Chordwise spacing aerodynamic detuning for unstalled supersonic flutter stability enhancement

Journal of Sound and Vibration (1987) 115(3), 483-497

CHORDWISE FOR

SPACING

UNSTALLED STABILITY S.

FLEETER

AERODYNAMIC SUPERSONIC

DETUNING FLUTTER

ENHANCEMENT AND

D.

HOYNIAK

Thermal Sciences and Propulsion Center, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, U.S.A. (Received 17 April 1986, and

in

revised form 2 September 1986)

Unstalled supersonic flutter is a significant problem in the development of advanced gas turbines because it restricts the high speed operating range of the engine. A new approach to passive control of unstalled supersonic flutter is aerodynamic detuning, defined as designed passage-to-passage differences in the unsteady aerodynamics of a blade row. In this paper, a mathematical model is developed to predict the unstalled torsion mode stability of an aerodynamically detuned turbomachine rotor operating in a supersonic inlet flow field with a subsonic axial component, with the aerodynamic detuning accomplished by alternate chordwise spacing of adjacent rotor blades. The unsteady aerodynamic moments acting on the blading are calculated in terms of influence coefficients. The stability enhancement associated with this alternate chordwise aerodynamic detuning is demonstrated utilizing an unstable twelve bladed rotor based on Verdon’s Cascade B flow geometry. This model and unstable baseline rotor configuration are then used to show that axial spacing detuning leads to greater flutter stability enhancement than does circumferential spacing aerodynamic detuning. Finally, the trade-offs between structural damping, alternate chordwise aerodynamic detuning, and alternate circumferential aerodynamic detuning are considered. 1. INTRODUCTION Unstalled supersonic flutter is a significant problem in the development of advanced gas turbines because it restricts the high speed operating range of the engine. In the flow regime associated with this type of flutter, the outer span of the blading is in a supersonic inlet flow field with a subsonic axial component. To remove this flutter from the engine operating range, an aerodynamic performance penalty is paid and part-span shrouds are incorporated into the rotor design. Aerodynamic detuning is a new approach to passive flutter control of turbomachine rotors which offers the potential of extending the flutter free operating range without resorting to part-span shrouds. Aerodynamic detuning is defined as designed passage-topassage differences in the unsteady aerodynamic flow field of a blade row. Thus, aerodynamic detuning affects the primary driving mechanism for flutter-the unsteady aerodynamic loading on the individual blades. This results in the blading not responding in a classical traveling wave mode typical of the flutter behavior of a conventional aerodyspaced rotor. To date, only one mechanism namically tuned rotor: i.e., a uniformly for aerodynamic detuning has been considered: alternate circumferential rotor blade spacing [ 11. In this paper, a new type of aerodynamic detuning is considered-alternate chordwise spacing of adjacent rotor blades. A mathematical model is developed to predict the torsion mode unstalled supersonic flutter stability of an aerodynamically detuned turbomachine 483 0022-460X/87/120483+

15 !%03.00/0

0

1987 Academic

Press Inc. (London)

Limited

S.

484

FLEETER

AND

D.

HOYNIAK

rotor, with the aerodynamic detuning accomplished by alternate chordwise spacing of adjacent rotor blades. The unsteady aerodynamic moments acting on the blades are calculated in terms of influence coefficients in a manner which enables the torsion mode stability of a chordwise aerodynamically detuned rotor as well as a conventional aerodynamically tuned rotor to be determined. The stability enhancement associated with alternate axial-circumferential aerodynamic detuning is then demonstrated utilizing an unstable twelve bladed rotor based on Verdon’s Cascade B flow geometry. In addition, the trade-offs between structural damping, alternate chordwise aerodynamic detuning, and alternate circumferential aerodynamic detuning are considered. 2. UNSTEADY AERODYNAMIC MODEL The unstalled supersonic flutter characteristics of conventional aerodynamically tuned rotors are determined from mathematical models based on a flat plate airfoil cascade embedded in a supersonic inlet flow with a subsonic axial component normal to the locus of blade leading edges (see Figure 1). The fluid is assumed to be an inviscid, perfect gas, with the flow isentropic, adiabatic, irrotational, and two dimensional. The isentropic assumption implies that the entropy produced by shock waves is negligible. In addition, the shocks are assumed to be weak and are thus modeled as Mach waves. The unsteady wakes are thin vortex sheets which emanate from the trailing edges of the blades and extend infinitely far downstream.

