Definition, transformation-formulae and measurements of tipvane angles

Okt. 1987

.

flrli

and measurements

of tipvane

angles

A. Bruining

T

U

De Ift

Institute for Windenergy Delft University of Technology

A. Bruining

Delft University Press,

1987

Bibliotheek TU Delft 1111111111111111111111111111111111 C 0003828065

0756

740

9

Stevinweg 1 2628cN Delft The Netherlands

By order of:

Delft University of Technology Institute for Windenergy

Kluyverweg 1 2629HS Delft The Netherlands

Report IW-R510 October 1987

Carried out within the Dutch Development Program for Wind Energy (NOW-2); by order of the Management Office for Energy Research PEO; financed by the Ministry of Economie Àffairs.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Bruining, A.

Definition, transformation-formulae and measurements of tipvane angles / A. Bruining. - Delft : Delftse

Universitaire Pers. - 111., fig., tab.

Carried out within the Dutch Development Program for Wind Energy (NOW-2); by order of the the Management Office for Energy Research PEO. - Report IW-R510.

ISBN 90-6275-424-4

SISO 653.2 UDC 533:621.548 NUGI 834 trefw.: windmolens; aerodynamica.

copyright~by Delft University Press.

No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means, without written permission from Delft University Press.

C.ntents

Summary .1. Symbol~

2. Introduction

3.

Coordinate systems

3.1 The tipvane reference coordinate system 3.2 The local tipvane coordinate system 4. Tipvane angle types

4.1 Aerodynamic angles ·_{4.2 Construction angles}

4.2.1 Construction angles general 4.2.2 Angles Kolibrie mounting parts 4.3 Wind tunnel model angles

4.3.1 Towing tank model, R

=

0.18 m tunnel 4.3.2 Wind tunnel model with R

₌

0.36 m 4.3.3 Wind tunnel model with R

₌

0.468 m 4.4 Production angles

model

5.

_{Aerodynamic angles with blade flapping angle included} 6.Conversion between the different tipvane angle types

6.1 Conversion to the aerodynamic angles

6.1.1 Conversion from the construction angles to the aerodynamic angles, general

6.1.2 Conversion from the Kolibrie mounting part angles to the aerodynamic angles

6.1.3 Convers ion from the R

= 0.18 m wind tunnel

model angles to the aerodynamic an~les 6.1.4 Convers ion from the R

= 0.36 m wind tunnel

model angles to the aerodynamic angles

6.1.5 Conversion from the production angles to the aerodynamic angles pag.

4

5 9

10

10 10 11 11 11 11 14 15 15 15 15

19

21 23 23 23 23 24 24 24 6.2 Convers ion from the aerodynamic angles ₂₅ 6.2.1 Convers ion from the aerodynamic angles to the 25

6.2.2 Conversion from the aerodynamic angles to the 25 production angles

6.3

Conversion of aerodynamic angles from zero to

non-zero flapping angle ~

7.

Measurement of the aerodynamic tipvane angles

7.1 Definition of the measurement angles

7.2

Measuring method

7.3

Calculations of the aerodynamic tipvane angles

from the measurement angles

7.4

FACT rotor angles

8.

Wagging effect

9.

Airfoil section curving

9.1

Introduction 9.2 Tipvane curving

9.3

Mistakes in curving

9.3.1

a-effect

9.3.2

.

r-effect

9.3.3

A-effect Tables Figures 26

27

27

27

27

29 30

31

31

32

32

32

32

32

34 35

Summary

Tbis report contains the theoritical background of the different angle systems used to define the attitude of the tipvane in the

3-dimensional space.

The angle system is based on the Euler system.

Different Euler angle types were used for the various wind tunnel- , towing tank- and full-scale tipvane modeIs.

For these different angle types the definitions are given in vector notation. From these vector notations the transformation formulae could easily be derived. _{The influence of a rotor blade flapping angle}

on the tipvane angles is included.

Also the method to measure the attitude of the tipvane and

transformation from these measured angles to the Euler angle system is olltlined.

Finally some side effects are described on the angle of attack of the tipvane due to rotation, translation and curving of the tipvane.

