The linear aeroelastic equations for one curved blade of a vertical axis wind turbine in state vector form are presented. The method is based on a simple integrating matrix scheme together with the transfer matrix idea. The method is proposed as a convenient way of solving the associated eigenvalue problem for general support conditions

Nitzsche, F.

English

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AEROELASTIC ANALYSIS OF A 
TROPOSKIEN-TYPE WIND TURBINE BLADE 
F. Nitzsche 
Departmen t of Aeronautics and Astronautics 
Stanford Un i versity 
Stanford, CA 94305 
Grant NGL 05- 020- 243 
July 1981 
NI\SI\ 
National Aeronautics and 
Space Administration 
langley Research Center 
Hampton, Virginia 23665 
UG 18 1 81 
t/\,'JGLL t '< ~ :,EARCH CENTEIt 
LlBRA~Y, NAS/\ 
tv ~'Antl<il\l' \'i1~l"11~'4 
https://ntrs.nasa.gov/search.jsp?R=19810023086 2020-03-21T11:14:21+00:00Z
AEROELASTIC ANALYSIS OF A TROPOSKIEN-TYPE WIND TURBINE BLADE 
F.Nitzsche 
Stanford University 
Department of Aeronautics & Astronautics 
Stanford, CA 94305 USA 
SUMMARY 
The linear aeroelastic equations for one curved blade of a vertical 
axis wind turbine are presented in state vector form. An attractive method, 
based on a simple integrating matrix scheme together with the transfer ma-
. trix idea, is proposed as a convenient way of solving the associated eigen-
value problem for general support conditions. 
I NTRODUCTI ON 
The troposkien shape has been frequentry invoked in structural mod-
eling of vertical axis wind turbine (VAWT) blades [1]. Although ~t is free 
of bending stress in the equilibrium position determined by a constant an-
gular velocity, tests have indicated that under certain conditions serious 
vibrations about the original ~hape may occur. They involve bath chordwise 
and flatwise bending as well as torsion of the blade. 
Few theoretical works have been published on the determination of 
natural frequencies and vibration mode shapes of such special but neverthe-
less "important kinds of blade geometry [2-4]. The tendency among investiga-
tors has been to deal with a more general geometry, which brings perhaps un-
necessary complexity into the problem and requires the use of strictly nu-
merical procedures for obtaining reliable results. Futherrnore, the shape of 
actual VAWT blades has been very close to the troposkien; good approxima-
tions like those suggested by Reis & Blackwell [5] are commonly used in prac-
tice. One can recall that the complicated flow field generated by VAWT al-
ready adds an important source of inaccuracy to the aerodynamic load esti-
mation 'and consequently to problems such as flutter speed determination. 
Therefore it is important for preliminary design purposes, to develop semi-
analytical methods based on the troposkien idealization, which will enable 
quick analyses and general understanding of the conditjons that may affect 
blade stability. The intention of this paper is to present an approach with 
the aforementioned characteristics. 
1 
"OPTIMUM" LINEARIZED EQUA~IONS OF ~1OTION 
The general second-degree nonlinear aeroelastic equations ' of motion 
for a slende~ nonuniform and extensible blade ~ere presented by Kaza & Kv-
artenik [6]. For physical understanding of the aeroelastic phenomena, how-
ever, such degree of complexity does not seem necessary. It ~ould be more 
interesting to seek a set of linear equations that allows the study of the 
relative importance of various parameters on the solution. These parameters 
include CG offset, shear deformations, rotary inertia and different kinds 
of blade support, along with the possibility of extending the model to a 
more realistic one containing the influence of tower and guy-cable stiff-
ness' as well as the other blades. 
