR) i hhr WjWisin { Cos do

Sh -

n2(

aij w 0) ES8sij - S 10sij - S21sij ] 1 1 2 2L ggrij

u 0 (tj i hhr WjWisin E do

GJ

J

0

2L3b S20sijmow o R

1

Lgarij u0

f(bR)3

ihar Aj W isin C on

Sggsij

El

2

=xxo n

L 4S25sij + w^m w2 S7sij

0

0 0 1

S -

ghsij

EIxxo S Lghrij = u r

2

fo

(bR)

3

L ahr WjAicos

_

do2sijm L 4 w 2 0 a0

0 0

2 1

Sggsijaw0) (S32sij - S44sij)

GJO

+ m Lab2 S43sij

o Rw

o L- 1-

f

(bbR)

3iW . A

isin ^ doagriju0 a

ahr j

GJoS7lsij

CwSahsij

m w23 b r20o R 10

4 _

. (_ +

S58sijS

+ S

1 Lgarij = u r2

0 a °

( bR)iaar

AjAi do (B5)

54sij 72sij)r 2 0 a0

SagsijGJ

oS 70sij n2

C w o)m w2 L 3 b r2r

1

? (S73sij - S59sij)

S =lsij

1f

0PC zz sin2t

u u+ EI Cos 2 C)Wj W. do

xx

0 o R a0a0

(B4)

GJ0

Saasij = m k2 L2w2

(S51sij + S69sij)o m0 0

+ (n 2(SI7si jw0)

r2- S60s i j

a0

fL hhrij uo C bR)2 ihhrWiWjO

cos2E do

f1

Lhgriju 0 O (bR)2 ihhrWjWi sin

cos do

fo1S 2sij (EIzz - EIxx)WjWi sin ^ cos & do

1S 3sij = Mhhsij M99 sij , 0 mWjWi do0

1 _ , , _

S 7sij = fo TcWjWi do

1

S8sij me(eH + n) Aj W i cos E dofo

1

S10sij = fo

meA j W i cos ^ do

1 , ,

S 20sij = fo

EB2E ' A j W i sin & do

12

1 „

S 21sij = 00

Tc eAAj W isin { do

S 25sij

1

= f(FI ZZ cos 2C + E ^ x s in2C) W 'V do

0

1

S 32sij =

_ _

m(n + e H )eA i i sin doU

1

S 43sij = 00

_EB2C'Ai w icos do

1

S 44sij = f0

_

Tc eAA j W isin C do

1 ,

S 51sij = AEJAj A Ido

0

/' 1 _

S 54sij = %hsijrao= J meWj A i cos do

S 57sij

1

= fo

_ _

Tc AAjAi do

S 58sij=

fo(n +

ieH)em1J A i cos C do

l

S59sij= f

0

_ _

(n + e H )emW j A isin & do

S 605i

J

1 _

(m( - l A.A. cos 2^ doJ i

0 2 1/

S 69sij =

1

foEBI{'2AjAi

_

d

1

S 70sij = f0

EB2 Wj A icos ^ d_,n

1 „

S 71sij = f0

E62 W j A isin ^ do

1

S 72sij = f0

_ _Tc eAWj A i cos C do

1 _ _

S 73sij = f0

TceAWjAi sin ^ do

1

M hasij =fo

meAjWicos ^ do

(B6)

1

M gasij = f meA j W i sin C do0

1 _

Maasij fo mmAi i do

fmc, ra =meWjAisin C do0

(B7)

Nondimensional Quantities

_ r - RHTI L = L

b=b/bR

h s hs/bR

9 s = 9s/bR

= two

FTxx = El xx /El xxo

ETZZ = EIZZ /El xxo

hJ = GJ/GJ0

EA = EA/EA0

m = m/m0

e = eA/bR

e = e/bR

e = RH/L

T c _ Tc /ma 2L2

FBI = EB1/GJ0L2

EB2 = EB2/GJ0L

km = km/km

0

k = km I m /k m

0

km = km /km2 2 0

13

5. Hodges, D. H., and Dowell, E. H., "Nonlinear

Equations of Motion for the Elastic Bending andTorsion of Twisted Nonuniform Rotor Blades,"NASA IN D-2818, December 1974.

6. Rosen, A., and Friedmann, P., "Nonlinear Equa-

tions of Equilibrium for Helicopter or Wind

Turbine Blades Undergoing Moderate Deforma-tion," NASA CR-159478, December 1978.

7. Kaza, K. R. V., and Kvaternik, R. G., "Non-

linear Aeroelastic Equations for Combined Flap-wise Bending, Chordwise Bending, Torsion andExtension of Twisted Nonuniform Rotor Blades inForward Flight, NASA TM-74059, August. 1977.

