The testing of Darrieus wind turbines has indicated that under certain conditions, serious vibrations of the blades can occur, involving flatwise bending, torsion, and chordwise bending. A theoretical method of predicting the aeroelastic stability of the coupled bending and torsional motion of such blades with a view to determining the cause of these vibrations, and a means of suppressing them was developed

Ham, N. D.

English

NASA Technical Reports Server

FLUTTER OF DARRIEUS WIND TURBINE BLADES
Norman D. Ham
Department of Aeronautics and Astronautics
Massachusetts Institute of Technology
Cambridge, Mass. 02139
ABSTRACT
The testing of Darrieus wind turbines has indicated that under certain
conditions, serious vibrations of the blades can occur, involving flatwise
bending, torsion, and chordwise bending. It is the purpose of this paper
to develop a theoretical method of predicting the aeroelastic stability of
the coupled bending and torsional motion of such blades with a view to
determining the cause of these vibrations and a means of suppressing them.
i. Introduction
The troposkien type of vertical-axis wind turbine, Reference i, embodies
flexible curved blades connected at each end to the top and bottom of a verti-
cal shaft which permits the blades to rotate about the vertical axis under the
action of the wind. The curve of each blade is ideally that of a troposkien;
i.e., the shape taken by a flexible cable of uniform density and cross section
whose ends are attached to two points on a vertical axis, when it is spun at
a constant angular velocity about the vertical axis. In this manner the bend-
ing stresses in the blades are eliminated.
The testing of such devices has indicated that under certain conditions,
serious vibrations of the blades can occur, involving flatwise bending, torsion,
and chordwise bending. It is the purpose of this report to develop a theoret-
ical method of predicting the aeroelastic stability of the coupled bending and
torsional motion of such blades with a view to determining the cause of these
vibrations and a means of suppressing them. The present analysis is an exten-
sion of that of Reference 2.
2o Flatwise Bending Vibration of the Ideal Troposkien
Consider the rotating troposkien-type rotor blade shown in Figure i.
The bending moment at x is given by
{Lyo+y%%;]_ 2 y(_)} m(_) (x-_) d_ - H[y(o) - y(x)]M_x)
where
= rotor rotation&l speed
m(_) = rotor blade mass at _ per unit of x-distance
H = blade tension force at mid-point between blade ends
Differentiating the bending moment equation twice with respect to x,
_2M(X)_x2 = [yo+Y(X)] _2m(x) - y(x) re(x) + H _2Y_x2
This work was sponsored by Sandia Laboratories, Albuquerque, New Mexico, for
the U.S. Energy Research and Development Administration.
77
https://ntrs.nasa.gov/search.jsp?R=19780011681 2020-03-20T12:55:13+00:00Z
From the theory of elasticity for the bending of slender beams,
M(x) = EI _2y
3x 2
Therefore
32M(x) 32 %--=-- [EI ]
3X 2 _x 2 _x
Then the blade bending equation of motion is
32 % 32Y
-- [EI ] - H-- - _2m(x) (yo+y) + m(x)y = 0
3x 2 _x _x 2
(1)
Note that m(x) dx = m(s) ds, where s = arc length along the blade, and
m(s) = m, a constant. Then
ds _x 2 1/2re(x) = m _x = m [i + ( ) ]
Equation (i) is the equation of the vibrating blade: the steady-state
case is described and solved in Reference i. For present purposes, consider
the equation which results when it is assumed that y = y(@):
-- [EI -H- - _2m(8) [yo+y(8)] + m(8)y(8) = 0
Z@2 _@2 3x _x _@2 3x
(2)
00
Let y = n=_l yn(@) gn(t)
-H
Substituting into Equation (2), there results
2
32 _ 3 Yn (_@)2 ._)e 2
[EI n_ 1 -- _x gn ] (_x)
_O2 302
_2yn 38 2
n=l 3@2 (_x) gn- _2m [Yo + nil Yn gn ] + m n[ 1 Yn g n = 0 (3)
78
Assume free vibration of the blade in the nth flatwise mode in a vacuum:
i_ t
gn = gn e n
i_ t
2- n 2
gn = -_ngn e = -_ng n
Substituting into the nth term of Equation (3), i.e., the equation for
free vibrations in the nth mode,
_2 2 2
-- [EI _ Yn (_@)2 (_@)2 _ Yn (_@)2 2
_@2 _@---2- _x gn ] _x - H -- gn - _2m [Yo+Yn gn ] = m w Yn g
_@2 _x n n
(4)
Then including all modes, substitute Equation (4) into Equation (3) to obtain
oo CO
m n_ 1 Yn gn + m n=_l 2 Yn gn = 0 (5)
This is the equation for flatwise free vibration in a vacuum of the rotating
ideal troposkien blade.
