-
Aeroelastic instability analysis of composite rotating blades
based on
loewys and theodorsens unsteady aerodynamics
*Touraj Farsadi1) and Altan Kayran2)
1), 2) METUWind centre, Department of Aerospace Engineering,
Middle East Technical University, Ankara, Turkey 1)
[email protected]
ABSTRACT
Classical aeroelastic stability analysis approach is presented
for the simplified composite blade model. For the purpose of the
study, the composite wind turbine blade is modeled as an elastic
cantilevered rotating thin-walled composite box beam with the
developed Circumferentially Asymmetric Stiffness (CAS) structural
model. Circumferentially Asymmetric Stiffness structural model
takes into account non-classical effects such as transverse shear,
material anisotropy and warping inhibition. For the aeroelastic
stability analysis, aerodynamic analysis approaches used in the
present study are based on classical Theodorsens theory and Loewys
returning wake method in the frequency domain in conjunction with
the CAS structural model. Hamiltons principle and extended Galerkin
method are used to obtain the coupled system of equations which are
then posed as an eigenvalue problem. Results show that Theoderson
and Loewy theories give close results in low rotor speeds but when
the rotor speed is increased, the difference between these two
methods becomes apparent. It is also shown that the fiber angle of
the CAS structural model affects the aeroelastic instability speed.
1. INTRODUCTION Wind energy conversion is a fast growing source
among renewable energies in the world. Wind turbines are getting
larger for past ten years to capture more energy from the wind.
Increasing the size of the turbines has resulted in a price
reduction for the electricity per kWh, but it has also created
problems associated with the length of the blades. One probable
problem area is the aeroelastic stability problems of long wind
turbine blades due to the increased bending and torsional
flexibility. Increased bending and torsional flexibility of long
wind turbine blades may cause torsional divergence and flapwise
bending-torsion flutter at high speeds. Stiffness tailoring is one
way to control the elastic deformation of blades made of composite
materials in a
1)
PhD Candidate 2)
Professor
mailto:[email protected]
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structural optimization framework. Tangler (2000) pointed out
that composite material could be used in the wind turbine blade to
efficiently sustain the complex loading for long blades. A
Thin-Walled Beam (TWB) is a slender structural element whose
distinctive geometric dimensions are all of different orders of
magnitude such that its thickness is small compared to the
cross-sectional dimensions, while its length greatly exceeds the
dimensions of its cross-section (Librescu 2006). Composite TWBs are
used in jet engines, tilt rotor aircrafts, helicopters, turbo
machinery, and in wind turbines as rotor blades, and their dynamic
behavior is the topic of many studies. In the case of the rotating
beam, Song (2001) performed the free vibration analysis of rotating
pre-twisted TWBs, incorporated flaplag elastic coupling and
adaptive capabilities using the Extended Galerkins method (EGM).
Sina (2011) investigated the rotation effects in eigenvalue
analysis of single cell-laminated composite TWB with closed
cross-section. The effects of rotation, ply angle, taper ratio,
slenderness, and hub ratios on natural frequencies and mode shapes
of rotating TWB with flaptwist elastic coupling are studied. In
wind turbine blades, for the aerodynamic solution, most codes use
the Blade Element Momentum (BEM) method, as described by Glauert
(1963). Blade element moment method is very fast and yields
accurate results, provided that reliable airfoil data exist. Since
wind turbines operate in unsteady flow environment for most of
their time, it is important for the analyst to recognize that many
of the tools to model unsteady aerodynamic effects on airfoils have
already been laid down. Results for incompressible, unsteady
airfoil problems are formulated in the frequency domain, primarily
by Theodorsen (1935) and Loewy (1957). An authoritative source
documenting these classical theories is given by Leishman (2000).
These solutions have the same root in unsteady thin-airfoil theory,
and give exact analytic solutions for airloads for different
forcing conditions. Classical flutter is defined as the violent
instability when the torsional mode of the blade couples with the
flap-wise bending mode, resulting in rapid growth of the amplitude
of the flap-wise and torsional motions (Politakis 2008 and Vatne
2011). Pitch-flap flutter and divergence are very similar to the
classical flutter and divergence of fixed wing airplanes. Of
course, the centrifugal action on the wind turbine blade is not
part of the classical flutter of fixed wing aircraft. It should be
noted that due to the centrifugal forces, the flapwise stiffness
effectively increases in rotor blades. Janetzke (1983) at the NASA
Lewis Research Center published the first paper directly related to
the aeroelasticity of the wind turbine. Their study explored the
possibility of whirl flutter and searched the effect of
pitch-flap-coupling on the teetering motion of a 2 blade wind
turbine. In 1998, Sandia Laboratories & National Renewable
Energy Laboratory published an article on the aeoelastic tailoring
of the wind turbine blade (Veers 1998). In this study, aeroelastic
property of the blades is utilized to shape the power curve and
reduce loads. Early and recent investigations of stability in wind
turbine blades address the issue of classical flutter for smaller
rotors (10m blades) (Lobitz 1998). Lobitz (2004) investigated the
flutter limit of a MW sized wind turbine blade based on isolated
blade stability analysis using quasi-steady and unsteady Theodorsen
aerodynamics. In this study, it is shown that the predicted flutter
speed of the blade using quasi-steady aerodynamics is lower than
the flutter speed obtained using unsteady aerodynamics. Hansen
(2004) indicated that the flutter rotational speed
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is in the realm of two times the operational speed of the rotor.
