SANDIA REPORT SAND82-0672 UnlimitedRelease UC-60PrintedFebruary 1982

Reprinted July, 1983

Reprinted November 1984

4

Aeroelastic Stability Analysisof a Darrieus Wind Turbine

David Popelka

PreparedbySandiaNationallLaboratoriesAlbuquerque,New Mexico87185andLivermore,California94550fortheUnitedSatesDepartmentofEnergyunderContractDE-AC04-76DPO0789

SF2900Q(8-81)

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SAND82-0672

AEROELASTIC STABILITY ANALYSIS OF ADARRIEUS WIND TURBINE*

David Popelka**Member Technical Staff

Applied Mechanics Division 111Sandia National Laboratories

Albuquerque, NM 87185

February, 1982

Abstract

An aeroelastic stability analysis has been developed forpredicting flutter instabilities on vertical axis windturbines. This report describes the analytical model andmathematical formulation of the problem as well as the physicalmechanism that creates flutter in Darrieus turbines.Theoretical results are compared with measured experimentaldata from flutter tests of the Sandia 2 Meter turbine. Basedon this comparison, the analysis appears to be an adequatedesign evaluation tool.

* This work was performed at Sandia National Laboratories andwas supported by the U.S. Department of Energy undercontract number DE-AC04-76DPO0789.

** Currently Dynamics Specialist at Bell Helicopter Textron,Ft. Worth, Texas.

Table of Contents

Page

I. Introduction. . . . . . . . . . . . . . . . . . . . . 5

II. Mathematical Formulation of the Flutter Problem . . . 5

III. Description of the Flutter Model. . . . . . . . . . . 8

IV. Derivation of the Equations of Motion . . . . . . . . 9

v. Derivation of Aerodynamic Loads . . . . . . . . . . . 17

VI. Flutter Test Program. . . . . . . . . . . . . . . . . 23

VII. Correlation of Theory and Experiment. . . . . . . . . 24

VIII. Conclusions. . . . . . . . . . . . . . . . . . . . .26

IX. Recommendations for Future Work . . . . . . . . . . . 27

x. References. . . . . . . . . . . . . . . . . . . . . .28

4

Introduction—

Tests of a scale model Darrieus wind turbine have shown

that under certain conditions, the turbine may experience a

flutter instability. Although flutter has not been observed on

a full scale turbine, an analysis is required that will

guarantee the stability of a future design.

A survey was made of the currently available aeroelastic

stability analyses which are listed in Refs. 1-5. The majority

of these analyses are complex, exact treatments of the

aeroelastic problem. Although these analyses will probably

yield accurate results, they are cumbersome to implement and

the physical understanding of the flutter problem is easily

lost. Ref. 2 is a simplified method which revealed that mass

balance of the blade cross section was not important for

flutter stability; a finding which has had tremendous impact on

blade manufacturing costs.

In this report, an approach was chosen that can give

physical insight as well as reasonable numerical results. Test

data from the Sandia 2 Meter turbine proved invaluable in

guiding the development of the analysis along a path that would

yield a maximum of understanding with a minimum amount of

mathematical complexity.

Mathematical Formulation of the Flutter Problem

The modal analysis method was used as the basis for the

flutter analysis. This approach is used frequently for

5

analysis of helicopter blade response and was used by Ref. 1

for the vertical axis wind turbine. This method allows the

analyst the freedom to describe the structural characteristics

of the turbine in minute detail by using NASTRAN finite element

modeling techniques to generate the necessary natural

frequencies and mode shapes. The general equations of motion

for the aeroelastic analysis of the turbine are as follows:

[M]{x] + [D]{;} + [K]{x} = -Q2[KM] {x} - 2Q[C1 {~} + [AK]{x}

where

{x} =

[M] =

[D] =

[K] =

[KM] =

[c] =

[AK] =

[AD] =

[AM] =

S2=

{F(t)} =

(1)

+ [AD]{;} + [AM]{x} + {F(t)}

structural displacement response

structural mass matrix

structural damping matrix (diagonal matrix)

structural stiffness matrix including centrifugalstiffening

centrifugal softening matrix

Coriolis matrix

aerodynamic stiffness matrix

aerodynamic damping matrix

aerodynamic mass matrix

turbine rotational speed

time dependent external forces

The forces F(t) consist of harmonic aerodynamic forcing

functions which are independent of the structural response.

These forces do not affect the stability of the turbine and are

therefore neglected.

