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(1)

DYNAMIC RESPONSE OF WINDTURBINE

TO YAWED WIND

by

PAPER Nr. :55

Akira AZUMA and Shigeru SAITO Institute of Interdisciplinary Research Faculty of Engineering, The University of Tokyo

Tokyo, Japan Fumitaka NAKAMURA Toyota Motor Corporation

Aichi, Japan

(2)

DYNAMIC RESPONSE OF WINDTURBINE TO YAWED WIND

By

Akira AZUMA and Shigeru SAITO University of Tokyo

Fumitaka NAKAMURA Toyota Motor Corporation

SUMMARY

Dynamic response of a two-bladed windturbine to yawed wind is

analyzed by means of the local circulation method. The dynamic system

is considered to consist of blade deformation, rotor rotational motion and yawing motion of the windturbine. The amplitude of the 2-P vibra-tion in the bending moment and the rotor torque are more significant in the change of wind direction than in that of wind speed. The exemplified windturbine can follow the change of wind direction with

fairely small response time. The inertial forces and moments are

much smaller than the aerodynamic components because of the high rigidity of the present rotor.

1. Introduction

As precisely explained in Ref. 1, the aerodynamic forces and moments acting on horizontal-axis or propeller type windturbine are appreciably influenced by the change of wind direction with respect to the rotor axis. Thus the power coefficient based on the power of inflow or the mechanical efficiency of the windturbine is, as shown in Fig. la, strongly deteriorated by the yawing angle of the rotor. Similarly, as shown in Fig. lb, the blade bending moment also

fluctuates severely during one revolution.

Usually the speed and direction of wind cannot be controlled artificially and vary from site to site and time to time. Therefore, the rotor plane is adjusted to be normal to the wind velocity by making yawing motion of the rotor shaft around a vertical axis prepared by a swivel and a tail fin. In small windturbines as shown in Fig. 2, when the wind direction does not coincide with the shaft axis, the

restoring moment to make the above yawing motion can usually be generated by the aerodynamic force acting on the tail fin which is, like weather vane, installed on an opposite side of the rotor shaft. Then the lateral or side slip angle and the resulted yawing motion will bring various fluctuations on the aerodynamic and inertial forces and moments of the windturbine.

2. Method of Analysis

Aerodynamic Forces and Moments

(3)

By referring to Fig. 2 and by assuming that (i) the preconing angl Sp, the lead-:lag, flapping, and torsional deformations (v, w and ~) are small, and (ii) the aerodynamic forces and moments may be given by quasi-steady treatment because of small reduced frequency such as K=0.03, the relation between the airload ~ and the circulation r can be given by ~ = -pu cc 1 2 2 ~ = pur where u

=J

u~

+ uz p UT - Rn{]JsinljJ +x+

;/rut}

uP = Rn{A - SPJJcoslJ! - w/Rn} a = tan-1(Up/UT) - (8 + tjl) ]J {Vsin('l' - '¥) + 9r'i'}/RQ w A = {Vcos ('!' - '!') - v.}/Rn w ~

I

and where the wind speed V may be a function of height h.

(1)

(2)

(3) (4)

If the deformations and yawing motion are specified, then equation (1) can be solved by the LCM. Usually the effects of the rate of deformation of the blade on the aerodynamic force, which are given by the final term of the expression of UT and UP' may be

neglected as small quantities.

Then the aerodynamic forces and moment (about the elastic axis) at radius r and azimuth \jJ can be given by

dFA/dr = d (UT/U) ~(Up/U)

dFA/dr = ~ (UT/U) + d(Up/U) (5)

dMAx/dr m8 + (dFAy/dr)eA,z + (dFA /dr)eA z , y.

Inertial Forces and Moments

The inertial forces and moments of a blade element are given by

(6) dF1/dr

(4)

and dF 12/dr dM1/dr = dM1/dr = d2w d2'1' d'l' . 2 -m[{dt2-rdt;cos~+2nrdts1n4+rSPQ } d2<j> d2'1' . d'l' +e {-2--;sm~-2Q-cos~}] y dt dt dt d2<j> d2'1' d'l' -I {-2--;sin~}-1 {-2Q-cos~+<j>Q2} X dt dt Z dt 2 2 +I {-znd w +q,n2}-r

{Q

2

-2n~}

y dtdr yz dtdr 2 d2 {d w 'I' d'!' . 2} -m[e y dlt 2-r~2cos~+ZrQ~s1n~+rS Q t t p 2 +e

