University of Massachusetts - Amherst

ScholarWorks@UMass Amherst Wind Energy Center Reports

UMass Wind Energy Center

1979

Wind Turbine Tower Wake Interface J. Turnberg Duane E. Cromack

Follow this and additional works at: http://scholarworks.umass.edu/windenergy_report Turnberg, J. and Cromack, Duane E., "Wind Turbine Tower Wake Interface" (1979). Wind Energy Center Reports. 23. http://scholarworks.umass.edu/windenergy_report/23

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WIND TURBINE TOWER WAKE INTERFERENCE

Technical Report

3. Turnberg and D. Cromack Energy A1 t e r n a t i v e s Program U n i v e r s i t y of Massachusetts Amherst, Massachusetts 01003

August, 1979

Prepared f o r Rockwell I n t e r n a t i o n a l Corporation Energy Systems Group Rocky F l a t s P l a n t Wind Systems Program P.O. Box 464 Golden, CO 80401 As a p a r t o f t h e U.S. DEPARTMENT OF ENERGY D I V I S I O N OF DISTRIBUTED SOLAR TECHNOLOGY FEDERAL WIND Ei4ERGY PROGRAM

DISCLAIMER This report was prepared a s an account of work sponsored by the United S t a t e s government. Neither the United S t a t e s nor the United S t a t e s Department of Energy, nor any of t h e i r employees, makes any warranty, express or imp1 i e d , o r assumes any legal 1 i a b i l i ty o r res p o n s i b i l i t y f o r the accuracy, completeness, or usefulness of any information, apparatus, product, o r process disc1 osed, o r represents t h a t i t s use would not infringe privately owned r i g h t s . Reference herein t o any s p e c i f i c commercial product, process o r service by trade name, mark, manufacturer, o r otherwise, does not necessarily c o n s t i t u t e o r imply i t s endorselllent, recommendation, or favoring by the United S t a t e s government o r any agency thereof. The views and opinions of authors expressed herein do not necessarily s t a t e o r r e f l e c t those of the United S t a t e s government or any agency thereof. PATENT STATUS This technical report i s being transmitted advance of DOE patent clearance and no f u r t h e r dissemination o r publ ication shall be made of the report without prior approval of the DOE Patent Counsel. TECHNICAL STATUS This technical report i s being transmitted in advance of DOE review and no f u r t h e r dissemination or publ ication shall be made of the report without prior approval of the DOE ?roject/Program Manager.

EXECUTIVE SUMMARY The response o f a wind t u r b i n e when t h e blades t r a v e l through t h e wake o f i t s supporting tower i s an important consideratdon i n t h e design o f a wind energy conversion system.

This tower induced f l o w

p e r t u r b a t i o n , corr~nionlyknown as tower shadow, has t h e c y c l i c e f f e c t o f unloading a blade f o r a s h o r t p e r i o d o f time w i t h each r o t o r r e v o l u -

A p e r i o d i c f o r c e o f t h i s n a t u r e has t h e c a p a b i l i t y o f e x c i t i n g

tion.

v i b r a t o r y responses and e x h i b i t i n g a f a t i g u e a f f e c t on t h e l o n g range operation o f the turbine. For t h i s study, t h e response o f t h e r o t o r t o t h e unsteady l o a d i n g i s examined using two a n a l y t i c a l models t h a t deal w i t h an i s o l a t e d t u r b i n e blade.

One model assumes t h e blade t o be r i g i d and hinged a t t h e hub,

w h i l e t h e o t h e r model assumes a f l e x i b l e blade c a n t i l e v e r e d a t t h e hub. Two approaches were chosen because each has c e r t a i n advantages.

The

r i g i d model i s simple and l i n e a r i z e d y e t o f f e r s i n s i g h t i n t o t h e problem, w h i l e t h e f l e x i b l e blade model i n c l u d e s many non-1 i n e a r terms and provides an in-depth a n a l y s i s o f t h e blade motion.

Each model i s solved t o

i d e n t i f y general trends t h a t occur under normal wind t u r b i n e operation. The wake t h a t p e r t u r b s t h e blade i s q u i t e complex because a wind t u r b i n e p i p e tower i s a c y l i n d r i c a l b l u f f body t h a t produces a wake w i t h f e a t u r e s common t o most b l u f f bodies. giving i t a variable structure.

The wake i s g e n e r a l l y unstable

Flow features change w i t h windspeed, tower

diameter, turbulence, and a host of o t h e r physical parameters.

It i s not

f e a s i b l e t o account f o r a l l aspects of t h e complex wake flow; thus, a simple wake model i s used as an approximation.

The main f e a t u r e s o f t h e

wake r e q u i r e d t o preserve t h e n a t u r e o f t h e blade i n t e r a c t i o n a r e t h e wake w i d t h and l o s s o f wind speed.

These f e a t u r e s a r e approximated by

using a r e c t a n g u l a r v e l o c i t y decrement o c c u r i n g behind t h e tower. T h i s tower shadow model i s shown i n F i g u r e 1. has s t r e n g t h ( w ) , w i d t h ) blade passage.

The v e l o c i t y decrement

and i t m a i n t a i n s t h e p e r i o d i c frequency o f

A simple nio~iientuna n a l y s i s o f t h e wake v e l o c i t y y i e l d s

the result,

f o r e s t i m a t i n g t h e v e l o c i t y decrement.

Since t h e Reynolds Number i s u s u a l l y

high, t h e l a r g e s t v e l o c i t y decrement p e r m i t t e d by t h i s model i s wo = .5, meaning t h e windspeed behind t h e tower i s h a l f t h e f r e e stream v e l o c i t y . I n modeling t h e t u r b i n e so t h a t t h e tower shadow a f f e c t i s c l e a r l y portrayed, i t i s necessary t o i s o l a t e t h e wake-blade i n t e r a c t i o n from t h e many o t h e r unsteady v a r i a b l e s ,

The v a r i a b l e s t h a t w i l l be neglected i n

t h e f o r c e system a r e changes i n wind speed and r o t a t i o n a l speed o f t h e r o t o r , wind shear, and g r a v i t y .

T h i s leaves a system t h a t i s p e r i o d i c a l l y

perturbed by t h e tower wake. One model used f o r t h e simp1 i f i e d a n a l y s i s c o n s i s t s o f a r i g i d slender beam attached t o t h e r o t o r hub by a hinge-spring.

T h i s model i s known as

an o f f - s e t hinge model and has been used e x t e n s i v e l y f o r helocopter s t u d i e s as w e l l as having been s u c c e s s f u l l y adopted t o wind t u r b i n e s i n many r e c e n t s t u d i e s [7].

The model i s shown i n F i g u r e 2,

The governing equation o f motion f o r t h e i s o l a t e d blade represents a p e r i o d i c a l l y f o r c e d s i n g l e degree o f freedom system..

The governing

3 FICUIiE 3 . 5

TOWER SHADOW MODEL

RECTANGULAR PilLSE APPROXI M A T I O N

_t

I

--

ccr,

7

I

:

-LL-L 540

720

AZIMUTH

ANGLE

4 Mgure 4.3 Blade Flapping Diagrsm

@ = Flanping angle = Coning angle

K= Hinge spring constant e= Hinge o f f s e t

~ i = r& i I n e r t i a l force Fc= ( e ~ + rcos(Q+3))a2dm Centrifugal force

equation i s g i v e n by t h e f o l l o w i n g expressions:

where:

-.

vo - nR, l / t i p - s p e e d r a t i o

- v

i

Q =

induced v e l o c i t y r a t i o

r/R;

s t a t i o n span

4 Y = pC1uCR 1

.d B = -.'

@I

d$

Oo

e

P

R'

;

L O C ~ ~i n Sertia

t e n

f l a p p i n g speed

= blade t w i s t = blade p i t c h

w($) = wake s t r e n g t h and

MC

=

2

n (r sin

A

+ cosh s i n l )

2 2 2 = R ( E cos A + cos A n

u

-

2 B sin A) + 2

with

n K

B E

= r o t a t i o n a l speed

= hinge spring constant = hinge o f f s e t c o n s t a n t

A = coning angle

I = mass moment o f i n e r t i a

An e q u i v a l e n t wake i s determined by s e t t i n g t h e shaded area behind t h e tower equal t o t h e area o f a s e c t o r swept by t h e blade;

This d e f i n i t i o n f o r t h e wake assumes t h a t t h e v e l o c i t y change depends o n l y on t h e blade asimuth angle, so t h e wake a c t s instantaneously over t h e e n t i r e blade when t h e shadow i s encountered. As an example o f t h e blade response, a s o l u t i o n i s determined f o r t h e steady s t a t e o p e r a t i o n o f t h e NF-I i s a 9 m/s (20 mph) wind. r o t a t i o n , t h e tower shadow d e f i c i t occurs between (64~+ 180') and

During (64~- 180°),

b u t t h e r e s u l t i n g response i s n o t s i g n i f i c a n t u n t i l t h e blade begins i t s ascent from t h e bottom o f r o t a t i o n .

The t u r b i n e blade follows an o s c i l l a t i n g

p a t h as i t r o t a t e s about t h e wind s h a f t .

The o s c i l l a t i n g

pattern i s similar

f o r a l l windspeeds because t h e damping remains l e s s c r i t i c a l .

'The blade r o o t

bending moment f o r t h e example case i l l u s t r a t e s t h e response (Figure 3 ) . O s c i l l a t i o n s o f t h e blade r o o t bending moments a r e t h e most important f e a t u r e o f t h e response.

These o s c i l l a t i o n s a r e b e s t described by t h e i r

maximum (!Imax) and minimum values (Mmi n ) .

F i g u r e 4 shows t h e maximum,

minimum and steady moments encountered over t h e e n t i r e o p e r a t i n g range o f wind speeds f o r WF-I.

The magnitude of t h e steady r o o t moment drops q u i c k l y

when t h e o p e r a t i o n a l mode i s changed t o constant r o t a t i o n a l speed. s u b t l e change occurs i n t h e magnitude o f t h e o s c i l l a t i o n s .

A more

I f t h e steady

moment i s removed from t h e response, a c l e a r p i c t u r e o f t h e tower shadow perturbation resul t s (Figure 5 ) .

The flatwise moment v a r i a t i o n increases

a t a f a s t e r r a t e under constant r o t a t i o n a l speed ( r e g i o n 111) operation, than would have occured if constant t i p - s p e e d - r a t i o has been maintained.

7

FIGURE 4.7

R I G I D PREDICTION

8

F I G U R E 4.8 O P E R A T I N G RANGE

9

FIGURE 4.9

CYCLIC LOADS

I n a d d i t i o n t o blade r o o t bending moment v a r i a t i o n s , t h e wake c o n t r i butes t o t h e yaw motion experienced by t h e t u r b i n e .

During h i g h winds,

WF-I has been observed t o o s c i l l a t e about a p o s i t i o n s l i g h t l y yawed away from t h e wind d i r e c t i o n .

A motion o f t h i s n a t u r e i s i n d i c a t e d by t h e

p r e d i c t e d shadow data when t h e blade moments f o r t h e e n t i r e r o t o r a r e resolved about t h e yaw a x i s .

An example o f t h e r e s u l t i n g yaw moments

o c c u r i n g i n a 20 m/s (44 mph) wind a r e shown i n F i g u r e 6.

The yaw

moment has a frequency o f t h r e e times t h e r o t a t i o n a l speed w i t h an amp1 i t u d e v a r i a t i o n about a p o s i t i v e mean yaw moment. The previous simple r i g i d blade model i s n o t adequate f o r an a n a l y s i s o f t h e f o r c e d i s t r i b u t i o n along t h e blade.

The r i g i d model i s u s e f u l f o r

determining many dynamic a f f e c t s caused by t h e tower shadow, b u t t h e r i g i d model l a c k s t h e a b i l i t y t o handle blade f l e x i b i l i t y and a complex geometry.

A wind t u r b i n e blade i s a non-uniform non-homogenious beam and

t h e e n t i r e motion of t h e blade i s needed f o r a d e t a i l e d a n a l y s i s o f l o a d i n g and moments. The equation of motion f o r a d i f f e r e n t i a l element o f a f l e x i b l e r o t o r blade i s ;

a

7 az [

2 2 a x ( az 7 Ixy + az I

R

where;

G

2 m r zdz

=

= blade t e n s i o n

z E = E l a s t i c modulus

2

1-

( Ga z % I +a t ~ ~ Y= F --aaz

11

FIGURE 4.1 1 YAW

MOMENTS

M = 1 ineal mass

Ixu' Iyy'

I

xy

=

aero moments of inertia

Fx = aerodynamic and centrifugal loads F Y'

I t i s evident by examination of these equations that the blade motion

i s coupled in the lag and flapping planes.

There i s no closed form

solution for the expression, so an approximate method for solution i s required.

A modal analysis i s chosen as the preferred solution technique

since the equations are uncoupled in the modal frame of reference. For this model, tower shadow i s represented by a rectangular pulse that i s both a function of azimuth angle and blade radius.

Therefore, the

velocity deficit i s applied gradually starting a t the blade root as the blade encounters the wake. Rated conditions were also chosen to show the typical response of the blade when tower shadow perturbation i s disrupting the flow.

Figure 7

shows the blade root bending moment prediction for the flexible blade. The blade response has many similarities to the rigid blade analysis in t h a t the shadow response occurs a f t e r the blade passes behind the tower

and the recovery from the shadow indicates a damped oscillation.

Bending

moments are not severe because the tower shadow i s applied and removed gradually.

The gradual loading of the blade i s believed t o be a realistic

model of the physical situation. Part of the output from the solution of the equations of motion includes the steady-state forces that would exist for a uniform flow field.

The

13

FIGURE 5.8 MODAL

PREDICTION

maximum bending moments on t h e blade occur between t h e .5 and .7 blade radius stations.

The s t r e s s o c c u r r i n g on t h i s s e c t i o n o f t h e blade should

be a maximum because t h e cross-sectional area decreases towards t h e t i p . Figure 8 shows t h e a f f e c t t h a t pre-coning t h e blade 10 degrees has on t h e bending moment d i s t r i b u t i o n .