6

To utilize modal analysis techniques, the solution to Eq. 1

is taken to be a linear combination of the normal modes of the

following equation:

[lI]{R} + [K]{x} = O (2)

This equation represents the structural properties of the

turbine and is solved by finite element techniques using the

NASTRAN Code. The normal modes contain geometric stiffness

effects resulting from centrifugal loading.

If @i are the mode shapes of Eq. 2, these modes can be

assembled into the modal matrix [+], and the physical

response {x} related to the modal response {q} as follows:

{x} = [@l{q}

Substituting this equation into Eq. 1 and then

premultiplying by [4]T yields:

[GIl{tj} + [2cuNGIl{d} + [GIwN2] {q} = -2’Q[$l T[cl[l$l {@..Q2[@lT[KM] [@l{q}

+ [@]T[AK][@]{q} + [$]TIADIII$I {G}

+ [4dT[AMl[$l {q} (3)

where:

{q} = modal response coordinates

[GI] = diagonal generalized mass matrix

[2EUNGI] = diagonal modal damping matrix

[GIUN2] = diagonal generalized stiffness matrix

E = structural damping factor

‘N =natural frequency

Q = turbine RPM

7

For very small values of structural damping or

proportional damping, the left hand side of Eq. 3

decoupled as discussed in Ref. 6. Since all terms

for

s completely

in Eq. 3 are

functions of the modal response coordinates, the right hand

side can be combined with the left hand side yielding:

[GI-AMB]{q} + [2EUNGI + CB - ADB]{~}+ [GIwN2 - AKB + KMB]{q} = O (4)

where:

AMB = [@] T[AM][@]

ADB = [@]T[AD][4]

AKB = [@] T[AK][@]

CB = m@lT[cl[41

KMB = Q2[@]T[KM][$]

Standard eigenvalue routines are used to solve this system

of equations for the complex eigenvalues, ~ = o + iw. The real

part of the eigenvalue, O, determines the stability of the

mode. A positive value indicates an instability and a negative

value indicates a stable configuration. The imaginary

component, iw, contains the flutter frequency.

Description of the Flutter Model

A typical NASTRAN model for the flutter analysis is shown

in Fig. 1. A single blade is used with a tower that has the

proportional torsional stiffness for that blade. The drive

train is modeled by a torsional spring which represents the low

speed shaft stiffness. The troposkein shaped blade is

represented by a series of straight beam elements. To simplify

the calculation of aerodynamic forces, each beam element

contains an additional intermediate node as shown in Fig.1.The distributed aerodynamic loads are computed for the beam

element and these forces are lumped to the intermediate node

which also contains the concentrated mass properties of the

beam element. The cross section of the turbine blade in Fig. 1

shows the relative location of the midchord, elastic axis, and

center of mass. The flutter instability involves an

interaction between the two modes shown in Fig. 1. These two

modes do not include tower translation, so the top and bottom

of the tower are pin jointed.

~rivation of the Equations of Motion

The equations of motion for the wind turbine are derived

from Newton’s laws using the following conditions:

1.

2.

All equations are expressed in the rotating coordinate

system. This eliminates time-dependent coefficients that

are present if the equations are derived in the fixed

coordinate system.

The turbine blade dynamics are based on concentrated mass

particles connected by massless elastic beams..

Fig. 2 illustrates the coordinate systems and degrees of

freedom needed to define the blade motion. The “R” coordinate

system represents the rotating coordinate system aligned with

the undeformed tower and blades. For convenience, at each mass

particle on the blade, a local “l” coordinate system is defined

parallel to the “R” system. The “2” coordinate system is

aligned with the local curvature of the blade, and is related

to the “l” coordinate system by the angle y as follows:

(5)

Coordinate system “3” follows the blade section during

elastic deformations. The relation between the “3” coordinate

system and the “2” coordinate system is given as:

(6)

The rotations 4X, #Iy and 6Z are transformed to the “R”

coordinate system as shown below:

where ~xr, ~yr’ and glzr are the rotations about the

irjrkr axes.

Referring to Fig. 3, the equations for dynamic equilibrium

of a elemental blade section can be written. In the limiting

case of an infinitesimal blade section length, a point mass

results with the following equations of motion:

r, - r2 sm=-~— aero (Force Equilibrium) (7)

(Moment Equilibrium) (8)

10

The underlined components of the above equation are due to

the dynamics of blade motion and are derived below.