{-~

¥+(v+o )n2}] z t y d 3w ~ 2dw d2'1' 2 -Iy{-dt 2dr-ZQdt+n dr+dt2cos~+SpQ } { d 3v 2 dv} { 2} -I -~ -Q - -m[e -r<j>Q yz dt dr dr y +e { 2Qddv -rn2}] z t

I

=

-I {d2v -2ndv}-I { d3w +2Qd<j> dMiz dr z dt 2 dr yz dt 2dr dr -2Q2 dw

-~·cos~-S

Q2} dr dt P -m[e {-2Qddv+rQ2}-e {r<j>Q2}]. y t z

Yawing Moment and Torque

(6)

(7)

By referring to Fig. 1, the external yawing moment about the vertical axis Mljl, which includes the inertial components of the rotor, is comprised of the hub moment ~· the moment caused by the horizontal force Hand the moment generated by the tail-fin force Lt as follows:

M'l'

=

-~-H.Q. r -L t t .Q. (8) where

~

~JR{(dFA

2

/dr+dF

12

/dr)rcos~

0 +(dMAx/dr+dM

1

x/dr)sin~}dr R H

~J

0

{(dFAy/dr+dF

1

y/dr)sin~

+(dFix/dr)cos~}dr (9)

(5)

and where E specify to take the summation for all b blades.

'

Similarly, the torque about the rotor shaft

Qh

is comprised of the rotor torque

Q

which also includes the inertial ¥orque as well as the aerodynamic torque and the torque ~ generated by the mechanical torque for driving an installed load.

(10)

where

( 11) Q

~

fr{(dFA/dr+dF1/dr)r

0

Elastic Equations of Motion of the Blade

The elastic deformation of a blade, the pretwist of which is

far larger than that of the helicopter-rotor blade, can be written by the generalized force balance for the generalized coordinate qj

(12)

where the gene<alized mass Mj and the generalized fo<ce

Q.

in

non-dimensional form a<e given by J

MJ.

~J'

[{I (J).+iii(e

w

-ev.)}iji.

XJ y j Z J J

D

-Hii(w.H~

4).)w.+Ci'

w~+I v~)w~

J Y J J Y J Y Z J J

-Hii(v.-e $.)v.+(I

v~+I w~)vj']dx J Z ] J Z J Y Z J

-[(I w~+I v~)w

+Cf

v~+I

w')v.J'

YJ YZJ j Z J yzj ] 0

q

3•

=J

1

[{dMA y /dx-(I +me yz z y

6

)+2(I y

~'+I ~')

yz 0

+I

~sinw+2I ¥cosw-me (-x~cosw

X Z y

+2x¥sinW+xSP)}(j)j

(6)

+{dF A/ dx-m:x:flp +m(xi¥cosljl-2x'i'sin1);)

+me (i¥sin1);+2tcos1);) }w. -{iiie x+2I y . . J z y ~

-Zme

V-I ~cosljl}w!

z y J

+{dFA· /dx+mS -me

~sinlj;}v.

y y z J

-{me

x~ZI

$-Zme

~-I ~cosljl}v!]dx

y yz y yz J

+[{me x+21 $-2me ~-I ~coslj;}w.

z y z y J

•• 1

+{me x+2I $-2me

v-I

'l'cosljl}v .]

y yz y yz J

0

(14)

and where (--)'s are nondimensionalized quantities of ( ) and the subscript j shows jth mode.

Equations of Yawing Motion and Driving Motion

In the present example, the windturbine has a degree of freedom around the vertical axis. Then the torque about the rotor shaft is affected by the yawing motion of the windturbine as well as the bending deformations of the blade. The following nondimensional equations are established: I 'I' ip =

i1,y

If.li'i=Q-q;, 3. Method of Computation (15) ( 16)

For solving the above nonlinear equations of motion, (11), (15) and (16) of the rotor dynamic system in yawing motion, the calculus of fimite differences has been applied. The external forces and moments were calculated at the time of one step before. The time step was 0.031 second, which was equivalent to time of blade passing over 10 degrees of azimuth angle, for the calculation of the airloading, and was 0.0031 second for the calculation of the blade deformation and yawing motion.

The loading torque was assumed empirically to operate in pro-portion to the square power of the rotor rotational speed,

(7)

4. Results of Computation

Geometrical dimensions of an exemplified windturbine and the elastic characteristics of the blade in nondimensional form are given in Table I and II. The windturbine is under development and is used to be a heat generator for agricultural purpose.