Recently, Farsadi (2016) demonstrated the aeroelastic stability
analysis of wind turbine blades using the thin walled beam
structural model and Wagners function in a coupled analysis
framework both in frequency and time domain. In the present study,
a unified model of aeroelastic systems in incompressible flow
regime using aerodynamic states based on the theories of Theodorsen
and Loewy is presented. The main goal of the study is to compare
the effect of using Theodorsens unsteady aerodynamics and Loewys
method on the aeroelastic stability characteristics of wind turbine
blades. It should be noted that Theodorsens aerodynamics neglects
the effect of the wake formed by the preceding blade, whereas
Loewys method includes the effect of returning wake of the
preceding blade. Using the two different aerodynamic models,
numerical results are obtained for the aeroelastic stability of
wind turbine blades modeled as rotating composite TWBs with the
previously developed Circumferentially Asymmetric Stiffness (CAS)
structural model. Circumferentially asymmetric stiffness structural
model takes into account a group of non-classical effects such as
the transverse shear, the material anisotropy and warping
inhibition. Hamiltons principle and the extended Galerkin method
are used to obtain the coupled linear governing system of dynamic
equations. Governing equations are posed as eigenvalue problems
involving Theodorsen and Loewys unsteady aerodynamics in the
frequency domain. U-g method is used to determine the
instability limit of the wind turbine blade. 2. ROTATING THIN
WALLED BEAM The analysis of rotating blade structures is more
complex than that of their nonrotating counterparts. In the
rotating case, in addition to the accelerations resulting from
elastic structural deformations, the centrifugal and Coriolis
accelerations have to be included in the modeling. In this present
study, the structural model is similar to those developed by
Librescu (2006) and Sina (2011). Inertial reference coordinate is
attached to the hub center, while the local coordinate system is
fixed to the blade and used to define the complex cross-sectional
properties, as shown in Figs. 1, 2.
Fig. 1 Schematic description of the rotating blade structure
simulated by
"Rotating Thin-Walled Composite Beam"
Fig. 2 Cross section of the TWB and the
displacement field
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Notice that for the thin walled beam theory considered herein,
the six kinematic
variables, u0(z,t), v0(z,t), w0(z,t), x(z,t), y(z,t), (z,t)
defined with respect to the rotating local coordinate system in
Fig. 2, represent 1-D displacement measures. a(s) and rn(s) shown
in Fig. 2 are the perpendicular distances from the shear center to
the normal and to the tangent of the mid-line beam contour and they
are defined in Eqs. (4) and (5). The present structural model of
the rotating thin-walled composite box beam, shown in Fig. 3,
includes some non-classical effects such as material anisotropy,
transverse shear, primary and secondary warping inhibition,
non-uniform torsional model and rotary moment of inertia. Also,
Circumferentially Asymmetric Stiffness (CAS) method is chosen among
various lay-up methods in order to induce bending-twisting coupling
in the composite rotor blade, as shown in Fig. 3. CAS ply-angle
configuration, achievable via the usual filament winding
technology, results in an exact decoupling between the extension -
twist on one hand and the flap shear elastic coupling on the other
hand. It should be noted that in the present study, thin walled
beam is taken as non-tapered beam with no pre-twist in order to not
to complicate the equations further. However, solution method based
on extended Galerkin method is general and also applicable to
tapered and pre-twisted beams.
Fig. 3 Thin walled- composite beam and
the CAS configuration
Fig. 4 Schematic description of the wing structure and
aerodynamic and structural
coordinates
Figs. 3 and 4 show that two coordinate systems exist, (x,y,z) as
the local
coordinate associated with the blade and (n,s,z) used to define
complex cross-section
profiles. The angular velocity of the rotor is assumed to be
constant in the global y
direction. The position vector " R " of a point in the deformed
rotating beam, measured from the centre of the hub, can be
expressed as:
= + +D
= = + +
D = + +
0
0 0 ,u
u
R R R
R R k R xi yj zk
ui vj wk
(1)
-
Considering angular velocity () and Eq. (1), "R&" and
"R&&" can be written as:
0
2 20
( ) ( )
2 ( ) 2 ( )
R u R z w i vj w x u k
R u w x u i vj w u R z w k
= +W + + + + - W + = + W- + W + + - W- + + W
& & &&
&& & && &&&& &
where " R0" and " Ru " are the hub radius and the undeformed
position vector of a beam point, respectively, and "" represents
the displacement vector, whose components are defined in Eq.
(1).
The 3-D displacement quantities ( ), ,u v w are assumed to be:
0
0
'0
( , , , ) ( , ) ( , )
( , , , ) ( , ) ( , )
( , , , ) ( , ) ( , ) ( ) ( , ) ( ) ( , ) ( ) ( )x y w
u x y z t u z t y z t
v x y z t v z t x z t
dx dyw x y z t w z t z t y s n z t x s n z t F s na s
ds ds
= -
= +
= + - + + - +
f
f
q q f
where
'0
'0
( , ) ( , ) ( , )
( , ) ( , ) ( , )
( ) ( ) ( )
x yz
y xz
z t z t v z t
z t z t u z t
dy dxa s y s x s
ds ds
q g
q g
= -
= -
= - -
The primary Warping Function ( wF ) is expressed as (Librescu
2006 and Sina 2011):
0
( )
( )[ ( ) ] , , ( ) ( ) ( )
( )
ns
Cw n n
C
r sds
h s dy dxF r s ds r s x s y s
ds ds ds
h s
= - Y Y = = -
The axial strains associated with the displacement field are
given by Librescu (2006) and Sina (2011):
0
0 ' ' ' ''0
' ' ''
( , , , ) ( , , ) ( , , )
( , , ) ( , ) ( , ) ( ) ( , ) ( ) ( , ) ( )
( , , ) ( , ) ( , ) ( , ) ( )
nzz zz zz
zz x y w
nzz y x
n s z t s z t n s z t
s z t w z t z t y s z t x s z t F s
dy dxs z t z t z t z t a s
ds ds
e e e
e q q f
e q q f
= +
= + + -
= - -
where prime denotes derivative with respect to the z coordinate.