Figure

1. Flat plate airfoil

cascade

in a supersonic

inlet Row with a subsonic

leading

edge profile

The physical domain thus contains an infinite number of characteristics across which the perturbation velocities as well as the velocity derivatives are discontinuous. These lines of finite discontinuities originate at the leading and trailing edges of the airfoils. Also, the unsteady convected vortex wakes which originate at the airfoil trailing edges influence every blade above them in the cascade. This is in contrast to the supersonic isolated airfoil where the wake cannot influence an upstream airfoil. As a result, the wakes must be considered in the calculation of the exit flow field. The unsteady aerodynamics are analyzed by considering the small perturbation harmonic torsion mode oscillations of the airfoil cascade in a classical traveling wave mode, i.e., with a constant interblade phase angle, p, between adjacent airfoils [2-61. Of particular

UNSTALLED

SUPERSONIC

FLUTTER

485

STABILITY

interest herein are the analyses which utilize a finite cascade representation of a semiinfinite cascade. In these, the cascade periodicity is enforced by stacking sufficient numbers of uniformly spaced single airfoils until convergence in the unsteady flow field is achieved. For the aerodynamically detuned, alternate chordwise spaced rotor, an analogous unsteady aerodynamic model is utilized. In particular, the unsteady aerodynamics associated with the small perturbation torsion mode harmonic oscillations of an alternately chordwise spaced two-dimensional flat plate airfoil cascade embedded in an inviscid, supersonic inlet flow field with a subsonic axial component are considered. The analysis of this type of aerodynamically detuned cascade configuration is most easily accomplished by utilizing a finite cascade representation of the semi-infinite cascade (see Figure 2). As seen, this alternate chordwise aerodynamically detuned cascade is composed of two distinct flow passages and two separate sets of airfoils. For convenience, these airfoil sets are termed the set of odd numbered airfoils and the set of even numbered airfoils. Thus, two passage periodicity is required for this detuned cascade: i.e., the periodic cascade unsteady flow field is achieved by stacking sufficient numbers of two non-uniform flow passages, or two airfoils, one from each set. /

Figure

2. Finite cascade

representation

for alternative

chordwise

,

aerodynamic

detuning.

Each of the individual sets of airfoils has a constant spacing equal to S,. Thus, the sets of even and odd numbered airfoils can be considered as individual cascades of uniformly spaced airfoils, each with twice the spacing of the associated baseline uniformly spaced cascade. An interblade phase angle for this aerodynamically detuned cascade configuration can be defined. In particular, each set of airfoils is individually assumed to be executing harmonic torsional oscillations with a constant aerodynamically detuned interblade phase angle, Pd, between adjacent airfoils of each set (see Figure 2). Thus, this detuned cascade interblade phase angle is twice that for the corresponding baseline uniformly spaced cascade. Figure 3 schematically depicts the aerodynamically detuned cascade geometry. R,, and R, denote the reference airfoils for the set of odd numbered and the set of even numbered airfoils, respectively. The alternate chordwise aerodynamic detuning is specified by the parameter R,,,, , the axial distance between the leading edges of the displaced airfoils as a percentage of the airfoil chordlength. In addition, circumferential spacing aerodynamic detuning is accounted for through the parameter P,,o,ma,,the airfoil normal direction spacing as a percentage of the flow passage between the airfoils. Thus, P$,,,, = 0% and P normal= 50% correspond to a uniformly spaced tuned cascade configuration.

486

S. FLEETER

AND

D.

HOYNIAK

Figure 3. Alternate chordwise aerodynamic detuning cascade geometry.

The unsteady continuity and Euler equations are linearized by assuming that the unsteady perturbations due to the harmonic airfoil oscillations are small as compared to the uniform throughflow. Thus, the boundary conditions, which require the unsteady flow to be tangent to the blade and the normal velocity to be continuous across the wake, are applied on the mean positions of the oscillating airfoils. The formulation of the linearized differential equations describing the unsteady perturbation quantities for the finite aerodynamically detuned cascade is based on the method of characteristics analysis of the finite uniformly spaced airfoil cascade of Brix and Platzer [7]. In particular the dependent variables are the non-dimensional chordwise, normal, and sonic perturbation velocities, U, u and a, respectively (a list of nomenclature is given in the Appendix). For harmonic motion at a frequency w, the non-dimensional continuity, momentum, and irrotational flow equations are

aulax+JMk.-1

av/ay+M&aa/ax+ikM~a=O,

au/ax + da/ax + iku = 0,

au/ax-JM&-1

avfax=O.