1. Symbols b bI I b -m B c

*

c Cl Cl I I c -m

vector in the B-direction _{of the local}

tipvane coordinate system (span wise direction)

vector in the B-direction of the local tipvane coordinate system that is rotated over A and _w

e

_w

vector in the B-direction of the local tipvane coordinate system that is rotated over r _{m m m}

,e

and A or A , _a r _a

,e

_a and A .

m

x-axis of the local tipvane coordinate system

vector in the C-direction of the local tipvane coordinate system (chord wise direction)

vector in the C-direction of the local tipvane coordinate system that is rotated over

e

c

vector in the C-direction of the local tipvane coordinate system that is rotated over A and r

a a

vector in the C-direction of the local tipvane coordinate system that is rotated over r and

e

p p

vector in the C-direction of the local tipvane coordinate system th at is rotated over r , _{m m m}

e

and A or A , _a r _a

,e

_a and A

c

i k n

*

n n' n" n'" N R

z-axis of the local tipvane coordinate system

uni't vector in the X-direction of the tipvane reference coordinate system

unit vector in the Y-direction of the tipvane reference coordinate system

unit vector in the Z-direction of the tipvane reference coordinate system

vector in the N-direction of the local tipvane coordinate system (normal on the tipvane surf ace )

vector in the N-direction of the local tipvane coordinate system that is rotated over

e

c

vector in the N-direction of the local tipvane coordinate system that is rotated over A and r

a a

vector in the N-direction of the local tipvane coordinate system that is rotated over A and

e

w w

vector in the N-direction of the local tipvane coordinate system th at is rotated over 'Y _p and

e

_p

Y-axis of the loc al tipvane coordinate system

U vb hor vb vert vC hor vc _vert X y Z AVC h or

Ae

undisturbed wind velocity

_(mis)

tipvane measuremen_{t angles defined in}

section 7.1 _(degrees)

tipvane measurem_{ent angle defined in}

section 7.1 _(degrees)

tipvane measurement angle defined in

section 7.1 _(degrees)

tipvane measurement angle defined in

section 7.1 _(degrees)

x-axis of the tipvane reference coordinate system

y-axis of the tipvane reference coordinate system

z-axis of the tipvane reference coordinate system

flapping angle

tilt angle of the tipvane

displacement of the tipvane due to the wagging effect

difference in the tipvane measurement angle vC

h I denotes accuracy, see

or section

_7.3

change of the incidence angle

_e

_,

see fig. 17 (degrees) (degrees) (m) (degrees) (degrees)

a

a

'

correction A Q Indices a c .''- :; k m w

incidence angle of the tipvane (degrees) , ' AQ 180 : '" . .. '

(= -_R

* - )

_n correction of the incidence _'. angle of the tipvane due to shifting of

the tipvane (degrees)

sweep angle of the tipyane (degrees)

tipvane offset angle, see fig. 16

angular speed of th~ rotor (.radjs)

,,".

indicates the aerod~amic ang;t~s

indicates the construction,ang:Les

indicates the Kolibrie,mountin:g partangIes

indicates the wind tunnel model angles R

=

0.36 m and R

=

0.468 m

indicates the (towing tank) model angles

2. Introduction

The position of the tipvane in a coordinate system is fixed by three

angles: the sweep. the tilt and the incidence angle.

'rhe system used to define the angles is the Euler system. In the Euler

pefinition the coordinate system is rotated over the first angle. The

se~ond _{angleis defined inthis rotated coordinate system. The third} angle is then defined in a coordinate system that is rotated over the

fiFSt two angles.

The sequence in which the angles are introduced is very important. For

thè same position of the tipvane the values of the three Euler angles are different if the sequence of rotation is different.

The choice of the systems depends on the specific purpose. For

example. for aerodynamic calculation a different choice is convenient than for the constI.'uction of models.

Different Euler tipvane angle systems that have been used for tipvane

. _{measureilients are described in this report. Also the relations between}

3.

Coordinate systems

All coordinate systems are Cartesian and right hand orientated. 3.1 The tipvane reference coordinate system

The definition of the reference coordinate system is given in fig. 1. The axis of this Cartesian coordinate system are X, Y, Z with the unit vectors

!,J

and~. ( ~ =

!

x

J )

The origin of the X, Y, Z reference system is in the middle of the mid span chord of the tipvane.

The X-axis is parallel with the rotor shaft and positive in the down wind direction. The Y~axis is radially orientated and positive in the outward direction.

3.2 The local tipvane coordinate system

The definition of the local tipvane coordinate system is given in fig.

2.

This is together with the tipvane rotated over the three Euler angles. The axis are B, N and C with the unit vectors Q, g, .and~. (~= b x, ~)

The origin of the local system is the same as the origin of the tipvane reference coordinate system.

When the three tipvane angles are zero, i.e.: A = 0

r

=

0

e

= 0

then the local tipvane coordinate system coincides with the tipvane reference coordinate system.