C e n t r i f u g a l r e l i e f reduces t h e t o t a l

moment by more than h a l f , which i s a s i g n i f i c a n t r e d u c t i o n o f t h e steady a p p l i e d loads. I n summary, both models i n d i c a t e t h a t t h e tower wake imparts a f o r c e t h a t causes t h e blade t o have a damped o s c i l l a t o r y motion w i t h l a r g e d e f l e c t i o n amplitudes occuring on t h e upswing o f t h e blade (g > 180"). The major discrepancy between t h e two model p r e d i c t i o n s i n v o l v e s t h e magnitude o f t h e r e s ~ ~ l t i nforces. g

Larger c y c l i c f o r c e s a r e always p r e d i c t e d

by t h e simple r i g i d model because t h e shadow i s assumed t o encompass t h e e n t i r e blade instantaneously, whi 1e t h e complex model assumes a gradual a p p l i c a t i o n o f t h e shadow. O f t h e two approaches, t h e r i g i d system solved by computer code RIGID

proved t o be e a s i e r and l e s s time consuming than i t s f l e x i b l e c o u n t e r p a r t solved by computer code DYNAMICS.

Since t h e simple model p r e d i c t s a more

d r a s t i c response, i t serves t o make conservation estimates o f t h e blade loading.

The more complex model serves t h e purpose of d e f i n i n g a d e t a i l e d

l o a d i n g d i s t r i b u t i o n along t h e blade.

For design appl i c a t i o n s , t h e simple

system w i l l i n d i c a t e problem areas and t h e complex system w i l l d e f i n e t h e loads a t those problem areas.

15

FIGURE

5.6

CENTRIFUGAL RELIEF

ABSTRACT The design o f a wind t u r b i n e i n v o l v e s t h e combination o f many parameters, one o f which i s t h e determination o f t h e dynamic l o a d cases a f f e c t i n g t h e blades. random f l u c t u a t i o n s .

The dynamic loads i n c l u d e many p e r i o d i c and

O f these loads, t h e c y c l i c l o a d i n g o f t h e blade

as i t passes through t h e wake o f t h e wind t u r b i n e s supporting tower i s t h e s u b j e c t o f t h i s paper. The tower wake and/or shadow causes a change i n t h e d e f l e c t i o n p a t t e r n o f t h e blade on a once per r e v o l u t i o n per blade basis.

Analytical

p r e d i c t i o n s developed f o r t h i s p r o j e c t show t h a t t h e blade e x h i b i t s an o s c i l l a t o r y motion.

The amplitude o f o s c i l l a t i o n ranges from a maximum

on t h e upswing o f t h e blade t o near zero amplitude immediately b e f o r e t h e tower wake i s encountered on t h e downswing. The magnitude o f t h e tower induced l o a d v a r i a t i o n i s an e s s e n t i a l p a r t o f a wind t u r b i n e design because c y c l i c l o a d v a r i a t i o n s have a f a t i g u i n g e f f e c t on s t r u c t u r a l components t h a t must be included i n t h e design process.

Therefore, t h e enclosed a n a l y s i s o f f e r s a procedure f o r

p r e d i c t i n g t h e wind t u r b i n e blade response t o tower shadow f o r use i n p r e l i m i n a r y design a p p l i c a t i o n s .

iii

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . ii ABSTKACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii TABLEOFCONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . i v LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i i CHAPTERI: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1 EXECUTIVE SUYbNRY

CHAPTER 2: 2.1 2.2 2.3 CHAPTER 3: 3.1 3.2 3.3 CHAPTER 4: 4.1 4.2 4.3 4.4 4.5 CHAPTER 5: 5.1 5.2 5.3 5.4

THE UNIVERSITY OF MASSACHUSETTS WIND FURNACE I

......

3

.................... 3 ................... 6 ................ 7 FLOld BEHIND A PIPE TOWER . . . . . . . . . . . . . . . . . 15 The I d e a l klake . . . . . . . . . . . . . . . . . . . . . . 15 Complications w i t h a Wake A n a l y s i s . . . . . . . . . . . . 20 Wake Model . . . . . . . . . . . . . . . . . . . . . . . . 22 R I G I D BLADE MODEL . . . . . . . . . . . . . . . . . . . . 27 Rationale . . . . . . . . . . . . . . . . . . . . . . . . . 27 The O f f - S e t Hinge Model . . . . . . . . . . . . . . . . . . 28 Aerodynamic Loads . . . . . . . . . . . . . . . . . . . . . 32 S o l u t i o n o f t h e Governing Equation . . . . . . . . . . . . 38 Analysis of Wind Furnace I . . . . . . . . . . . . . . . . 41

Operational Aspects S t r u c t u r a l Parameters V i b r a t i o n a l Considerations

COMBINED LEAD-LAG AND FLAPPI'IVG RESPONSE OF A WIND TURBINE ROTORBLADE

. . . . . . . . . . . . . . . . . . . 51 R a t i o n a l e . . . . . . . . . . . . . . . . . . . . . . . . . 51 Modal Equations o f Motion . . . . . . . . . . . . . . . . . 54 Aerodynamic Loads . . . . . . . . . . . . . . . . . . . . . 58

. . . . . . . . . . . . . . . . 64 CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . 75 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . 76 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 A n a l y s i s o f Wind Furnace I

A.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Theorem of Southwell . . . . . . . . . . . . . . . . . . . 80 Program South . . . . . . . . . . . . . . . . . . . . . . . 82 Flow Chart f o r Program South . . . . . . . . . . . . . . . 83 Program L i s t i n g . . . . . . . . . . . . . . . . . . . . . . 84

A.5

Terminal Session

APPENDIX A A.1 A.2

A.3

APPENDIX B

.

. . . . . . . . . . . . . . . . . . . . . 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

. . . . . . . . . . . . . . . . . . . . 86 APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 C.1 Program Rigid . . . . . . . . . . . . . . . . . . . . . . . 90 C.2 Program Rigid Flow Chart . . . . . . . . . . . . . . . . . 91 C.3 Program L i s t i n g . . . . . . . . . . . . . . . . . . . . . . 92 C.4 Terminal Session . . . . . . . . . . . . . . . . . . . . . 94 APPENDIX D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 D .1 Program Dynamics . . . . . . . . . . . . . . . . . . . . . 96 D.2 Program Dynamics Flow Chart . . . . . . . . . . . . . . . . 98 D .3 Program Listing . . . . . . . . . . . . . . . . . . . . . .100 D.4 Terminal Session . . . . . . . . . . . . . . . . . . . . .102 . APPENDIX E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B 1

Aerodynamic Forces

E.1 E.2

Function Icond Function Icond Flow Chart Program Listing

E.3

APPENDIX F F.1 F.2 F.3

. . . . . . . . . . . . . . . . . . . . . .108 . . . . . . . . . . . . . . . . .109 . . . . . . . . . . . . . . . . . . . . . -110

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . -111 Function Aero . . . . . . . . . . . . . . . . . . . . . . -111

. . . . . . . . . . . . . . . . -112 . . . . . . . . . . . . . . . . . . . . . .113 APPENDIX G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 G.l Function Bending . . . . . . . . . . . . . . . . . . . . .114 6.2 Function Bending Flow Chart . . . . . . . . . . . . . . . .115 6 . 3 ProgramListing . . . . . . . . . . . . . . . . . . . . . .116 APPENDIX H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 H . 1 Function Data . . . . . . . . . . . . . . . . . . . . . . .117 . H.2 Program L i s t i n g . . . . . . . . . . . . . . . . . . . . . 118 . . . . . . . . . . . . . . . . . . . . . H.3 Terminal Session 119 Function Aero Flow Chart Program L i s t i n g

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 I .1 Tower Shadow . . . . . . . . . . . . . . . . . . . . . . 121 1.2 20" Diameter Shroud . . . . . . . . . . . . . . . . . . . i20 1.3 30" Diameter Shroud . . . . . . . . . . . . . . . . . . . 121

APPENDIX I

LIST OF FIGURES 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

..................... ............. ................ ....................... ..................... Wake C a t e g o r i e s . . . . . . . . . . . . . . . . . . . . . . . . Reynolds Number Range . . . . . . . . . . . . . . . . . . . . . Mean Flow Vectors . . . . . . . . . . . . . . . . . . . . . . . Wake I n t e r f e r e n c e . . . . . . . . . . . . . . . . . . . . . . . Tower Shadow Model . . . . . . . . . . . . . . . . . . . . . . Coordinate Systems . . .

P r e d i c t e d Performance Rotor RPM As A Function of Wind Speed D e s c r i p t i o n of Blade Components Modal Coordinates WF-I Cambell Diagram

Comparison of Mode Shapes Blade Flapping Diagram Blade Element Diagram . . Shadow Model Blade Tip D e f l e c t i o n . Moment P r e d i c t i o n Operating Range C y c l i c Loads Shadow Width Yaw Moments

.

...... . .... ..... ...... ...... ....... 5.1 D i f f e r e n t i a l Element . . . . . . . . . . . . . . . . . . . . . 5.2 BladeElement Diagram . . . . . . . . . . . . . . . . . . . . . 5.3 L i f t and Drag Curve . . . . . . . . . . . . . . . . . . . . . . 5.4 ShadowModel . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Moment D i s t r i b u t i o n . . . . . . . . . . . . . . . . . . . . . . 5.6 C e n t r i f u g a l R e l i e f . . . . . . . . . . . . . . . . . . . . . . 5.7 Blade Tip Motion . . . . . . . . . . . . . . . . . . . . . . . 5.8 Modal P r e d i c t i o n . . . . . . . . . . . . . . . . . . . . . . . 5.9 Operating Moments . . . . . . . . . . . . . . . . . . . . . . . . 5.10ShadowWidth . . . . . . . . . . . . . . . . . . . . . . . . .

52 59 61 62 66 67

68 69 71 72 73

................... 1.1 Towershroud . . . . . . . . . . . . . . . . . . . . . . . . . 122 1.2 Tower Dimension . . . . . . . . . . . . . . . . . . . . . . . . 123 1.3 Tower Dimension Detail . . . . . . . . . . . . . . . . . . . . 124 I .4 20" Diameter Shroud . . . . . . . . . . . . . . . . . . . . . . 125 1.5 Hinge Detail . . . . . . . . . . . . . . . . . . . . . . . . . 126 I .6 Upper Shroud Assembly . . . . . . . . . . . . . . . . . . . . . 127 I .7 30" Diameter Shroud . . . . . . . . . . . . . . . . . . . . . . 128

5.11 S e n s i t i v i t y o f Parameters

LIST OF TABLES 2.1 2.2

Blade Design Mode Shapes

.......................... ..........................

8 10

INTRODUCT ION The response o f wind t u r b i n e when t h e blades t r a v e l through t h e wake o f i t s s u p p o r t i n g tower i s an i m p o r t a n t c o n s i d e r a t i o n i n t h e design o f a wind energy conversion system.

T h i s tower induced flow

p e r t u r b a t i o n , commonly known as tower shadow, has t h e c y c l i c e f f e c t o f unloading a b l a d e f o r a s h o r t p e r i o d o f time w i t h each r o t o r rev01 u-

A p e r i o d i c f o r c e o f t h i s n a t u r e has t h e c a p a b i l i t y of e x c i t i n g

tion.

v i b r a t o r y responses and e x h i b i t i n g a f a t i g u e a f f e c t on t h e l o n g range operation o f the turbine.

Therefore,

t h e tower shadow must be taken

i n t o account t o assure s t r u c t u r a l i n t e g r i t y . The e x t e n t o f tower i n t e r f e r e n c e i s p r i m a r i l y determined by t h e geometry o f t h e support.

There a r e two major types o f support s t r u c t u r e s

commonly employed, t h e t r u s s tower and the p o l e tower.

L i t t l e has

been accomplished t o q u a l i f y t h e a f f e c t o f a p o l e tower wake d i s t u r bance.

The m a j o r i t y o f a v a i l a b l e data has been produced by t h e Depart-

ment of Energy f o r t h e t r u s s towers t h a t support t h e i r l a r g e demonstration systems.

The wake behind a t r u s s tower i s r e l a t i v e l y s t a b l e and can

be p r e d i c t e d u s i n g P r a n d t l ' s m i x i n g l e n g t h theory.

The wake behind

a p o l e tower i s o n l y we1 1 behaved f o r low wind speeds.

A t moderate

and h i g h wind speeds, t h e wake becomes unstable because t h e near wake is

i n t h e r e g i o n of v o r t e x formation.

Even w i t h o u t t h e added compli-

c a t i o n of blade motion, present t h e o r y cannot p r e d i c t t h e d e t a i l e d near wake s t r u c t u r e f o r a n y t h i n g b u t low wind speeds.

The purpose o f t h i s p r o j e c t i s t o e v a l u a t e t h e a f f e c t o f tower shadow on t h e t u r b i n e r o t o r .

A wind t u r b i n e r o t o r blade i s d r i v e n by

f o r c e s t h a t have i n e r t i a l , e l a s t i c , and aerodynamic o r i g i n s .

O f these

f o r c e s , t h e aerodynamic f o r c e i s t h e r e s u l t o f t h e dynamic r e a c t i o n o f t h e blade t o t h e a i r .

Therefore,

f l o w p e r t u b a t i o n s caused by t h e tower r e s u l t

i n unsteady f o r c e s t h a t produce blade motion. The response o f t h e r o t o r t o t h e unsteady l o a d i n g i s examined using two a n a l y t i c a l models t h a t deal w i t h an i s o l a t e d t u r b i n e blade.

One model

assumes t h e blade i s r i g i d and hinged t o t h e hub, w h i l e t h e o t h e r model assumes a f l e x i b l e blade c a n t i l e v e r e d i n t o the hub. chosen because each has advantages.

Two approaches were

The r i g i d model i s simple and l i n e a r i z e d

so t h a t i t o f f e r s i n s i g h t i n t o t h e problem, w h i l e t h e f l e x i b l e blade model i n c l u d e s many non-1 i n e a r terms t h a t complicate t h e system.

Each model i s

s o l v e d t o i d e n t i f y general t r e n d s t h a t occur w i t h normal wind t u r b i n e operation. I t i s hoped t h a t t h e r e s u l t s o f t h i s research w i l l g i v e a designer

i n s i g h t i n t o t h e tower shadow phenomenon t o a s s i s t him w i t h design decisions, because an understanding o f tower shadow i s v i t a l t o t h e c o s t , s a f e t y , and r e l i a b i l i t y c o n d i t i o n s necessary f o r t h e economic g e n e r a t i o n o f power from t h e wind.

C H A P T E R

2

THE UNIVERSITY OF MASSACHUSETTS WIND FURNACE I 2.1

Operational Aspects To f a c i l i t a t e t h e i n v e s t i g a t i o n o f tower shadow, t h e wind t u r b i n e

o p e r a t i n g a t t h e U n i v e r s i t y o f Massachusetts i s used as a t y p i c a l example of a h i g h speed h o r i z o n t a l a x i s w i n d m i l l . t h e Wind Furnace I (WF-I),

T h i s wind t u r b i n e , known as

i s used f o r t h e space h e a t i n g o f a home,

Solar H a b i t a t I. WF-I has a downwind c o n f i g u r a t i o n w i t h a t h r e e bladed 9.9 m (32.5 f t . ) diameter r o t o r , and t h e torque developed by t h e r o t o r

i s t r a n s m i t t e d t o a 31.5 kW a.c. synchronous generator.

