Direct Use of Unsteady Aerodynamic Pressures in the Flutter Analysis of Mistuned Blades

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Direct Use of Unsteady Aerodynamic Pressures in the Flutter Analysis of Mistuned Blades

A. Srinivasan, G. Tavares

To cite this version:

A. Srinivasan, G. Tavares. Direct Use of Unsteady Aerodynamic Pressures in the Flutter Anal- ysis of Mistuned Blades. Journal de Physique III, EDP Sciences, 1995, 5 (10), pp.1587-1597.

�10.1051/jp3:1995212�. �jpa-00249403�

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J. Phys. III France 5 (1995) 1587-1597 OCTOBER1995, PAGE1587

Classification Physics Abstracts

46.30M

Direct Use of Unsteady Aerodynamic Pressures in the Flutter

Analysis of Mistuned Blades

A-V- Srinivasan (~) ^and ^G-G- ^Tavares (~)

(~ Department of Mechanical Engineering, University of Connecticut, Storrs, Connecticut, USA (~) ^Pratt ^and Whitney Aircraft, East Hartford, Connecticut, ^USA

(Received 6 December 1994, revised 27 June 1995, accepted 17 July 1995)

Abstract. An aeroelastic stability analysis of _a cascade of engine blades coupled only through aerodynamics is developed. The unique feature of the analysis is the direct _use of unsteady aerodynamic _pressures, rather than lifts and moments, in calculating the susceptibil, ity of _a cascade to flutter. The approach developed here is realistic and relevant for analysis

of low aspect ratio blades. However, in the calculations presented in this paper, the surface is assumed to be divided into equal elemental _areas. The formulation leads _to _a complex eigen-

value problem, the solution of which determines the susceptibility of the cascade to flutter. The

eigenvalues of _an assembly of alternately mistuned blades, operating at high reduced frequencies,

appear ^to be very sensitive to the level of mistuned frequencies. The locus of eigenvalues shows

a strong tendency to split _even for very small percentage differences between the frequencies of the two sets of blades. Further, blades with identical frequencies, but alternately mistuned

mode shapes, operating at high reduced frequencies show _a tendency towards instability.

Nomenclature

A ₌ Unsteady aerodynamic forces acting on a blade

C ₌ Normalizing coefficients in the representation of blade modes

h

= Vibratory displacement perpendicular to the local chord

K = Stiffness

I = Mesh point _on _a ^blade

m = Mass distribution N

= Number of blades

P = Unsteady aerodynamic force acting on a

blade element

q = Physical displacements of blade

s = Vibratory displacement along the local chord

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fl ₌ Interblade phase angle

~ = Generalized coordinate

uJ = Natural frequency of individual blades

# = Eigenvector ^of ^individual blade motion 4l = Matrix of eigenvectors

Subscripts

p = pth mode of vibration

r = rth blade

Introduction

The continuing need to compute accurately flutter susceptibility of engine blades arises from the trend to introduce _more low aspect ^ratio blades in advanced engine designs. The vibratory mode shapes of these blades _are obviously plate-type with significant variations in the deformation pattern along a chord. Clearly beam representations are unsuitable for describing

such deformation. Similarly lifts and moments which represent integrated effects of pressure distribution _are also not adequate. Aerodynamic analyses and corresponding codes typically

lead to unsteady _pressures on a blade profile and the general procedure in the past has been to calculate lifts and moments by integration of the pressure profiles. Such _an approach is

clearly unsuitable in application to low aspect ratio blades. It is also unnecessary because the

unsteady _pressures _can be directly used in _a formulation to compute ^flutter speeds. This paper is _an attempt at direct _use of unsteady _pressures and the formulation developed here leads to

the familiar complex eigenvalue problem, the solution of which determines the conditions at which _an assembly of mistuned blades is _prone to flutter. Previous attempts to _use unsteady

pressures ^to calculate flutter of blade assemblies have been restricted to the work _per cycle approach applied to _a system ^of ^tuned ^blades.