The tangential shear strain components can be defined as (Librescu
2006 and Sina 2011):
0 '
0 ' '0 0
' '0 0
( , , ) ( , , ) 2 ( , )
( , , ) [ ( , ) ( , )] [ ( , ) ( , )]
( , , ) [ ( , ) ( , )] [ ( , ) ( , )]
Csz sz
sz y x
nz y x
As z t s z t z t
dx dys z t u z t z t v z t z t
ds dsdy dx
s z t u z t z t v z t z tds ds
g g fb
g q q
g q q
= +
= + + +
= + - +
(2)
(3)
(4)
(5)
(6)
(7)
-
where 0 0,zz sze g are the normal strain and the in-plane shear
strain components at
the mid-surface of the thin-walled box beam, respectively. The
kinetic energy, the strain energy and the work of external forces
are calculated using the strain-displacement relationships,
sectional effective stiffness matrix and external forces. By
substituting these expressions into the Hamiltons Principle, given
by Eq. (8), the governing system of equations can be obtained. The
rotating thin-walled composite box beam theory produces a linear
relationship between the sectional structural loads and the strain
measures.
Hamilton`s principle and variational formulation can be written
as shown in Eq. (8) 2
1
1 2 0 0 0( ) 0 , 0
t
x y
t
T V W dt at t t t u v w- + = = = = = = = = d d d d d d dq dq
df
The kinetic energy of the system is given by Eq. (9).
T R R dndsdzd r d= ? & & The potential energy of the
system is given by Eq. (10).
( )10 ( )
1[ ]
2
L N
ij ij zz zz sz sz nz nx kkC h k
V dndsdz dndsdz=
= = + +
d s de d s e s e s e
The work of the external forces is given by Eq. (11).
[ ]00
( , ) ( , ) ( , ) ,
L
ae aeW L z t v z t T z t dzd d df= +
Where Lae and Tae are the unsteady aerodynamic lift and pitching
moment about the reference axis. Substituting Eqs. (2) to (7) into
Eqs. (9) and (10) and using Hamiltons principle, one gets Eqs. (12)
and (13) for the variation of the potential and kinetic energy.
0
0 0 0 0 0
0
0
Lz y x y x y x w z r
x z y z
z y y x x w w z r
x z
T w M Q M Q B M TV dz
Q T u u Q T v v
T w M M B B M T
Q T u
d dq dq f dfd
d d
d dq dq df f df
?+ - + - + + + + = - + + + + ?+ + - + + + +
+
0 0 0 0
L
y zu Q T v vd d
+ +
2 2
1 0 0 0 0 1 0 0 1 0 0 0 0 0
2 25 15 4 5 4
2 24 5 4 4 5 10 18
2 2
2
2
y y y x x x
x
b u w u u bv v b w u R z w w
T b b b b b
b b b b b b b
d d d
d q q dq q q f dq
f q f f
+ W- W + + - W- + + W + = - + - W + + - W + W +
+ - W - - W - + - W
& && &&&& &
&& && &
&& & &&0
210 18 0
L
L
dz
b b
f df
f f df
-
+ - W
&&
(9)
(10)
(11)
(8)
(13)
(10)
(11)
(12)
(13)
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The Euler Lagrange equations can be obtained from Eqs. (8), (12)
and (13) and they are given by Eq. (14).
20 0 1 0 0 0: 2 0x zu Q Tu b u w ud
+ - + W- W = &&&
0 0 1: 0y zv Q Tv bvd+ - =
20 1 0 0 1 0 0: T 2 0yw b w u b R z wd - - W + W + + =
&& &
25 15: M 0y y x y yQ b bdq q q
- - + - W = &&
24 14 4: 2 0x x y x xM Q b b bdq q q f
- - + - W - W = && &
2 24 5 4 5 4 10 18: B 2 0w z r xM T b b b b b b bdf f f f q f
f
+ + - + + - W + W+ + - W = && & &&
In Eq. (14), one-dimensional stress measures, , , , , , ,z r x y
y x zT T Q Q M M M and wB
can be defined in terms of stress resultants and stress couples
as:
( , )z zzT z t N ds= ( , ) ( , )r z pT z t T z t I=
( , )x sz zndx dy
Q z t N N dsds ds
= +
( , )y sz zndy dx
Q z t N N dsds ds
= -
( , )x zz zzdx
M z t yN L dsds
= -
( , )y zz zzdy
M z t xN L dsds
= +
( , ) 2z sz szM z t N L dsy= +
( , ) ( ) ( )w w zz zzB z t F s N a s L ds= +
where , ,zz sz znN N N are the stress resultants and szL and zzL
are the stress couples defined by Eq. (16).