The e n t i r e

r o t o r - g e n e r a t o r system i s mounted atop a 18.3 m (60 f t . ) stayed p o l e mast. The t u r b i n e has a design c a p a c i t y o f 25 kW a t 1 1 . 7 m/s (26 mph) wind speed. Operation o f t h e WF-I i s defined by f o u r regions: ( 2 ) constant t i p - s p e e d - r a t i o ;

down.

( 1 ) start-up;

( 3 ) constant r o t a t i o n a l speed; and ( 4 ) s h u t

These o p e r a t i o n a l regions a r e shown i n Figure 2.1,

t i o n of power o u t p u t as a f u n c t i o n o f windspeed.

a representa-

Power p r o d u c t i o n i s

c o n t r o l l e d w i t h changes i n blade p i t c h angle and generator e x c i t a t i o n . P i t c h angle and e x c i t a t i o n a r e s e t by a pre-programmed microprocessor based on r o t o r speed.

Figure 2.2 shows t h e p i t c h changes t h a t occur from

s t a r t - u p t o shut down as a f u n c t i o n o f wind speed. I n r e g i o n I,s t a r t - u p , t h e blades a r e p i t c h e d t o 40' t o produce maximum s t a r t - u p torque.

The wind speed must be s u f f i c i e n t t o t u r n t h e

r o t o r a t 40 rpm b e f o r e t h e blades a r e p i t c h e d t o 6.0" f o r r e g i o n I 1 operation.

Throughout r e g i o n 11, t h e aerodynamic c o n d i t i o n s needed t o

e x t r a c t t h e maximum power from t h e wind a r e maintained by keeping t h e

3 BLADES 9 . 9 m . DIA.

WIND

SPEED

CM/S)

PREDICTED PERFORMANCE

FIGURE 2.1

/

tip-speed-ratio a t a constant value of 7.5.

Tip-speed-ratio i s the r a t i o

of r o t o r speed t o wi nd speed ;

where:

Q

=

r o t o r rotational speed

R = r o t o r radius

Vo = wind speed

The generator excitation i s used d u r i n g constant tip-speed operation t o control the rotor rpm so t h a t t h e power will follow a cubic relationship w i t h increasing wind speed.

When the rotational speed reaches 167 r p m a t rated conditions, region I11 operation begins.

7he r o t o r speed and power remain constant

throughout region I11 because the blades a r e pitched t o " s p i l l the wind."

This control of speed and power i s maintained until the turbine

i s s h u t down.

Region IVY shut-down, i s broughtabout by pitching the

blades to an angle of 90" o r f u l l feather. 2.2

Structural Parameters The rotor blades a r e the primary components affected by the tower

shadow.

As e l a s t i c s t r u c t u r e s , the blades a r e continually subjected t o

a cyclic un-loading a s they pass behind the tower.

Sicne the un-loading

pattern i s periodic, f a t i g u e and resonance problems will be a f a c t o r i n any design.

Therefore, the blade s t r u c t u r a l properties must be known

before an analysis of the tower shadow perturbation can progress, since the aerodynamic, e l a s t i c , and i n e r t i a l forces depend on the geometry and material composition of the blade.

The wind furnace blades are designed t o have a twist and chord distribution near the optimum as predicted by aerodynamic theory and they are designed with an NACA 4415 airfoil section.

The blades are

constructed of glass reinforced plastic ( G R P ) molded into three structural The

members, the spar, skin, and trailing edge stiffener (Figure 2.3).

blade root stock i s a steel sleeve which serves as a bearing support to cantilever the blade into the hub a t a coning angle of ten degrees. The blade design i s summarized in Table 2.1 [ 2

1.

These design para-

meters are then used as input to the structural computer codes, MOMENTS, and FREQ, developed by Perkins [ 21.

The output of these computer codes

includes the following blade section structural properties:

the mass

distribution, mode shapes, and natural frequencies whose values are given in Table 2.2 with the modal coordinate system (Figure 2.4).

A

mode shape i s the orientation of the blade when i t oscillates a t a natural frequency.

For a flexible system there are an infinite number

of natural frequencies, b u t only the lowest few frequencies are important to the response of a system. 2.3 Vibrational Considerations The natural frequencies and blade properties presented in the previous section are sufficient to perform an elementary vibration analysis of possible resonant conditions.

Resonance occurs when the frequency of

the applied force system coincides with one of the natural frequencies of the body.

A t the resonant condition, large amplitudes may develope

causing a loss in structural i'ntegrity.

To determine the resonant

frequencies, i t i s necessary to identify cases where the blades' natural

TABLE 2.1 BLADE DESIGN

WF-1

r/Radius (Station11 0)

Chord (ft)

6

= 2.2 x 10 p s i

Eskin

=

Espar

6 4.4 x 10 p s i

Twist (degrees)

GSkin=

(Radius = 16.25 f t ) L.E. t o Spar Web (ft)

6

. 5 x 10 p s i

Skin Thickness (in)

Spar Thickness (in)

'skin = .0555

1b 3

pspar= -0501

3

6

Gspar= . 3 x 10 p s i

in

lb

~n

Web Thickness (in)

9

FIG. 2 . 3 DESCRIPTION

OF

BLADE

COMPONENTS

TYPICAL SECTION (NACA 4415)

SPAR

SPAR

1

BLADE

r SKIN

TRAILING EDGE ST1 F F E N ER

STOCK

\

Table

Ma55

2.2

r ~ I 5 T R X E U T I O N( K G )

10+050 6,123

NfiTURaL

4,766

FREQUENCY

28 + 430

4,016

3,463

(RaDIaNS/SECOND)

64 I 450

F'igure 2.4 KODAL COORDINATES

2,696

1*99?

1,473

1,025

0.527

frequency c o i n c i d e s w i t h t h e p e r i o d i c f o r c e o f t h e tower shadow. P r e v i o u s l y determi ned blade n a t u r a l frequencies were developed f o r a n o n - r o t a t i n g system. r o t a t i n g system.

These frequencies w i l l n o t be t h e same f o r a

The c e n t r i f u g a l f o r c e r e s u l t i n g from r o t a t i o n s t i f f e n s

t h e blade and thus r a i s e s t h e n a t u r a l frequency.

The increase i n n a t u r a l

frequency i s determined w i t h t h e a i d o f theTheorem o f Southwell, which i s discussed i n Appendix A.

B r i e f l y , t h i s Theorem s t a t e s t h a t t h e

frequency i s d i v i d e d i n t o two p a r t s , t h e n o n - r o t a t i n g e f f e c t and t h e rotational effect.

where:

w

These two p a r t s a r e combined u s i n g equation 2.2:

n

=

n o n - r o t a t i n g frequency

n

=

r o t a t i o n a l speed

a = Southwell c o e f f i c i e n t .

The Southwell c o e f f i c i e n t ( a ) i s found u s i n g t h e expression;

R = r o t o r radius

where: R

0

0

= hub r a d i u s = mode shape

m = mass per u n i t l e n g t h

The Southwell c o e f f i c i e n t s f o r t h e f i r s t t h r e e frequencies f o r t h e Wind Furnace blades have t h e values;

These values a r e used i n c o n j u n c t i o n w i t h eq. 2.2 f o r t h e e v a l u a t i o n o f t h e n a t u r a l frequency a t any r o t a t i o n a l speed. Force frequencies a r e now needed t o complete t h e frequency a n a l y s i s . The un-loading o f t h e r o t o r behind t h e tower produces a p e r i o d i c f o r c e t h a t has a primary harmonic component equal t o t h e r o t a t i o n a l speed

( 1 P ) on each blade and a t h r e e per r e v o l u t i o n (3P) harmonic on t h e r o t o r . These two harmonics a r e l a r g e s t i n magnitude, b u t they are n o t t h e o n l y components produced by t h e e x c i t a t i o n .

The n a t u r e o f t h e tower shadow

i s p e r i o d i c and u n l i k e a sinusodal disturbance, a p e r i o d i c f o r c e system The f o r c e d frequencies occur a t i n t e g e r

can e x c i t e many frequencies. mu1 t i p l es o f t h e r o t o r speed ; =

where:

nn

n = integer

I n general t h e h i g h harmonics have n e g l i g i b l e e f f e c t s on a system.

A comparison o f t h e frequencies i s c l e a r l y shown by t h e f a n diagram (Campbell p l o t ) i n F i g u r e 2.5.

The s t r a i g h t l i n e s extending from t h e o r i g i n

i n d i c a t e t h e v a r i o u s harmonics o f r o t o r speed and t h e curved l i n e s represent t h e n a t u r a l frequencies.

Each i n t e r s e c t i o n o f a harmonic l i n e w i t h

a n a t u r a l frequency l i n e represents resonance.

Since t h e occurance o f

resonance i s g e n e r a l l y unavoidable, i t s e f f e c t must be minimized.

The

s a f e s t r e g i o n t o havean occurance of resonance i s when t h e r o t o r speed i s v a r i a b l e , since t h e l a r g e amplitudes associated w i t h resonance occur when t h e frequencies c o i n c i d e f o r extended periods o f time.

Operational

FIGURE 2.5 WF- I

CAMBELL PLOT

ROTOR SPEED RPM I

I

0

5

I

I

I

15 20 W I N D SPEED MPH 10

I

25

experience with the Wind Furnace indicates t h a t the many resonant conditions occuring in region I1 operation do not a f f e c t the structure of the system.

I t should be emphasized that resonant conditions occuring

a t rated rpm will generally damage the turbine since the frequencies coincide f o r prolonged time periods.

From this point, the analysis will

involve the development of a wind turbine model t h a t predicts the load variation induced by the tower shadow.

C H A P T E R

3

FLOW BEHIND A PIPE TOWER 3.1

The I d e a l Wake

A wind t u r b i n e p i p e tower i s a c y l i n d r i c a l b l u f f body t h a t e x h i b i t s a wake w i t h f e a t u r e s common t o most b l u f f bodies.

One f e a t u r e i s t h a t

t h e wake c l o s e t o t h e tower i s s t r i k i n g l y d i f f e r e n t from t h a t e x i s t i n g f a r downstream.

The far-wake o c c u r i n g more than 100 tower-diameters

downstream i s s t a b l e and p r e d i c t a b l e .

Prandtls' mixing length theory

serves as t h e a n a l y t i c a l technique used t o p r e d i c t t h e v e l o c i t y d i s t r i b u t i o n i n t h e far-wake. variable structure.

The near-wake i s g e n e r a l l y u n s t a b l e g i v i n g i t a

Flow f e a t u r e s change w i t h windspeed, tower diameter,

turbulence, and a h o s t o f o t h e r p h y s i c a l parameters.

The remainder o f

t h i s chapter concentrates on t h e near-wake because i t i s t h i s r e g i o n I n o r d e r t o discuss t h e wake, i t i s

t h a t represents the tower shadow.

convenient t o d i v i d e t h e f l o w i n t o classes t h a t have s i m i l a r p r o p e r t i e s . There a r e g e n e r a l l y f o u r c l a s s i f i c a t i o n s g i v e n t o t h e wake: subcri t i c a l

, critical,

and s u p e r c r i t i c a l [3].

slow viscous,

Each c l a s s i s e x p e r i m e n t a l l y

determined and i d e n t i f i e d by a Reynold's Number regime, where t h e Reynold's Number i s d e f i n e d as t h e r a t i o o f i n e r t i a l t o viscous f o r c e s :

where:

V

0

= windspeed

D

= tower diameter

v

= kinematic v i s c o s i t y

When wind speeds a r e low, t h e wake i s c l a s s i f i e d as t h a t o f slow viscous f l o w (o < R

e

< 40).

Under these low Reynolds Numbers, t h e wake 15

i s s t a b l e because viscous f o r c e s dominate t h e f l o w .

Although t h e wake

i s s t a b l e , gradual changes occur throughout t h e slow viscous regime. For Reynolds Numbers up t o f i v e , t h e f l o w stays attached t o t h e tower. Above a Reynolds Number o f f i v e , t h e boundary l a y e r separates from t h e s u r f a c e of t h e tower c r e a t i n g two v o r t i c i e s .

These v o r t i c i e s a r e s i d e

by s i d e and remain s t a t i o n a r y behind t h e c y l i n d r i c a l tower.

These

s t a t i o n a r y v o r t i c e s begin o s c i l l a t i n g as t h e Reynolds Number increases beyond 40, because t h e f l u i d i n e r t i a l f o r c e s now have a g r e a t e r dominance over t h e f l o w .

F u r t h e r increases i n t h e i n e r t i a o f t h e f l o w causes t h e

v o r t i c i e s t o p e r i o d i c a l l y l e a v e t h e c y l i n d e r one a t a time from a l t e r n a t e sides.

This p e r i o d i c v o r t e x shedding i s t h e dominant f e a t u r e o f t h e sub-

5 c r i t i c a l wake regime (40 < R < 1 . 5 ~ 1 0 ). The parameter used t o describe e t h e p e r i o d i c p r o d u c t i o n o f v o r t i c i t y i s t h e Strouhal Number ( S ), d e f i n e d t as:

where:

f = frequency of v o r t e x p r o d u c t i o n (Hz)

and has a value i n t h e neighborhood o f .21 throughout t h e s u b c r i t i c a l regime.

I n t h i s region, t h e f l u i d boundary l a y e r separates from t h e

s u r f a c e of t h e c y l i n d e r a t approximately 82" from t h e up-wind s t a g n a t i o n point.

Since s e p a r a t i o n occurs on t h e f r o n t p o r t i o n of t h e tower, t h e

wake d i r e c t l y behind t h e tower i s wider than t h e diameter o f t h e tower. The wide wake of t h e s u b c r i t i c a l regime continues u n t i l t h e boundary 1ayer becomes t u r b u l e n t . T r a n s i t i o n t o turbulence t r i g g e r s the c r i t i c a l region o f f l o w 5 6 (1 .5x10 < R < 1 . 5 ~ 1 0 ) Turbulence t r a n s f e r s momentum i n t o t h e e boundary l a y e r causing t h e f l u i d t o r e - a t t a c h a f t e r i n i t i a l separation.

.

Therefore,

t h e p o i n t o f s e p a r a t i o n moves t o t h e back s i d e o f t h e tower.