The B-axis is in span wise direction and positive in down wind direction.

The N-axis as perpendicular to the B and C axis and positive in the outward direction (over pressure side of the tipvane).

The C-axis is in chord wise direction, positive towards the trailing edge.

4.

Tipvane angle types

Fig.

3

_{gives the positive directions of the}

₃

_{tipvane angles}

A

(sweep), r (tilt) and

_e

(incidence).

The positive directions of A and r _{are in agreement with a right hand}

orientated coordinate system. A positive increment of

e

gives a

positive increment of a, the angle of attack of the tipvane.

Table 1 gives an overview of the different angles types. (pag.

34)

4.1

The aerodynamic angles

The sequence of rotation for the 3 _{Euler angles is in the case of the}

"aerodynamic" angles:

1. A , _{i.e. rotation around the N-axis of the local system}

a

2. r , _{i.e. rotation around the C-axis of the local system}

a

3.

e

_a

,

_{i.e. rotation around the B-axis of the local system.}

The last rotation is

e

_a

.

_{This has the advantage that a change in}

_e

a corresponds with the same change in a.

For all the aerodynamic calculations this set of aerodynamic tipvane angles is used.

Af ter the rotation over A _a and r _a the components of the vectors Q, ~'

and~' _{(see fig. 4) are, expressed in the}

_!'

j and ~ _{vectors of the}

tipvane reference coordinate system:

i cos

r

cos A a a b

₌

j sin

r

_a k

-

cos

r

sin A a a i

-

sin

r

cos A a a n'

₌

_.i

cos

r

_a k sin

r

sin A a a and

i sin A

a c'

₌

j 0

k cos A

a

The last rotation is the rotation ij _{about the work line of}

a b in the

!! I , c' plane (see fig. 5) •

The vee tors n and c expressed in n ' _-

_,

c' become: n

₌

n ' cos ij _{c' sin} ij

a a

c

₌

n ' sin ij _{+ c' cos} ij

.

a a ,-.

And expressed in

i,

_i

and k:

i

-

sin 'Y cos A cos ij

-

_{sin A sin} ij

a a a a a

n

₌

_i

cos 'Y cos ij

a a

k sin 'Y sin A cos ij

-

_cos A sin ij

a a a a a

i

-

sin 'Y cos A sin ij _{+ sin A cos} ij

a a a a a

c

_{= i}

cos 'Y sin ij

a a

k sin 1 sin A sin

e

+ cos A cos

e

a a a a a

4.2 Construction angles

4.2.1 Construction angles general

The only difference with the aerodynamic angles is the different sequence of rotation of the Euler angles. The sequence of rotation of the

3

Euler angles is:

1. ij

c' rotation around the B-axis of the local system 2. A

c' rotation around the N-axis of the local system

3.

r

_c' rotation around the C-axis of the local system.

The construction angles were used during the manufacturing process of the mounting parts of the FACT-rotor.

The first rotation is the

e

rotation about the

!

vector: in the

J.

~

*c *

plane. This gives the

Q.

n • c vector. Vector b coincides with vector i. See fig. 6.

*

*

Th~ vectors n and c are:

*

n

₌

_J

cos

e

k sin

e

-*

c c c

₌

_J

sin

e

_c + k cos

e

_c

*

Af ter the rotation

e

the next rotation A is carried out in the ~. n

*

c c

and c system. The last rotation is r . See fig.

7.

c

*

The vectors

Q.

n and ~ become. expressed in

!.

g and c

*

(comparable with

Q.

n' and c' in the aerodynamic tipvane angle system):

i cos A cos

r

-*

c c b

₌

_-

n sin

r

*

c c

-

sin A cos r c c i

-

sin

r

cos A

-*

c c n

₌

n

_-

cos

r

*

c c sin

r

sin A c c and i sin A

-*

c c = _-n 0

*

c cos A c

Expressed in the

!.

J

and k components of the reference coordinate system the vectors

Q.

n and c of the local tipvane coordinate systems

are:

i cos A cos r

c c

b =

_J

cos

e

_c sin

r

_c

-

sin

e

_c sin A _c cos

r

_c

k

-

sin

e

sin

r

-

cos

e

sin A cos

r

c c c c c

i

-

sin r cos A

n

_{= i}

cos

e

cos 'Y + sin

e

sin 'Y . _{sin /Ic} ," '/

C C C c

k

-

sin

e

cos 'Y + cos

e

sin 'Y sin A

c c c c c and i sin A o -e ;.i c

_{= i}

sin

e

_c cos A c k cos

e

_c cos A c

4.2.2 Angles of the Kolibrie mounting parts

The sequence of the Euler angles is the same as for the copstruction angles:

1.

e

_k _.'