3. Flatwise Bending Vibration of the Approximate Troposkien
Reference 3 describes an approximation to the troposkien shape consisting
of _ev=_ght and circular-arc segments. In this instance, the origin of the axis
system shown in Figure 1 is located at the center of the circular-arc segment,
and the distance Yo of the origin to the axis of rotation is determined by the
geometry of the straight segments.
Reference 4 suggests the following approximate mode shapes for the fiatwise
vibration of non-rotating circular arcs of radius R with pinned ends:
(n_^
Yn = R sin _--}@ (6)
This expression was shown in Reference 2 to be an approximate solution to
the eigenvalue equation describing the vibration of rotating circular arcs, for
the case _ = _ and yo = 0.
79
The vibration frequencies are given by the expression
2 2 _2
= _ + K (7)
n n n
o
where _ and K must be determined experimentally, or numerically by finite-
n n
o 2
element analysis. For the special case of Reference 2, K = n - i.
n
Then Equations (5), (6), (7) describe approximately the flatwise free
bending vibration in a vacuum of the circular-arc segment of the rotating
approximate troposkien blade.
4. Torsion/Chordwise Bending Vibration of the A_proximate Tro_oskien
Reference 5 considers the torsion/chordwise bending vibration of non-rotating
circular arcs. The forces due to rotation are shown in Figure 2. Adding these
forces to the equations of Reference 5, in the nomenclature of the present analysis
the torsion/chordwise bending equations become, for the circular-arc segment of
the approximate troposkien blade,
,v mR 4
z - RS'' - k (z'' + RS'') EI 2 (z - z_ 2 2_ H z'').... --_ (8)mR
Z'' - R8 + k(RS'' + z'') = 0 (9)
where 8(8) = blade torsional displacement
z(8) = blade chordwise bending displacement
R = radius of circular arc
k = GK/EI 2
GK = torsional stiffness of blade section
EI 2 = chordwise bending stiffness of blade section
m = rotor blade mass per unit length of blade
Note that flatwise and torsion/chordwise bending are coupled dynamically
only by the Coriolis forces 2_ym and 2_zm.
8O
Let z = n=_l Zn(e) f (t)
n
= n_ 1 8n(@) fn(t)
Substituting into Equation (8), and neglecting the Coriolis coupling term
at present, there results
,vf "B' (n n' "B1 Zn - R _ f - k Z__ z f + R _ ' f )n n 1 n n 1 n n 1 n n
mR 4
EI 2 (n_l Zn fn _ _2 n_I_ Zn f H2 n_l= Z''n fn ) (i0)
n mR
Assume free vibration of the blade in the nth torsion/chordwise bending mode in
a vacuum:
i_ t
f = f e n
n n
i_ t
n -2
n n n n n
Substituting into the nth term of Equation (i0), i.e., the equation for free
vibrations in the nth mode,
EI
mR 4' [zlVn fn - RB''n fn - k(Z''n fn + RB''n fn )] + _2m Zn fn + --z_RZ''n fn
9
=-m z _- f (ii)
n n n
Then including all modes, substitute Equation (ii) into Equation (!0) to obtain
m z + m _ z f = 0 (12)
n i n n nl n n n
This is _^_**=equation foz _ torsion/chordwise free vibration in a vacuum of the
rotating approximate troposkien blade, not including Coriolis coupling with
flatwise vibration of the blade.