Modes of Deformations

As shown in Fig. 3, since the eigenvalues of the blade deforma-tions are almost insensitive for the change of the rotational speed of the rotor, the eigenvalues and the modes of deformation are treated as

constant values specified at the normal operation state. Effects of Wind Shear

As shown in Fig. 4, the windturbine is considered to operate in the wind speed of 8.0 m/s at the rotor hub, for the case of (i) uniform flow or (ii) sheared flow of V = V10(h/lO)l/6. Shown in Fig. 5 are

the torque variation, lOO{Q-Q(1/J=O)}/Q(1jJ=O), and the bending moment

variation, {MB-MB(1jJ-O}/MB(1jJ=O), in comparison between the above two cases. It can be seen that the effect of wind shear is obvious specifically in the bending moment variation.

Effects of Yawed Wind

Here let us assume for simplicity that the yawing motion of the rotor system and the rotor speed are constrained or fixed to their initial values. Shown in Fig. 6 is the torque change caused by the yawed wind (~w=45°) in comparison with that of normal wind (~w=0°).

It must be notified that the level of effective torque or torque output is appreciablly reduced (about 35 percent) by the diagonal angle of the wind and the torque variation of twice per revolution (2P) is observed.

The torque variation and the bending moment variation are shown in Fig. 7. They have peak values of 7 and 25 percent respectively compared with those of the no yawed angle (~w=0°).

Effects of Yawing Motion

Let us consider here two examples.such that (i) the wind speed is abruptly increased in a step form, 6V = O.l5V, and (ii) the wind direction is suddenly changed from ~w=0° to ~w=30°. Here the yawing motion of the rotor system about the vertical axis and the rotor speed are considered free.

(i) The results of the former case are shown in Fig. 8 for the variations of rotor speed lOO{n-Q(1jJ=O)}/Q(1jJ=O), the torque variation, the yaw angle,~. the variation of normal force, 100{N-N(1/J=O)}/N(1/J=O),

and the variation of bending moment. As the wind speed increases,

every quantity increases gradually and approaches to each final value. It is interesting to find that a small yawing motion is induced by the

(8)

motion is, in the present example, aboutfive second or 5 revolutions of the rotor.

(ii) The results of the latter case are shown in Fig. 9. At the initial stage the effective torque'is reduced by 0.5 percent and is, then, recovered to the initial value because the yawing motion reduces the diagonal angle of the rotor with respect to the wind,

~ ~ ~w· During this period the 2P variations of aerodynamic and

inertial forces and moments are predominant. Their peak-to-peak values

are 10 percent in the torque, 5 percent in the normal force and 8 percent in the bending moment. This fact is important for the design of structural configuration and of material selection of the blade, drive shaft, gear trains and tower, all of which are under influence of the above exciting forces and moments.

Although the angular rate and acceleration of the yawing motion are predominant, the inertial effects on the rotor dynamics are not so significant that the inertial forces and moments acting on the

blades are much smaller than the aerodynamic components. This is

because the blade of the exemplified rotor have high rigidity and thus the blade .deformation is very small.

The time constant of the second in the present example. turbine to follow the change of frequency of which is more than 5. Conclusion

damped yawing motion is about three This enables the exemplified wind-wind direction, the predominant the above time constant.

By appling the local circulation method (LCM) to the aerodyoamic analysis of the rotor of a two-bladed windturbine, the dynamic response of the system, which was comprised of the blade elastic deformations, the rotor driving motion and the yawing motion of the windturbine, was

analyzed. The following facts were drawn: (i) The yawed angle of the

wind reduces the mean values of the aerodynamic forces and moments, but it induces the vibratory change in the above every quantity.

(ii) The change of wind speed including the wind shear has almost no effect on the vibratory change in the forces and moments, but it

induces a yawing motion slightly. (iii) The change of wind direction

affects strongly the yawing motion and the vibratory change in the

aerodynamic and inertial forces and moments. (iv) The vibratory change

has 2P variation at the early stage of the motion and is attenuated by the yawing motion.