Re-attachment o f t h e boundary l a y e r i s accompanied by t h e d i m i n i s h i n g w i d t h of t h e wake and an increase i n t h e Strouhal Number ( S

t

.44).

When

t h e Reynolds Number a t t a i n s a h i g h enough value, s e p a r a t i o n and subsequent re-attachment o f t h e boundary i s no l o n g e r present, so t h e f l o w i s c l a s s i f i e d as s u p e r c r i t i c a l (R

6

> 1 . 5 ~ 1 0) .

I n t h i s f l o w regime, s e p a r a t i o n occurs e +120° from t h e up-stream s t a g n a t i o n p o i n t . The wake i s o n l y m e a t about wider than i t was f o r t h e c r i t i c a l regime and t h e Strouhal Number decreases t o an average value o f .28.

The drag c o e f f i c i e n t a c t i n g on a c y l i n d e r i s

a good i n d i c a t o r o f t h e f o u r classes o f flow, where drag c o e f f i c i e n t i s d e f i n e d as:

where:

Fd = drag f o r c e on t h e body p

= f l u i d density

V

= f l u i d velocity

A

= c y l i n d e r area

Figure 3.1 shows the drag c o e f f i c i e n t p l o t t e d as a f u n c t i o n o f t h e Reynolds Number.

A t low Reynolds numbers, h i g h drag c o e f f i c i e n t s i n d i c a t e

t h e slow viscous regime.

The drag c o e f f i c i e n t then decreases and l e v e l s

o u t i n t h e v i s c i n i t y o f u n i t y throughout t h e s u b c r i t i c a l regime and then drops d r a s t i c a l l y i n t h e c r i t i c a l regime. When a Reynolds Number versus wind speed curve ( F i g u r e 3.2)

is

developed f o r an assortment o f tower diameters, i t becomes apparent t h a t t h e tower Reynolds Number i s g e n e r a l l y high. t h e v i c i n i t y o f t h e c r i t i c a l f l o w regime.

The wake, t h e r e f o r e , i s i n

F i g u r e 3.3 i s an example o f

F I G U R E 3.1

REYNCLDS I'JUiCtilER

WAKE

CATEGOEIES

FIGURE 32

a time average of the velocity in the wake near the c r i t i c a l regime [4]. Because tlie averaging process masks the unstable nature of the wake, i t has Inany features common t o the flow occuring a t Reynolds Numbers l e s s t h a n five.

One notable feature in the figure i s the region of stagnant f l u i d

extending 1.2 di ameters behind the tower.

This stagnent region o r tower

shadow would impart a strong impulse t o the blades i f they were to pass through.

In the more developed flow downstream, the shadow i s l e s s pro-

nounced.

A t present, the WF-I blades r o t a t e through the more developed

flow since the tower i s -254 rn (10 i n . ) in diameter.

If a .762 m (30 i n . )

tower were i n s t a l l e d , the blades would travel through the stagnent region. The plane of rotor rotation f o r the present .254 m (10 i n . ) and the .762 m (30 i n . ) tower are superinlposed on Figure 3.3 t o emphasize the a f f e c t t h a t

changing the diameter has on the flow.

Plans f o r changing the WF-I tower

have been developed and Appendix I outlines the procedure. 3.2

Corrlplications with a Wake Analysis The categorization of the wake i n t o d i s t i n c t groups, identified by ranges

in Reynolds Number, i s only useful as a rough approximation.

The four discrete

regimes were developed using standard experimental conditions f o r the flow around cylindars.

When conditions s t r a y from the experimental standard,

transitions between discrete categories occur a t d i f f e r e n t Reynolds Numbers. The t r a n s i t i o n from subcri t i c a l t o c r i t i c a l flow i s particularly s e n s i t i v e t o deviations.

I t i s unfortunate t h a t the c r i t i c a l region i s s e n s i t i v e ,

because the tower wake occurs in the range of c r i t i c a l Reynolds Numbers f o r normal operation and a tower exposed t o the environment i s f a r from standard conditions. The wake s t r u c t u r e , occuring in the c r i t i c a l region, i s d ~ t~o ea transformation of the boundary layer from laminar t o turbulent.

This

CV

Lii

E

t r a n s i t i o n i s determined by p h y s i c a l f e a t u r e s o f t h e wind and t h e tower. Tower roughness and wind t u r b u l e n c e have a major e f f e c t on t h e c r i t i c a l Reynolds Number.

A rough tower and/or t u r b u l e n t wind t r i g g e r s t h e t r a n s i -

t i o n from t h e s u b c r i t i c a l r e g i o n a t l o w e r Reynolds Nurr~bers w h i l e h i g h e r Reynolds Numbers a r e needed i f t h e tower i s smooth and/or wind i s steady. Also, p r o t r u s i o n s from t h e tower c o n t r i b u t e t o t h e wake.

The o r i e n t a t i o n

o f guy w i r e s , rungs, e t c . c r e a t e disturbances t h a t may widen o r o t h e r w i s e s i g n i f i c a n t l y a f f e c t t h e wake i n an u n p r e d i c t a b l e manner. The r o t o r adds i t s own c o n t r i b u t i o n t o t h e wake because i t slows t h e approaching wind and i n t e r f e r e s w i t h wake f o r m a t i o n behind t h e tower. The r e s u l t o f downstream i n t e r f e r e n c e i s i n d i c a t e d i n F i g u r e 3.4,

which

p o r t r a y s t h e a f f e c t t h a t a s h o r t s p l i t t e r p l a t e has on t h e wake o f a cylinder.

As t h e p l a t e i n moved downstream, t h e Stroukal Number and base

) change.

pressure c o e f f i c i e n t ( C

The drop i n v o r t e x shedding frequency

PS

and increase i n pressure occur because t h e p l a t e d i s t u r b s t h e f o r m a t i o n o f the v o r t i c i e s . formation,

When t h e p l a t e i s moved beyond t h e r e g i o n o f v o r t e x

t h e c o e f f i c i e n t s a b r u p t l y r e t u r n t o t h e i r normal values [5].

The r o t o r may a f f e c t t h e wake i n a s i m i l a r manner s i n c e i t a c t s i n t h i s region o f vortex formation. These e x t e r n a l parameters combine, c r e a t i n g a system which i s n o t ameanabl e t o a complete a n a l y t i c a l o r experimental a n a l y s i s .

Therefore,

t h e n e x t s e c t i o n serves t o reduce t h i s complex s i t u a t i o n i n t o a simple model r e p r e s e n t i n g t h e wake and tower shadow e f f e c t . 3.3

Wake Model Since i t i s n o t f e a s i b l e t o account f o r a l l aspects o f t h e complex

wake flow, a simple wake model i s used as an approximation.

The main

Figure 3 . 4 Wake interference.

-

f e a t u r e s o f t h e wake r e q u i r e d t o preserve t h e n a t u r e o f t h e wake blade i n t e r a c t i o n are t h e wake w i d t h and l o s s o f wind speed.

These

f e a t u r e s a r e approximated by using a r e c t a n g u l a r v e l o c i t y decrement This tower shadow model i s shown i n

o c c u r i n g behind t h e tower. F i g u r e 3.5.

The v e l o c i t y d e c r e r ~ e n thas s t r e n g t h (w0) , w i d t h ( 6 ) and

i t m a i n t a i n s t h e p e r i o d i c frequency o f blade passage.

The r e c t a n g u l a r

p u l s e i s chosen because i t i s simple and adaptable t o a n a l y t i c a l o r

A p e r i o d i c p u l s e o f t h i s form can be modeled w i t h a

numerical methods. Fourier series.

T h i s s e r i e s has t h e form:

($) = a,

a

+

n= 1 where:

a

WO

0

an $

n

cos nY

6

= --

2.n

2w -- - O n

n 6 sin 2

= azimuth angle

One approximate method o f c a l c u l a t i n g t h e v e l o c i t y decrement and shadow w i d t h i s by assuming t h a t viscous e f f e c t s a r e n e g l i g i b l e and t h a t t h e tower i s a semi-permiable membrane. t h e same wake as p r e v i o u s l y modeled.

T h i s assumed s t r u c t u r e produces

The wake w i d t h i s t h e same as t h e

membrane width, which has a value equal t o t h e tower diameter.

The

v e l o c i t y decrement i s found u s i n g t h e momentum theorem i n c o n j u n c t i o n w i t h experimental data f o r t h e drag as a c y l i n d e r . When a c o n t r o l volume i s e s t a b l i s h e d around t h e tower, t h e momentun equation f o r t h e system has t h e form:

F

D

= P Q

(Vo

-

V)

(3.5)

FICU!?E 3 . 5

TOWER SHADOW MODEL

RECTANGULAR PULSE APPROX I MATION

1 --F

, ,

I

:

-LL-Jz 540

720

AZIMUTH ANGLE

where:

FT = 1/2 P C P

V D i s tower drag f o r c e / u n i t l e n g t h D = density o f a i r

Q

= V D i s the flow rate/unit length 0

v = wake v e l o c i t y Re-arranging equation (3.4) y i e l d s t h e simple r e s u l t f o r t h e v e l o c i t y decrement as : n

Since t h e Reynolds Number i s u s u a l l y high, t h e drag c o e f f i c i e n t never has a value g r e a t e r than one. p e r m i t t e d by t h i s model i s w

0

Therefore, t h e l a r g e s t v e l o c i t y decrement = .5, meaning t h e windspeed behind t h e

tower i s h a l f t h e f r e e stream v e l o c i t y .

This sirr~plemodel w i l l serve

as t h e wake r e p r e s e n t a t i o n f o r t h e e s t i m a t i o n o f t h e tower shadow e f f e c t performed i n t h e f o l 1owing chapter.

CHAPTER

4

RIGID BLADE MODEL 4.1

Rational The dyanmic response o f a wind t u r b i n e r o t o r i s a complex problem

and tower shadow i s o n l y one f a c e t .

For a p r e l i m i n a r y a n a l y s i s o f t h e

tower shadow e f f e c t , t h i s chapter presents a method t o s i r r ~ p l i f yt h e r o t o r system and c l a r i f y t h e dynamics.

I n general, t h e wind t u r b i n e has

many degrees o f freedom t h a t a r e s e t i n t o motion by a complex f o r c e system.

The f o r c e s t h a t a c t on t h e r o t o r a r e aerodynamic, g r a v i t a t i o n a l ,

and i n e r t i a l .

O f these forces, t h e aerodynamic and i n e r t i a l f o r c e s c o n t r i -

b u t e t o t h e blade response from tower shadow.

Aerodynamic f o r c e s a r e

due t o t h e i n t e r a c t i o n o f t h e a i r on t h e t u r b i n e blades and t h e i n e r t i a l f o r c e s a r e t h e r e s u l t o f blade motion. I d e a l l y , t h e aerodynamic f o r c e s would be steady i f t h e wind acted u n i f o r m l y over t h e e n t i r e r o t o r a t a constant speed w i t h no g r a v i t a t i o n a l affects.

But t h i s i d e a l s i t u a t i s n never e x i s t s .

Aside from g r a v i t y ,

wind v a r i a t i o n s l i k e gusts, shear, and tower shadow r e s u l t i n an unsteady l o a d i n g c o n d i t i o n on t h e r o t o r which may have damaging r e s u l t s .

The

p o t e n t i a l l y damaging e f f e c t s o f tower shadow have a1 ready been proven by o p e r a t i o n a l experience w i t h t h e NASA MOD-0 wind t u r b i n e .

With t h e

i n i t i a l tower c o n f i g u r a t i o n , t h e tower shadow r e s u l t e d i n excessive blade f l a p p i n g and consequent m a t e r i a l f a t i g u i n g .

Subsequent removal o f t h e

s t a i r s from w i t h i n t h e tower s t r u c t u r e s i g n i f i c a n t l y reduced t h e tower shadow [ 6

I.

I n modeling t h e t u r b i n e so t h a t t h e tower shadow e f f e c t i s c l e a r l y portrayed, i t i s necessary t o i s o l a t e t h e wake-blade i n t e r a c t i o n from t h e 27

The v a r i a b l e s t h a t w i l l be neglected i n

many o t h e r unsteady v a r i a b l e s .

t h e f o r c e system a r e changes i n wind speed and r o t a t i o n a l speed of t h e r o t o r , wind shear, and g r a v i t y . perturbed by t h e tower wake.

T h i s leaves a system t h a t i s p e r i o d i c a l l y

Simp1 i f i c a t i o n s i n t h e s t r u c t u r a l aspects

o f t h e r o t o r a r e a l s o r e q u i r e d when modeling t h e t u r b i n e ,

The r o t o r blade

i s assumed t o a c t l i k e a r i g i d slender beam w i t h motion i n t h e plane of r o t a t i o n (edgwi se) uncoupled from t h a t perpendicular t o t h e plane o f rotation (flatwise). neglected

Of these, t h e edgewise motion i s small and w i l l be

s i n c e t h e aerodynamic requirements o f t h e blade produce a

s t r u c t u r e t h a t has small edgewise forces and a l a r g e edgewise s t i f f n e s s . Three c o o r d i n a t e systems a r e used t o d e s c r i b e t h e t u r b i n e blade motion (Figure 4.1 ) .

The XYZ system t h a t i s attached t o t h e hub and

r o t a t e s w i t h a constant speed (n).

An X ' Y ' Z ' system attached t o t h e blade

r o o t and i n c l i n e d by t h e coning angle (A). The blade i s then l o c a t e d by t h e XYZ system t h a t i s f i x e d t o t h e blade, so t h a t i t moves through t h e f l o p p i n g angle ( 6 ) measured from t h e ( X ' Y ' Z ' ) blade r o o t system. 4.2

The Off-Set Hinge Model The s p e c i f i c model used f o r t h e s i m p l i f i e d a n a l y s i s c o n s i s t s o f

a r i g i d slender beam attached t o t h e r o t o r hub by a hinge-spring.

This

model i s known as an o f f - s e t hinge model and has been used e x t e n s i v e l y f o r h e l i c o p t e r s t u d i e s as w e l l as having been s u c c e s s f u l l y adopted t o wind t u r b i n e s i n many r e c e n t s t u d i e s [ 7

1.

The model g i v e s a good npproxi

mation f o r t h e lowest mode o f blade v i b r a t i o n i n flapping.

This approxi-

mation o f t h e motion i s shown i n F i g u r e 4.2 which d i s p l a y s t h e lowest mode shape f o r t h e WF-I and t h e hinged b l a d e motion.

-

Coordinate

X Y Z systemr

Systems

Rotor hub caordinate system, i t r o t a t e s a t the machine rpm,

x y z system:

Blade root coordinate system, i t i s ' i n c l i n e d a t the coning angle

/

/

I

x y z aystemt

)(.