2 2 2
2 2 2
, 1, , , 1, , 1,
t t t
zz zz zz sz sz sz nz nz
t t t
N L n dn N L n dn N n dns s s
- - -
= = = ?
The reduced mass terms, ,i pb I in Eqs. (14) and (15) are
defined in the work Sina
(2011) Boundary conditions at two edges of the blade can also be
obtained from Hamiltons principle as:
(14-a)
(15)
(14-b)
(14-c)
(14-d)
(14-e)
(14-f)
(16)
-
0 0
0 0
0
210 18
0 0
0 0
0 0
0 0
0 0
0 0
0 0
x z
y z
z
x x
y y
w z r
w
u or Q T u
v or Q T v
w or T
or M
or M
or B M T b b
or B
d
d
d
dq
dq
df f f f
df
= + =
= + =
= =
= =
= =
= + + + + - W = = =
&&
In Eq. (17), the left-hand side boundary conditions are called
as essential boundary conditions and the right-hand side ones are
called as natural boundary conditions. Constitutive relations for a
general orthotropic material can be written as:
11 12 13 16
12 22 23 26
13 23 33 36
44 45
45 55
16 26 36 66
0 0
0 0
0 0
0 0 0 0
0 0 0 0
0 0
ss ss
zz zz
nn nn
zn zn
sn sn
sz sz
C C C C
C C C C
C C C C
C C
C C
C C C C
s e
s e
s e
s g
s g
s g
? ? ? ? ? ? ? ? ? ? ?= ? ? ? ?
where ijC represent stiffness coefficients.
The stress resultants " `N s " and stress couples " `L s " can
be reduced to the expressions given by Eqs. (19) and (20).
0
011 12 13 14
'21 22 23 24
244 45 55
zz
zz sz
sz
nzz
nz nz
N K K K K
N K K K K
N A A A
e
g
f
e
g
= = -
0
041 42 43 44
'51 52 53 54
zz
zz sz
sz
nzz
L K K K K
L K K K K
e
g
f
e
=
Where reduced stiffness coefficients ( )ijK are defined in
Farsadi (2016).
Taking into account the present CAS structural configuration,
the entire system of
equations splits exactly into flap / torsion / vertical
transverse shear 0, , xv f q , and
extension / lateral bending / lateral transverse shear0 0, , yu
w q
, respectively as shown in
Eq. (14).
(19)
(20)
(17)
(18)
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3. UNSTEADY AERODYNAMIC MODELS
3.1 Theodorsens unsteady aerodynamics The aerodynamics for a
single blade is similar to that of a fixed wing with a free stream
velocity that varies linearly from the root to the tip, assuming
that the shed wake of the preceding blade dies out sufficiently
fast so that the oncoming blade will encounter essentially still
air (Lobitz 2004). Based on Theodorsen aerodynamics, for the
general motion of thin airfoil of chord
length 2b undergoing a combination of pitching and plunging
motion in a flow of steady
velocity U, unsteady lift and pitching moment about the
reference axis are given by Eqs. (21) and (22), respectively.
2 2
3 2
1( ) 2 ( )( ( ) ]
2
1 1( ( ) ( ) )
2 8
ae rel rel rel
noncirculatorycirculatory
ae rel
noncirculatory
L b v ba U bU C k U b a v
T b av U a b a
pr f f pr f f
pr f f
= - + + + + - +
= - - - + +
&& & &&& &14444444444244444444443
14444444444444444244444444444444443
& &&&&1 2
2 1 12 ( ) ( )( ( ) )2 2rel rel
circulatory
U b a C k v U b apr f f+ + + + - &&44444444444444444
444444444444444443 14444444444444444444244444444444444444443
The first term in Eq. (21) is the non-circulatory or apparent
mass part, which
results from the flow acceleration effect. The second group of
terms is the circulatory components arising from the generation of
circulation about the airfoil. Theodorsens
function, also named as lift deficiency function, C(k) is a
complex-valued function which
depends on the reduced frequency k, ( b/ U). Theodorsens
function is defined in terms of Hankel functions by Eq. (22).
= +
(2)1
(2) (2)1 0
( )H
C kH iH
3.2 Loewys unsteady aerodynamics Loewy (1957) postulated a two
dimensional model representing the aerodynamics of an oscillating
rotary wing operating at low inflow. Loewys rotary wing unsteady
aerodynamics theory is an incompressible theory, somewhat similar
to the Theodorsens theory, but main difference is that in Loewy
approach, the effect of the spiral returning wake behind the rotor
is taken into account approximately. Fig. 5 is a schematic of
Loewys two-dimensional model that is used to determine the effects
of previously shed wakes on the lift deficiency function. Loewy
assumed that there are infinite number of wakes behind the
reference blade and applied the Biot-Savart law to each layer of
shed vorticity to add together the effects on the differential
downwash equation. Two indices that are used to account for the
vorticity shed by a given wake are n, which indicates the
revolution number of the reference blade, and q which indicates the
blade number that wake belongs to.
(22)
(21)
-
Fig. 5 Loewys unsteady aerodynamic model for multi-blade rotor
system (Couch 2003)
Loewys theory considers only steady simple harmonic motion which
has taken place over an infinite period of time. Therefore, Loewys
theory assumes infinite number of wake layers behind the rotor.