For a rotating dynamic system, the position, velocity and

acceleration of a particle of mass is given by:

~=rP

+.; + ~xrP P..

i==: + ~xr + 2WX; + ~x~xrP P P P

where rP

is the position vector of the concentrated mass and

(9)

u is the angular velocity of the rotating coordinate system.

Referring to Figure 2, the position vector of the center of

mass of a blade section is:

T=(r + Ux) i. +Uj +Uk~ ~ + egj3o r yr

(10)

The angular velocity of the blade section is

‘3= $xi2 + ~yj2 + $zk2 + Qkr

(11)

Transforming the angular velocity to the “3” coordinate

system yields,

‘3 = (?X - Q.siny - @yOcosy)i3 + (@zL!siny + ~y + @xQcosy)j3

+ (-Q$ysiny + $Z + QcoSy)k3

Beginning with Eq. 9 in coordinate system “3” and using

Eqs. 5 and 6 to transform to the “R” coordinate system, the

particle velocity expressed in the “R” system is:

.

v= ; = [fix- ‘w - eg(fl + $Zr)]iv r

(12)+ [t + Q(rO + UX) - egfl$zr]jr + [tiz + eg ~

Y xr]kr

11

The acceleration of the particle in the “R” system is given

by:

.. ..: =;=[U - 2L?C - (rO + UX)L?2 + (!22$ - ~zr)egli

x Y zrr (13)

+ [u + 2Qtix - Q2U - 2f2~zreg - Q2egJjrY Y

+ [uz + ~xreglkr

The inertia force, ma, is written in matrix form as follows:

1m 000 0 -meg

m: = OmOOOO

10 0 m meg O 0.

[

-m f22

o

0

0

-mQ 2

0

0

0

0

0

0

0 1m02eg

o

0

..Ux

uY

Uz..+Xr

iyr

izr

[

o - 2mQ 000 0

+ 2mQ O 000 -2megQ

00 000 0.

The angular momentum of the blade section is given by:

[PJ = [IS] {u3}

(14)

(15)

12

where [13] iS the inertia matrix for the blade section and

{~~l}is the angular velocity of the section. The rate of

change of angular momentum is given by:

s= [j] + {U3}X[P31 (16)

The inertia matrix for a blade section is expressed in the

blade coordinate system as:

[

wI 00xx

[13] = O Iyy O

0 0 Izz.

The products of inertia are assumed to be small and have

been neglected to simplify the equations. Substituting Eq. 11

into Eq. 16 and transforming to the “R” coordinate system

results in the following equations in matrix form:

13

nk-1

w

-Kt+%

0

I

NwE!I

*N

La0uNN

u0

+00

-xxu1

NN

!2;ou:.F1m0+Nc

1

NN

00

:0

07i-+

mov

00

0

00

0,

+

l-(

.2mN

I+N0

00

+0

00

0

●Nmouxu%

00

0I

+

0000

.00II

.irA

The center of mass offset from the elastic axis creates

a moment about the elastic axis. The mass offset expressed

in the “R” coordinate system is:

agr = eg[(-f$zcosy + $Xsiny)ir + jr + ($zsiny + $XCOsy)kr] (18)

The moment due to the mass offset is:

[

ir jr kr

egr xm~= eg(-+zcosy + $Xsiny) eg

1

e9(@zsiny + @xCoSy) (19)

m;x m=Y

m~z

Substituting Eq. 14 into Eq. 19 yields:

[

00 meg 2meg O 0.- —{?g xma=

r000 000

-meg O 0 00 meg 2

1

..Ux..‘YUz

5XX

$Yr

;Zr

+

[

o 00 22m~ eg

1-mroQ2eg O

+ o 00 2-mrofleg 00

mf/2eg O 0 0 00

(20)

Ux

uYUz

@xr

‘yr

‘$zr

The complete equations of motion are assembled from

ecjuations 14, 17, and 20 and are written as follows:

15

‘XR

‘YR

‘ZR

‘XR

‘YR

‘ZR ‘1m o

0 m

o 0

0 0

0 0

-meg o

“o -2m~

?mfl o

0 0

0 0

0 0

0 2megfl

20-roil

o -mn2

0 0

0 0

0 0

mf12eg O

0

0

m

meg

o

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

meg

2 2 2meg + IXXCOS y + Izzsin Y

o

sinycosy(Iz, - Ixx)

o

0

0

0

-5)[(sin2y- COS2Y) (Ixx - Izz) - IYYI

o

0

0

0

0 -meg

o 0

0 0

0 sinycosy(Izz - Ixx)