1. Analysis

Consider _an assembly of engine blades that _are different from each other and coupled only through aerodynamics, _as shown in Figure I. The difference _among the blades _may arise due either to different frequencies and/or mode shapes and/or damping in each blade of the

assembly. Let the vibration of the rth blade in its pth mode be represented _as hi ^~

51) 14»lr

= Cl ^~2»

S2j,

And for the kth blade vibrating in its ith mode

hi, ^~

St,

(§ii)k

" C~ ~2,

s2,

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N°10 FLUTTER ANALYSIS OF WIDE-CHORD ENGINE BLADES 1589

blade indices

~ ~ N-I N 2

~ deformed blade

,

' blade

element areas

Fig. I. Cascade of cantilevered blades.

where C~ are normalizing coefficients, and _s and h _are displacements along and perpendicular

to the local chord of each blade.

Consider the vibrations of kth blade only. The governing differential equation is

imi~iql~ ₊ iKiiql~ ₊ iAl~

= ° Ii)

where matrix [m] represents ^the mass distribution, (q) matrix represents ^the physical displacements, [K] the stiffness matrix, and [A] ^the unsteady aerodynamic forces pertaining to the kth blade.

The solution of the eigenvalue problem with (A) ₌ 0 gives the frequencies and mode shapes for each natural mode, I-e-, _uJ~ and (#~) for each mode I

= 1,2,3,. of the kth blade.

Let

14~i ⁼ 1<ii i<~i 1<31 1<pi

for _a total of _p natural modes

iqi ₌ 14lii~i

where the number of generalized coordinates _~ corresponds to the total number of "blade alone" modes used in the calculation.

14l~'i~imi~14li~lil~ + i4l~'i~iKi~i4li~l~l~ + i4l~'i~lAl~ ₌ °

k ₌ 1, 2, 3,

,

N (number of blades)

If m$ is the modal _mass in ith mode of kth blade, then the generalized _mass _can be shown to be

[4l~]~[m]~[4l]~ ₌ iii and the corresponding generalized ^stiffness to be [4l~]~[K]~[4l]~ ₌ _[uJ(]~.

If _one chooses

~ l

c~ ⁼ ~

/~

then for the kth blade, the governing equation of motion ii) reduces to

(~)~xi + (~Jj)~(~)~ ⁺ (4l~)~(A)~

" ° (2)

Structural Aerodynamic

k=1,2,3,. ,N

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Phi ^~ list llh2

(A)~

=

~s2 (3)

1~h3 lls3

where Phi

" force normal _to _a chord _at elemental _area I of blade k and Psi

= force parallel

to _a chord at elemental _area I of blade k.

~'t N

" ~(~~'~,to~t _~ ~~'~tQ~t)~~ (~)

n=1

where Aa

= (Ax)(Ar); Ax and Ar _are elemental lengths along the chord and _span respectively.

It must be noted that each AP above is obtained upon summation _on all interblade phase angles fir, _r

= 1,2, 3,.. N, I-e-,

N

~~_~~p~ fl -27rr/N

~~~~~ ~~~~'~'~~~

~

The first term in equation (4) represents ^the _pressure along ^h at the lth mesh point region _on

the kth blade, due to unit motion along ^h direction at the lth mesh point regiofl of nth blade alone.

The expression similar _to (4) for vibration along the chord is

N

~~t _" ~(~~~~to~t _~ ~~~/t~~t)~~ (~)

n=1

Clearly, the implication of the representation ⁱⁿ equations (4) and (5) is the twc-dimensional strip theory approach. That is why the influence of motion _at elemental _area I of _a blade _on

elemental _area j ^{of all} ^blades for I # j has been neglected.

For simplicity, if _we _assume Aa to be the _same at each mesh point of each blade (this assumption need not be valid when calculating an actual fan blade), then

llhl ~~l

Psi AP)f,~ AP)/~ q]1

llh2 ~l~$~i ~l~$~i ~~2

~~

~

~ ~~'~,2 ~~'/2 ^~'

~pkn ~pkn

~ ^sh,2 ^ss,2 (fi)

Phi ^~

~ ~l~(f~~ ~l~(/~ ~~t

Psi A~("~ AP$"~ qlt

' AP)j[~ AP)/~

~hL ~ll~~L ~~~/L _~~L

~sL _q~L

Thus,

N

lAl~

" A~ ~l*l~~l~l~

n=1

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N°10 FLUTTER ANALYSIS OF WIDE-CHORD ENGINE BLADES _I59I

Then the equation ^of motion (2) _can ^be rewritten _as

N

(~)~Xl ~ (~J~j~Xp(~)~Xl ^~ ^~~ ~(~~j~X2L[f~~X2L(~~LXp(~)~Xl " (°) (~)

n=1

k=1,2,3,. ,N Here, NP ₌ number of elemental _areas _on each blade. Let

iBl~" ₌ 14~~'l~i*l~"14ll" (8)

Let _us examine _an eleInent of [B] ⁱⁿ ^order to understand the physical meaning of [B]. ^The

three matrices _on the RHS of equation (8) _are shown below when only two blade modes _are included. Recall

/lli ^~

Siz

(~j~)~ =

C~ /l21

~

S21