(la)

(lb, c)

The flow tangency boundary condition requires that the normal perturbation velocity component, v, is equal to the normal velocity of the airfoil surfaces on the mean position of the oscillating airfoils. For an airfoil cascade executing harmonic torsional motions about an elastic axis located at x0 as measured from the leading edge, the dimensionless normal perturbation velocity component on the nth airfoil is v,(x, Y,~,t) = -1y,{l +(x-xx,)ik}

ei(“+“p),

where Y,~denotes the mean position of the airfoils, k is the reduced frequency, p is the interblade phase angle, and (Y,,denotes the amplitude of oscillation of the nth airfoil. To complete the specification of the mathematical model for the case of supersonic inlet flow with a subsonic axial component, the following conditions must also be satisfied. There can be no upstream propagation of disturbances in supersonic flow; pressure disturbances must be bounded at an infinite distance from their source; and disturbance waves impinging on blade surfaces must be reflected, while those impinging on wake surfaces must be transmitted through the wake. Also, the pressure and normal velocity across the wake surfaces are continuous, although the tangential velocity can be discontinuous. Solutions are obtained by the method of characteristics, which identifies characteristic paths along which the partial differential equations can be rewritten in total differential

UNSTALLED

form. The compatibility

SUPERSONIC

FLUTTER

STABILITY

487

equations are

where the subscripts 5, 71,and str indicate that the relation is valid along the left or right running Mach lines or the streamline direction, respectively. Thus, the formulation of the unsteady aerodynamic mathematical model for the aerodynamically detuned cascade is complete. At the intersection points of the characteristics, equations (3) are a system of three differential equations in three unknowns, with the appropriate boundary conditions specified in equation (2). A finite difference method is used to solve these equations at each point in the flow field. The unknown chordwise, normal, and sonic dimensionless perturbation velocities, u, 21,and a, in each of the two periodic flow passages of the semi-infinite cascade are determined by using this scheme in conjunction with a two airfoil passage stacking technique. The dimensionless unsteady perturbation pressure distributions on the surfaces of a reference airfoil from each set in the periodic detuned cascade, airfoils R, and R,, are defined by these perturbation velocities. The non-dimensional unsteady aerodynamic moment acting on a reference airfoil from each set, MR, are then calculated by integrating the unsteady perturbation pressure difference across the chordline, 1 MR, =

I0

Ap(x, y,, t)(x - x,Jdx eiw’= CaaaR e’“‘,

(4)

where Ri is an index which denotes the two reference airfoils R, and R, and C,, is the unsteady aerodynamic moment coefficient. 3. INFLUENCE COEFFICIENT TECHNIQUE The boundary conditions specified in equation (2) require that the airfoils oscillate with equal amplitudes, a situation not appropriate for the detuned airfoil cascade. In addition, the application of this analysis is unnecessarily costly because the complete periodic perturbation flow field must be recalculated, not only for every new cascade geometry and flow condition, but also for each interblade phase angle value considered for a particular cascade and flow condition. These limitations are easily rectified by calculating the unsteady aerodynamic moments on each of the two reference airfoils in terms of influence coefficients: i.e.,

+{&(C&)R,,.R,

+&(&)R,,,R,

+*

“+ciR<(c~)R,,.R<,+*

”

Here, [CILIR,,,R,, denotes the influence coefficient on the reference airfoil, R, or R,, associated with the motion of airfoil number n. Physically it represents the unsteady aerodynamic moment acting on the fixed reference airfoil, R,, or R,, due to unit amplitude motion of airfoil number n. The two groups of bracketed terms in equation (5) are associated with the motion of the sets of odd numbered airfoils and even numbered airfoils, respectively. The complex

S.

488

FLEETER

AND

D.