IYY

o

0

2 2 2meg + Ixxsin Y + IzzcOS

o 0

0 -2mfleg

o 0

n[sin2y - COS2Y)(IXX - Izz) - Iyyl o

0 -252cosysiny(Ixx- Iz=)

2Qsinycosy(Ixx - Izz) o

0 mf12eg

o 0

0 0

22meg f/ + S22[1~xsin2y+ I~zc~s2y - I -mrof12eg

YY ]o

-mroi22eg Q2[(COS2Y - sin2Y) (1== - Ixx)l o

0 0 0

uxuY

Uz

4X=

$Yr

izr

Ux

uYu=

$xr

‘yr

‘$zr

(21)

The first portion of Eq. 21 is the mass matrix for the

blade section, which is represented by [M] in Eq. 1. The

second matrix in Eq. 21 is the Coriolis matrix, which appears

as [C] in Eq. 1. The last portion of Eq. 21 is the centrifugal

softening matrix, or [KM] in Eq. 1.

For most turbine analyses, the blade inertias IXX, IYY,

‘Zz are not included in the NASTRAN model and are therefore

not used in the flutter analysis. However, they are included

in this derivation for completeness.

Derivation of Aerodynamic Loads

The aerodynamic loads acting on the turbine blades are

derived using unsteady aerodynamic theory as discussed in Refs.

7-1o. These references cite Theodorsen ’s original work on a

two-dimensional airfoil oscillating in a steady air stream.

This theory accounts for all possible motions of the blade

section that will produce aerodynamic loads.

Several simplifying assumptions, listed below, are made for

the airload calculations:

1) Two-dimensional strip theory assumed applicable

2) No stall considered

3) Chord line and zero-lift line assumed to coincide

4) No inflow permitted through the turbine

5) Blade relative wind velocity assumed constant during

turbine rotation

6) Aerodynamic center located at quarter chord point

17

7) Turbine wake not modeled

8) Small angles assumed, linearized aerodynamic equations

As outlined in Ref. 10, the unsteady lift, moment, and drag

acting on a blade section are given by the following equations:

{[L=~PbV2CK# + (1 - 2a) ;6 1+2e- _b2a0+~;

+i V* v }(22)

..

1M = & pb2v2 = + (1 + 2a)2 b2 “-

V*$CKfi+(l/8+a)-e

V*

-[a - Lj + 2(k-a2)CK]~ 6 - (1 + 2a)CK61

~ = ~ pV2C Cdo

(23)

(24)

where:

a = nondimensional distance from elastic axis tomidchord (positive if elastic axis is aft ofmidchord, expressed as a fraction of b)

ao = lift curve slope

b = 1/2 chord length

c = chord length

Cdo = drag coefficient for airfoil

CK = Theodorsen ’s lift deficiency function

D = drag of airfoil (per unit span)

h vertical translation of the airfoil at elastic axis= (positive up)

L = unsteady lift at elastic axis (per unit span)

M = unsteady moment at elastic axis (per unit span)

v = relative wind velocity for blade section, == Vo + G!ro

Vo = wind velocity

e = pitch rotation of the airfoil (positive nose-up)

D = air density

18

The expressions for 6 and h are obtained from Eqs. 12 and 13

ar~d are then resolved into the “2” coordinate system to give:

A = (fixsiny - !JUysiny + tizCOSy) k2

.,,h = (Uxsiny - 2.QGysiny- (r. + Ux) Q2siny + U

(25)

z COSY) k2

The geometric pitch of the airfoil is given by the

fc]llowing equations:

6=(C$ ~r cosy - @zr siny)i2

‘5 = (fxr Cosy - $Zr siny) i2(26)

..e = (ixr Cosy - ;Zr siny) i2

Eqs. 25 and 26 are substituted into Eqs. 22 and 23 to give

the lift and moment acting on the blade section. In

Eq. 22, the lift can be separated into two components. The

first component is the circulatory lift, Lc, which depends on

the value of CK. This component of the lift vector acts

perpendicular to the relative wind and is the result of

circulation produced by the lifting airfoil. The second

component is the noncirculatory lift, LNC, which includes the

“apparent mass” of the air stream and acts perpendicular to the

chord line of the airfoil.