8]
Reference 2 suggests the following approximate mode shapes for the
torsion/chordwise bending vibration of non-rotating circular arcs for the
case e = _ and Yo = 0:
z = R sin n8
n
8 n = sin n8
C.
These expressions were shown in Reference 2 to be approximate solutions to the
eigenvalue equation describing the vibration of rotating circular arcs. Note
that they satisfy only the pinned-end boundary condition, and are therefore
even more approximate for other end conditions, such as the elastic restraint
of the curved segment by the straight segments of the approximate troposkien
blade.
For the case _ _ 7, the following approximate mode shapes will be used,
as for the flatwise modes:
z n = R sin (_)@ (13)
= sin (n_)_@8n
C.
The vibration frequencies are given by the expression
(14)
_2 -2 _2= _ + _, (15)
n n n
o
where _ and K must be determined experimentally, or numerically by finite-
n n
o 2_1.
element analysis. For the special case of Reference 2, K = n
n
Then Equations (12), (13), and (14) describe approximately the torsion/
chordwise bending free vibration in a vacuum of the circular-arc segment of the
rotating approximate troposkien blade.
82
Note that Equation (9) expresses the structural coupling between torsion
and chordwise bending of the circular arc. Substitution of Equations (13) and
(14) into Equation (9) yields the nth mode structural coupling coefficient for
circular arcs having pinned ends,
R8 (n_) 2
Cn=--n_--
n
l+k
l+k (n_) 2
(16)
This approximate relation was investigated experimentalij in Reference 6 and
found to be valid for the troposkien blade with pinned ends for the case n = 2,
if an effective value _eff = 2.8 is used to account for the flexibility of the
straight segments of the blade.
For the case of clamped ends, the approximate nth mode structural coupl_ng
coefficient for circular arcs with clamped ends is shown in Reference 6 to be
2
Cn = [ 0.75 (n_) l+k
sin2(nn)__ @ 2.25] _-- l+k(_)2
where the effective value _eff = 2.8 again applies.
Equation (16) is believed to be a reasonable approximation for the structural
coupling coefficient of practical blade configurations in the absence of a more
precise determination by finite element analysis of the blade.
5. Blade Flutter Equations
Adding the Corio!is forces due to blade bending, and the blade aerodynamic
lift force, as shown in Figures 2 and 3, Equations (5) and (12) become
2 _ = _ __
_ m v 5 + _ m _ v Q + Z. 2r_-_ z f 1 dL sin6 (17)
n=l "n _n n_l n -n -n n=l n n _ u0
_ _2
+ _i m Zn _ + _ m z f = 0 (18)
-n_ 1 2m_ Yn gn n n n=l n n n
83
where chordwise aerodynamic forces are neglected.
Using Equations (6) and (13), multiplying Equations (17) and (18) by
n_
R sin(_--)8, and integrating from 0 to _, the coupled equations of motion in
the nth mode are obtained:
In gn + I 2n n gn + 2In_ _n = o sin6 sin(_)8 dL_de
f -2
-2ln_ + I + I _ fgn n n n n n
= 0
= R2where I _ m sin2( )@ d@
n o
(n_ sin(_)8 dOand noting that °_ sin _--)@ = 0.
Following Reference 7 and Figure 3, the aerodynamic lift for the wind
turbine rotating in still air at constant rotational speed _ is
1 dL = _ 1 pac 2 _ 1
R d8 8 _Yl - _ pac_YlC(k) [9 sin6 + _yl 8 + (0.5c - XA)_]
where p = air density
a = blade section lift curve slope
c = blade chord
8 = blade twist angle (postive nose down) (see Fig. 2)
Yl = outer blade undisturbed shape = Yo + R sin_
C(k) = Theodorsen's lift deficiency function
x A = distance from section aerodynamic center to blade elastic
axis, positive when A.C. is forward.
