Aeroelastic Stability Analysis of a Darrieus Wind Turbine · [GI] = diagonal generalized mass...
Transcript of Aeroelastic Stability Analysis of a Darrieus Wind Turbine · [GI] = diagonal generalized mass...
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)
lamedby SandiaNationalLaboratories,operatedfortbeUnitedStatesDepartmentofEnergybySendIaCorporation.NOTICE Thisreportwee reperedeeenaccountofworksponeoredbyan
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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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