Figure 4 shows the aerodynamic loads acting on a typical

blade cross section. The lift, drag, and moment can be

resolved into the blade coordinate system which yields:

19

F= = (Lc + Da + LNC)k3

F = (Lca - D)j3Y

Mx = Mi3

(27)

The angle of attack, ~, is composed of the geometric pitch

angle, e, plus the induced angle of attack, B. The induced

angle of attack is the angle due to blade motion, and is the

ratio of the blade vertical velocity to the horizontal

velocity. The equation for u, is as follows:

a=(f) ~rcosy - @zrsiny - [(fixsiny - QU siny + fizCOSy)]/vY (28)

Substituting Eqs. 22, 23, 24, and 28 into Eq. 27 yields the

aerodynamic forces in the blade coordinate system. These

forces are then resolved into the “R” coordinate system by

using Eqs. 5 and 6.

The final expression for the aerodynamic loads are written

in matrix form as follows:

20

[

-A1bSY2

o

-AlbSyCy

-A1b2aSYCY

o

1Alb2aSy2

[

22 + A3SY

‘A1CK2VSY

I-A1CK2VSICY + A,sycy

[

-Alb(l + 2a)CKVSYCY

o

Alb(l + 2a)CKVSY2

A12blEY2

0

-A12bOSyCy

Al b2a2flSYCY

0

-Alb2a2QSy2

A1cK2vnsY’ - A3flSy2

o

A1CK2WSYCY - A3QSYCY

A4flSYCY

o

-A4i2SY2

-AlbSYCY

o

-llbCY2

-Ab2aCY2

0

L1b2aSYCY

-A12VCKCYSY + A3CYSY

o

-?..2VCKCY2 + A.CY 21

-Alb (

nlb(l

3

+ 2a) CKVCY2

0

+ 2a)CKVCYSY

o

0

0

0

0

0

-A1b2aCYSY

o

-A1b2aCY 2

-A6bCY2

o

A6bSyCy

A1bVCYSY + A7SYCY

o

AlbVCY 2+ A7CY2

A5b2VCY2

o

-A5b2VSYCY

AlCK2V2SYCY - A3VSYCY

-A,brOf12sYcy

A CK2V2CY2 - A “CY21 3

+ A3V

A4VCY2

~ b2ar *25Y210

-A4VCYSY

o

0

0-00

0

0

0

0

0

0

0

2Alb=ofi ‘Ycy

o

-A1brOQ2SY2

o

0

0

Alb2aSy

![1

2 Ux

o uY

A1b2aSyCy Uz

A6bSyC y $Xr

o tyr

-A6bSy2 izr

-A1bVSy2

- A7SY2

11r+o ily

-AlbVSy Cy - A7Sy Cy (JZ

-A5b2”Syq $xr

o JyrA5b2VSy 2

bzr

-A 3V - A12CKV2SY2 + A3VSy12 U*

22Albron SY

II

‘Y‘A1CK2V2SyCy + A3VSy Cy u

z-A4VCYSY oxr

o‘yr

A4VSY2‘$Zr

where:

‘1 = pba OAL /2.

‘3 = 1/2 pV2b CdoAL

‘4 = AI(l + 2a) VCKb

‘5 = A,[a - 1/2 + 2(1/4 - a2)CK]

‘6 = A,(l/8 + a2)b2

‘7 = AICK(l - 2a)bV

Sy = siny

Cy = Cosy

The first matrix in Eq. 29 is the aerodynamic mass matrix

corresponding to [AM] in Eq. 1. The second matrix is the

aerodynamic damping matrix which is represented by [AD] in

Eq. 1. The last portion is the aerodynamic stiffness matrix

denoted by [AK]. Note that neither [AD] nor [AK] are symmetric

because the aerodynamic forces do not constitute a conservative

system.

The aerodynamic loads in Eq. 29 include the Theodorsen

Function, CK, which is a measure of the unsteadiness of the.

flow field. The parameter, CK, is a complex number which

alters the phase angle between the airfoil oscillation and the

resultant aerodynamic forces. Its value is dependent on the

reduced frequency, or Strouhal number. For the flutter

instabilities observed on the VAWT, the Strouhal number is very

low which renders CK equal to unity. This corresponds to a

“quasi-steady” flow field which appears to be an adequate

representation of the aerodynamic loads for the VAWT flutter

problem.

22

Flutter Test Program—

To verify the accuracy of the flutter analysis, a flutter

test program was conducted. This test program utilized the

Sandia 2 Meter turbine with several sets of aluminum blades.