2. _Ak .'.

3.

'Yk

Due to amistake the rotation 'Yk is not introduced around the swept chord vector C but around thenon swept ~ (i.e. around an axis which

does not include rotation over the angle A k).

This gives for the vee tors

Q.

g and ~. including the rotation over the non swept vector c

i _{cos Ak cos 'Yk}

b

_{= i}

cos

e

_{k cos Ak sin 'Y - sin k}

e

_{k sin Ak} k

-

sin

e

k cos Ak sin 'Y - cos k

e

k sin Ak i

-

_{sin 'Yk}

n

_{= i}

cos

e

_{k cos 'Yk} .. i

k

-

sin

e

k cos 'Yk i _{sin Ak cos 'Yk}

c

_{= i}

cos

e

_{k sin Ak sin} 'Yk + sin

e

_k cos Ak

k

-

sin

e

4.3

Wind tunnel model angles

4.3

.

1

Towing tank model

e

=

0.18 m tunnel model The$e angles are indexed with w.

The sequence of rotation of the 3 Euler angles is:

1. A w 2.

e

w

3.

r w

Thi$ set of angles is used for the towing tank / wind tunnel model. The difference with the aerodynamic angles is that the sequence of Y

and

e

is interchanged. Af ter the setting of the first two angles the components of the vectors

2",

g", ~ (see fig.

8)

are, when expressed in the

!,

j, k vectors of the tipvane referençe coordinate system:

i cos A w b' ,

₌

_J.

₀ k - sin A w i - sin

e

sin A w w n' ,

₌

_J.

_cos

_e

w k - sin

e

cos A w w i cos

e

sin A w w c

=

j sin

e

w k cos

e

cos A w w

The last rotation is the rotation over the angle Y about the c vector w

in the

2",

g" plane (see fig. 9).

The vectors b and n expressed in b" and n" become:

b

=

b" cos Y +. n" sin Y

w w

b" sin Y + n" cos Y

w w

b

₌

b cos A n ' sin

e

sin A c' cos

e

sin A

-m m a m a m

n

₌

n ' cos

e

C' sin

e

a a

c

₌

b sin A + n' sin

e

cos A + c' cos

e

cos A

.

-m m a m a m

And expressed in the

i.

.i

andk components:

i cos r cos A cos A + sin r cos A sin

e

sin A

a a m a a a m

sin A cos

e

sin A

a f;l m

b

₌

_i

sin r cos A

-

cos r sin

e

sin A

m a m a q .ffi ,

k

-

cos

..,

sin A cos A

-

sin

..,

sin A sin

e

sin A

a a m a a a m

cos A cos

e

sin A

a a m

i

-

sin r cos A cos

e

-

sin A sin

e

a a a a a

n

₌

_.i

cos r cos

e

a a

k sin r sin A cos

e

-

cos A sin

e

a a a a a

and

i cos r cos A sin A

-

sin

..,

cos A sin

e

cos A

a a m a a a m

+ sin A cos

e

cos A

a a m

c

₌

_.i

sin r sin A + cos r sin

e

cos A

-m a m a a m

k

-

cos r sin A sin A + sin '( sin A sin

e

cos A

a a m a a a m

+ cos A cos

e

cos A

a a m

A

=

0 for the R

=

0.36 m wind tunnel model. The expressions then a

become:

i cos r cos A + sin r sin

e

sin A

a m a a m

b

₌

_.i

sin r cos A

-

cos '( _sin

e

sin A

-m a m a a m

k

-

cos

e

sin A

i - sin '( cos

e

a a n

₌

_.1

cos

r

cos

e

a a k - sin

e

a and

i cos

r

sin A - sin r sin

e

cos A

a m a a m

C

=

_J

sin r sin A + cos r sin.

e

cos A

~m a m a a _m

k cos

e

cos A

a m

And written with the correct index m of the R

₌

0,36

mmodel (the Euler sequence is 1: r

m' 2:

e

m'

3:

A ): m

i cos r cos A + sin

r

sin

e

sin A

m m m m m

b

₌

_.1

sin

r

cos A - cos

r

sin

e

sin A

m m m m m m k - COS

e

sin A m m i - sin '( cos

e

m m n

₌

_.1

cos

r

_m cos

e

_m k - sin

e

_m

i cos

r

sin A - sin

r

sin

e

cos A

m m m m m

c

₌

_J

sin r sin A + cos r sin

e

cos A

-m m m m m m

k cos

e

cos A

m m

4.4

Production angles

These angles are indexed with p. The sequence of rotation of the

3

Euler angles is:

1- r

p 2. A

P

3.

e

_p

The difference with the aerodynamic angles is th at r and A are

interchanged. Accidentally this sequence was used by the production of the tipvanes for the FACT rotor.