F'ixed t o the blade, and i n c l i n e d t o the x y z system by the flapping zngle

t

Blade azimuth meesured f r o m tbe top o f rotation,

F I G U R E 42

COMPARISON OF MODE SHAPES

ACTUAL I S T FLAPPING

MODEL FLAPPING

The governing equation f o r t h e f r e e motion o f t h i s system i s found using t h e free-body diagram of blade f o r c e s shown i n F i g u r e 4.3.

When

moments a r e resolved about t h e hinge spring, t h e equation takes t h e form ;

..

2

2

2

2

IB t eR rcgn M [ ~ c o sx + s i n A ] + I n [B(cos 1 - s i n I] +

(4.1)

cosx s i n x ] + K B = 0 B

I = mass moment of i n e r t i a o f a blade

where:

M = mass of a blade R = radius o f the turbine e = hinge e f f e c t r = c e n t e r o f g r a v i t y measured from t h e blade r o o t cg k = hinge s p r i n g s t i f f n e s s e

n

= r o t a t i o n a l speed

x

= coning angle

B = f l o p angle

H i g e r o r d e r terms i n B have been neglected s i n c e t h i s angle i s g e n e r a l l y small [ 71.

The equation o f motion i s more convenient when t h e o f f s e t

hinge constant;

i s introduced.

.. B

Then eq. 4.1 becomes;

+ ~ [2 (ECOSI r +

2 cos x

-

2 sin A) +

B

+ r2 ( d i n 1 + cosh s i n x ) (4.3)

The s o l u t i o n o f eq. 4.3 represents t h e motion o f a f r e e l y v i b r a t i n g wind t u r b i n e blade w i t h a n a t u r a l frequency (un) given by: w

2

=

n

n ( ~ c o sA + cos

2

2

i- s i n i )

+

6 I

The n a t u r a l frequency e x h i b i t s t h e r o t a t i n g and n o n - r o t a t i ng components discussed i n s e c t i o n 2.3, W

L

n

=uL

L

rotating

+

non-rotating

a t r e s t , t h e t u r b i n e blade has a n a t u r a l frequency o f

W

2 non-rotating

- I

Blade r o t a t i o n a l speed causes t h e blades t o s t i f f e n , which increases t h e n a t u r a l frequency by t h e amount

W

2 rotating

=

2

n (Ecosh2 + cos

2

A

-

2 s i n A)

A steady c e n t r i f u g a l f o r c e i s a l s o produced when a coned blade r o t a t e s .

This

c e n t r i f u g a l f o r c e creates a moment given by, 2

MC = n ( € s i n A + c o s i sin^) t h a t p u l l s t h e blade towards t h e plane o f r o t a t i o n .

The a c t i o n of t h i s

moment serves t o r e l i e v e t h e aerodynamic moments t h a t d e f l e c t t h e blade away from t h e plane o f r o t a t i o n . 4.3

Aerodynamic Loads

A wind t u r b i n e i s d r i v e n by t h e dynamic r e a c t i o n o f t h e blades t o the air.

A blade a c t s as a l i f t i n g surface because i t posses an a i r f o i l

Mgure 4.3 Blade Flapping Diagmm

@

-

napping angle

1\ = Coning angle

K= Yinge spring c o n s t a n t e= Hinge o f f s e t

~ = r&$ I n e r t i a l force Fc- ( e ~ + cos(@+?l))fi2dm r Centrifugal f o r c e

cross-section.

The rnagni tude o f the 1i f t f o r c e depends on t h e blade

o r i e n t a t i o n and the square of t h e a i r v e l o c i t y a c t i n g p a r a l l e l (U ) and P perpendicular (UT) t o the plane o f r o t a t i o n . Figure 4.4 shows the geometry o f the forces and v e l o c i t i e s a c t i n g on a blade cross-section used t o express the equation f o r l i f t ( L ) per u n i t length;

Since t h e tower shadow changes, the perpendicular v e l o c i t y encountered by t h e blade, i t changes the l i f t and creates blade motions.

Calcula-

t i o n o f t h e v e l o c i t y v a r i a t i o n s i s essential t o t h e f o r c e evaluation. The s i m p l i f i e d wake model developed i n s e c t i o n 3 . 4 defines the v e l o c i t y d e f i c i t e created by the tower.

Equation 3 . 3 i s used d i r e c t l y

i f t h e wake width ( 6 ) i s replaced by an equivalent azimuth arc l e n g t h ( ~ J I ) . An equivalent arc l e n g t h i s determined by s e t t i n g t h e shaded area behind the tower equal t o t h e area o f a sector swept by blade (Figure 4.5) y i e l d i n g the expression:

Therefore, equation 3 . 3 i s r e - w i r t t e n as;

This d e f i n i t i o n f o r t h e wake assumes t h a t t h e v e l o c i t y change depends o n l y on the blade azimuth angle, so t h e wake a c t s instantaneously over the e n t i r e blade when the shadow i s encountered.

-

Figure

.

4.4 Blade Element Diagram

I

( L i f t on Blade Element)

$

f

= v e l o c i t y perpendicular t o t h e r o t o r plane

Ut * v e l o c i t y t a n g e n t i a l t o t h e b l a d e element

(This v e l o c i t y i s p r i m a r i l y due t o r o t a t i o n Qr) VR =

J u t 2 , + up2

-

r e s u l t a n t t o t a l v e l o c i t y a t b l a d e element

U

-2

4

= b l a d e element a n g l e = tan-'

8

= b l a d e element p i t c h angle

a

= b l a d e element angle of a t t a c k

ut

= l i f t f o r c e p e r u n i t span

FIGURE 4.5 S I i A DOW

MODEL

A s i m p l i f i e d equation f o r t h e l i f t f o r c e v a r i a t i o n due t o v e l o c i t y changes i s developed i n Appendix G and presented below by equation 4.11.

where:

-

p,

-.R, ,

l/tip-speed r a t i o

i induced v e l o c i t y r a t i o 'i = - QR * ' T,

s t a t i o n span

= r/R;

4 "IaCR ; Lack's i n e r t i a term = -46' =

eo

e

P

* d+

=

A. n y flipping

speed

= blade t w i s t = blade p i t c h

The v a r i a b l e v e l o c i t y components i n t h e l i f t equation a r e t h e tower shadow and b l ade moti ons

.

An i n t e g r a t i o n o f t h e f l a t w i s e c o n t r i b u t i o n of l i f t from t h e blade r o o t t o t h e blade t i p determines t h e blade r o o t bending moments ( M MBA =

jR

L cosfl

COSB

rdr

0

-

['

BA

).

LR~,~,

0

The r e s u l t o f the i n t e g r a t i o n y i e l d s ;

which i s combined w i t h t h e f r e e v i b r a t i o n equation (eq. 4.3) t o o b t a i n t h e governing equation f o r t h e forced motion o f a wind t u r b i n e blade.

The motion implied by e q u a t i o n (4.15) and the s o l u t i o n f o r t h e e x p r e s s i o n w i l l be d i s c u s s e d i n t h e next s e c t i o n . 4.4

S o l u t i o n of the Governing Equation The governing e q u a t i o n of motion f o r the i s o l a t e d b l a d e r e p r e s e n t s

a p e r i o d i c a l l y f o r c e d s i n g l e degree o f freedom system.

Equations of

t h i s t y p e have been s t u d i e d e x t e n s i v e l y i n many v i b r a t i o n s t e x t s and t h i s s e c t i o n w i l l draw from methods developed f o r v i b r a t i o n a n a l y s i s t o examine t h e tower shadow response [ 8

1.

The c l a s s i c a l form f o r the

d i f f e r e n t i a l e q u a t i o n o f motion of a s i n g l e degree o f freedom system i s given by;

where:

= LO

damping r a t i o

n = n a t u r a l frequency

M(t) = a p p l i e d moment and i t i s useful t o a r r a n g e e q u a t i o n (4.15) i n t o this form. In the time domain, the i s o l a t e d blade e q u a t i o n i s expressed by;

I t i s i n t e r e s t i o g t o note t h a t t h e system damping has an aerodynamic

o r i g i n , as i n d i c a t e d by t h e c o e f f i c i e n t o f t h e flapping v e l o c i t y

(k).

The magnitude o f t h e damping i s r e l a t e d t o t h e Lock Number (y) and subsequent motion o f t h e blade a f t e r i t i s perturbed depends o f t h e amount o f aerodynamic damping.

I f t h e damping i s g r e a t e r than o r equal t o a

c r i t i c a l amount, t h e blade w i l l n o t o s c i l l a t e a f t e r i t has been disturbed. The damping r a t i o ( C ) serves as an i n d i c a t o r f o r subsequent blade motion s i n c e i t i s t h e r a t i o o f t h e a c t u a l damping d i v i d e d by t h e c r i t i c a l damping.

For t h i s system, t h e damping r a t i o i s given by;

When t h i s r a t i o i s l e s s than one, t h e blade o s c i l l a t e s .

I n general, t h e

wind t u r b i n e blades w i l l have a dampiug r a t i o l e s s than one i n d i c a t i n g t h a t t h e blade e x h i b i t s a damped o s c i l l a t i o n a f t e r the tower shadow has been encountered.

The frequency o f the damped o s c i l l a t i o n i s expressed

as;

and the amplitude o f o s c i l a t i o n decays e x p o n e n t i a l l y , because t h e blade behaves l i k e a damped s i n g l e degree o f freedom system.

A steady moment (MST) and t h e p e r i o d i c moment (Mp) comprise t h e t o t a l moments a c t i n g on t h e blade r o o t .

The steady moment, given by,

has no a f f e c t on t h e blade motion.

I t o n l y serves t o i n i t i a l l y d e f l e c t

t h e blade by t h e amount,

and t h e proper choice o f coning angle r e s u l t s i n zero i n i t i a l d e f l e c t i o n f o r t h e blade.

Blade motions are the r e s u l t o f t h e p e r i o d i c moment caused

by t h e tower shadow,

where t h e tower shadow has been represented by a F o u r i e r Series. t h a t t h e c o e f f i c i e n t s o f equation ( -16) have been defined,

Now

a solution

can be obtained. The steady s t a t e s o l u t i o n f o r t h e blade f l a p p i n g d e f l e c t i o n i s expressed by a series, where each term i n t h e s e r i e s i s t h e c o n t r i b u t i o n t o the d e f l e c t i o n made by each harmonic o f t h e F o u r i e r tower shadow representation.

M

B = B

0

Therefore,

+y

the t o t a l response o f t h e blade i s g i v e n by;

wo 2 +

a,

C

4nWn

n=1

I 2wo -nsJI n~ s i n cos (n$-Bn) i I

2

(wn -no

2

R

!

+ ( 2 ,nnn)2

where t h e phase angle (On) between c o n t r i b u t i o n s i s ; 0,

= arctan [

2

Wn

Wn

"2]

-nR

i

A p r a c t i c a l e v a l u a t i o n o f t h e response u s i n g equation (4.24) r e q u i r e s the truncation o f the i n f i n i t e series.

The number o f terms necessary

t o assure t h e d e s i r e d accuracy o f t h e s o l u t i o n depends on t h e w i d t h o f I f (6$) were c l o s e t o

t h e r e c t a n g u l a r p u l s e (6q).

IT,

t h e s e r i e s would

converge r a p i d l y , b u t t h i s i s n o t t h e s i t u a t i o n behind a p i p e tower. The wake produced by a p i p e tower i s narrow, so t h a t t h e s o l u t i o n does n o t converge r a p i d l y t o a steady d e f l e c t i o n .

Since t h e closed form s o l u t i o n

o f t h e equation o f motion converges a t a slow r a t e , numerical techniques a r e necessary.

Computer code RIGID, l i s t e d i n Appendix

C,

i s used t o

s o l v e t h e equation o f motion u s i n g E u l e r s ' time stepping i n t e g r a t i o n . 4.5

A n a l y s i s o f Wind Furnace I The equations presented by t h e previous s e c t i o n w i l l be used f o r

t h e a n a l y s i s o f t h e WF-I.

Since t h e WF-I does n o t have t h e s p e c i f i c

geometry assumed by t h e o f f - s e t hinge model, an e q u i v a l e n t blade must be developed.

The e q u i v a l e n t system r e t a i n s t h e f o l l o w i n g t u r b i n e

characteristics: Blade r a d i u s

R = 4.95 m (16.25 f t . )

Hub r a d i u s

RH = 0.495 rn (19.25 i n . )

Blade mass

M

N a t u r a l Frequency Coning a n g l e

= 15.44 kg (34 l b s . ) on =

h =

25 rod/sec

10°

The slender r i g i d blades r e q u i r e d by t h e model have mass moment o f i n e r t i a equal t o ; I=

1 7 m

(R-R,,)~ = 102.15 kgm2

This moment of i n e r t i a i s then used with t h e b l a d e s ' n a t u r a l frequency t o o b t a i n a hinge s p r i n g c o n s t a n t ;

k~

n-m

= 63843 radius

= w '1

n

Aerodynamic loads a r e determined from mean values f o r t h e blade chord and t w i s t d i s t r i b u t i o n , because t h e blades have been modeled w i t h a The e q u i v a l e n t chord has the value;

c o n s t a n t chord and l i n e a r twist. @a

C

Cn = -263 m n

c =

and t h e e q u i v a l e n t twist i s c a l c u l a t e d i n a s i m i l a r manner a s ; a,

2

e

o

=

' e n n

= .262 r a d i u s

As an example of the blade response, a s o l u t i o n i s determined for the s t e a d y s t a t e o p e r a t i o n of t h e WF-I i s a 9 m/s (20 mph) wind.

Equa-

t i o n 4.16, the governing equation of motion has c o e f f i c i e n t s equal t o

the q u a n t i t i e s ; w

n

c

= 28.814 r a d / s e c = 0.345

MST = 2569 n-m

P

=

4919.515 n-rn

and a s o l u t i o n f o r t h e equation of motion i s performed by computer code RIGID.

The blade t i p d e f l e c t i o n due t o a ,254 m ( 1 0 " ) shadow width i n

a 9 m/s (20 mph) wind i s shown i n Figure 4.6. moment a l s o shows t h e response ( ~ i g u r e4 . 7 ) .

The blade r o o t bending Moments w i l l be used

throughout the c h a p t e r because the blade stress i s dependent on the

magnitude of the bending moments. d e f i c i t occur between

(6$ +

During rotation, the twoer shadow

180°) and ( 6 $ - 180°), b u t the resulting

response i s not significant until the blade begins i t s ascent from the bottom of rotation.