Whe~ gravity effects are neglected, the troposkien can be seen as 
the plane curve descr ibed by a light rope rotating at constant angular' veloc-
ity. Its shape is known in closed analytic form [1]. Working in the same 
fashion as Ashley [4], small perturbations from this equilibrium shape are 
assumed. After adding torsion to chordwise and flatwise bending and by as-
suming simple harmonic motion in time, the final order of ,the governing sys-
tem of differential equations is found to be 12. However, the choice of the 
dependent variables remains open. If, as in reference 4, the three cartesian 
components of the linear displacement related to the bending, the rotation 
related to the t orsion and the incremental tension are selected, a set of 
coupled linear partial differential equations subject to one geometric con-
straint equation is obtained. Asymptotic solutions may be useful, but only 
:simplified versions of the pr~blem could be carried out ~ithout a 'great deal 
of difficulty. An integrating matrix approach, like the one suggested by 
Hunter [7] , could be employed in order to solve the eigenvalue problem for 
the free vibration analysis. The choice of the spatial independent variable 
as the distance along the blade introduces technical difficulties however, 
'and the method fails. A more "natural" spatial independent variable is dis-
covered to be the angle formed by the tangent to the undeformed shape and 
the axis of rotation. Even then, however, an unnecessary degree of algebra-
ic difficulty is introduced. Without discarding the possibility of the so-
lutions by the aforementioned means, the Reissncr's variational principle 
of the elasticity suggests that the best dependent variables are likely to 
'be the l2xl state vector of "natural" quantities connected with both equi-
librium and deformation of the blade: bending moments, torque, shears, bend-
'ing slopes and displacements. The integrating matrix [7] together with the 
trans,fer matrix method [8], is expected to constitute an efficient and pow-
~rful tool for dealing with the vibration of such beams subjected to gener-
al boundary conditions. 
2 
• 
____ J 
AEROELASTIC ANALYSIS OF THE BLADE 
Assuming small perturbations from the equilibr ium shape is equiva-
lent to cons t ructing a linear field of deformations on a prestressed, un-
twisted, plane troposkien- curved rod. The governing equations for a three-
dimensionally curved r od under such a general initial stress configuration 
were obtained by different authors. Taking Nair & Hegemier's derivation [9 ] , 
based on the principle of virtual work, neglecting cross section warping 
and introduci ng t he conditions defined by the troposkien hypothesis, one 
obtains a set of 1 2 linear first-order ordinary differential equations with 
nonconstant coefficients . These govern the 12 perturbation quantities cho-
sen to be the dependent va riables : 
Qi + KoT + f l = 0 Ml + KoM3 - Q2 - TOX! + ml = 0 
. Q,' + f = 0 M' + Q - T X + m = 0 2 2 2 I 02 2 
T' - KoQI + f3 = .0 M' - K M + m = 0 3 0 I . 3 
u
l + KoW - X2 - QI/GA 0 X~ - M2/EI~~ == 0 
v' + Xl - Q2/GA = 0 a' - K X - M /GJ == 0 
o I 3 
w' - K U - T/EA[I-e2(EA/EI ») - 1 + eM lEI [1-e2(EA/EI )]-1 ~ 0 
o nn I nn nn 
Xl' + K a - MI/EI [1- e 2 (EA/EI ) ]-1 + eT/EI [1-e2 (EA/EI ) ]-1 == 0 
o fln flfl flY) flY) . 
where Figure- l together with the list below, define the nomenclature used: 
( )'= differen tia tion with respect to s; s= spatial independent variable, 
taken as di s t ance along the locus of the section shear centers (elastic ax-
is); QI-2= interna l shear components; T= tension along the blade; MI - 2 - 3= internal moment components; u,v,w= elastic axis displacements; XI _2 ,d= -ro-
tations "about t he elastic axis (a= angle of attack); K :::I lo·c-ai initial cur-
vature, given by the troposkien; T = local initial tengion, given by the 
troposkien ; e= CG offset, constantOalong the blade; EI ,EI = bending ri-
:gidities; EA= longitudinal rigidity; GJ= torsional rigiaity?TlGA= effective 
shear rigidit y ; fl_2_3,ml_2_3= -external loads (forces and moments) of both 
inertial and aerodynamic origin. 
It be came evident during the development that some extensibility 
should be a llowed in this formulation in order to get the most rational s et 
of equations . I f inextensibility is assumed by simply letting EA become i n-
finitely large, important information is lost. Shear deformation is also in-
cluded for compl e t eness . 