Af ter the first two angles have been set. the components of the vee tors

_Q.

_{!!' " •}

e'"

(see fig. 11) expressed in the

i.

_.1

andk vee tors of the tipvane reference coordinate system are:

i cos A cos 1 P P b

₌

_.1

cos A _p sin 1 P k - sin A p i ·- sin 1 _p n" ,

₌

_.1

_cos 1 P k 0 i sin A cos 1 P P c" ,

₌

_.1

_{sin A sin 1} p _P k cos A P

The last rotation is 9 _p about the b in the n" , _-

.

c" , plane (see fig. 12) .

~

The vee tors n and c expressed in n" , and c'" become:

n

₌

c" , sin 9 _p + n'" cos 9 _p ... ~ <t

C

=

c" , cos

e

+ n" , sin

e

p p

And expressed in

_i.

..i

and ~

i

-

sin A _p cos 1 _p sin 9 _p

-

sin 1 _p cos

e

_p n

₌

_{..i -}

sin A _p sin 1 _p sin

e

_p + cos 1 _p cos

e

P k

-

cos A _p sin 9 _p

i sin A cos 1 cos

e

-

sin 1 sin

e

p p p p p

c

-

_..i

sin A sin 1 cos

e

+ cos 1 sin

e

p p p p _P

5.

Aerodynamic angles with blade flapping angle incluqed

The flapping angle ~ rotates the i and j vee tors of the tipvane

reference coordinate system. The k vector is only translated a little

down wind parallel to itself. See fig.

13.

The vectors in the flapped tipvane reference coordinate system are

cal led i~, j~ and ~~

'rhe components i~, j~ and ~~ of the vectors Q~, g~ and ~~ in the

flapped tipvane coordinate system X~, Y~, z~ have the same length as

the com~onents

i,

j and ~ of Q, g and ~ in the unflapped tipvane reference coordinate system X, Y, Z. See fig. 14.

The components

i,

j and ~ of the flapped vee tors Q~, g~ and ~~ in the

unflapped tipvane reference coordinate system are given by the

relation~:

i

=

...., i~ cos ~ + j~ sin ~

j

= -

i~ sin ~ + j~ cos ~

k

=

Vector b has the components, expressed in the aerodynamic tipvane angles: i cos Y cos A a a b

₌

j sin y a k - cos y _{sin A} a a

and with the flapping angle ~:

i~ cos Y cos A i cos Y cos A cos ~ + sin Y

a~ a~ a a a

.!2~

=

_j~ sin Y

=

j - cos Y cos A sin _~ + sin Y

a~ a a a

~~

- cos Y sin A k - cos y _{sin A}

a~ a~ a a

sin cos

~

Similarly for g~ _{and ~~:}

i (

-

_sin A _a sin

e

_a

-

sin Y cos A cos

e )

cos~

a a a

+ (cos Y cos

e )

sin~

a a

g~ =

..i ..

- sin A sin

e

- sin Y cos A cos

e )

sin~

a a a a a

+ ( cos Y cos

e )

cos~

a a

k ~. cos A~ sin

e

.

.

+ sin Y sin A cos

e

a a a a

i sin A cos

e

-

sin Y cos A sin

e )

cos~

a a a a a

+ ( cos Y sin

e )

. sin~

a a

~~ =

_..i

-

( sin A cos

e

-

sin Y cos A sin

e )

sin~

a a a a a

+ ( cos "'( . sin

e )

_cos~

a a

k cos A cos

e

+ sin Y sin 'A sin

e

-6.

Convers~on between the different tipvane angle types

The expressions for the conversion between the different tipvane angle types can be obtained by comparing the

!.

J

and ~ components of the

g.

n and c vectors. Thes~ components for the different angle types are derived in section

4.

In this way

9

equations are obtained with

3

unknown angles. Of these 6 equations are redundant. because the axis are orthogonal. Therefore the

3

most convenient equations are used.