Loewy has shown that the unsteady aerodynamic lift and moment can
be written in a form identical to the classical Theodorsen theory,
except that the lift deficiency function must be replaced by the
more complicated lift deficiency function denoted by , ,C k h m .
In the new definition of the lift deficiency
function, k is the reduced frequency and h is the wake spacing,
which is a function of the period of the rotor revolution and the
inflow velocity, and m is frequency ratio. Derivation of Loewys
function is similar to the Theodorsens function except the terms
which account for the vorticity generated by the reference blade
and subsequent blades in previous revolutions. Loewys function in
terms of Hankel functions is given by Eq. (23) (Loewy 1957).
2
1 1
2 2
1 0 1 0
2 , ,, ,
2 , ,
H k J k W k h mC k h m
H k iH k J k iJ k W k h m
where
1
2 2, , 1 , ,
kh im kr UW k h m e m h
Qb bQ
In Eq. (24), r is radial position of the airfoil, R is the
radius of the blade, Q
is the number of blades, b is the semi-chord, is the rotational
velocity of the rotor, U is the inflow velocity. Comparing Loewys
function with the Theodorsens function shows that additional terms
in the numerator and the denominator are scaled
by the weighting factor W. For large wake spacing, weighting
factor aproaches zero
(24)
(23)
(24)
-
and the Loewys function reduces to the Theodorsen function, as
expected. Fig. 6
compares the real parts and imaginary parts of the Theodorsens
and the Loewys
functions for U=15 m/s, =4 RPM and Q=3, respectively. The
limitations of the Loewy lift deficiency function are that it is
restricted to a rotary blade such as wind turbine blade with not
high inflow velocity. For the case investigated in Fig. 6 Loewy
function can be used in a reasonable reduced frequency range
between 0 and 1. It can be seen
that when k 1 which corresponds to high inflow, the effects of
any shed layer of vorticity is negligible, and Theoderson and Loewy
functions give the same results.
Fig. 6 Comparison of real and imaginary parts of Theodorsens and
Loewys
functions versus reduced frequency for r/R=0.1,
U=15m/s,Q=3,=4RPM
Loewy's function changes the problem from that of a thin airfoil
in harmonic motion to that of a rotary wing, undergoing harmonic
motion. For high inflow, Loewys function approaches the Theodorsens
function. For the high inflow case, it would be expected that all
shed vorticity beyond a small fraction of a revolution would be too
far below the reference blade to have a significant effect. The
most important result from Loewy is that the wake geometry and
phasing is the primary cause of unsteady rotor loading. 4.
GOVERNING SYSTEM EQUATIONS OF THE AEROELASTIC SYSTEM
In the present study, an anisotropic, rotating thin walled
composite box beam is used in order to study the effect of
employing the Theodorsens function and Loewys returning wake on the
aeroelastic stability of the wind turbine blade in incompressible
flow. For the derivation of the governing system of equations, the
kinetic energy, the strain energy and the work by external forces
of the wind turbine blade structure are calculated using the strain
displacement relations, the sectional effective stiffness matrix
and the external loads. Furthermore, the assumption of small
deformations and small strain theory results in the linear
relationship between the cross sectional external loads and the
strain measures. The present rotating thin walled composite box
beam
-
model, by means of the Hamilton`s principle and the variational
calculus, yields two sets of governing system of equations. The
first set is elastically coupled by the "flap / torsion / vertical
transverse shear" deformation modes and the second set is
elastically coupled by the "extension / lateral bending / lateral
transverse shear" deformation modes. In this study, the second set
is not taken into account. The equations of motion corresponding to
the first set, are given by Eq. (25).
'' ' '''0 55 0 56 0 1 0
'' '' ' '' 233 37 55 0 56 4 14
''' '' '''' '' '' 265 0 66 73 77 4 5 4 5
10 18
0: ( ) ( )
0: ( ) ( )( )
0: ( ) ( ) ( ) ( )
( )(
x z ae
x x x x x
x x r ae
v a v a T v L bv
a a a v a b b
a v a a a T b b T b b
b b
d q f
dq q f q f q q
df q f q f f f f
f
= + + + + =
= + - + - = + - W
= + - + + + + - W + = + -
+
&&
&&
&&
&& 2 )f- W
where the unsteady aerodynamic lift Lae and the unsteady
aerodynamic pitching moment Tae are defined in Eq. (21).
Centrifugal force that appears in the flap and the
torsion equations as Tz and Tr represent centrifugal stiffening
expressions. Tr plays the role of torsional stiffness induced by
the centrifugal force field.
The boundary conditions at the root of the blade (z=0) are given
by:
'0 0 0x
dv alsodzfq f f= = = = =
At the tip of the blade (z=L), from the Hamiltonss principle,
boundary conditions
come out as:
' ' ''' 20 55 0 56 1
' '33 37
'' ' ''' ' ' ' 265 0 66 73 77 10 18 1
' ' ''56 0 66
0: ( ) ( ) 0
0: 0
0: ( ) ( ) ( ) 0
0: ( ) 0
x
x x
x x p
x
v a v a b R z
a a
a v a a a b b b R z I
a v a
d q f
dq q f
df q f q f f f
df q f
= + - + W =
= + =
= + - + + + + + W =
= + - =
&&
where Tz , Tr and R(z) are given by 2 2
0
21
4 50 2
1( ) ( ) ( ),2
( , ) ( ),
2, ,
( , )
z
p pmp pm ph
r z ph
R z R L z L z
T z t b R z
I II b b I I
m mT z t T I
b
= - + -
= W
= + = =+
=
and aij and bij and m0 , m2 in Eqs. (25-28) are defined in
Librescu (2006).