(9)

AC c CG d EI , EI , Eiyz y z (e , e ) y z (e ,

e )

y z (eA ,y , eA ,z ) (en, ez;) (eA,n' eA,

~)

EA FA

(FAx' FAy' FAz) (FAx' FAy' F ) Az (FIx' Fly' Flz) (Fix' Fry' Flz) GJ H h (I , I y' I z' Iyz) X

(I ,

1 y'

I ,

Iyz) X z I'¥ = = Nomenclature aerodynamic center

lift slope of tail fin number of blades lift coefficient blade chord center of grativity section drag banding rigidity

CG position from EA in (x, y, z) coordinate (ey, e )/R z

AC position from EA in (x, y, z) coordinate CG position from EA in ( 1;, n, ~) coordinate AC position from EA in ( 1;, n, ~) coordinate

elastic axis

feathering axis

aerodynamic forces in (x, y, z) coordinate

(FAx, FAy'

FAz)/~~~

inertial forces in (x, y, z) coordinate

(Fix' Fly'

Flz)/~~~

torsional rigidity

horizontal force in the x-axis height

inertial mements in (x, y, z) coordinate (I , I , I , I x y z yz ) /K R

-o

2

inertial moment of windturbine without rotor about Y-axis

I'l'/MR2

inertial moment of windturbine without rotor about Z-axis

\JIMR

2

radius of gyration

coefficient of loaded torque in Eq. (16) lift of tail fin

section lift

(10)

Jl,t M

(MAx, MAy' MAz) (MAx, MAy' MAz)

MB

~

m N r v =

distance betwen Z-axis to tail fin total mass of windturbine

aerodynamic moments in (x, y, z) coodinate (MAx, MAy'

MAz)/~llil~

flatwise bending moment mass of a blade

inertial mements in (x, y, z) coodinate (Mix' Miy'

Miz)/~llil~

nond~ensionalized j-th generalized mass

total moments in (X, Y, Z) coodinate total yawing moment about Y-axis

section mass mR/~

aerodynamic pitching moment (positive for head-up)

normal force

torque Q/M(llilo)>

nondimensional generalized force loaded torque

Q /M(llilo) 2

m

torque about Z-axis, see Eq. (10)

nondimensional generalzed coordinate rotor radius

spanwise position of a blade

rotor disc area fin area

time

resultant velocity

=~U~

+

U~

perpendicular velocity against wing section tangential velocity against wing section wind velocity

wind velocity specified at heigh h (h

=

10 m, 21 m)

(11)

w w. J X (X, Y, Z) (x, y, z) a. a.t 13

sP

r

0 y' 0 z

6 , 6

y z

(!;,

n, I;) n nt 8 :\ )1 p l: $

~j

w lJi

no

Q

IT

w.

J

w.

J =

blade deformation along z axis

j th flapwise mode

r/R

coordinate fixed in the windturbine coordinate fixed in and rotating with the rotor hub

angle of attack

angle of attack of tail fin flapping angle of blade preconing angle

circulation on a blade

distances between EA and FA along y and z axes

o

y /R,

o

z /R

local coordinate fixed on a blade

efficiency or local axis fixed on the blade decrement of dynamic pressure at tail fin pitch angle positive for pitch down

inflow ratio positive for downflow advance ratio

air density

sununation

torsional blade deformation

j th torsional mode yaw angle

wind direction about z-axis azimuth angle of a blade

initial rotor rotational speed rotor rotational speed

QjQO

j th natural frequency

w.!Qo

(12)

Supercripts: ( )

~/rl

dt 0

(

/

~ dx

=

Rd( ) dr ( - ) nondimensionalized value of ( ) REFERENCES

1) Nasu, Ken-ichi and Azuma, Akira: An Experimental Verification of

the Local Circulation Method for a Horizontal Axis Wind Turbine. The 18th Intersociety Energy Conversion Engineering Conference, Orlando, Florida, August 21-26, 1983, pp. 245-252.

2) Azuma, A., Nasu, K. and Hayashi, T.: An Extension of the Local Momentum Theory to Rotors Operating in a Twisted Flow Field. Vertica, Vol. 7, No. 1, 1983, pp. 45-59.

Table I Dimensions of a windturbine blade

Items Dimensions

Rotor radius, R 7m

Blade mass, M b 122kg

No. of blades, b 2

Preconing angle, f3p O.Odeg

Collective pitch angle, Oo 1.6deg

Wing section NACA 4418

Coefficient of loaded torque, km 6.23

Reynolds number, R e 8X 10'

Tail fin area, S t 15 .39m'

Distance, lr 7.0m

(13)

Table II Characteristics of a windturbine blade

Station Chord Pretwist Mass per Ely Elz Eiyz GJ lx ly lz lyz k\ .