These test were conducted in the following manner:

1. The non-rotating blade frequencies were measured for

comparison with the NASTRAN model of the turbine. The

modal damping of the turbine was also measured but it was

difficult to get a repeatable value. Generally, the modal

damping varied from .1% to .4% critical. A value of .35%

was used in the analysis.

2. The turbine speed was set and an impulse was applied to the

turbine through the brake system to trigger the flutter

instability. A torque meter was used to record the

resulting torque oscillation which increased during a

flutter instability and decayed to a stable configuration.

The mechanism that creates flutter is clearly shown in slow

motion films of the flutter instability. The instability is

due to coupling between the flatwise bending and torsion mode

shown in Fig. 1. The flatwise bending mode involves radial

motion of the blade which creates Coriolis forces that amplify

the response of the torsion mode. The resulting elastic

deflections cause changes in the aerodynamic forces which add

energy to the vibrating blade and create flutter.

23

Correlation of Theory and Experiment

The flutter analysis was verified by comparing the

theoretical results with measured test data. Fig. 5 shows the

flutter stability for the two meter turbine with three aluminum

blades (NACA 0012, CHORD = 2.91 in) and a truss tower. This

figure shows the variation in modal damping and modal frequency

with turbine speed. The modal damping curve was calculated

with zero structural damping. This damping curve is very

shallow which means that small changes in the structural

damping have a large influence on the flutter speed. The

theoretical flutter speed was found by adding .35% structural

damping to the modal damping curve, giving 850 RPM as the

flutter speed. This is in fair agreement with the measured

flutter speed of 745 RPM. The calculated flutter frequency,

18.5 Hz, agrees well with the measured flutter frequency, 18 Hz.

The turbine was modified by replacing the truss tower with

a torsionally stiff pipe tower. The flutter instability did

not occur in tests up to 900 RPM. Theoretical results agree

with this data.

Wind speed effects were evaluated by testing the turbine in

25 mph winds. The turbine configuration consisted of three

aluminum blades (NACA 0012, CHORD = 2.91 in) and a truss

tower. Fig. 6 shows the measured flutter speed to be in the

range of 705-720 RPM at a flutter frequency of 18 Hz. The

theoretical results indicate flutter at 695 RPM at a frequency

of 16.5 Hz. The theory predicts that the wind velocity has a

24

larger influence on the flutter speed than the test results

indicate. This is probably due to the simplifying assumptions

in the aerodynamic load calculations.

A larger set of blades (NACA 0012, CHORD = 3.47 in) was

installed on the two meter turbine with the truss tower as

shown in Fig. 7. This set of blades did not flutter up to

speeds of 1050 RPM. The analysis, however, predicts flutter at

1000 RPM. This disagrees with the test results, but since

flutter was not observed in the test, the magnitude of the

theoretical error is unknown.

Fig. 8 displays the flutter results for the small blades

(NACA 0015, CHORD = 2.31 in). Flutter was noted at 777 RPM at

a frequency of 18.5 Hz. Theoretically the flutter speed is 865

RPM at a frequency of 18.7 Hz, which correlates well with the

test data.

The stability of the 17 Meter Sandia turbine is shown in

Fig. 9. The calculated flutter speed is 176 RPM at 4.5 Hz,

which is well above the 50 RPM operating speed.

These results indicate that the analysis is capable of

assessing the effect of turbine design changes on flutter

speed. The stability trends are predicted accurately by the

program and the numerical results are sufficiently accurate to

establish confidence in the analysis.

25

Conclusions

The analysis developed here has shown to be a useful tool

for understanding and predicting flutter of Darrieus, vertical

axis wind turbines. Additional test data is needed to fully

verify the accuracy of the analysis and to establish error

bounds. The analysis is potentially weak in the areas of

aerodynamic force calculation since several simplifying

assumptions have been made. A more complex aerodynamic model

would improve the predictive capability of the model.

Several observations have been made based on the results of

the analysis and tests:

1.

2.

3.

4.

5.

Flutter is a result of aeroelastic coupling of two primary

blade modes; the first flatwise bending mode and the first

torsion mode.

Flutter does not require that two blade modes be in

resonance. The frequencies of the. flatwise mode and the

torsion mode do not converge in the operating RPM range.

However, increasing the separation of the flatwise mode and

the torsion mode increases the flutter RPM.

Tower and drive train torsional stiffness affect the

torsion mode frequency which affects the flutter RPM.

Flutter occurs at a frequency very near the flatwise mode

frequency.