6.1

Conversion to the aerodynamic angles

6.1.1 Conversion from the construction angles to the aerodynamic angles (general)

The j component of vector

g

gives:

sin 1 = sin 1 cos ij _{- sin} ij _{sin A cos 1}

a c c c c c

j component of c:

sin ij _{= sin} ij _{cos A / cos 1}

a c c a

i component of b:

cos A _a

=

cos A cos 1 / cos 1 _{c c a}

6.1.2 Conversion from the Kolibrie mounting part angles to the aerodynamic angles

The

1

component of vector b gives:

k component of b:

sin A

=

a

6.1

.

3

Conversion from the R =

0.18

m wind tunnel model angles to the aerodynamic angles

The j component of vector b gives sin Y

=

sin Y cos

e

a w w k component of b:

sin A = ( cos r sin A + sin

e

cos A sin Y ) / cos r

a w w w w w a

j component of c:

sin

e

= sin

e /

cos r

a w

a

6.1.4

Conversion from the R

=

0.36

m wind tunnel model angles to the aerodynamic angles

The j component of vector b gives:

sin 1

₌

sin 1 cos A - cos 1 sin

e

sin A

a m m m m m

j component of n:

cos

e

₌

cos 1 cos

e

/ cos Y

a m m a

k component of b:

6.1.5

Conversion from the production angles to the aerodynamic angles The

i

component of vector b gives:

sin r

₌

cos A sin r

a p p

k component of b:

sin A

=

sin A / cos r

a p a

i

component of c: sin ij =

a sin A sin p r p cos ij p + cos r p sin ij p ) / cos r a

6.2

Convers ion from the aerodynamic angles

6.2.1

Convers ion from the aerodynamic angles .to the construction

angles

The i component of vector c gives:

sin A

=

sin A cos ij _{- sin r cos A sin} ij

c a a a a a

i component of b:

cos r

₌

cos r cos A / cos A

c a a c

i

component of c:

sin ij

₌

_{cos r sin} ij _{/ cos A}

c a a c

6.2.2

Convers ion from the aerod;ynamic angles to the 2roduction angles

The k component of vector b gives:

sin A

=

cos r sin A

1

component or'b: I"

. i.: ... .. ,;' I

sin 1 _p

-

sin 1

_/

'cos A'

a p !: f k component of c: . .1: '\ cos e

=

p cos A a cos e +a 'sin 1 a sin A a sin e ) / cos a A p

6.3

Convers ion from the unflapped aerodYrlamit ariglés to the

aerodynamic angles including aflapping angle ~

. ' \

Comparison of the expressions . for' ~, n· and c wi th '~~ ~ !!~ arid ~~. gi "es·

the conversion formulaè.

The

1

component of vector b gives:

sin 1

-

s{n 1 . cos' .~

_-

_{cos 1 cos A sih~'}

a~ a a a

. ~: k component of b:

sin A

=

sin A cos 1

_/

cos y

a~ a a a~

j, component. of c: ..•. . " '

.' , .· •. i.··}

sin e

-

( ·sin· e cos ,1 cos~

-

sin A cos e sin~ .

a~ a . a a 'a

-+ sin"r _a cos A sin ij sin~ )

_/

cos 1

a a _{a~ ,}

.' ~.

• t ::"

';':,

"l -, These 3 expressions give for _~

₌

0 the unflapped angles: I .. i'" \.

1

=

1 a~ a A

=

A a~ a e

₌

e

.

..

':: .. a~ a

.

7.

Measurement of the aerodxnamic tipvane angles

The '~erodynamic ang1es cannot be measured direct1y ~n a convenient

w~,becau~e the t~pvane reference coordinate system is not fixed with

respe.ct to the ground. It rotates with the rotor.

For this purpose an indirect method was deve1oped. Via a simple

conv~rsion the aerodynamic angles can be calculated.

1.1

Definidon of the meas\,lrement angles

The 1ine frQm the

~ ~hord

point of the tipvane at the mounting part

toward .. the rotor shaft is positioned exactly hori~ontal. The angle

between the vert~cal vector y and the span vector

Q

is then called

vb

h or . The anll'le between v and the chord vector c is vC""r - . - h or . The same

ang~es

are measured once again with the 1ine from the

~

chord point of tipvane at the mounting part now positioned exact1y vertical. These

.' angiea at'e denoted vb

vert and vCvert respectivel~. See fig. 15 .

. vb_{h or}._. ' vb _ver t' vC_{h or} ,vc _ver t are called ··the measurement ang1es. _{. .}

7.2

Meaaurement m~thod

lngeneral it is impossible to position exactly the1ine between the

~

.ehord point of the tipvane and the shaft hori~ontal1y end vertically, because the tipvane is a 1itUe bit shifted forwards. Therefore, the

trailing ~dge of the rotor b1ade or tube are used for the positioning.