HOYNIAK

amplitude of harmonic oscillation for the set of odd numbered airfoils is denoted by GR,,exp (i(wr + np,)), where Pd defines the constant interblade phase angle between the sequential odd numbered airfoils. Similarly, the set of even numbered airfoils are assumed to oscillate with the complex amplitude C?R,,exp (i(wt+ np,)), where Pd is the same interblade phase angle as utilized for the even numbered airfoils. The phase difference between the motions of the sets of odd and even numbered airfoils is accounted for in that GR,,and iR,. denote complex amplitudes. Thus, the unsteady moments acting on the two reference airfoils can be written in matrix form as

ECM’IR,, [CM’IR,, I[ [CM’].,. [CM*IR~,

GR,, GR<,

1

(6)

iw,

e

’

where

[CM’IR~,,R~, = [c~],,..‘,+eiPd[C~]R,,,R,+. ’‘+ei”N-“‘21P’[C~]R,,,R,, [CM*]R,,,R, = [C~]R,,,R~+eelp~‘[C’,],,,R~+’ ’.+ei”N-3”21P’f[C~-‘]R,,,R,. The terms [ CM1]R,,,R, describe the influence that the set of odd numbered airfoils has on the unsteady moment developed on reference airfoils R, and R,, respectively. [CM*]R,,,R, represents the effect that the set of even numbered airfoils has on these two reference airfoils. 4. EQUATIONS OF MOTION The equations describing the single degree of freedom torsional motion of the two reference airfoils of the non-uniformly spaced cascade are developed by considering typical airfoil sections, depicted in Figure 4. Positive torsional displacements are defined as clockwise motion such that the blade is in a leading edge up configuration. The elastic restoring forces are modeled by linear torsional springs at the elastic axis location, with the inertial properties of the airfoils represented by the mass moment of inertia about the elastic axis. The equations of motion are then determined, by using Lagrange’s technique, as I,,

(7)

djR,+(1i2ig)z~RWZU~,(YR,=MR,r

Figure 4. Single degree of freedom

torsion model for chordwise

detuned

cascade.

UNSTALLED

SUPERSONIC

FLUTTER

489

STABILITY

where the subscript R, is an index denoting the reference airfoils R, and R,; g denotes the structural damping coefficient, and the undamped natural frequencies are W_ = m. For harmonic motion of the reference airfoils, and by utilizing the unsteady aerodynamic moments defined in equation (6), the single degree of freedom torsion mode equations of motion can be expressed in non-dimensional matrix form as

&‘+R,,

(8)

where p,,,= CL rSK (,,(, =

wR,:,,&

0.0

Ia,< ,,/,/mR,,,,,b’,

+

[CM21Rc,,,

d,

-

e.0=

(1

+

2i!?)pRe

@aR, ,,/, /%,

,,,, riRc,,,dR

Y =

The stability of the cascade is determined exponent (iw):

“Ji/W’,

,,,,, %

PR,,,,

wg

=

=

p *iv.

vb2,

reference frequency.

by relating the frequency

w/o0 = l/A=

ITIR..,,,l

ratio, -y, to the

(9)

Thus a positive value for p, the real part of (w/w,), corresponds to an unstable cascade configuration. The mathematical model specified by equation (8) is appropriate for both aerodynamically tuned and detuned cascades, with the aerodynamic detuning accounted for through the influence coefficients [ CM’]R?,, and [ CM2]Re,,R,,.

5. RESULTS

To demonstrate the effects of chordwise aerodynamic detuning on supersonic unstalled torsion mode flutter, the aeroelastic stability model specified by equation (8) has been applied to a 12 bladed rotor. Verdon’s Cascade B flow geometry [8], depicted in Figure 5, is selected as the baseline airfoil cascade flow and geometric configuration. This cascade has a supersonic inlet flow field with a subsonic axial component and is characterized by an inlet Mach number of 1.281, a stagger angle equal to 63*4”, a solidity of 1.497, and a midchord elastic axis location. The airfoil mass ratio is 193.776 and the radius of gyration, r,,, is 0.3957.

Figure

5. Verdon’s

cascade

B flow geometry.

490

S. FLEETER

AND

D. HOYNIAK

To verify the formulation of this aerodynamically detuned finite cascade model, the limiting case of a uniformly spaced cascade configuration has been considered. In particular, both this detuned finite cascade analysis based on an influence coefficient technique and the uniformly spaced infinite cascade analysis of Adamczyk and Goldstein [6] have been applied to Verdon’s uniformly spaced Cascade B configuration. The real and imaginary parts of the unsteady aerodynamic moment coefficient, C,, defined in equation (4), predicted by using these two models, are presented in Figure 6. Two interblade phase angles are associated with each point: Pd refers to the detuned interblade phase angle whereas p is the conventional uniformly spaced cascade interblade phase angle. As seen, there is excellent correlation between these two analyses for all interblade phase angle values. 1 Or

0.0 -

0.6-

-0,6-

Figure 6. Correlation of finite cascade model and infinite cascade analysis. 0, Adamczyk and Goldstein infinite cascade; A, influence coefficient, finite cascade; M = 1.281; k = 0.5; C/S = 1.497; 6 = 63.4”.