In order to solve the set of equations given by Eq. (25),
Extended Galerkin Method (EGM) is employed. For this purpose, the
aerodynamic and the structural
(26)
(27)
(28)
(25)
(27)
(28)
-
variables are separated in time and spatial coordinate z and the
assumed mode shapes are defined by Eq. (29).
11
22
33
10
2
3
( , ) ( ) ( )( , ) ( ) ( )
( , ) ( ) ( ) ( , ) ( ) ( )
( , ) ( ) ( ) ( , ) ( ) ( )
TTBBv v
T Tx x x BB
T TBB
B z t z q tv z t z q t
z t z q t B z t z q t
z t z q t B z t z q tf f
q
f
==
= =
= =
YY
Y Y
Y Y
where
,1 ,2 , ,1 ,2 ,
, 1 , 2 ,2 , 1 , 2
,2 1 ,2 2 ,3
1 1,3 1 1,3 2 1,4
2 2,4 1 1,4 2 1,5
... ...
...
... ,
...
...
T T
v v v v N v v v v NT
N v N N N v NT
x x N x N x NT
B B N B N B NT
B B N B N B N
q q q q
q q qf f f f f+ + + +
+ +
+ +
+ +
= = = == = =
Y Y Y Y
Y Y Y Y
Y Y Y Y
Y Y Y Y
Y Y Y Y
,2
,2 1 ,2 2 ,3
1 1,3 1 1,3 2 1,4
2 2,4 1 1,4 2 1,5
...
...
...
...
T
NT
x x N x N x NT
B B N B N B NT
B B N B N B N
q
q q q q
q q q q
q q q q
f
+ +
+ +
+ +
= = =
In Eq. (30), N is the degree of the polynomial assumed for
admissible function
and N is taken as 5 in the current study because of the
convergence achieved. " iY "
are the so-called admissible functions which satisfy the
geometric boundary conditions of the problem. Admissible functions
are assumed in form of five degree polynomial and the coefficients
of the polynomial are considered as one as shown in Eq. (31).
= + + + +2 3 4 5i z z z z zY
It should be noted that based on the convergence study related
to the order of the polynomial in Eq. (31), convergence in the
results are obtained for a polynomial of degree five. The state
vector or the column matrix of time dependent variables is defined
by Eq. (32).
1 2 3
TT T T T T Tv x B B B
q q q q q q qf =
Applying Galerkin method to the governing equations given by Eq.
(25), after some manipulations, the general equations of the
aeroelastic system can be obtained in terms of time dependent
variables as shown in Eq. (33).
( ) ( ) ( ) 0s ae ae s aeM M q C q K K q+ + + + =&&
&
where " Ms " and "Mae " are the structural and aerodynamic mass
matrices and "Cae" is the aerodynamic damping matrix and " Ks" and
"Kae" are the structural and aerodynamic stiffness matrices,
respectively. In order to the solve the governing system of
equations given by Eq. (33) based
on Theodorsens and Loewys unsteady aerodynamics in frequency
domain, the U-g
(33)
(30)
(29)
(32)
(31)
(29)
-
method is used. In the U-g method, an eigenvalue problem is
built and solved by the addition of an artificial damping term. In
this method, to determine the aeroelastic instability boundary,
damping versus speed curves are plotted, although damping values
determined are physically meaningless except around the flutter
boundary where the damping value is equal to zero.
In the U-g method, adding the artificial structural damping is a
tricky way to eliminate some complex terms that come from the
existence of the reduced frequency
in aerodynamic matrices. By adding the artificial damping g to
the structure, the governing differential equation in time domain
can be rewritten as:
{( ) ( ) ( (1 ) ) } 0s ae s ae s aeM M q C C q K ig K q+ + + + +
+ =&& &
Damping of the aeroelastic system is zero at the instability
boundary. Thus, assuming undamped harmonic oscillation of the
blade, one can express the time dependent variables as in Eq.
(35).
w= i tq qe
By substituting Eq. (35) into Eq. (34), one gets Eq. (36).
2{ ( ) ( ) ( (1 ) )} 0s ae s ae s aeM M C C i K ig K qw w- + + +
+ + + =
Dividing two sides of Eq. (36) by 2U and defining the reduced
frequency as in Eq.
(37), Eq. (36) can be re-written as Eq. (38).
w
=b
kU
2
2 2
(1 )( )
s s ae ae ae
ig k kK M M C i K
bU b
+= + - -
Where, 2
ae aeK U K= and
ae aeC UC=
It should be noted that U appearing in Eqs. (37-39) is defined
as the relative velocity at the blade tip due to the free stream
velocity and rotational speed of the blade. Eq. (38) can further be
written as Eq. (40).
[ ( ) ] 0s
A k K ql+ =
Where,
(34)
(35)
(36)
(37)
)
(31)
(38)
(39)
(40)
-
2
2
2
( ) ( )
(1 )
s ae ae ae
k kA k M M C i K
bbig
Ul
= + - -
+=
The real and imaginary part of l can be used to calculate the
velocity and the artificial damping g based on Eq. (42) as:
1 Im( ),
Re( ) Re( )U g
l
l l= =
In the U-g method, by iterating over the reduced frequency k,
aerodynamic matrices can be calculated and eigenvalue problem can
be solved for each reduced frequency. Since artificial damping does
not exist in real life, solution of the eigenvalue problem is
accurate only at the in instability boundary. Instability velocity
is calculated
by iterating over the reduced frequency k until the artificial
damping introduced becomes zero.