The turbine blade follows an oscillating path a s i t

rotates about the wind shaft.

This osciallating pattern i s similar for

a l l windspeeds because the damping r a t i o remains l e s s than unity. Oscillations of the blade root bending moments are the ~iiostimportant feature of the response. t h e i r maximum (Mmax)

These oscillations are best described by

and minimum values (Mmin). Figure 4.8 shows the

maximum, nl-inimum and steady moments encountered over the e n t i r e operating range of w-ind speeds f o r WF-I.

The magnitude of the steady root moment

drops quickly when the operational mode i s changed to constant rotational speed.

A more subtle change occurs in the magnitude of the oscillations.

If the steady moment i s removed from the response, a clear picture of the tower shadow pertabation resul t s (Figure 4 . 9 ) .

The flatwise moment

variation increases a t a f a s t e r r a t e under constant rotational speed (region I I I ) operation, then would have occured i f constant t i p-speedr a t i o had been maintained. Increases in the tower diameter also change the root bending moment due t o changes i n the tower shadow.

Figure 4.10 shows the a f f e c t of

tower diameter on the moment variation i n a 9 m/s (20 mph) wind.

The

response due to the amount of blockage i s small f o r a narrow tower and increases to a maximum as the tower diameter i s enlarged.

Increasing

moment variation occurs because the response i s a function of how long the blade remains shaded and the strength of the blockage. the tower will

In the l i m i t ,

shade the e n t i r e rotor and the periodic component of the

bending moment will converge to steady value.

Convergence occurs about

F I G U R E 4.8

O P E R A T I N G GANGE

WIND SPEED

M/S

F I G U R E 4.10 SHADOW W l DTH

SHADOW

WIDTH

M

t h e steady moment due t o wind speed minus t h e tower shadow d e f i c i t e . I n a d d i t i o n t o blade r o o t bending moment v a r i a t i o n s , t h e wake c o n t r i b u t e s t o t h e yaw motion experienced by t h e t u r b i n e .

During h i g h

winds, WF-I has been observed t o o s c i l l a t e about a p o s i t i o n s l i g h t l y yawed away from t h e wind d i r e c t i o n [9 1.

A motion o f t h i s n a t u r e i s

i n d i c a t e d by t h e p r e d i c t e d shadow data when t h e blade moments f o r t h e e n t i r e r o t o r a r e resolved about t h e yaw a x i s .

An example o f t h e r e s u l t i n g

yaw moments occuring i n a 20 m/s (44 mph) wind a r e shown i n F i g u r e 4.11. The yaw moment has a frequency o f t h r e e times t h e r o t a t i o n a l speed w i t h an amplitude v a r i a t i o n about a p o s i t i v e mean yaw moment.

Experimental

data has been c o l l e c t e d a t low windspeeds t h a t v e r i f y t h e frequency o f t h e tower shadow p e r t u r b a t i o n on t h e yaw c h a r a c t e r i s t i c s o f t h e t u r b i n e [ 9 I n summary, t h e o f f - s e t hinge r e p r e s e n t a t i o n o f a wind t u r n i n e o f f e r s a simple technique t o i n d i c a t e elementary a f f e c t s o f tower shadow on t h e r o t o r dynamics.

The n e x t s t e p i n t h e a n a l y s i s i s t h e i n c l u s i o n o f a

non-uniform f l e x i b l e blade i n t o t h e model so t h a t t h e bending moment d i s t r i b u t i o n along t h e blade can be determined. a n a l y s i s w i l l be presented i n t h e n e x t chapter,

T h i s more i n v o l v e d

1.

FIGURE 4.1 I YAW

MOMENTS

W-N

lN3NOW

CHAPTER

5

COMBINED LEAD-LAG AND FLAPPING RESPONSE OF A WIND TURBINE ROTOR BLADE 5.1

Rational The simple r i g i d blade model o f t h e previous chapter i s n o t adequate

f o r an a n a l y s i s o f t h e f o r c e d i s t r i b u t i o n along t h e blade.

The r i g i d

model i s u s e f u l f o r determining many dynamic e f f e c t s caused by t h e tower shadow, b u t t h e r i g i d model l a c k s t h e a b i l i t y t o handle blade f l e x i b i l i t y and a complex geometry.

A wind t u r b i n e blade i s a non-uniform non-homo-

genious beam and t h e e n t i r e motion o f t h e blade i s needed f o r a d e t a i l e d a n a l y s i s o f l o a d i n g and moments. I he equation o f motion f o r a d i f f e r e n t i a l element o f t h e r o t o r

blade i s found u s i n g t h e same coordinate system developed f o r t h e r i g i d blade model.

To simp1 i f y t h e d e r i v a t i o n o f t h e equation o'f motion,

t o r s i o n a l e f f e c t s a r e neglected. edgewise motion.

The f o r c e s and moments a c t i n g on a blade element a r e

shown i n F i g u r e 5.1. force ( V ) ,

T h i s leaves o n l y coupled f l a t w i s e and

These forces and moments a r e t h e l o c a l shear

t h e bending moment ( M ) , t h e aerodynamic l o a d (F), and t h e

c e n t r i f u g a l t e n s i o n (G).

Force e q u i l ibrium on t h e element r e q u i r e s t h a t ;

I n t e g r a t i n g expression 5.1 w i t h r e s p e c t t o z y i e l d s f o r t h e c e n t r i f u g a l t e n s i o n G;

FIGURE 5.1

DIFFERENTIAL ELEMENT

R

2 m r zdz

z The moment e q u i l i b r i u m f o r t h e element r e q u i r e s t h a t

D i f f e r e n t i a t i n g t h e moment equations w i t h r e s p e c t t o z gives;

A s u b s t i t u t i o n from equations 5.2 and 5.3 i n t o 5.7 and 5.8 y i e l d s ;

Bending moments i n equations 5.9 and 5.10 a r e g i v e n by t h e E u l e r - B e r n o u l l i theory o f bending [lo]. t o t h e displacement by;

For small displacements, t h e moment i s r e l a t e d

When t h e bending moments a r e s u b s t i t u t e d i n t o t h e equations o f motion 5.10),

(5.9,

the following results; 9

I t i s e v i d e n t by examination o f these equations t h a t t h e blade motion

i s coupled i n t h e l e a d - l a g and f l a p p i n g planes.

There i s no

closed form

s o l u t i o n f o r t h e expression, so an approximate method i s required.

A

modal a n a l y s i s i s chosen as t h e p r e f e r r e d s o l u t i o n technique since t h e equations a r e uncoupled i n t h e modal frame o f reference.

The d e r i v a t i o n

o f t h e uncoupled form i s c a r r i e d o u t i n t h e n e x t s e c t i o n and t h e s o l u t i o n i s obtained u s i n g computer code DYNAMICS which may be found i n Appendix D. 5.2

Modal Equations o f Motion Modal a n a l y s i s i s based on t h e assumption t h a t t h e response o f a

system i s determined by t h e l i n e a r combination o f t h e orthogonal mode shape.

The mode shape represents t h e d e f l e c t i o n c o n f i g u r a t i o n o f t h e

system when i t v i b r a t e s a t a n a t u r a l frequency.

I n o t h e r words, t h e

mode shapes and n a t u r a l frequencies a r e s o l u t i o n s t o t h e f r e e v i b r a t i o n equation.

A f u r t h e r e x p l a n a t i o n o f modes and t h e i r orthogonal p r o p e r t i e s

can be found i n most v i b r a t i o n s t e x t s [Ill. The combination o f modes comprizing t h e response o f t h e blade i s represented by;

where rn(x,y)

i s a mode shape and g n ( t ) i s a modal amp1 i t u d e .

This

equation i s conveniently expressed i n v e c t o r s u b s c r i p t n o t a t i o n by;

where t h e summation i s i m p l i e d by repeated s u b s c r i p t s .

This n o t a t i o n w i l l

be used throughout t h e chapter. The governing equation o f motion (eq. 5.13) has t h e f o l l o w i n g form i n subscript notation;

where:

=

Ai i

Iyy Ixyi

Ixy

Bii

-

IG 1G .

IXX

0

1

0'

I f t h e r i g h t hand s i d e o f equation (5.15) i s zero, t h e equation representing

t h e f r e e v i b r a t i o n o f t h e blade r e s u l t s ;

When a system o s c i l l a t e s i n a normal mode (g. . ) w i t h a n a t u r a l frequency 1J

(uj), every p a r t o f the system o s c i l l a t e s i n phase o r antiphase w i t h every

o t h e r p a r t o f the system.

Thus, t h e t y p i c a l displacement i s expressed

'i

0 I. J. s i n w J. t

=

(5.17)

There are an i n f i n i t e number o f these s o l u t i o n s f o r t h e f r e e l y v i b r a t i n g blade.

Since t h e equations o f motion (5.15 and 5.16) and t h e i r s o l u t i o n s

(5.14 and 5.17) are known, t h e mqdal equation can be determined. The modal equation o f ~ i i o t i o nrepresents the d i f f e r e n c e between the forced motion and t h e f r e e motion. governing equation

(Aii

A s u b s t i t u t i o n o f eq. 5.14 i n t o the

generates;

O..")" JJ

g . + (B.. @ . . ' ) I g

J

11

IJ

j

+ Cii

0..

JJ

'g'j

= Fi

A s u b s t i t u t i o n o f eq. 5.17 i n t o the f r e e v i b r a t i o n equation y i e l d s ;

5.18 and 5.19 are p r e - m u l t i p l i e d by t h e transpose o f the mode

When eq. shape

(O..) J1

and subtracted from each other, a modal equation f o r the

d i f f e r e n t i a l element r e s u l t s ;

~ 1 the 1 coupled terms have been eliminated from the modal equation because of the o r t h o g o n a l i t y property of the mode shapes. condition stipulates that;

and

0.. Dij J1

=

0 for j # j

The orthogonal

When eq. 5.20 i s integrated from the blade root t o the t i p , the modal equation for the entire blade results as;

The f i r s t integral i s named the modal mass;

and represents a diagonal mass matrix because of the orthononality conditions.

The second integral i s the generalized force;

For an unconed turbine blade, the generalized force i s composed of the aerodynamic forces on the blade, b u t i f coning i s present, i t must include the additional centrifugal force. Therefore, the additional centrifugal component; Fi = MnZ 2 tan

x

i s added t o the aerodynamic loads t o obtain the generalized force as a coned turbine bl ade. Therefore, the uncoupled modal equation;

and assumed response

represent t h e t o t a l motion o f t h e wind t u r b i n e blade.

An advantage o f t h e

modal equation i s t h a t a s u f f i c i e n t l y accurate s o l u t i o n i s obtained when o n l y t h e f i r s t few modes o f v i b r a t i o n a r e included i n t h e a n a l y s i s . U s u a l l y t h e modal equation o f motion (5.27) can be solved i n t h e modal coordinate system.

However, f o r t h e wind t u r b i n e r o t o r blades,

t h e aerodynamic loads depend on blade v e l o c i t i e s ; therefore, t h e modal equations cannot be i n t e g r a t e d d i r e c t l y .

Thus, t h e s o l u t i o n i n v o l v e s t h e

transformation between t h e modal coordinates and t h e physical coordinates t o c a l c u l a t e t h e aerodynamic loading and t h e generalized forces.

Computer

code DYNAMICS l i s t e d i n Appendix C i s used t o s o l v e t h e modal equations f o r a wind t u r b i n e generator. 5.3

Aerodynamic Loads I n Chapter 4, a s i m p l i f i e d expression f o r t h e aerodynamic loading on

t h e r o t o r was presented.

This expression excluded many higher order

terms so t h a t an a n a l y t i c a l i n t e g r a t i o n would be possible.

Since numerical

techniques a r e used f o r t h e modal analysis, t h e neglected v a r i a b l e s can be included i n t h e f o r c e system.

Blade element theory i s r e t a i n e d f o r t h e

determination o f t h e aerodynamic forces, b u t t h e blade element diagram shown i n Figure 4.4 i s modified t o i n c l u d e drag ( F i g u r e 5.2). Both li f t and drag depend on t h e r e l a t i v e wind ( V R ) and t h e angle o f a t t a c k (a).

The magnitude o f these f o r c e s are given by t h e equations;

Up L perpendicular velocity Ut = tangential velocity

vr

=

ut2 + up2

= resultant

9 = blade element angle 8 = blade p i t c h angle

a = angle o f attack

L = lift D = drag

velocity

and

The l i f t and drag c o e f f i c i e n t s , Ce and CD, a r e e x p e r i m e n t a l l y determined q u a n t i t i e s t h a t depend on t h e a i r f o i l shape.

These q u a n t i t i e s a r e

g e n e r a l l y expressed g r a p h i c a l l y as shown i n F i g u r e 5.3, which shows t h e c h a r a c t e r i s t i c s o f t h e NACA 4415 a i r f o i l used i n t h e c o n s t r u c t i o n o f t h e WF-I blades [121. The r e l a t i v e v e l o c i t y a c t i n g on t h e a i r f o i l i s composed o f components perpendicular (U ) and para1 l e l (UT) t o t h e plane o f r o t a t i o n . These P p a r a l l e l and perpendicular v e l o c i t i e s a c t i n g on an element o f t h e blade a r e g i v e n by;

u

where:

P

= Vo ( 1

-

(a+w($))

- i

(5.31 )

Yo = wind speed a = axial interference factor b = radial interference factor w($) = tower shadow f a c t o r u = x blade v e l o c i t y

i

= y blade v e l o c i t y

For t h i s , modal tower shadow i s represented by t h e r e c t a n g u l a r p u l s e o f s e c t i o n 3.4,

b u t i t i s b o t h a f u n c t i o n o f azimuth angle and blade radius.

Therefore, t h e v e l o c i t y d e f i c i t i s a p p l i e d g r a d u a l l y s t a r t i n g a t t h e blade r o o t as t h e blade encounteres t h e wake. shadow approximation.

F i g u r e 5.4 d i s p l a y s t h i s tower

FIGURE -.

SHADOW

'

5&

MODEL

Rotor induced velocities are indicated by the axial and radial interference factors.

No attempt i s made to calculate these quantities

f o r the dynamic system.