~~~~

hit ^~ h12 ^~

Sii S12

j@~jk ^~ ~k ck

I 2

hLi hL2

8Li 8L2 _2LXL

j~Tjkj~jknfljn_(C)(hit_C~(h12 ⁸¹ⁱ ^liLi ^Li)~j

812 hL2 8L2)~

~pkn ~pkn

hh,I hs,I

~pkn ~pkn

~~'~ aP))[ _aPjj~ _[li _[12

~pkn ~pkn ^~~ ^~~

~ sh,2 ss,2 ~k ~k (g)

~pkn ~pkn ^I ²

~~$~ _~~$~~ hit ~12

sh,t ss,t ~n ~n

~pkn ~pkn ^Li ^L2

hh,L hs,L

~pkn ~pkn

sh,L ss,L

Thus, [B]~" for 2 modes is a 2 _x 2 matrix. Note that the second of the three matrices represents the effect of aerodynamic interaction between the blades k and _n.

Multiplying the first two matrices of equation (8) _we get ^the ^elements ^{of the} ^matrix as shown below:

Ii, I) _" C'(h~i~~'~,I ~ ~~i~~~fi)

(1,2)

" C~(h'i~~'~i _~ ~)i~~~li)

(2,1)

" C~(~)2~~'fi ~ ~'2~~~fi)

(2, 2) C~(h'2~ll~~i _~ ~'2~~~fi)

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(~, 3)

" C)(h~i~~~f2 ~ ~~i~~~f2)

Thus,

B~"(I,I)

" C)C[

~ (h)l~~'~,I ~ ~)i~~$~l)h/1

~ (h)l~~'/1 _~ _5~l ~~~l_)8~l

~ ~ (~~i~~'/L _~ ~~l~~~~L)~ii)

~~~(l,₂₎

" C~C~((h~l~~~~1 ~ S~l~~~~l)h~2 ~

Thus, the general element B(" _can be shown to be

where the _summation xtends over all _elemental areas. _Each^lement of [B] _thus epresents the

work done by the nth lade,

ibrating in its jth ^ode, on the kth blade vibrating

in blades.

lil~ ₊ _iwil~l~l~ + Aa (iBl~~i~l"

= i°I (lo)

k=1,2,3,. ,N

If _we _assume

©~ = C~/G

©j = Cj li

AP

= AP/mou~(

where _MO is the frequency at which flutter susceptibility is being examined and where _mo is

some reference _mass, then

ES

" W~A~/

Where

~~~ ~l~'~~ ~~~~ ~~~~~ ~~~~ ~~~~ ^~~

~

~~

141~ + ildil~i~l~ + ldi(ill~"i~l"

= i°1 l12)

k=1,2,3,. ,N

Equation (12) is valid for each k. Therefore the governing equation of motion for all blades

can now be written, I-e-,

lfl ₊ _[HI lzl + tdl[Al lzl

= 1°1 l13)

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O-10

O PresentAnalysis

U Reference2

_

f 0.08 _-3 Aerodynamic and

t O-M

~ ~~~~"'~°~ ~~~~'~

c

~y ^-i

IE

~ ^o

# 0.02

w ~ Numbers indicate

" predominant nodal

diameter 3

2

-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0

Eigenvalue (Real Pan)

Fig- 2- Comparison of analyses.