5. NUMERICAL RESULTS In the present study, fiberglass/epoxy is
taken as the material of the blade. The geometric and material
specifications of the rotating thin-walled composite box beam with
the CAS lay-up are given in Tables 1 and 2. Thin-walled beam has
rectangular cross-section and constant cell wall thickness in order
to demonstrate the effect of using Theodorsens and Loewys lift
deficiency functions on the aeroelastic instability boundary.
Table 1 Geometric and material properties of the rotating
composite TWB
Geometric Properties Material Properties
Length, L m 40 1E GPa 17
Width, 2w m 1.4 2 3E E GPa 3.5
Depth, 2d m 0.5 12 13G G GPa 0.8
Wall thickness, h m 0.02 23G GPa 0.65
Density, 3kg / m 1950 Poissons ratio, 0.28
(41)
(42)
) (31)
(43)
-
Figs. 7-18 investigate effects of the fiber angle, rotation
speed and number of blades (Q = 1,2 and 3) on flutter speeds.
Iterative eigenvalue solution is performed for two different fiber
angles (0o and -150) and for two different fixed rotational speeds,
5 rpm and 10 rpm, for determining the flutter speeds. Unsteady
aerodynamic models of Theoderson and Loewy are used in the unified
aero elastic system introduced in the previous section. Loewy used
the Biot-Savart law instead of potential flow theory to account for
the layers of shed vorticity beneath the reference rotor blade
caused by the reference blade and other blades in previous
revolutions. It can be seen that the flutter speed obtained via
Theodorsens function does not depend on number of blades.
Theodorsens function gives same flutter speed for different wind
turbine type with two or three blades. The difference in prediction
of the flutter speed between Theodorsens and Loewys theories
increases by increasing the number of blades. As the modern wind
turbines have three blades, utilizing Loewy theory is more
reasonable than Theodersons theory. Aeroelasticity of the isolated
Wind turbine blade with three blades as popular wind turbine type
are comprehensively investigated in continue. For the 0o fiber
angle case, Figs. 11 and 12 show the variation of the artificial
damping with the relative velocity at the blade tip, given by Eq.
(44), for the flapwise bending and torsional deformation modes. In
these figures a is a distance between the aerodynamic center and
the elastic center. In Figs. 11 and 12, it is noticed that when the
rotational speed of the blade is increased, flutter speed
determined by employing Loewys lift deficiency function in the
aeroelastic system of equations separates from the flutter speed
determined by the use of Theodorsens lift deficiency function. With
the Theodorsens lift deficiency function, the flutter speed for the
rotational speed of 10 RPM is obtained as 29.5 m/s and the use of
Loewys lift deficiency function gives a flutter speed of 36 m/s.
When the rotational speed is decreased, as in the case of 5 RPM,
flutter speeds obtained are almost close by the use of either lift
deficiency functions in the unsteady aerodynamic model.
2 2
inf( )
low at r LU U r
== + W
Table 2 Gross properties chosen for the MW size wind Turbine
blade
Rating 1.5 MW Blade number 3 Rotor diameter 60 m
Blade chord 3.5 m Maximum Tip speed 50 m/s
(44)
-
Fig. 7 U-g diagram For = 0
0, = 5 rpm, a=0.3,
Q=1
Fig. 8 U-g diagram For = 0
0, = 10 rpm, a=0.3,
Q=1
Fig. 9 U-g diagram For = 0
0, = 5 rpm, a=0.3,
Q=2
Fig. 10 U-g diagram For = 0
0, = 10 rpm, a=0.3,
Q=2
Fig. 11 U-g diagram For = 00, = 5 rpm, a=0.3,
Q=3
Fig. 12 U-g diagram For = 0
0, = 10 rpm, a=0.3,
Q=3
Torsional
mode
Torsional
mode
Flapwise bending
mode Flapwise bending
mode
Flapwise bending
mode
Torsional
mode
Flapwise bending
mode
Torsional
mode
Flapwise bending
mode
Torsional
mode
Flapwise bending
mode
Torsional
mode
-
Figs. 17 and 18 present the U-g plots for fiber angle of -15o.
Similar to the 0o case, when the rotational speed is increased,
flutter speed determined by employing Loewys lift deficiency
function in the aeroelastic system of equations separates from the
flutter speed determined by the use of Theodorsens lift deficiency
function. For the rotational speed of 10 rpm, the use of
Theodorsens unsteady aerodynamic theory in the aeroelastic system
gives a flutter speed of 32 m/s and Loewys theory gives a flutter
speed of 39 m/s. For the rotational speed of 5 rpm, the difference
in flutter speeds calculated by the two theories are decreased and
they are 31.5 for Theodersons theory and 35 m/s for Loewys
theory.
For both fiber angle and rotational speed cases, Table 3
summarizes the flutter
characteristics of the wind turbine blade. In Table 3, m is the
ratio of oscillatory flutter frequency to the rotational speed and
h is the wake spacing, shown in Fig. 5. Results presented in Table
3 show that high wake spacing, such as 2.22 or 2.4, corresponds to
high inflow, and the effect of any shed layer of vorticity on the
flutter speed is negligible. It is concluded that when the spacing
between the wake spirals is small, as would the case for high
rotational speed or low inflow case, flutter speed increases.