The stability of the baseline rotor at a reduced frequency of 1.1 is shown in Figure 7, which presents the root locus of the eigenvalues. As seen, this baseline rotor is unstable for forward traveling waves characterized by interblade phase angles of 30”, 60”, 90”, and 120”. The stability enhancement due to alternate chordwise spacing aerodynamic detuning is demonstrated in Figure 8. As seen, a rotor with 10% chordwise detuning, while offering increased stability as compared to the baseline rotor, is still unstable for forward traveling waves characterized by interblade phase angles of 30”, 60” and 90”. Further increasing the alternate chordwise detuning to 20% results in a stable rotor configuration for all interblade phase angle values. The effect of incorporating alternate circumferential spacing aerodynamic detuning into the baseline unstable rotor at a reduced frequency of 1.1 is considered in Figure 9. As seen, at this reduced frequency value, the introduction of up to *50% circumferential spacing aerodynamic detuning into the baseline rotor design does not lead to a stable rotor configuration.

UNSTALLED

SUPERSONIC

FLUTTER

104

STABILITY

491

I /

Stable

1 Unstable I I

-%

I I

-06OO

103 -BOO

102 -

:

102oO

I

-? 1.01

I

87

3 E 100 @ g 5 0.99

3

-Go0

I

I

lb

0.90-

I& I

,$/

09?-

.090e

I 120°

O-96 I

0.95 -0.06

Figure 7. Stability

of baseline

-004

uniformly

spaced

I

I

12 bladed

I

002

0

-0.02 Real (w/wa)

k = 1.1; Pnw,,,o, = 50%; P,,G,, = 0%.

rotor. Baseline:

1041.03102-,101‘; IOO-

3 s

1099-

096-

0.970.96

-

0.951

-006

Figure

8. Stability

enhancement

1

1 -0.04

due to alternate

1

1 1 -0.02 Real (w/q,)

chordwise

1

0

aerodynamic

1

1 0.02

detuning.

k = 1.1; P ,,,,,),,‘,, = 50%.

p\,,,,: 0, 0%; 0, 10%; +, 20%.

To have an aggressive aerodynamic rotor design, i.e., a design not unduly limited by an overly conservative flutter stability criterion, it is necessary to account for the structural damping inherent in the system. Also, it is difficult to ascertain the degree of instability of the rotor configurations from the root locus plots of Figures 7,8 and 9. Thus, the effect of structural damping on stability for the unstable baseline and unstable aerodynamically detuned rotor configurations has also been considered: i.e., the level of structural damping required for stability of an unstable rotor is determined.

S. FLEETER

492

AND

D. HOYNIAK

Sf~ble

A

103-

A

1 oz-

0 O

0

“0

0

0

I

A lo 3

IOl-

0

I I I ’ I I

o

3

BI

g loo.c_ g

m

5

I / / 1

0

/m

0.99-

0

oA

0

0.98-

0

0.97

/

0

I A0

Alo On/&b

0.96095 -006

I

I -0 04

I

I -0.02

I

I 0

1

I 0.02

Real (who) Figure 9. Effect of alternate circumferential aerodynamic detuning on stability. k = 1.1; P,,,,, = 0%. P,,,,n,lo,: 0, 75%; 0, 65%; A, 60%.

Figure 10 shows the effect of structural damping on the stability of the baseline rotor. As seen, this unstable baseline rotor requires a structural damping value equal to 0.008 for stability. The influence of structural damping on the unstable alternate chordwise and alternate circumferential aerodynamically detuned rotors at a reduced frequency of 1.1 is presented in Figures 11, 12 and 13. An unstable rotor with 10% chordwise spacing lC4

r

Figure 10. Effect of structural damping on stability. k = 1.1; P,,,,nl,o, = 35%; P,,‘,).,= 0%. g: ?,?0.0; 0,0.004; +, 0.008.