The analysis proceeds by grouping the dependent variables in a 12xl 
state vector y, which contains generalized internal forces YF and gener a l -
ized ~isPlacements YD. The external loads are al so collected in a l2xl vec-
tor p. T T 
Y == t yFI ynt = {Ql MI Q2 M2 T M31 U Xl v X2 w a I 
p = {f I .0 } T = {f 1 ml f 2 m2 f 3 m3 1 0 0 0 0 0 0 l T 
where f = f(A)+ f (J). respectively its aerodynamic and inertial components. 
3 
Hamilton 's principle is involked to evaluate f as function of the 
generalized displacements. By taking both the translational and. rotjtio,nal 
parts of the kinetic energy, coriolis and rotary inertia effects are a~to­
matically included . After the rigid blade inertial load is subtracted from 
the equations, the following approximations appear to be significant to 
bring all inertial terms into a convenient form: 
ff~2d~dn «Jfn2d~dn ~ f f(~2+ n2)d~dn 
Here coordinates ~ and n are defined in Figure-I, and p is the blade mass _ 
per .unit of volume. 
As a first approximation, any reasonable aerodynamic theory is ade-
quate to complete the aeroelastic problem formulation. The quasisteady strip 
theory can be employed without a great deal of difficulty. The author cites 
Kaza & Kvaternik's derivation as appropriate [6]. 
Simple harmonic motion is assumed in order to eliminate the time de- . 
pendence: this approach is known to be suitable for analyzing both free vi-
bration and flutter. It follows that the inertial (J) and aerodynamic (A) 
' loads can be written as: 
p = [ J + A ] Y = ( [ 00 
where J FD and ~D are 6x6 submatrices dependent on the harmonic frequency. 
In Appendix A the nondimensional forms of the equations are presented for 
the free vibration problem in particular. 
SOLUTION OF THE AEROELASTIC PROBLEM 
The main objective of this paper is to propose a convenient method 
for solving the foregoing aeroelastic problem. Numerical results are expect-
ed to be published in a later work. Therefore, the author will now de-
scribe how to obtain the eigenvalue problem associated with either free vi-
bration or flutter analysis. 
Lehman [10] presents a modified version of an integrating matrix 
scheme apparently first used by Hunter [7] which can be adapted for obtain-
ing transfer ~trices relating state vectors at different stations of aone-
. dimensional elastic structure. Appendix B gives a detailed derivation of the 
global transfer matrix Un relating state vectors at the two ends o f the 
blade: 0 
T~e enforcement of any physicallY possible set of boundary condi-
tions · at y and y will lead to the determination of the co.rresponding eigen-
o n frequencies (as well as airspeed in the case of flutter). Such eigenfrequen-
cies are related to the complex roots of the so called frequency determinant 
associated with the particular set of selected boundary conditions. The ac-
curacy of the method relies on the number of discretization points takendur-
ing the integration process, as well as on the basic polyno.mial from which 
4 
• 
the integrating matrix is constructed (cf. ref. 10) • The transfer 'tIia'trix capa-
bility for dealing with inhomogeneous boundary conditions, through the in-
troduction of a new transfer matrix of very simple form expressing the 10-" 
'ca1 inhomogeneity, is what really makes this approach attractive. The con-
struction of more elaborate models of VAlIT ~ including for example guy wires 
and tower stiffness, seems possible without difficulty. Spring type bound-
ary conditions are easily handled by a single matrix multiplication of the 
form: 
Here Un+l is the new 
trix. 0 
un+1 = K Un 
o 0 
globa~ . trans~er matrix and K is the spring transfer ma-
CONCLUDING REMARKS 
A semi-analytical method based on a~ integrating matrLx scheme was 
proposed for obtaining the transfer matrix of a troposkien-curved blade~ 
Such an approach is expected to be very efficient to study both free vibra-
·tions and flutter of VAlIT • 
5 
---- -_._ ... _-------------
Y = 
APPENDIX A 
NONDIMENSIONAL EQUATIONS FOR FREE VIBRATION IN STATE VECTOR FORM 
-Q1 
~ 
Q2 
M2 
T 
.M3 
[~FF S = 
SDF 
[ : J = 
u 
Xl 
v 
w 
y' = [ S + J ] y(s) 
~FD] 
~DD 
- f2xl2 J FD 
o . 