They are assumed to be equivalent t o the inter-

ference factors obtained from a steady s t a t e momentum analysis of the wind turbine,

Momentum theory determines the axial and radial inter-

ference factors by iterating through the f o l l owing equations unti 1 they converge upon the proper values [13]. tan 0

- a-

1+a

-

- b- -

=

"0 I-a - I-a fir F I 7 m '

B c (C, cos 8n

CD sin

0)

r sin2@

B c (C, sin

I -b

0+ 0

- CD cos 0)

8 n r sin (ilcos

0

In these equations ( b ) i s the number of blades, ( C ) i s the chord length, and (0) i s the relative pitch angle. Once the velocity components are determined, they yield the angle of the attack ( a ) , since;

and

The angle of attack i s then used w i t h a i r f o i l data (Figure 5.2) and eqs. 5.29 and 5.30 to determine the l i f t and drag on the blade element. The l i f t and drag forces are resolved into components parallel and perpendicular to the plane of rotation as;

Fx F~

= -D cos

0+

L sin g

(5.381

= D s i n 0 + L cos 0

(5.39)

t o serve as the aerodynamic i n p u t t o t h e generalized forces defined by eq. 5.25. Analysis o f Wind Furnace I

5.4

An extensive analysis o f t h e WF-I blades using computer code DYNAMICS has been performed and the r e s u l t s are presented i n t h i s section.

The

data necessary f o r the analysis has been l i s t e d i n the previous sections, b u t an example of the i n p u t i s condensed below f o r review and ease of reading.

.

DATA W I N D SPEEb ( M / S ) * * * * * * * * * * * * 9.000 TIP SPEED R ~ T ~ ~ * * * * * * * * * 7.500 * * * * PITCH A N G L E ( n E G ) + . + + + + + + . . + -66000 C O N I N G AJetGLE ~ ~ E G ~ * * * * * *106000 * * * * S H A D O W WIr'TH ( M ) * + + + * + * * * * + * 06254 5 H A V O W STRENGTH V/V,,,,,,,,, 0,500 BLADE R A D I U S ( M ) + + + + + + + + + + * *4r953 STAT'IOCt SPCIPt

Oe100

06200

06300

CHORD n I S T R U B U T I O N

06411

06445

06400

06500

06311

0.259

(M)

06384

TWIST r ~ I S T R I P U T I O N ( D E G )

456000 256600 156700 10+400 7.400 AXIAL

INTERFERENCE FACTOR

06090

06140

06180

06200

0.210

RCIDICIL IbtTERFEREl4CE FCICTOR

06140 0+050 06027 0.016 nnss DISTRIBUTION ( K G ) 10.050 66123 4.766 46016 N n T U R A L FREOUENCY

3e463

(RAnICINS/SECOND)

28 430 X

O*Oll

646450 .

MODAL COORDINATES

O*OOO

06000 '06010 -01020 -0.040 0.000 0+010 0.050 0.110 0.180 O*OOO -O+OIO -06020 -06040 -06060 Y

OVAL 06000

o*ooo O*OOO

COORPINATES

06010 06000 06030

06030 0.090 0*000 O * O O O 06090 0,180

0.170

o*ooo 01270

.

-..

S I u n i t s a r e employed by the program and i t should be noted t h a t a l l angles inputed i n t o the computer code a r e i n radians.

The use o f degrees i n t h e

o u t p u t i s f o r simple i d e n t i f i c a t i o n . P a r t of the output from the program DYNAMICS includes the steadys t a t e forces t h a t would e x i s t f o r a uniform f l o w f i e l d .

Rated c o n d i t i o n s

were choosen f o r the t y p i c a l example of t h e forces and moments.

F i g u r e 5.5

shows a comparison between t h e f l a t w i s e and edgewise moment d i s t r i b u t i o n . I n t h e previous chapter, edgewise motion was neglected a l t o g e t h e r and the magnitudes of the moments i n d i c a t e t h a t t h e assumption was reasonable. The maximum bending moments on the blade occur between t h e .5 and .7 blade stations.

The s t r e s s o c c u r r i n g on t h i s s e c t i o n o f t h e blade should be

a maximu~iibecause the cross-sectional area decreases towards t h e t i p . F i g u r e 5.6 shows t h e a f f e c t t h a t pre-coning t h e blade 10 degrees has on t h e bending moment d i s t r i b u t i on.

C e n t r i f u g a l re1 i e f reduces t h e t o t a l

moment by more than h a l f , which i s a s i g n i f i c a n t r e d u c t i o n o f t h e steady appl i e d 1oads. Rated c o n d i t i o n s were a1 so chosen t o show the t y p i c a l response of t h e blade when tower shadow p e r t u r b a t i o n i s d i s r u p t i n g the f l o w .

F i g u r e 5.7

shows t h e blade r o o t bending moments and F i g u r e 5.8 shows t h e t i p d e f l e c t i o n . The blade response has many s i m i l a r i t i e s t o t h e r i g i d blade a n a l y s i s i n t h a t the shadow response occurs a f t e r t h e blade passes behind the tower and t h e recovery from the shadow i n d i c a t e s a damped o s c i l l a t i o n .

Bending

moments a r e n o t severe because t h e tower shadow i s a p p l i e d and removed gradually.

The gradual l o a d i n g o f t h e blade i s b e l i e v e d t o be a r e a l i s t i c

model o f t h e physical s i t u a t i o n .

A1 so the damped n a t u r a l frequency of

t h e t r u e geometry i s l e s s than e x i s t s f o r an assumed constant chord blade.

1

MOMENT

IW U

r l L 4.4

OlSTRI BUTION

F IGURE

5.6

CENTRIFUGAL EELIE F

FIGURE 5.7 BLADE T I P

MOTION

FIGURE 5.8 MODAL

PREDICTION

The damping i s indicated by the number of oscillations a blade makes during a complete rev01 ution. Over the entire operating range, tower shadow causes a cyclic response. The magnitude of the oscillations a r e shown i n Figure 5.9, which shows the maximum (Mmax), minimum (Mmin) and steaty (MSt) moments f o r the WF-I blade.

The steady bending moments increase i n region I1 operation and

decrease i n region 111, while the cycl i c tower shadow moment variation increases steadily w i t h wind speed.

The magnitude of the moment variation

a1 so increases when the wake width becomes larger a s shown i n Figure 5.10. A doubling of the shadow w i d t h i s accompanied by a doubling of the cyclic

moment variation.

Therefore, the comnon sense approach of small tower

causing fewer problems applies to the tower shadow predictions. To examine the a f f e c t t h a t important parameters have on the dynamic model, a s e n s i t i v i t y plot i s developed.

Figure 5.11 shows the sensitivity

of the model to changes i n coning angle, wind speed, shdow width, and shadow strength.

The ordinate i s defined as; Parameter val ue - -P Standard Parameter value Po

and the abssica i s defined i n terms of the variation in cyclic bending moments, where the variation i s defined a s the maximum blade root bending moment m i n u s the minimum blade root bending moment; Variation of Moment -Standard Variation of Moment Mvo Standard conditions a r e defined f o r the WF-I as; Wind speed = 9 m/s (20 mph)

F I G U R E 5.9

F I G U R E 5.10

SHADOW

WIDTH

M

F JGURE 5.11

SENSITIVITY OF PARAlLlETER.5

Coning angle = 10" Shadow w i d t h = .254 m (10") Shadow s t r e n g t h = .5 The r e s u l t s o f the s e n s i t i v i t y t e s t i n d i c a t e t h a t coning angle does n o t e f f e c t t h e c y c l i c l o a d i n g p a t t e r n and t h a t windspeed i s a dominant v a r i a b l e f o r h i g h windspeeds.

Both windspeed and shadow w i d t h have a decreased

a f f e c t when they have small values. The physical s i g n i f i c a n c e o f the decreasing a f f e c t i s t h a t t h e moment v a r i a t i o n i s approaching a steady value o t h e r than zero.

For t h e shadow w i d t h t h e 1 i m i t i n g case i s an impulsive

load, and f o r t h e wind speed, i t i s the moment v a r i a t i o n t h a t occurs a t cut-in velocity.

I n conclusion o f t h i s chapter, i t should be noted

t h a t t h e wake s t r u c t u r e has been defined by a simple model and t h i s model may be changed a t any time i f more d e t a i l i s known about t h e wake o r blade response

.

C H A P T E R

6

CONCLU'IONS AND RECOMMENDATIONS 6.1

Conclusions The purpose of t h i s study was the evaluate the a f f e c t o f tower shadow

on the wind t u r b i n e r o t o r . of r o t o r blade response.

Two models were developed f o r t h e assessment One model i n v o l v e d a simple r i g i d blade w h i l e

t h e o t h e r modeled a complex f l e x i b l e blade.

Both models i n d i c a t e d t h a t

t h e tower wake imparts a f o r c e t h a t causes the blade t o have a damped o s c i l l a t o r y motion w i t h l a r g e d e f l e c t i o n amplitudes o c c u r i n g on t h e upswing o f t h e b l a d e

(Y

>

180").

The major discrepancy between t h e two model

t h e magnitude of t h e r e s u l t i n g forces.

predictions involves

Larger c y c l i c forces a r e always

p r e d i c t e d by t h e simple r i g i d model because the shadow i s assumed t o encompass t h e e n t i r e b l a d e instantaneously, w h i l e t h e complex model assumes a gradual appl i c a t i o n o f t h e shadow.

O f the two approaches, t h e r i g i d system solved by corrlputer code R I G I D proved t o be e a s i e r and l e s s time consuming than i t s f l e x i b l e c o u n t e r p a r t solved by computer code DYNAMICS.

Since t h e simple model p r e d i c t s a more

d r a s t i c response, i t serves t o make conservative estimates o f t h e blade loading.

The more corr~plexmods1 serves t h e purpose of d e f i n i n g a d e t a i l e d

l o a d i n g d i s t r i b u t i o n along t h e blade.

For design appl i c a t i o n s , t h e simple

system w i l l i n d i c a t e problem areas and t h e complex system w i l l d e f i n e the loads a t those problem

areas.

The computer codes have been documented i n t h e appendices t o f a c i l i t a t e t h e i r use.

These codes many

be extended t o i n c l u d e force v a r i a t i o n due

t o wind shear and g r a v i t y by program m o d i f i c a t i o n s t h a t change t h e a p p l i e d loads o r wind f i e l d , n o t t h e s o l u t i o n technique.

75

Since the s o l u t i o n technique

has been successful 6.2

.

Kecomnendations A1 though t h e model s p r e d i c t s o l u t i o n s f o r t h e wake b l a d e i n t e r a c t i o n ,

experimental i n f o r m a t i o n i s needed t o c o n f i r m o r r e f u t e t h e p r e d i c t i o n s . Experimental a n a l y s i s should i n c l u d e b o t h frequency and amplitude a n a l y s i s of t h e blade motion.

A frequency a n a l y s i s i s needed t o v e r i f y t h e blade n a t u r a l frequencies and resonant c o n d i t i o n s .

O f t h e p o s s i b l e resonant c o n d i t i o n s , t h e 3~

forcing

harmonic o c c u r i n g i n a 7.6 m/s (17 mph) wind speed should be p a r t i c u l a r l y s t r o n g because t h e blades n a t u r a l frequency c o i n c i d e s w i t h t h i s f o r c i n g frequency.

A power s p e c t r a l d e n s i t y a n a l y s i s should y i e l d most o f t h e frequency

information, since the a p p l i c a t i o n o f t h i s analysis i s t o e s t a b l i s h the frequency composition o f data [I61. Experimental data w i l l a l s o y i e l d i n f o r m a t i o n about t h e b l a d e motion and wake s t r u c t u r e .

The b l a d e motson may n o t be apparent f r o m t h e data

because t h e data w i l l c o n t a i n a l a r g e amount o f extraneous i n f o r m a t i o n . Much o f t h e extraneous data w i l l have frequency components l e s s than o r equal t o 2 %

T h i s 1 ow frequeccy i n f o r m a t i o n can be e l i m i n a t e d by h i g h pass

f i l t e r i n g techniques.

Once t h e extraneous a f f e c t s have been removed, t r e n d s

o f blade motion may be i d e n t i f i e d . For t h e d e t e r m i n a t i o n o f t h e wake s t r u c t u r e , i t w i l l be necessary t o vary t h e tower diameter. t h e tower w i t h shrouds.

T h i s v a r i a t i o n can be performed by encompassing The shrouds n o t o n l y change t h e wake w i d t h b u t

they w i l l a l t e r t h e wake s t r e n g t h which i s r e l a t e d t o t h e r a t i o o f X/D, where X i s t h e d i s t a n c e from t h e plane o f r o t a t i o n t o t h e tower a x i s and

D i s t h e tower diameter.

A method o f condensing t h e l a r g e q u a n t i t y o f

information i s w i t h the use of non-dimensional parameters.

Cyclic moment

variations may be non-dimensionalized by

2(M&-Mmin) M -M

-

--

max min

M

Mave

and the windspeed can be expressed i n terms of a Reynolds nurnber;

The non-dimensional terms allow a p l o t of the non-dimensional moment versus Reynolds number f o r various X/D r a t i o s .

This type of experimental analysis

should f a c i l i t a t e the i d e n t i f i c a t i o n of the wake and the v e r i f i c a t i o n of

the proposed analytical predictions.

REFERENCES -1.

Savino, Joseph M., Wagner, Lee H,, and Nash, Mary, Wake C h a r a c t e r i s t i c s o f a Tower f o r t h e DOE-NASA Mod-1 Wind Turbine, A p r i l 19/8, DOE/NASA/ 1028-78/17.

2.

Perkins, F.W;, and Cromack, D.E., Wind T u r b i n e Blade S t r e s s A n a l y s i s and N a t u r a l Frequencies, UM-WF-TR-78-8.

3.

T r i c h e t , P i e r r e , Study o f t h e Flow Around a C y l i n d e r and o f i t s Near Wake, NASA TT-16844, 1975.

4.

Cantwell, B r i a n Joseph, A F l y i n g Hot Wire Study o f t h e T u r b u l e n t Near Wake o f a C i r c u l a r C y l i n d e r a t a Reynolds Number o f 140,000, Ph.D. D i s s e r t a t i o n , C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a , 1975.

5.

Roskko, Anotal, "On t h e Wake and Drag o f B l u f f Bodies," Aeronautical Sciences, Vol 22, 1955.

6.

Wind T u r b i n e S t r u c t u r a l Dynamics, NASA Conference P u b l i c a t i o n 2034, DOE P u b l i c a t i o n CONF-771148, 1977.

7.

Stoddard, F o r r e s t S., S t r u c t u r a l Dynamics, S t a b i l i t y and C o n t r o l of High Aspect R a t i o Wind Turbines, Energy A l t e r n a t i v e s Program, U n i v e r s i t y o f Massachusetts, Amherst, Mass., UM-WF-TR-78-11.

8.

Demarogonas, Andrew D., 1976.

9.

Cohen, Richard, Yaw Dynamics, Energy A l t e r n a t i v e s Program, U n i v e r s i t y o f Massachusetts, Amherst, Mass., UM-WF-TR-79.