UNSTALLED

SUPERSONIC

FLUTTER

493

STABILITY

1047 Stable I Unstable 1031.02-

1

‘,O’-

F ICKI? g & 0,990.90

0.97 i

-0-06

-004

0

-DO2 Real Wool

002

Figure 11. Structural damping required for stability of a rotor incorporating detuning. k = 1.1, P,,,,_, = 50%; P,,,,, = 10%. g: Cl, 0.001; +, 0.003.

1.04-

alternate

chordwise

aerodynamic

S$ble\ Unstable

I.03 1.021.01 a 7 3 IGOf ? g 0.99+5 0,900.97O-96 0.951

-0.06

1

1 -0.04

’

’ -0.02

’

’ 0

I

I

0.02

Real (w&I Figure 12. Structural damping required for stability of a rotor incorporating alternate circumferential =O%. g:Cl,0.0;0, 0.004; +, 0.008. namic detuning. Baseline: k = 1.1; P,,,,,,r,u, = 50%; P,,",,

aerody-

detuning requires structural damping of 0.003 for stability (see Figure 11). Structural damping values of 0.008 and O-007 are required to stabilize rotors which incorporate circumferential spacing detuning (see Figures 12 and 13). A significant parameter with regard to flutter of turbomachine rotors is the reduced frequency, k. To demonstrate the effect of reduced frequency on unstalled supersonic

S. FLEETER

AND

D.

HOYNIAK

1.03-

1-02-

^o l-01 B 3 ;: l,OOi e 2 0.99 H 0.90 -

o-97 -

0.96 0,951

1

-006

1 -004

1

1 -0.02 Real (who)

1

’

’

0

I 002

Figure 13. Structural damping required for stability of a rotor incorporating alternate namic detuning. k = 1.1; P,,, ,,,,a, = 65%; P,,,,, =O%. g: Cl, 0.0; 0, 0.005; +, 0.007.

circumferential

aerody-

flutter, the baseline rotor is further destabilized by decreasing the reduced frequency value for this baseline rotor configuration, as shown in Figure 14. At a reduced frequency of l-0, this rotor is unstable for forward traveling waves characterized by interblade phase angles of 30”, 60”, 90”, 120” and 150”. The large degree of instability of this reduced frequency 1-O rotor is evidenced by its requiring a structural damping value equal to 0.016 for stability, as shown in Figure 15. 104r

L-Unstable z3 1.01

-3 -

t -1020°

z

I

=‘l-00

x0.99

5

0,98 097

I I

I

t

096

t

0.951 -006

Figure

14. Stability

of uniformly

-0.04

spaced

-0.02 Real (w/a-J

rotor with k = 1.0. Baseline:

k = 1.0; I’_,<,, ,,,“, = 50%; P, ,“,, = 0%.

UNSTALLED

SUPERSONIC

FLUTTER

495

STABILITY

::::02 -006

-0.04

Real LJ/WO) Figure

15. Structural

damping

required

for stability of a uniformly x, 0.016.

spaced

rotor

with

k = 1.0.k = 1.0;

Pnomm,= 50%; P,,,,, =O%. g: 0, 0.0;0, 0.012;

Figures 16 and 17 show the effect of alternate chordwise spacing aerodynamic detuning on the stability of this rotor at a reduced frequency of 1.0. Chordwise aerodynamic detuning, while increasing the stability of the rotor, does not result in a stable design, as shown in Figure 16. To stabilize a rotor with 20% chordwise detuning at a reduced frequency of l-0, a structural damping of O-007 is required, as shown in Figure 17.

-0-08

Figure

16. Stability

enhancement

-004

-002 Real (w/we)

due to alternate

P,,,,,,,1,,, = 50%. P,,,,,: A, 0%; 0, IO%, Cl,25%.

chordwise

0

002

aerodynamic

detuning

with k = 1 ,O. k = 1.0;

496

S

FLEETER

AND

D. HOYNIAK I

104-

StobJe

j Unntoble

1-03-

;j P l.OO3 f .go99E c-( 098-

-006

-0.04

-002 Reol (who)

0

0.02

Figure 17. Structural damping required for stability of a rotor incorporating alternate chordwise aerodynamic detuning with k = 1.0. k = 1.0; P ““,,,,ol = 50%; P,,,,, = 25%. g: 0, 0.0; 0, 0.005; x, 0.007.