12x12 
sym 
SFF 
a
3 
. • 
2 
r aa2 · 
-T 
=-S == DD 
'S = 
FD 
• -I( 
0 
. lIe • • -I( 
0 
-lIe • 
I( 
0 
. I( 
0 6x6 
a 12x1 a4 6x6 6x6 
J = [men2) [- (Zi2+ !) ~M B'M - !"M 4> "M + u.;;;,,1'1 ~ "M + r2 [-w2,C + ..[W 4> ] J FD \ V 2 2 .1 2 a 3 
. . 
cI> = 
. 1 
6x6 ~~2] 
6x6 
cI>11 = [e2~ -~2~] cI>21 = [-:: : ] "cI>31 -[: ~J -.o2</> Co2</> 
cI>12 = r~2~ e2~] cI>22 = [ ~~ ~1 '~32 = [: ;J Co2</> .0 2</> all 2x2 matrices 
r 
1 -1 
x 1 1 1 a 
1 1 
. .. 1'1 = B = 'c = r 1 
-1 a 
1 1 
l sym x 1 1 a 6x6 -'6x6 ox6 
6 
• 
------_ .. _-
• 
APPENDIX A 
where the new symbols are: a ~ EI Ic 2GAo a = [l-(x Ir )2)-1. a =EI lEI 
I nn ' 2 a a '3 nn ~~ 
a4= Elnn/GJ; GA= shear rigidity; (-)= dimensionless quantities: (force)/V, . 
(moment)/Vc, (displacement)/c; '( )= diagonal matrix; I= identity matrix; 
0= null matrix; ( )T= transpose matrix; (.)=zero element; x ~ dimension-
. a .. 
less cross section static unbalance (in units of chord); r c dimensionless 
a 
cross section radius of gyration (in units of chord); 1= blade length; c= 
blade chord; m= blade total mass; V= vertical component of the initial ten-
sion (constant for a gravity-free troposkien); n= angular velocity of the 
VAtfT; w= harmonic frequency; ~= angle defining the troposkien shape (Fig-
ure-2); y= shear coefficient; s= sll; ~ = lK ; ~= wIn; h ~sin x· ~ = cos x 
o 0 x' x 
.i= yCI"; sym= symmetric matrix • 
7 
-- - - -.-- ----- -- .---_._---
APPENDIX -B 
TRANSFER MATRIX DERIVATION WITH THE USE OF THE INTEGRATING MATRIX 
In state vector form, the nondimensiona1 aeroe1astic equations can 
be written as: 
y' c [ S + J + A J y ~ Z Y 
where S, J and A are 12x12 previously defined matrices 
dimensionless state vector of the dependent variables. 
operator L (cf.ref .11) is applied, one gets: 
y=LZy+k 
(B-1) 
and y is the 12x1 
When the integral 
(B-2) 
where k is a constant vector, dependent upon the boundary conditions.Pre-
multiplying equation B-2 by a boundary condition matrix D (cf.ref .7),the 
state vector at one end is isolated since, by their intrigsic structure, 
D L = O. Therefore y ~ k and B-2 can be re,rritten as: 
o 0 y = ( I - L Z ) Y - (B-3) 
o 
One notices that the term in parentheses is, by definition the inverse of 
the transfer matrix U. Hence, 
y = U.y = ( I - L Z )-1 Y 
o 0 
(B-4) 
where I stands for the unit matrix. In particular, premultiplying U by an-
other boundary condition matrix at the opposite end D , yields: 
n 
- - n -y = D U Y = U Y (B-S) 
n n 0 0 0 
that is, the relation between state vectors at the two ends of the blade. 
The transfer matrix Un can be seen as a product of intermediate transfer 
matrices relating con~itions at adjacent points along the integration 
path. The manipulation of most of these matrices is facilitated by their 
block structure and spars~ness. 
8 
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REFERENCES 
1. Blackwell, B. F. and Reis, G.E., "Blade Shape for a Troposkien Type of Ver-
tical Axis Wind Turbine,"Sandia Laboratories Report SAND 74-0154,Apr.1974. 