Wind velocity reduces the flutter RPM, but for operational

wind speeds, the effect is relatively small.

26

Recommendations for Future Work

The flutter problem should be investigated further with a

comprehensive program of testing and theoretical

investigations. It is recommended that the following problem

areas be addressed:

1.

2.

3.

4.

Determine the effect of chordwise mass balance on the

flutter stability. This will involve the fabrication of

blades with significant chordwise center of gravity

offset. Ref. 2 predicts that mass offset has little

influence on flutter stability. This should be verified

with test data and results from the analysis.

Study the effect of blade frequency placement and the role

of tower torsional stiffness on flutter stability.

Evaluate the effect of wind velocity on flutter stability

in greater detail. Very high, short duration wind gusts

may reduce the flutter RPM.

Modify the aerodynamic load model to account for stall,

dynamic inflow, turbine wake effects, and periodic

variation in relative wind velocity.

Acknowledgement

The author would like to acknowledge the technical support

on this project from R. C. Reuter, D. W. Lobitz, W. N.

Sullivan, J. R. Koteras, and T. M. Leonard. Assistance during

the test program was provided by M. H. Worstell, J. D.

Burkhardt, and L. H. Wilhelmi.

27

References

1.

2.

3.

4.

5.

6.

7.

8.

9.

A. J. Vollan, “The Aeroelastic Behavior of LargeDarrius-Type Wind Energy Converters Derived from theBehavior of a 5.5 Meter Rotor,” Dornier System GMBH, GermanFederal Republic

Norman D. Ham, “Aeroelastic Analysis of the Troposkien-TypeWind Turine,” Sandia Laboratories, SAND 77-0026, April,1977.

Krishna Rao Kaza, Raymond G. Kvaternik, “AeroelasticEquations of Motion of a Darrius Vertical Axis Wind TurbineBlade,” NASA TM-79295, December, 1979.

J. Wendell, “Aeroelastic Stability of Wind Turbine RotorBlades,” MIT Aeroelastic and Structures ResearchLaboratory, Department of Aeronautics and Astronautics,Cambridge, Massachusetts, September, 1978.

William Warmbrodt and Peritz Friedman, “Coupled Rotor/TowerAeroelastic Analysis of Large Horizontal Axis WindTurbine,” AIAA Journal, Vol 18, No. 9, September, 1980.

Leonard Meirovitch, Analytical Methods in Vibrations, TheMacmillan Co., Collier-Macmillan Limited, London, 1969.

Raymond L. Blisplinghoff, Holt Ashley, Robert L. Halfman,Aeroelasticity, Addison-Wesley Publishing Company, Inc.,l?eadlng Massachusetts, 1955.

Norman Abramson, &n Introduction to the Dynamics ofAirplanes, Dover Publicfiions, In~~, New York, 1958.

Y. C. Funa. An Introduction to the Theorv ofAeroelast~~i~~, Dover Publications Inc. ,”New York, 1969.

10. Robert Scanlan and Robert Rosenbaum, Aircraft Vibration andFlutter, Dover Publications, New York.

28

TYPICAL BEAM ELEMENT

AERODYNAMICLOAD RESULTANTS

DISTRIBUTEDAERODYNAMIC LOADS

LUMPED MASS AND INERTIAAT INTERMEDIATE NODE

()