6. SUMMARY

AND

CONCLUSIONS

A mathematical model has been developed to predict the torsion mode stability of an aerodynamically detuned supersonic turbomachine rotor, with the aerodynamic detuning accomplished by alternate chordwise spacing of adjacent blades. The unstalled supersonic torsion mode flutter stability enhancement associated with alternate chordwise aerodynamic detuning was demonstrated by applying this model to an unstable twelve bladed rotor based on Verdon’s Cascade B flow geometry. This model and unstable baseline rotor configuration were also used to show that alternate axial spacing aerodynamic detuning offers significantly increased stability enhancement as compared to alternate circumferential aerodynamic detuning. In addition, the effect of structural damping, i.e., the structural damping required for stability, was considered. This showed that at a reduced frequency of 1.1, 20% chordwise aerodynamic detuning was equivalent to a structural damping of 0.008: i.e., either structural damping of 0.008 or the introduction of 20% chordwise spacing aerodynamic detuning, was required to stabilize the baseline rotor. Finally, the trade-offs between structural damping, alternate chordwise aerodynamic detuning, and alternate circumferential aerodynamic detuning were examined. It was found that alternate chordwise detuning decreased the structural damping required for stability of the baseline unstable rotor by more than a factor of 2, from O-016 to 0.007 with 20% chordwise detuning at a reduced frequency of 1.0, and from 0.008 to 0.003 with 10% chordwise detuning at a reduced frequency of 1.1. With *30% circumferential aerodynamic detuning, structural damping of 0.007 and 0.008 were required for stability eat a reduced frequency of 1.1.

REFERENCES 1. D. HOYNIAK

and S.FLEETER 1985 American Society ofMechanical Engineers Paper 85-GT-192.

Aerodynamic detuning analysis of an unstalled supersonic turbofan cascade.

UNSTALLED

SUPERSONIC

FLUTTER

STABILITY

497

1977 American Society of Mechanical Engineers Papers 77-GT-44 and 77-GT-45. Furtherdevelopmentsin the aerodynamic analysis of an unsteady supersonic cascade, Parts 1

2. J. M. VERDON

and 2. 3. M. K. KUROSAKA 1974Journal of Engineering for Power 96, 13-31. On the unsteady supersonic cascade with a subsonic leading edge-an exact first order theory-Parts 1 and 2. 4. J. M. VERDON and J. E. MCCUNE 1975 American Institute of Aeronautics and Astronautics Journal 13, 193-201. Unsteady supersonic cascade in subsonic axial flow. 5. J. E. CARUTHERS and R. E. RIFFEL 1980 Journal of Sound and Vibration 71, 171-183. Aerodynamic analysis of a supersonic cascade vibrating in a complex mode. 6. J. J. ADAMCZYK and M. E. GOLDSTEIN 1979American Institute of Aeronautics and Astronautics Journal 16, 1248-1254. Unsteady flow in a supersonic cascade with subsonic leading-edge locus. 7. C. W. BRIX JR. and M. F. PLATZER 1974 American Institute of Aernonautics and Astronautics Paper 74- 14. Theoretical investigation of supersonic flow past oscillating cascades with subsonic leading-edge locus. 8. J. M. VERDON 1973 Journal of Applied Mechanics 40, 667-671. The unsteady aerodynamics of a finite supersonic cascade with subsonic axial flow.

APPENDIX:

NOMENCLATURE

dimensionless perturbation sonic velocity airfoil chord C unsteady aerodynamic moment coefficient influence coefficient of airfoil n G-f structural damping g I mass moment of inertia k reduced frequency, k = oC/ U, K spring constant M dimensionless unsteady aerodynamic moment cascade inlet Mach number MCC Pnormol airfoil normal direction spacing PJILlT, airfoil leading edge location S airfoil spacing cascade solidity CIS U dimensionless perturbation chordwise velocity free stream velocity urn V dimensionless perturbation normal velocity a amplitude of oscillation ,. complex oscillatory amplitude interblade phase angle p” ” oscillatory radian frequency reference radian frequency WO cascade stagger angle matrix a

C

(Ioi

Subscripts d

n R

R, R,,

detuned cascade airfoil number reference airfoil reference for set of even numbered airfoils of detuned cascade reference for set of odd numbered airfoils of detuned cascade