Neglecting the _ terms since c << R, multiplying by R sin(n_)8 and dividing
I _2, there results
n
84
__ d"1 I W sin_ sin )8 _ d8
I _2 o
n
where C
n
RB
n
= -- as before.
z
n
= - m_g n - m_Cnf n
Then
m_ = _ C (k)
+ (n_)
o_(Yo/R sin_) sin2_ sin2 _ - @ d8
2(_)@ d@
ms : _ C-7_
o_(Yo/R + sin6) 2 sin6 sin2(_)@ d8
o_Sin 2 n_(_-)8 de
pacR
where 7 =
m
C(k---_= typical value of C(k) evaluated at Yl = Yo + R sine (see Appendix),
and the contributions of modes other than the nth are neglected.
Dividing Equations (23) and (24) by I _2 and including the aerodynamic
n
terms,
g__n gn 2
_2.+ m_ _-- + V n gn + 2 _ + msCnf n = 0
(21)
gn fn _2 =
- 2 _-- + f_-_ + n fn 0
(22)
where
2 1/2
n
V ___nnn _ = [( ) + K ]n
Gn 2 1/2
_n=._ -= [( ) +_ In
These are the flutter equations for the wind turbine rotating in still air.
85
5, Calculation of Flutter Boundaries
Assume the following solutions to Equations (21) and (22):
gn = gn ev_t
V_tf =_ e
n n
where v = _ + i _. Assume at present that C(k) = i.
Substitution of Equations (23) and (24) in Equations (21) and (22)
results in two coupled homogeneous algebraic equations. For a solution to
exist, the determinant of coefficients must vanish; i.e.,
(23)
(24)
( 2 + m.V + v 2) (2V + msC n)g n
- 2_ [2 + 521
n
= 0
Expanding the determinant yields the characteristic equation of the system:
AV 4 + BV 3 + CV 2 + DV + E = 0
where the coefficients are combinations of the system parameters. Then by
Routh's criteria for system stability
A, B, C, D, E > 0 (25)
and BCD - AD 2 - B2E > 0 (26)
-2
It is now possible to determine the values of V
n
required to satisfy conditions
(25) and (26) for a given blade configuration. Since
2
n
2 (__)= + KVn n
86
and
2
n
,_2 = (._._) +n n
the values of rotational speed at which flutter or divergence will occur (_F)
are then specified for given non-rotating flatwise and torsional frequencies
and
n n
o o
In the present case, the characteristic equation is
4 V 2 (m_5_ V 2 5 2V + m.V 3 + (4 + + _2)V2 + + 2m 6 Cn)V + = 0g n n n n
Then conditions (25) and (26) determine the rotational speed at flutter, _F"
Applying condition (25), flutter occurs when
-2 m6
V = - 2 -- C (27)
n m. n
g
Applying condition (26), flutter occurs when
2 C 2 4m 8 m. C - m S m. C U 2
52 = 2m8 n - 9 n
n n (28)
n 2m 2 - m 8 m. Cg g n
6. Application to an Existing Blade
The theory developed above is now applied to the preliminary prediction
of the flutter of the blades of the Sandia seventeen-meter wind turbine. For
this turbine the parameters for the geometry of the circular arc portion of the
blade are
Yo = 9, !0 ft.
R = 18.33 ft.
= 2 rad.
For this case _t can. be shown that m E = 1.575 m.g
Equation (16) for n = 2, k = .0512, _eff = 2.8 (Ref. 6):
C =-4.2
n
Also from
87
Applying Equation (27), flutter occurs when
-2
V = 13.2
n
From Reference (8), for the seventeen-meter blade,
-2 (324)2
_2 = _ + 7.70
-2
Equating and solving the two expressions for v2'
= 138 rpm
Applying Equation (28) flutter occurs when
-2 2
_2 = 13.2 + 0.768 _2
From Reference (8), for the seventeen-meter blade,
2 215 2
_)2 = (_--) + 6.76
Then = (215)2
-2 18.4 + 0.768V2
-2
Equating and solving the two expressions for 92'
= 80.4 rpm
Here Equation (28) is evidently the critical case.