No. unit length

'ti-I"R2 - - MbW - - R'

,;n

o/R Or (deg) m/Mb , MbR,Q2 MbRl.Qz MbRJ.Q: MbR3.QZ MbR2 MI>R2

I 0.10 0.043 27.9 0.210 1422.0X!O-' 382.2XJo-• 2137. 4XJo-• 33.72XJo-• 5.447X!O-' 14.77XW' 39.7XJo-• 22.2XJO-• 1.06!XIO-' 2 0.14 0.051 21.2 0.180 881.6X!O-' 393. 1x1o-• 1655.0X!O-' 16.86XW' 5.186XIO-' 9.488Xlo-• 42.37XW' 17.8!XIO-• 1.490X!O-' 3 0.21 0.109 14.0 0.183 360.2XIO-' 299.1XIO-' 850.5X!O-' 21.62XIO-' 4.630X!O-' 4.976XW' 41.32XW' II. 75XJo-• 6.796X!o-•

4 0.29 0.135 9.4 0.174 173.5X!O-' 223. 7xJo-• 452. 7X!o-· 27. 71X!O-' 1.410X!O-' I. 0!5X Jo-• 13.09XIO-' 2.649XJo-• 10.5!X10-' 5 0.36 0.113 6.6 0.177 73.60X!O-' 124.8X1o-• 191 .8x!O-' 18.32X!o-• 0.979XIO-' 0.545XW' 9.244Xlo-• 1.421XIO-' 7 .3!0XIO-• 6 0.43 0.096 4.6 0.186 4!.13X!O-' 84.07XIO-' !00.3XIO-' l!.80X!O-' 0.679XJO-' 0.3!7XIO-• 6.47Xlo-• 0. 772X!O-' 5.286XW' 7 0.50 0.083 3.0 0.180 25.93x1o-• 60.65X!O-' 55.83X!o-• 8.429X!o-• 0.453Xl0-' 0.186X!O-' 4.35XW' 0.400X!O-• 4.oooxw-• 8 0.57 0.074 1.9 0.162 15.61XIO-' 39.56X!O-' 29.04XIO-' 5.057XIO-' 0.290X!O-' O.l!OX!O-' 2.79XW' 0.205X!O-• 3.122XIO-' 9 0.64 o:066 1.0 0.137 9. 798XJo-• 26.26Xlo-• 15.27XIO-' 3.078XJo-• 0.181XIO-' 0.065X!O-' !.744XIO-• O.IOIX!O-• 2.490X!O-' 10 0.71 0.059 0.3 0.104 5.552X!O-' 15.43X1o-• 7 .146X Jo-• !.979XIO-' 0.102XIO-' 0.036XW' 0.988X!o-• 0.0458Xlo-• 2.020X!O-' 11 0.79 0.054 -0.3 0.071 3.019XW' 8.606XW' 3.l13X!O-' 1.246XIO-' 0.0567XIO-' 0.0!9XW' 0.584XW' 0.0199XW' !.673XW' 12 0.86 0.050 -0.8 0.042 1.496X10-' 4.334X!o-• 1.202X1o-• 0.660XIO-' 0.0283X!O-' 0.009XW' 0.273XIO-• 0.0075XIO-' 1.429XIO-• 13 0.93 0.046 -1.2 0.031 0. 783XW' 2.293XJo-• 0.481X10-' 0.513XIO-' 0.0!24XIO-' 0.004XIO-' 0.120X10-' 0.0025X!O-' 1.224XW'

(14)

N cr:

"

>

Q. ~IN .__ a 0 II

>'"

0 z w 0 u.. u.. w 0.2 0.1 0 m 2 <l z 0 ;::: .,; :::J 1-0 :::J ...J u.. 1-z w 2 ---PRESENT METHOD EXPERIMENT 0 • • 0 0 0 'l'w=O deg. & A A 'l'w=30 deg.

'l'w=45 deg . 5

TIP SPEED RATIO, RQ/V (a) On the efficiency

VERTICAL AXIS BLADE

A WIND VELOCITY

r

RO~ORSHA~~~~NE

YAWING ANGLE, 'Pw 0 WIND VELOCITY PITCH

~~V

ANGLE, a~ 6=7.1deg. 10

0.1 'ZIMUTH ANGLE, o/= J,'Qdt

WIND

~ITY

. . . HORIZONTAL AXIS

0 0 0 PRESENT METHOD EXPERIMENT cOO 0 oo. _9!>.00~'" 0 0 0·~---~~---~~~---~~0~----~2,.~~ 0 0 0

AZIMUTH ANGLE, 'I'

0 00 2 oO 00 Oo

; -Or···

e = 4.6 deg. V= 7.97m/s 0=84.8 rad/s 'Pw=45 deg.