8. Kaza, K. R. V., "Nonlinear Aeroelastic Equa-

tions of Motion of Twisted Nonuniform, Flexible

Horizontal-Axis Wind Turbine Blades," NASACR-159502, July 1980.

9. Kaza, K. R. V., and Kvaternik, R. G., "Aero-

elastic Equations of Motion of a DarrieusVertical-Axis Wind Turbine Blade," NASATM-79295, December 1979.

10. Kaza, K. R. V., and Kvaternik, R. G., "Non-

linear Flap-Lag-Axial Equations of a RotatingBeam," AIAA Journal, Vol. 15, No. 6, June 1977,pp. 871-874.

11. Smith, S. N., "Discrete Frequency Souna Genera-tion in Axial Flow Turbomachines," ARC-R&M No.3709, 1973.

12. Adamczyk, J. J., and Goldstein, M. E.,

"Unsteady Flow in Supersonic Cascade withSubsonic Leading-Edge Locus," AIAA Journal,Vol. 16, No. 12, December 1978, pp. 1248-1254.

13. Lane, F., "System Mode Shapes in the Flutter of

Compressor Blade Rows," Journal of the Aeronau-tical Sciences, Vol. 23, No. 1, Jan. 1956,pp. 54-56.

rQ = k m /bR0 0

moA0°m

u o = m a/nobR

k A = kAlbR

L -RT - RH

at = at Wo

arLan

(B7)

References

Kaza, Krishna Rao V. and Kielb, Robert E.,

"Effects of Mistuning on-Bending-TorsionFlutter and Response of a Cascade in Incom-pressible Flow," AIAA Paper No. 81-0532presented at the AIAA Dynamics SpecialistsConference, Atlanta, Georgia, April 9-10, 1981(DOE/NASA/1028-29; NASA TM-81674), scheduled

for publication in the AIAA Journal, August orSeptember 1982.

2. Kielb, Robert E., and Kaza, Krishna Rao V.,

"Aeroelastic Characteristics of a Cascade ofMistuned Blades in Subsonic and Supersonic

Flows," ASME Paper No. 81-DET-122, presented atthe Eighth Biennial ASME Design EngineeringConference on Mechanical Vibrations and Noise,Hartford, Connecticut, September 20-23, 1981

(NASA TM-82631), accepted for publication inthe ASME Journal of Mechanical Design.

3. Kielb, Robert E., "Aeroelastic Characteristics

of a Mistuned Bladed-Disc Assembly," Ph.D.Dissertation, The Ohio State University,Columbus, Ohio 1981.

4. Houbolt, J. C., and Brooks, G. W., "Differen-

tial Equations of Motion for Combined FlapwiseBending, Chordwise Bending, and Torsion ofTwisted Nonuniform Rotor Blades," NACA Report1346, 1958.

J.

14

TABLE I. - BLADE SECTIONAL PROPERTIES

F

0.0199 0.1017 0.2372 0.4083 0.5917 0.7628 0.8983 0.9801

EI 0.9719 0.9665 0.7827 0.6072 0.3067 0.1633 0.1149 0.0839_xxEI Zz 112.4 124.8 152.4 139.4 122.7 119.3 127.7 124.1

GJ 0.0719 0.9665 0.7827 0.6072 0.3067 0.1633 0.1149 0.0839

EA 0.9974 1.0225 0.9711 0.9358 0.7641 0.6482 0.6038 0.5541

EB 1 0.4436 0.5338 0.5675 0.7275 0.6901 0.7694 0.9458 0.9734

EB2 0 0 0 0 0 0 0 0

e 0 0 0 0 0 0 0 0

TA0 0 0 0 0 0 0 0

k A 0.5859 0.6095 0.6265 0.6722 0.6972 0.7461 0.7995 0.8226

k m 1 0.0940 0.0925 0.0854 0.0767 0.0603 0.0478 0.0415 0.0370

k m 1.0103 1.0516 1.0816 1.1616 1.2060 1.2913 1.3840 1.42422

m 0.9974 1.0225 0.9711 0.9358 0.7641 0.6482 0.6038 0.5541

b 1.010 1.051 1.082 1.161 1.206 1.291 1.384 1.425

TABLE II. - REFERENCE QUANTITIES

N = 28 om = 4374 kg/m3

bR = 0.0946 m Oa = 1 kg/m3

R H = 0.3876 m ao = 340.3 m/sec

R T = 1.021 m Ao = 0.0034 m2

E = 1.23x10 11N/m2 r = 0.5774a0

G = 4.74410 10 N /m2 mo = varies, rad/sec

l xxo = 9.19x10- 8 m4Mh = M9 = Ma = 2

Jo = 3.676x10 -7 m 4= tan-1 [1.552(n + eH)]

15

LIND

V

MED

X^

YQ,y

ZQ

Figure 1. - Blade coordinate systems beforeand after deformation.