Final prediction of the flutter speed requires accurate determination of
the structural coupling coefficient C by finite element analysis.
n
7. Conclusions
(i) Coriolis forces provide the dominant coupling between flatwise bending
and torsion/chordwise bending of the blade. (Also see Reference 2).
(2) Blade flatwise bending velocity and blade torsional displacement
contribute the dominant aerodynamic forces acting on the blade.
88
(3)
(4)
Accurate determination of the structural coupling between blade
torsion and blade chordwise bending is essential to blade flutter
prediction.
The following parameters determine the rotational speed at flutter
for troposkien-type blades:
(a) blade geometry Yo" R, _ (see Figure i).
(b) blade torsion/chordwise bending structural coupling coefficient
C 2 for the second mode.
(c) blade rotating and non-rotating flatwise bending frequencies
_2 and _2 for the second mode.
o
(d) blade rotating and non-rotating torsion/chordwise bending
frequencies _2 and _2 for the second mode.
O
REFERENCES
i. Blackwell, B.F. and Reis, G.E., "Blade Shape for a Troposkien Type of
Vertical-Axis Wind Turbine", Sandia Laboratories Report SAND 74-0154,
April 1974.
2. Ham, N.D., "Aeroelastic Analysis of the Troposkien-Type Wind Turbine",
Sandia Laboratories Report SAND 77-0026, April 1977.
3. Reis, G.E. and Blackwell, B.F., "Practical Approximations to a Troposkien
by Straight-Line and Circular-Arc Segments", Sandia Laboratories Report
SAND 74-0100, March 1975.
4. Den Hartog, J.P., "The Lowest Natural Frequency of Circular Arcs", Phil.
Mag. 5 (Series 7), 400, 1928.
5. Ojalvo, I.U., "Coupled Twist-Bending Vibrations of Incomplete Elastic
Rings", International Journal of Mechanical Sciences, 4, pp. 53-72, 1962.
89
6. Ham, N.D., "An Investigation of the Structural Coupling Coefficient and
Torsional Frequency of Troposkien-Type Wind Turbine Blades with Application
to Blade Flutter Prediction", Sandia Laboratories Report in preparation, 1977.
7. Miller, R.H. and Ellis, C.W., "Blade Vibration and Flutter", Journal of the
American Helicopter Society, l, 3, July 1956.
8. Barzda, J., Kaman Aerospace Corp., Bloomfield, Conn., unpublished communication.
DISCUSSION
Comment: I would like to reemphasize the significance of Professor Ham's
conclusions that a) quarter-chord mass balancing of the turbine blades
is unimportant with regard to flutter speed location, and b) increasing
torsional stiffness is an effective method of increasing flutter speed.
The impact of the first conclusion on design and fabrication of low cost
blades is very favorable, and knowledge of the second conclusion is
valuable to the designer. At the same time, I would like to de-emphasize
the threat of encountering flutter in any particular blade design. Based
on Professor Ham's analysis, blade property evaluation, field experience,
and consultation with fabricators, it is relatively easy to design a
blade that will not flutter.
Q: Do you agree with R. Reuter that it is relatively easy to design
Darrieus blades which do not flutter in their operating regime?
A: Yes.
9O
Q: AmI correct in presuming that the forced response to periodic stalling
should also be independent of the chordwise c.g. of the blade sections,
and also of the chordwise shear center?
A: Yes.
Q: What were your assumptions relative to the aerodynamics; was the
free stream velocity zero? If not what of the resultant periodic
coefficient?
A: In the analysis the free stream velocity was assumedto be zero. If
this were not so, harmonic airloads would be generated and would
cause a forced blade response at various integral multiples of rotor
speed. However, in a linear analysis, this forced response is uncoupled
with any self-excited motion such as flutter. Experimentally, model
blade flutter occurred at the samerotor rotational speed whether or
not there was an incident wind.
91
YFigure I. - Blade geometry.
I/ 2,, >'
yo Z
--LC 
Figure 2. - Blade dynamics.
92
Rm
I
T
Figure 3. - Blade aerodynamics.
93


Flutter of Darrieus wind turbine blades

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server