(b) On the bending moment fluctuation

Figure 1. Effects of yawing angle of a windmill11

(15)

AZIMUTH ANGLE '1'=180' '

v

z

y FIN AREA S 1 DISC

ARE~:::w-:-~

70

.·· _.. A PLANE NORMAL TO SHAFT

(a) Coordinate systems

INDUCED VELOCITY, v1

JRQ~ ROTOR

POSITION, £, y---r---~

X

(b) Flow velocities, forces and moments in

yawing motion

\

~

z 6y

(c) Sectional cenfigurations and the sectional tift, drag and moment.

(Before deformation and ~p=O)

Figure 2. Schematic view of a windmill in

yawing motion.

(16)

Jw~;

~t

- /;

-1 w2=90.4 s-1 v

~fL ~

_ 1 w3=119.0 s-1

~t -==-=-=-=---=~

_ 1 w4=169.0 s-1 w

1f.

c:=~

0

=

v -1 w5=206.4 s-1

(a) Natural frequencies and modes at 00=5.71 s-1 I ~ ; 150 FIRST LEAD-LAG, v

b

FIRST TORSION, <j> z w

6

1001---~--ti!

SECOND FLAP, w

lJ._ •DESIGN .PO IT

<;!

50 a: FIRST FLAP, w :::>

~

z 0 _.-100 -=:::::;::::::.:--60 50 100 ROTOR REVOLUTION, RPM (b) Natural frequencis vs RPM

Figure 3. Characteristics of a windturbine blade.

E .c i-" I <9 w I 0 .c - I-I <9 w I 0 V=Bm/s

T

...

<J-•

l

E ~

"'

I

WIND SPEED (m/s), (a) Without wind shear

V" = V1o(h/1 0)116 v21=8m/s

I

I

v

V10=?.01m/s WIND SPEED (m/s) (b) With wind shear

(17)

40.00 0 0 ~ X 0 0 0 20.00 II

,.

'X

4o.oo!-'-·---Jo.oo

a

:::0.

0 WITNOUT WIND SHEAR- 0

II 20.00 -20.00 II

z

6

S!:

0.00'1--~---~---'-40.00 I

0 -WITH WIND SHEAR

a

II -20.00

z

a

-4o.oo:'n---,e.,---..*n---.,;F.

6

0.0 5.0 10.0 15.0

TIME, t (sec) (a) Torque variation

40.00 0 0 ~ X 0 0 o 20.00 II ~

z

X m 8 4o.oot--~~~~~~~~~~~~~~--lo.oo

::;:

11 WITHOUT WIND SHEAR- :::0.

0

~ 20.00 -20.00 11

::;:

z

:::0. -40.00

:!

o

I

11 -WITH WIND SHEAR m

z

-20.00 ::;:

m

1 -

40.00:'-n---,e.,---,-k-r.---..-? "' 0.0 5.0 10.0 15.0

::;: TIME, t (sec)

(b) Bending moment variation

Figure 5. Effect the wind shear on the windturbine characteristics. ('Pw=O") 360.00 300.00

fi'

0, 240.00

"'-a

180.00 ui :::>

a

0: 0 f- 60.00 'Pw=O· 'Pw=45" V=8m/s Q0=5.71s-1 O.Ot,.----~,_---"*-.---,...,_ 0.0 5.0 10.0 15.0 TIME, t (SEC)

(18)

=

~ X Ci

"

o-g

Ci

"

o-c;

'

0 = Ci

"

z

s.

g X 0 0

~

]:::: i

o

40.00'1J\NV\I\ANJV\AI\J\NV\JV\NJV\AI\J\N'v 0 00 0

II WITHOUT WIND SHEAR-

J

:=,

~ 20.00 -20 00

1

0 0.00 -40 00 0

II I

,_ -WITH WIND SHEAR

g_

6

-2o.oo

-I 0 -40.00:'-n---F~--~,--k-,~--~ "" 0.0 5.0 10.0 15.0 0 0 X 40.00 0 20.00 II 2:

-l

0.00 ;:,_

'if

-20.00 2:

:ff

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(b) Berlding morrient variation Figure 7. Effect the wind shear on the windturbine

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