We _are _now ready to cast the eigenvalue problem. Let

lzl = I£le~~~ lI4)

Then

I-l~iIi ₊ _[Hi ₊ WiiAi)izi

" i°1 lls)

([HI + UJilAi)lTl ₌ >~lTl (16)

which is of the complex eigenvalue form

lAllTl

= >lzl. (ii)

2. Numerical Analyses and Discussion of Results

Equation (Ii) is the Inost familiar form to which flutter formulations have been reduced by several investigators in the past. ^The ^usual approach is to begin with _an assumed frequency

of vibration which will _serve _as the analysis frequency. Unsteady aerodynamic _pressures _are computed _at this frequency. For tuned systems ^the analysis frequency is the _saIne _as the blade frequency and the resulting eigenvalue is the _same _as the analysis frequency unless the aerodynamic ^forces _cause an appreciable change. For _mass ratios of relevance to engine blades, the departure ^froIn ^the analysis frequency is usually negligible. However the choice of analysis frequency for mistuned systems is not always clear and _may require several iterations.

Therefore it is _necessary to note the analysis frequency and verify if the eigenvalues computed

for the mistuned system depart too far from the assumed analysis frequency. In any case, the nature of the eigenvalues determines the _extent of susceptibility of _an assembly to flutter. Also

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30

* Numbm indlcwe Ap vdum at %C-.0127, 25

Mode4

' 20

£#

~ ₁₅

Mode 3

lS§ _/

io /

,

Mode1 ,'~

Mode 2

5 ° (11.31,.A6, Al,.3@

o

-~ -w

x/c

Fig. 3. Unsteady aerodynamic _pressures along the chord at 50~ _span.

the eigenvectors represent the distribution of amplitudes of blades around the _rotor for each

eigenvalue.

Figure I shows the system ^of ^blades cantilevered at their _root in _a nominal disk considered to be rigid. Thus _any coupling _among the blades is through aerodynamics only. The analysis

is _a logical extension of the formulation developed in reference iii and _was verified by making comparisons ^of ^results obtained for _a simple 12 bladed system ^with corresponding results obtained in reference [2]. ^This comparison is shown in Figure 2 and found to be acceptable. The real values represent frequency of vibration and the imaginary values represent aerodynamic damping.

A typical blade with the following parameters _was chosen for further analysis: gap/chord

= I, stagger angle ₌ 45degrees, number of blades

= 12, ^Mach ^Number ₌ 0.8, _aspect ratio

= 0.6, and _mass ratio (blade masslaerodynamic mass)

= 150. Figure 3 is _a plot of the

magnitude of unsteady _pressures acting on the chord at 50$lo _span for the modes indicated at an interblade phase angle of -30°, computed using _an analysis developed by Smith (Ref. _[3])

and Inodified to account for chordwise modes. The Inodification itself is rather straightforward

as the formulation allows for _a general description of motion along the chord. The values at the _very beginning and the very end _are, of _course, unsteady _pressures close to the leading

and trailing edges respectively. The locus of eigenvalues of the tuned system corresponding to the first four modes of vibration of the blade _are shown in Figure 4. The analysis frequency for each mode corresponds to the natural frequency of the blades in that Inode. The higher modes (increased reduced frequencies) for this assembly _are clearly _more susceptible to flutter.

The combination of increased unsteady _pressures with increased reduced frequency (Fig. 3)

and the predominantly plate-like mode shapes, along with other parameters, ^result ⁱⁿ ^reduced aerodynamic damping. It would therefore _appear that low aspect ^ratio ^blades ^with ^mode

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0.2

£ Mode 2

$~ ^{Mode I}

&

~f ^O-I

E #

3 Mode 3 fl

~ ~j

I .G

j~ ^#

Mode 4 (

£ 0

-1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2

Elgenvalue (Real Pan) 102

Fig. 4. Locus of eigenvalues for different modes of vibration for _a tuned system of blades.

0,02

$~

~'~~ Mistuned ,

+0,5%

£m

~

w

i~

?

~

w

-0.02 -0.01 0 0.01 0.02

EigenV#ue ~fleal Pan)

Fig. 5. Effect of mistuning _on flutter for Mn=0-8, 1= 2.6 (mode12.)

shapes typical of those shown in this analysis demand _a forInulation of the type presented

in this paper in order to capture ^the ^details ^of ^chordwise variation of both aerodynamic and structural features. The influence of reduced frequency _on the locus of eigenvalues is displayed

in Figure 5.