The angle that is introduced by this offset is called ~. See fig. 16.

vb

h ,vch ,vc t and vc t of the tipvanes of

. or or ver ver the. experimental

wind tU;t"bine in Hoek van Holland are measured with the assistanee of a

woo4en clam~ and a plumb 1ine.

7.3

Calculations of the aerodynamic tipvane ang1es from the

measure!ent ang1es

When a~l the vectors are expressed in the tipvane reference coordinate

sys~èm the ang1e between two vectors can be ca1culated by the dot

Vector b and c are expressed in A _{a' a} 1 and a

.

a

.

i cos 1 cos A a a b

₌

_.i

sin 1 a k - cos 1 sin A a a

i sin A cos 9 - sin 1 cos A sin 9

a a a a a

c

₌

_J.

cos 1 sin 9

a a

k cos A cos 9 + sin 1 sin A sin 9

a a a a a

In the horizontal position of the blade, the vector v is: v

₌

_.i

sin tjJ - ~ cos tjJ

{v.b)h _or

=

_{lvi Ibl cos (vbh} )

₌

sin tjJ sin 1

or a

+ cos 1jJ cos

(y.c)hor

=

_{,lvi Icl cos} , (vc_hor)

=

sin 1jJ cos 1 sin 9

a a

- cos 1jJ (cos A cos 9 + sin 1 sin

a a a

In the vertical position of the blade, the vector v is: v

₌

_J.

cos 1jJ +

and the angles:

(v.b) _ver t

=

(v.c) _ver t

=

k

lvi sin 1jJ

Ibl cos (vb ver t)

=

cos tjJ sin 1 a

- sin 1jJ cos '1\ Y sin A a a sin 9 ) a a 1 sin A a a sin lvi Icl cos (vcvert )

=

cos tjJ cos la sin aa

+ sin tjJ (cos A cos a + sin 1 sin A

a a a a 9 ) a

There are

4

relations with

4

unknowns: ,I. _'1', A _{a' a} 1 and a • _a

In the case that tjJ can be measured or calculated from the geometry 1 relation can be used for checking the measurement of the angles. If 1jJ

is non-zero an simple iterative procedure is used to calculate A , 1 a a

and

e .

The angle (v.C)h is used for checking by comparing the

a or

measured value of vC

h or with the calculated value of vCh or from A , Y a a

and

e

($ is known). The difference is denoted with AVC

h .

a or

7.4

FACT rotor angles

The geometry of the FACT-rotor blades implies a value of $ = O. This

gives very simple relations.

A

,r

and

e

are calculated by:

a a a

sin Y

=

cos (vb

vert)

a

sin 9

=

cos (vc_vert)

_/

cos r

a a

sin A

=

cos (vb

h ) / cos Y

a or a

The accuracy of the measured and calculated angles is in the order of

8.

Wagging effect

When the mounting position of the tipvane shifts forward or backward with no rotation the effective incidence angle changes, because the oncoming flow is circular. See fig. 17. If the tipvane shifts forward or backward the angle of attack decreases or increases respèctively.

The shifting of the tipvane occurs ,as an unavoidable by productwhen

e

is changed by a rotation around a centre that is not located on the chord of the tipvane but which is somewhat closer to the rotov axis. '

This gives a reduction of the

e

change. If the rotation point'of

e

should coincide with the rotor axis line, there would be no effective change of 9 at all.

If the displacement, generated by the rotation of the tipvane, is AQ

than the correction on 9 ,due to shifting of the tipvane is:

9 _{correc l.on} t" Af/. _R

*

180 11 (degrees)

with Afl.: amount of shift (backw.ards

=

positive).

a

and , are the same kind of angles. But the difference is correction

that the offset angle , is used in section

7.3

for the effect of the displacement of the tipvane by the measuring method for the

9. Airfoil section curving

9.1 Introduction

The airfoil section of the tipvane should be adapted to the rotating situation.

The oncoming flow to the tipvane is circular. If the airfoil section has the same curvature as the flow, the situation will be comparable with the original airfoil section in the parallel flow.

Until January '84 all tipvane airfoil sections of the wind tunnel models and Kolibrie tipvane airfoils were corrected by curving the chord line. Af ter January '84 the correction was made by curving the meanline.