Va

ZQ

Veff

Q ^a

w

ELASTIC AXIS

.*, eA

e^

--- CENTER OF GRAVITY

\\ --- AREA CENTROID

i ux

z

X^

Figure 2. - Coordinate systems of blade cross section.

50010e 9e 8e 7e 6e// /

400

300

CY

UzwwOf 200

100

0

5e

i

2^4e

3e

IT

2e

1B

/ — le

-T I I I

1000 2000 3000 4000 5000

ROTOR SPEED (rpm = 60 MiT)

Figure 3. - Campbell diagram of the fan blade.

1.04

I :^^

W

Q 1.02

ID

zW

1.00Q

z .98c^Q

REFERENCE

STATION

v 9010

0 83%

0 76%

q 68%

o

e°o

o ^v °

0 0 qv

0 n

0

ABEAM MODEL

° ^v

VA 0 °p° `^ °` o

A

0STABLE-" UNSTABLE

96 ' 1'

-.04 -.03 -.02 -.01 _ 0 .01REAL PART OF EIGENVALUE, µ

Figure 4. - Comparison of eigenvalues from beam model and typicalsection models: a o = 1409.6 radlsec, Q = 446.8 radlsec,Max = 0.55.

1.06

i:^h

LLj

Jazwc.^w

O

Qa.}

Qzc^Q

m

zUQ

QX

STABLE UNSTABLE00

° A1.04

°

° p°°

O O 00 0

1.02 0 OO

O 0O O O

00 O 0

1.00 00 O 00O 0 0 0 00

0 TUNED

98° MISTUNED

(ALTERNATING)

96-.04 -.03 -.02 -.01 0 .01

REAL PART OF EIGENVALUE, p

Figure 5. - Comparison of eigenvalues of tuned and mistunedcascade: wo = 1391.1 radlsec, Q = 446.8 radlsec, M ax = 0.55.

60 UNSTABLE

STABLEUNSTABLE

STABLE

t^ ~DTUNED DAMPED, h560.002-TUNED, ^aS 1 ' 2 =

UNDAMPED ^^ MISTUNED,DESIGN POINT UNDAMPED

52 \

h ^ ^48 UNSTAB+— h

STABLE

44

40

3000 3400 3800 4200 4600 5000

ROTATIONAL SPEED (rpm = 60Q/2TT)

Figure 6. - Comparison of tuned and mistuned flutter boundaries.

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No,NASA TM-82813

4. Title and Subtitle 5. Report Date

COUPLED BENDING-BENDING-TORSION FLUTTER OF A6. Performing Organization CodeMISTUNED CASCADE WITH NONUNIFORM BLADES

505-33-527. Author(s) 8. Performing Organization Report No.

Krishna Rao V. Kaza, The University of Toledo, Toledo, Ohio 43606and Lewis Research Center and Robert E. Kielb, Lewis Research Center

E-115610. Work Unit No.

9. Performing Organization Name and Address

National Aeronautics and Space Administration 11. Contract or Grant No.Lewis Research Center NSG-3139Cleveland, Ohio 44135 13. Type of Report and Period Covered

Technical Memorandum12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration 14. Sponsoring Agency Code

Washington, D. C. 20546

15, Supplementary Notes

Prepared for the Twenty-third Structures, Structural Dynamics, and Materials Conference

cosponsored by the AIAA, ASME, ASCE, and AHS, New Orleans, Louisiana, May 10-12, 1982.

16. Abstract

A set of aeroelastic equations describing the motion of an arbitrarily mistuned cascade withflexible, pretwisted, nonuniform blades is developed using an extended Hamilton's principle.The derivation of the equations has its basis in the geometric nonlinear theory of elasticity inwhich the elongations and shears are negligible compared to unity. A general expression for

foreshortening of a blade is derived and is explicitly used in the formulation. The blade aero-dynamic loading in the subsonic and supersonic flow regimes is obtained from two-dimensional,unsteady, cascade theories. The aerodynamic, inertial and structural coupling between thebending (in two planes) and torsional motions of the blade is included. The equations are usedto investigate the aeroelastic stability and to quantify the effect of frequency mistuning on flutterin turbofans. Results indicate that a moderate amount of intentional mistuning has enoughpotential to alleviate flutter problems in unshrouded, high-aspect-ratio turbofans.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Aeroelasticity Unclassified - unlimitedCascade flutter STAR Category 39

Mistuning

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price'

Unclassified Unclassified

For sale by the National Technical Information Service, Springfield, Virginia 22161

National Aeronautics andSpace Administration

Washington, D.C.20546

Official Business

Penalty for Private Use, $300

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