10.

R i v e l l o , Robert M., Theory and A n a l y s i s o f F l i g h t S t r u c t u r e s , McGrawH i l l , New York, 1969.

11.

Bigg, John M., I n t r o d u c t i o n t o S t r u c t u r a l Dynamics, McGraw-Hill, New York, 1964.

12.

Abbott, I r a A,, and Van Doenhoff, A l b e r t E., Dover P u b l i c a t i o n s , Inc., N e w York, 1959.

13.

Wilson, Robert E. and Lissaman, P e t e r B.S., o f Wind Power, NTIS, PB-238595, J u l y 1974.

14.

Bramwell, A.R.S., 1976.

15.

Shapiro, Jacob, P r i n c i p l e s o f H e l i c o p t e r Engineering, Temple Press L i m i t e d , Bowling Green Lane, London, 1955.

.

Journal of t h e

V i b r a t i o n Engineering, West Pub1i s h i n g Co.

.

,

Theory of Wing Sections, A p p l i e d Aerodynamics

H e l i c o p t e r Dynamics, John Wiley and Sons, New York,

16.

Bendat, J u l i u s S. and P i e r s o l , A l l e n G., Random Data: A n a l y s i s and Measurenient Procedure, John W i l e y and Sons, Inc., New York, 1971.

A P P E N D I X

A.l

A

Theorem of Southwell The Theorem of Southwell mentioned in section 2 . 3 i s discussed

here because of i t s usefullness i n accounting for the centrifugal stiffening of a rotating blade.

The Th.eorem states t h a t in an elastic

system, the spring forces can be divided into two parts, such t h a t the total potential energy i s the sum of two partial potential energies. Thus, the natural frequency ( w n ) of the blade can be approximated by;

where ol i s the natural frequency a t standstill and

w2

frequency of a blade having no bending resistance.

Centrifugal tension i s

i s the natural

the only stiffening component of up. Rayleigh's method i s used to establish the rotating frequency when the blade mode shape i s assumed to remain unchanged by blade rotation [14]. Therefore, if the non-rotating mode shape

( p i ) i s given, then the

maximum potential energy due to centrifugal loading i s ;

where G i s the centrifugal tension. R

G$

mfi

z

2

zdz

The maximum k i n e t i c energy f o r a system o s c i l l a t i n g a t a n a t u r a l frequency i s expressed by;

Now, t h e r o t a t i n g frequency o f t h e b l a d e i s g i v e n by equating t h e energies and s o l v i n g f o r w2;

When G i s r e p l a c e d by i t s i n t e g r a l d e f i n i t i o n , e q u a t i o n 5 i s re-arranged t o y i e l d the solution;

ry:o

j:

W 2 = -

i

mzdz 0

m pi2 dz

T h i s equation i s i n t h e form

where

a

i

i s t h e Southwell c o e f f i c i e n t g i v e n by

a

i

=

Go(1

mz dz

m

oi2

(

d'i2 ) dz dz

dz

(

df'i 2 ) dz

A.2

Program South The s o l u t i o n o f t h i s equation i s performed by f u n c t i o n SOUTH.

This

f u n c t i o n uses a data package t h a t c o n s i s t s o f a group o f s t o r e d v a r i a b l e s . The s t o r e d q u a n t i t i e s are shown i n t h e program l i s t i n g as underscored names, which have t h e f o l l o w i n g meaning;

R -

A D IUS, -

r o t o r radius

S P A C E, spacing between blade sections -

ST A T I0N, l i s t i n g of a l l s t a t i o n spans expressed as r / R M A 5

-

5, mass

per u n i t l e n g t h a t any s t a t i o n span

M 0 D EX, x coordinates o f t h e mode shape M 0 D E Y, -

y coordinates o f t h e mode shape

To r u n t h e program, type SOUTH and t h e computer w i l l r e t u r n t h e solution.

A.3

FLOW CHART -

FOR PROGRAM SOUTH

SOUTH

!

\

COMPUTE

,---

-

-& DM =

- .

I -

DETERMIqE THE UPPER lINTEGER VF EQ. 6 ( I N 1 ) I: 1

-

-

-

-

I

i

....

1

-..-.........

DETERMINE THE LOWER : OF EQ. 7 ( I N 2 ) 1

i INTEGER

........

r_-____.-~

r-.- . . . . . . . . .

j

. . . . . - . . - - .

IN 1 iIN 2 P R I N T THE SOUTHWELL COEFFICIANTS I-._ .. __ . - . . . .......

1

.,.

L:

END ) /

II

I I

1

A.4

PROGRAM LISTING

A.5

TERMINAL SESSION

SOUTH

3+00299629 2,912068217 5,126333881

A. 6

DATA RECORDS

RCIDIUS ------

4 953

SFCICE -----

0 4953

A P P E N D I X

B

Aerodynamic Forces

B.1

This approach f o r calculating the l i f t on a blade element follows the method presented by Stoddard, Structural Dynamics, S t a b i l i t y and

---Turbines, p. 59-66. Control of High Aspect Ratio Wind

To derive the

aerodynamic forces, we i s o l a t e a blade element d r a t radius r, and draw a vector diagram of the v e l o c i t i e s perpendicular and tangential t o t h e r o t o r plane.

This i s shown in Figure B.1.

The drag i s neglected since i t i s small compared t o 1 i f t and the l i f t can be represented a s ;

where:

p =

a i r density

- -dC, do

'Lo

= slope of the 1 i f t curve

C = chord a = angle of a t t a c k

Significant aerodynamic perturbdtions will depend on changes i n angle of a t t a c k a , so VR will be allowed t o remain constant.

A f u r t h e r assumption

i s made, saying t h a t VR has roughly the same magnitude a t Ut. VR

2

"Ut

*

1

( n r )2 and l i f t i s now;

Therefore,

Up

-

perpendicular v e l o c i t y

Ut = tangential v e l o c i t y

Vr =

ut2 + up2

=

resultant

9 = blade element angle 8 = blade p i t c h angle

a = angle o f attack

L = lift

velocity

and; a

= a r c t a n ((3

-

0 ) = (3

-

U

0 =

2-

u,

This gives;

We now assume l i n e a r t w i s t along t h e blade, so t o t a l p i t c h i s ;

= blade t w i s t

with:

e

= p i t c h measured a t t h e t i p P Therefore, the l i f t per u n i t l e n g t h for a l i n e a r l y t w i s t e d wind t u r b i n e

blade a t constant n i s ;

The v e l o c i t i e s can be w r i t t e n :

[Vo where:

"0

Vi

(1-w($))

-

vi]

cos B

= constant f r e e stream wind = a x i a l induced v e l o c i t y

w ( $ ) = tower shadow d e f i c i t

B

r

= f l a p p i n g angle = c o n t r i b u t i o n o f the flapping v e l o c i t y

-

ri

For small flapping angles, cos B

-

1.

Now the lift expression is

written;

or non-dimensionally;

where non-dimensional quantities are:

v

= A = 1 o' nR tip-speed-ratio v-- - - - non-dimensional i nduced vel oci ty

'

i'

QR

n

r

=-=

R

span station

c R4 Y =

I

Lock Number

In physical terms, the Lock Number

(y)

can be descibed as the ratio

of the aerodynamic moment due to a sudden increase of blade pitch to the centrifugal moment due to a sudden increase of a flapping angle 1151. Therefore, if the blade had infinite inertia, its motion would not be effected by changes in aerodynamic forces and the Lock Number would be zero.

APPENDIX C.l

C

Program RIGID This program i s used f o r the analysis of the rigid isolated blade

model.

T h e operation of the program i s simple since only one data f i l e

T h e input data f i l e i s a 5 by 3 matrix

i s needed f o r the input variables. arranged in the following order: Blade mass ( k g ) ,

Mass Moment of i n e r t i a ( k g m 2 ),

Center of gravity ( m )

Radius (m) ,

Hinge o f f s e t ( m ) ,

Natural frequency (rod/s)

Pitch (rod),

Twist

Coning Angle (rod)

Chord ( m ) ,

Windspeed (m/s) ,

T i p-speed-ratio

Snadow Strength,

Shadow w i d t h ( m ) ,

Axial interference

(rod),

The program commences by f i r s t computing and printing the important structural constants.

After the constants have been establ i shed, the

differential equation i s solved using Euler's forward stepping i n time integration scheme.

Integration proceeds u n t i l the transient solution

i s eliminated from the response.

The transient motion fades a f t e r four

complete cycles, or rotor revolutions.

Once integration i s complete, the

solution i s printed out. The program then asks i f you want an i n p u t record. responds with YES, the data record i s printed. w i t h NO, the program ends.

If the operator

If the operator responds

Following i s the flow chart f o r Program RIGID.

C.2

Program R I G I D Flow Chart

*

1

Enter Data F i l e

-

!

L

I n i t i a l ize Constants i n t h e Governing Equation

- - --.- -

-- -!

i I

ii i

I

1

1 Yes

i

d-

[ N--=N+i] 1

1

P r i n t Constants

1

Set L i m i t s : Tower Shadow, w l , w 2 P r i n t o u t , PT T o t a l Revolutions ,N1

/ Yes I

--ill_-

-

-

I I

b.

--

Increment Azmuth Angle T-T+DT

'.

M a t r i x , TOT

-'- I

.-

[

-3

P r i n t Data

.

I

C.3

Program L i s t i n g

VRIGIDCOJV v R I G I ~P L A ~ E ; D ; T ; T O T j K P ; E ; O M ; t J F S ; L O i M & ; M C ; M S T ; E r S T $ M P ; D R ; S S ; D S ; W 1 ; W 2 ; n T ;

-

--

. -- .

-

-

N1;N;Tl;FTl;FT

.--

D+BL&DE

KP+~Cl;2]XnC2;31*2

E+~ClilI~~C2;2~X~Cli31+~C1i21 O M C D C ~ ; ~ ] X ~ ~ C ~ ; ~ ] + ~ C ~ ; ~ I

~ F S ~ ( ( E X ~ O ~ C ~ ; ~ ] ) + ( ( ( ~ O ~ C ~ ; ~ ] ) R ~ ) - ( ~ O ~ ~ ~ ; ~

L0t(1*2x02x~C4;11xDC2;11*4~+~C1;21 ~ f i t ( ( ( ~ - D ~ ~ i 3 ~ ) ~ ~ ~ 4 ~ 3 l ~ 3 ~ - ~ ~ ~ MCt((E~~oDC3;33)+(20~t3;31~X10~~C3;33~X~t1;21X~~*2 MSTtMfi-MC BST+MST+DCl;2]XNFS

~ P C ( O , ~ X L . O X ~ C ~ ~ ~ ~ X O M * ~ ) + ~ C ~ ~ ~ I X ~ D R + L O X O M + ~ ~ X N F S * ~ , ~ F+(~xQRxNFS*O,~)+OM Q+NFS+OM*2 R+MST+DC~;~]XOM*~ S+MP+nC1;2]xOM*2 'WIND

(M/S),,,**,,+,,,*++*'II~O 3tnC4i23 ~ N G L . E + , , , , , , + , , , , + , + + + + + * ' I ~3 O+ ~ C 3 i l I X 1 8 0 + ~ 1

SPEED

'FTICH

'TIP5PEEP ~ 'ROTATIONAL.

~ T ~ O * * * * * * * * * , * * * * * * * ' ? 31t IO3 C 4 i 3 3 SPEED

(RPM),,,,,,,,,'?iO

3tOMX6O+(i)2

FREQUEl-lCY,,,,,,,,,,,,,,aIIO

'NATURfiL

'DfiMPING ~ f i ~ ~ ~ * , * * * + * 'K'fiMFEn MfiTURfiL. FREQUENCY,,,,,,,'ylo 'PERIOD 'LOCK ~

OF

3tNFS*0,5

* * * 3~t D * R* * * * * * ~ ? 1 0

~ ~ K ' F + ( ( ~ - D R R ~ ) * O , ~ ) X ~ ~ F S * O ,

DOMFED O S C I L A T I O N , , , , ' ? l o

3+02+DF

U ~ ~ E R , * * * * * * + * * * * * * * * * *34-0 **'?1O

'SHADOW

MOMENT

'STEADY

STOTE

ROOT

(N-M),,,,,,,,+,,,'?iO

'STEASBV

STOTE

DEFLECTION

I I

sS+~C5;13

~ s t ' ~ x n c 5 ; 2 l + ~ C1 23 ; w1+01-ns+z W2tOl+DS+2 Nl+5 D T t O l X i + i 8 0 PT1+o1x10+180 T l t o 2 T+.FT+Pl+O P+EfiT DB+DDP+M+O

MOMENT

3tMP

(N-M),'?lO (I3EG)

+

,'

10

3+MST 3tSSTXl80+01

L2:TOTi-1

L1:

5fT9B9DP9DDPpM

DE~+DB+DSIPXS~T

B ~ ~ - P + ( ( D P ~ + D E ) + ~ ) x D T D ~ P ~ + R - ( ( F x D E ~ ) + ( ~ x ~ ~ ) + S X S ~ X ( ~ ~ ) ) 5+Pl DP+DB1 SlDP+DDP1 T+T+DT FT+FT+DT +(PT

Compute Generalized Forces

P P r i n t Deflections

Subroutine BENDING C a l c u l a t e s Bending Moment D i s t r i b u t i o n

Print; Bending Monients C e n t r i f u g a l Moments Aerodynamic Moments

Assign V a r i a b l e s To M a t r i x Def

Program L i s t i n g

E.3

VINCO~Df~IV ~QEF+INCOMD;MX;MY;K;OM;F~;Q;T;MS~UP;UT;U;V;U~;V~;FA;FC;~>:;~~T';%~:;%Y~

MX+MODEs W 7 +ggqEr

O M + L ~ ~ ~ P ~ ~ ~ X ~ ~ ~ ~ + R A D I Q ~ F ~ + ( ( ~ ~ ~ ~ % ~ ) + ( ~ W ~ ~ ) X S O U T H ) Q+l+pMX T+(pMX)plr((D-2)P2)~1 M S + ( + / ( ( M X * ~ ) + M Y * ~ ) X ( ( ~ M : ~ : ) ~ M A S S ) X T X ~ ~ ~ E , E + ~ ) X F ~

uP+-lxwzfipx(i-exrnb) U T + ~ ~ ~ I ~ ~ ~ X ~ ~ ~ ~ ~ ~ ~ ~ O M X ( ~ + F : L I ~ I F I L )

N+O Fa+UP GERO U T

F

~

+

~

~

~

~

x

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~

M

~

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x

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