A correction of the incidence angle 9 is necessary - and through that also the angle of attack a is changed - ifthe mounting point of the tipvane does not coincide with the point which is kept fixed during the curving correction. The

~

chord position is used as the fixed point, when curving the airfoil section. See fig. 18a and fig. 18b. The correction of the incidence angle is given by:

9correction

=

AQ R

*

180 n (degrees)

where AQ is the distanee between the mounting point and the fixed point during curving.

If the mounting point is between the leading edg~ and the fixed point during curving, 9 t' is positive and the angle of attack a

correc 10n increases.

The fixed point during curving may be defined as the point where the chord, and af ter curving the tangent to the chord, is perpendicular to the radius.

The correction is of the same kind as is given by the wagging effect but ft is introduced in a slightly different way.

tipvane angles sequence of rotation description in

1

·2

₃

4

section aerodynamic angles A Y

e

4.1

a a a , construction angles

e

A Y

4.2.1

c c c Kolibrie angles 1) .-

e

4.2.2

k Ak Yk

towing tank model A

e

Y

4.3.1

w w w R

=

18

cm. wind tunnel R

₌

36

cm. Y

e

A

4.3.2

m m m model angles wind tunnel R

₌

46.8

cm. A Y

e

A

4.3.3

a a a m model angles production angles Y A

e

4.4

p p p

1) not Euler angles.

y

I

~

_.t;kL>

.

/

.~.

-1---' "-...

~ . ~

I

I

I

I

I

OR

..,;;?:::::=:::

N

I

I

n

~

rotor shaft ~

Fig. 2: Definition of the local tipvane coordinate system (position before rotation over the Euler-angles).

IB sweep angle·

incidence angle

Fig. 3: Definition of the 3 tipvane angles

A,

y

and

e

with their

1

cos 'Ia

1

cos 'Ia cos "a

\

!i

cos "a

Fig. 4: The~, ~', ~' vectors af ter rotation over

A

and y for a a the aerodynamic tipvane angles.

n'

~'cos

e

_a

Fig. 5: Rotation

e

in the ~I, Cl plane for the aerodynamic a

1

cos Sc

Fig. 6: Rotation of the first angle

e

of the construction tipvane

c angles.

~

f

•

Ac

~ cos

Fig. 7: The vectors of the local tipvane coordinate system in the

~, ~*, c* coordinate system (~, n*, ~* is rotated over

6

c).

Fig. 8: The vectors ~", E,.", ~ af ter rotation over Aw and 8

w for the wind tunnel model angles.

Fig. 9: Rotation y in the ~", bil plane for the wind tunnel

w

.oICO$ 9 a

... '

fm

Fig. 10: Rotation

e

and the fourth rotation

A

in the ~, ~' and c'

a ,", "'," m

F~g. 11: The vectors of the local tipvane coordinate system b, ~III,

Cl II af ter rotation over y and A for the production angles.

~cos _Sp

Fig. 12' Rota

tion

e

in tho. !'-" " c'" _{plano for tho}

production p ang1es .

\

\

I I \

y

Zj3

rotor shaft axis

, - /

...

I

~

/

~

I

~

I

I

Fig. 13: The change of the tipvane reference coordinate system due to the flapping angle

S.

v

rototiono/ direction

\

rotor

Clxis ttne

( b

•

v

Fig. 15: The horizontal and vertical position of the tipvane for measuring the tipvane angles.

v

=

J

sin l\J -k cos l\J

-

-

-rotiono

'

l direction

.

~

',r" . .

rotor axis \ine " ~ , wind di rection

k

NI

k

I

•

c

'!.=l

(osl\J+~sin 4J~

dirE~ctions

of loeal

~

(2)

flow at

ti pvane

Fig. 17: The wagging effect of the tipvane if the

e

"hinge"

curving point

=

mounting point

I

curv.d meen li n.

I

I

I

I

I

"I

I

Q

,---+-I

I

Fig. 18a: Uncurved and curved airfoil section at ~ chord point

mounting point

r

I .

o

I

.

Allo

MI

o

I

'

.

.

,

. 9correction

11

o

I

.

I

curved meon ti ne

Q7

_

o

_._._f_._.

curved meon lines locol flow curv.

y

.:" .

I

I

I

o

I

I

I

,--k

Q

.

.

I

down wind tipvone

oirfoil section '" '. "

points

upwind tipvone oirfoil section

-'

.

_,'-

.

Fig. 19: Increase of the local effective camber of the down wind tipvane airfoil section due to the y-effect.

curving point

local flow curve

I

I

I

I

I

I

I

Q.--h

I

1-

·

.

fiq. 20: Oecrease of the local anqle

Ot

attack of the down wind tipvane airfoil due to the A-effect.