Theodorsens unsteady aerodynamics theory cannot predict the
increase of the flutter speed for small wake spacing.
Fig. 13 U-g diagram For = -150, = 5 rpm, a=0.3,
Q=1
Fig. 14 U-g diagram For = -150, = 10 rpm, a=0.3,
Q=1
Flapwise bending
mode
Torsional
mode Torsional
mode
Flapwise bending
mode
-
Fig. 9 U-g diagram For = -150, = 5 rpm, a=0.3,
Q=2
Fig. 16 U-g diagram For = -15
0, = 10 rpm, a=0.3,
Q=2
Fig. 17 U-g diagram For = -15
0, = 5 rpm, a=0.3,
Q=3
Fig. 18 U-g diagram For = -15
0, = 10 rpm, a=0.3,
Q=3
Furthermore, it is seen that the frequency ratio m, which
measures the phase relationship between shed layers of vorticity,
in the Loewys lift deficiency function also affects the flutter
speeds. For the particular wind turbine blade studied, for both
fiber angle cases, it is seen that at the flutter boundary low
frequency ratio corresponds to higher flutter speeds. For both
fiber angle cases, flutter mode is identified as flapwise bending
mode. Loewy has shown that damping coefficient associated with the
flapwise bending mode has sharp drops at the integer values of the
frequency ratio (Loewy 1957). Viswanathan (1977) reported that at
low integer values of the frequency ratio, this drop is confined to
a small neighborhood near the integral values, but at higher values
of the frequency ratio, the width of low damping increases. Lower
flutter speed
Flapwise bending
mode Flapwise bending
mode
Flapwise bending
mode
Torsional
mode
Torsional
mode
Torsional
mode
Flapwise bending
mode
Torsional
mode
-
for higher value of the frequency ratio, given in Table 3 for
both fiber angle cases, could be due to the widening of the width
of the low flapwise damping.
Table 3. Effect of Theodorsens and Loewys unsteady aerodynamic
models and fiber angle on the flutter characteristics of the wind
turbine blade
Number of
blades
Fiber angle (Deg.)
Rotation speed (RPM)
Unsteady Aerodynamic
model
Flutter tip speed (relative velocity)
(m/s)
Inflow speed (m/s)
Flutter frequency
(Rad/s)
=
h
Q = 3
0
5 Theoderson 29 24.97 1.35 - -
Loewy 32 27.9 1.35 2.6 2.22
10 Theoderson 29.5 4.8 1.5 - -
Loewy 36 18.9 1.6 1.53 1.21
-15
5 Theoderson 31.5 27.36 1.6 - -
Loewy 35 31.3 1.6 3.07 2.4
10 Theoderson 32 9.36 1.7 - -
Loewy 39 24.17 1.8 1.73 1.34
Table 3 also shows that flutter speeds of the blade with
negative off-axis fiber angle are higher than the flutter speeds of
the on-axis fiber angle case. Both, Theodorsens and Loewys unsteady
aerodynamics, predict the same behaviour. The increase of the
flutter speed of the blade with the negative off-axis fiber angle
case is attributed to the positive effect of bending-twisting
coupling effect that the negative off-axis ply angles induces in
the blade. Figs. 3 and 4 show that when the ply angles are
negative, fibers are aligned towards the leading edge and when
bending occurs trailing edge deflects more than the stiffer leading
edge resulting in reduction in the effective angle of attack of the
blade section. Reduction in the effective angle of attack
ultimately causes increase of the flutter speed. 6. CONCLUSIONS In
the present study, Circumferentially Asymmetric Stiffness (CAS)
structural model of a rotating thin-walled composite box beam is
used as the simplified model of the wind turbine blade in
conjunction with Theodorsens and Loewys functions in unsteady
incompressible flow to study the aeroelastic instability of the
composite blade. Hamiltons principle and the extended Galerkin
method are used to obtain the coupled linear governing system of
dynamic aeroelastic equations. Frequency domain solutions are
performed for fixed rotational speeds to investigate the effect of
using Theodorsens and Loewys unsteady aerodynamics on the flutter
speeds of composite wind turbine blades. As a by-product of the
study, the effect of fiber angle of the CAS model on the flutter
speed is also show. Flutter speed results show that Theodorsens
lift deficiency function, while applicable to the fixed-wing case,
may not be as valid for rotary-wing aircraft for cases when the
effect of previously shed layers of vorticity should be considered.
In the present study, Theodorsens lift deficiency function is used
primarily to set a baseline for the flutter calculations so that
comparisons with the other lift deficiency functions can be made.
It is concluded that when the spacing between the wake spirals is
small, as would the case for high rotational speed or low inflow
case, flutter speed increases. Theodorsens unsteady aerodynamics
theory cannot predict
-
the increase of the flutter speed for small wake spacing. It is
also shown that flutter speeds of the blade with negative off-axis
fiber angle are higher than the flutter speeds of the on-axis fiber
angle case mainly due to the decrease of the effective angle of
attack due to the bending-twisting coupling induced by the negative
off-axis fiber angle. ACKNOWLEDGEMENTS This work was supported by
the METU Centre for Wind Energy and Scientific and Technological
Research Council of Turkey (TBTAK), Project No: 213M611. REFERENCES
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