2. Ham, N.D.,"Aeroelastic Analysis of a Troposkien-Type \-lind Turbine," 
Sandia Laboratories Report SAND 77-0026, Apr.1977. 
3. Ham, N.D.,"Flutter of Darrieus Hind Turbine Blades,"Proceedings of \-lind 
Turbine Structural Dynamics Workshop, NASA CP-2034/CONF-771148~977pp.77-91. 
4. Ashley, H.,"Use of Asymptotic .Methods in Vibration Analysis,"Proceedings 
of llind Turbine Struct.Dynam.Workshop,NASA CP-2034/CONF-77ll48,l977pp.39-52. 
5. Reis, G.E. and Blacktvell, B.F.,"Practical Approximations to aTroposkien 
by Straight-Line and Circular-Arc Segments,"Sandia Laboratories Report 
SAND 74-0100, Harch 1975. 
6. Kaza, K.R. V. and Kvatern·ik, R.G., "Aeroelastic Equations of Motion of a 
Darrieus Vertical-Axis Wind-Turbine Blade,"NASA TM-79295, Dec.1979. 
7. Hunter, W.F.,"Integrating-Matrix Method for Determining the Natural Vi-
bration Characteristics of Propeller Blades,"NASA TN D-6064, Dec.1970. 
8. Pestel, E.C. and Leckie, F.A.~ Matrix .Methods in Elastomechanics, Me. 
Graw-Hill, 1963. 
9. Nair, S. and Hegemier, G.,"Effect of Initial Stresses on the Small De-
f ormations of a Composite Rod, "AIAA Journal,V.16,No.3,Mar.1978,pp.2l2-2l7. 
10. Lehman, I .. , "A Hybrid State Vector Approach to Aeroelastic Analysis with 
Application to Composite Lifting Surfaces,"in AlAA/ASt:-lE/ASCE/AHS 22nd. 
Structures, Struct.Dynamics and Materials Conf.,Part 2, Apr.1981,pp.82l-3l • 
9 
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5=1 
Figure-2: Blade Element Definition 
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.. . . .. 
1. Report No. 2. Government Accession No. 
NASA CR- 165764 
4. Title and Subt itle 
Aeroelastic Analysis of a Troposkien- Type Wind Turbine 
Blade 
7. Author(s) 
F . Nitzsche 
9. Performing Organ izat ion Name and Address 
stanford University 
Stanf ord, CA 94305 
12. Sponsor ing Agency Name and Address 
National Aeronautics and Space Administration 
Washington, DC 20546 
v • j f 
3. Recipient 's Catalog No. 
5. Report Date 
July 1981 
6. Perform ing Organizat ion Code 
8 . Performing Organization Report No. 
10. Work Unit No. 
11 . Contract or Grant No . 
NGL 05- 020-243 
13. Type of Report and Per iod Covered 
Contractor Report 
14. Sponsor ing Agency Code 
15. Supplementary Notes L 1 T h· 1 M . t R V D tt J ang ey ec nlca onl or: . . ogge , r. 
Paper presented at International Colloquium on Wind Energy, 
Brighton, England, August 1981. 
16. Abstract 
The linear.aeroelastic equations for one curved blade of a vertical axis wind 
turbine are presented in state vector form . An attractive method, based on a 
simple integrating matrix scheme together with the transfer matrix idea, is 
proposed as a convenient way of solving the associated eigenvalue problem for 
general support conditions. 
17. Key Words (Suggested by Author(s)) 18. Distribution Statement 
Aeroelasticity 
Wind Turbine 
Troposkien 
19. Securi ty Classif . (of this report ) 
Unclassified 
20. Security Classif . (of this page) 
Unclassified 
Unclassified- Unlimited 
Subject Category 44 
21 . No. of Pages 
11 
22. Price" 
A02 
N-305 For sale by the Nat iona l Techn ica l Informat ion Serv ice, Spr ingf ield , Virginia 22161 


Aeroelastic analysis of a troposkien-type wind turbine blade

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