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0.008

§ Tuned A 0.I%

) _a 0.25%

11 0.006

° ~~~

$ ~jj

~

l

~i

0.004

II

0.002

-1.2 -0.8 -0A 0 0.4 O-B 1.2

Elgenvalue ~fleal Pan)

Fig- 6. Effect of mistunning _on flutter for Mn=0.8, 1

= 4.55 (mode (3).

With only 1/2$l alternate mistuning, the locus of eigenvalues ^for ^mode ² (reduced frequency

of 2.16) is split into two parts _as shown for this assembly of blades. Earlier investigations (Refs. [4,5]) ^have ^shown that for reduced frequencies less than I, such a split in the locus of

eigenvalues _occurs for mistuning levels far higher than 1/2$l. Therefore the reduced frequency

appears to play _a _very significant role in decoupling the blades. At _an _even higher reduced

frequency corresponding to mode 3 (I.e., 4.55) the splits in the locus ofeigenvalues of the tuned system begin to _occur at _as low _a departure as 0.I% difference in the frequencies and continue

as shown in Figure 6.

Another interesting feature of this approach is the study pertaining to an assembly with identical blade frequencies but with different mode shapes for _every other blade, I-e-, alternate

mistuning through differences in mode shapes. The locus corresponding to this condition is

split "vertically" _as shown in Figure 6. Modes 3 and 4 _were used _as mode shapes of blades and the analysis frequency corresponds to that of mode 3. It is hard to generalize with this limited study, but if the trend is any indication, then alternate mode shapes mistuning _can be less favorable because _one of the branches of the locus dips towards lower damping, a feature

totally unlike that observed in earlier studies with frequency mistuning.

The analysis presented here is realistic in that it _uses unsteady _pressures, rather than lifts

and moments to calculate flutter. This, _as it happens, is _more convenient in view of the fact

that unsteady aerodynamic analyses calculate _pressures. The _pressures need _to be integrated

to obtain lifts and moments if the analyses _use the latter. Not only is this step unnecessary but it is perhaps unrealistic especially for low aspect ratio blades.

Conclusions

This _paper is the first attempt to calculate _an eigen-solution for flutter susceptibility of _a mistuned bladed system using unsteady aerodynamic _pressures directly. In the past, flutter

analyses using unsteady aerodynamic _pressures have been limited to work/cycle approach applied to tuned systems ând eigen-solutions ôf ^mistuned systems ^have ^been ôbtained by

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using the _more familiar lift and moment coefficients of cascades. Unsteady aerodynamic codes used in industry compute pressures acting _on the surface of _a blade vibrating _as _a member of _a cascade and therefore it seemed that _an approach to utilize them _more directly in _a

flutter calculation _was deemed not only natural but also long overdue. The analysis presented here provides _a framework that _can be used to develop flutter codes that _are appropriate for application to low aspect ^ratio blades. Plate-like modes of such blades _can therefore be used efficiently in conjunction with distributed _pressures in _a consistent approach to flutter

calculation. Mistuning due either to differences in individual blade frequencies and for mode shapes _are treated by this analysis. Reduced frequency of _a vibrating assembly _appears to

play _a significant role in "splitting" the locus of eigenvalues with alternate blade frequency mistuning at levels of mistuning far lower than has been calculated in the past for medium to

high aspect ^ratio ^blades. ^The significance of this feature is being studied.

References

ill Srinivasan A-V- and Faburtmy J-A-, Cascade Flutter Analysis of Cantilevered Blades, J- Eng- Power106 (January, 1984).

[2] ^Whitehead D-S-, Torsional Flutter of Unstalled Cascade Blades at Zero Deflection, R & M N° 3429, Cambridge University Engineering Laboratory (England, 1964).

[3] Smith S-N-, Discrete Frequency Sound Generation in Axial Flow Turbomachines, A-R-C- R& M N° 3709 (1973)-

[4] ^Srinivasan A-V-, Influence of Mistuning _on Blade Torsional Flutter, R-80-914545-16, United

Technologies Research Center, East Hartford, CT, NASA CR-165137 (August, _1980)-

[5] Kielb R-E- and Kaza K-R-V-, Aeroelastic Characteristics of

a Cascade of Mistuned Blades in

Subsonic and Supersonic Flows, _paper presented at the Vibrations Conference (June I98I)-