(}NASTRAN MODEL

() BLADE CROSS SECTION

DRIVE TRAIN STIFFNESS

FLUTTER MODES

FLATWISE BENDING MODE TORS1ON MODE

Figure 1. Mathematical Modeling Technique for theFlutter Analysis

29

$ZR

BLADE ELEMENT COORDINATE SYSTEM xlylzl

Y1

‘o

Y~ROTATING COORDINATE SYSTEM, )(R YR ZR

jy2

\\ CENTER OF MASS

/’/

/ MIDCHORD ELASTIC AXIS

/<z,

//

DEFORMED BLADE

/ /INATE SYSTEM

//

t ‘z/2E<

BLADE AXISCOORDINATESYSTEM X2 Y2 22 “)p LJy? Uz =TRANSLATIONAL

DEGREES OF FREEDOM

@)(, @y, #z =ROTATIONAL DEGREESOF FREEDOM

Figure 2. Coordinate SystemsFlutter Analysis

and Degrees of Freedom used in the

30

ZR4

\

Pa ,~AERO

M2

AL

\‘w

MOMENT ABOUT ELASTIC AXIS

FORCE AT ELASTIC AXIS

ikiAERO = AERODYNAMIC

~AERO = AERODYNAMIC

mZ=lNERTIA FORCE AT CENTER OF MASS

6 = RATE OF CHANGE OF ANGULAR MOMENTUM VECTOR

~ 1, q2 = MOMENT RESULTANTS AT INBOARD AND OUTBOARDENDS OF SEGMENT

~ 1, ~z = FORCE RESULTANTS AT INBOARD AND C)I,J’T’BOARDENDS OF SEGMENT

Figure 3. Force and Moment Equilibrium for a Blade Element

Lc = CIRCULATORY LIFT

LNC = NON-CIRCULATORY LIFT

D = AERODYNAMIC DRAGM = AERODYNAMIC MOMENT

Vv = VERTICAL VELOCITYVH = HORIZONTAL VELOCITY

CY= ANGLE OF ATTACK= ~ + o

@ = GEOMETRIC PITCH

/? = INDUCED ANGLE OF ATTACKv~ = RELATIVE WIND VELOCITY

Figure 4. Aerodynamic Forces and Moments Acting on an Airfoil Section

32

-Ja

4

3

2

1

b“

o

-1

FLUTTER RESULTS

2 METER TURBINE

3 ALUMINUM BLADES

CHORD =2.9 1 in.,NACA 0012

TRUSS TOWERWIND SPEED = O● EXPERIMENTAL DATA

m THEORY u, TORSION/

00

/0

/h), FLATWISE ~ ~ -

0, FLATWISE _ _ ~ -THEORETICAL

- FLUTTER(WITH 0.35%STRUCTURAL

200 400 600 3

\

1000

TURBINE RPM

UNSTABLE

Figure 5. Comparison of Calculated and Measured Flutter Stabilityfor the 2 Meter Turbine (NACA 0012, CHORD = 2.91 in)

33

FLUTTER RESULTS

2 METER TURBINE

3 ALUMINUM BLADES

CHORD =2.9 1 in,NACA 0012

TRUSS TOWERWIND SPEED = 25 MPH● EXPERIMENT

///---

/“

u, TORSION

~~ u, FLATWISE

THEORETICAL FLUTTER\ (WITH 0.35%

STRUCTURAL DAMPING)

EXPERIMENT705-720 RPM

I .v

500 600 900

1

TURBINE RPM

UNSTABLEI O, FLATWISE

\

Figure 6. Effect of Wind Speed on Flutter Stability of the2 Meter Turbine (NACA 0012, CHORD = 2.91 in)

34

4

n

3

2!

I

c

FLUTTER RESULTS2 METER TURBINE3 ALUMINUM BLADESCHORD= 3.47 in, NACA 0012TRUSS TOWER /0WIND SPEED= O //-

/~, TORSION~ THEORY//

/-

01

//’ ‘,-

- _ u, FLATWISE

0“

/

MArE$

RP

Jr

1 1 I \

200 400 600 800

TURBINE RPM

Figure 7.

‘H;:u;%!w-(WITH 0.35%STRUCTURALDAMPING)

Comparison of Calculated and Measured Flutter Stabilityfor the 2 Meter Turbine (NACA 0012, CHORD = 3.47 in)

35

FLUTTER RESULTS2 METER TURBINE3 ALUMINUM BLADESCHORD =2.31 IN, NACA 0015TRUSS TOWERWIND SPEED = O● EXPERIMENTAL DATAm THEORY

00

(A), TORSION / H ‘

@-,--

00

/

30

4-e - THEORETICAL

--”e FLUTTERem.-n=865 RPM-

200 400 600 80\

1000

TURBINE RPM\

10 i

Figure 8. Comparison of Calculated and Measured Flutter Stabilityfor the 2 Meter Turbine (NACA 0015, CHORD - 2.31 in)

36

FLUTTER RESULTS17 METER TURBINECHORD = 24.24 h, NACA 00122 ALUMINUM BLADESPIPE TOWER /

WIND SPEED = O /0

/

---6’TORs’OY/ O, FLATWISE \/A

/ k~THEORE’TICA

/ f\

Y

FLUTTER/“’ 9=176 RPM

~~ ymH 0.355“JRA

r“FLA7 v!”)f O, TORSION

1 1 I .

8

6

4

2

I40 80 120 160

\200

TURBINE RPM

UNSTABLE

5

Figure 9. Flutter Stability Calculations for the 17 Meter SandiaTurbine

37

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