Page 436 - Wind Energy Handbook

410 COMPONENT DESIGN

2 V @P9 1 @P9
^ c c Y ¼ P9 þ ¼ 2P9 þ V (7:21)
2 2
2 2
2 2
Ù r Ù r @V Ù r @V
@F Y @F Y 1 @ 1 @P9
^ c c XY ¼ ¼ ¼ (ÙrF Y ) ¼ (7:22)
x
@ _ x @V Ùr @V Ùr @V

1 @F X
^ c c XY ¼ 2F X V (7:23)
Ùr @V
@F X @F X
^ c c X ¼ ¼ (7:24)
@ _ x @V
x
Equations (7.21) and (7.23) are derived from the equations
Ùr^ c Y þ V^ c YX ¼ 2F Y ¼ 2P9=Ùr
c
c
and

c
c
Ùr^ c XY þ V^ c X ¼ 2F X
which may be verified using Equations (7.17) to (7.20).
From Equation (7.21) it is clear that the damping coefficient in the in-plane
direction, ^ c Y , will always be negative when 2(P9=V) exceeds @P9=@V, and that a
c
negative power curve slope should be avoided if the size of the negative damping
is to be kept small.

Effect of blade twist

In the discussion so far, damping of vibrations in the out-of-plane and in-plane
directions only has been considered. In practice, blade twist will result in the
flapwise and edgewise vibrations taking place in directions rotated from the out-of-
plane and in-plane directions in the same sense as the blade twist, but by a lesser

amount (see Section 5.8.1). If we define x - and y -axes in the directions of the

flapwise and edgewise displacements, each making an angle of Ł to the x- and y-
axes respectively, as shown in Figure 7.16(c), then the edgewise damping coefficient
per unit length is given by:

2

2
^ c c ¼ ^ c Y cos Ł (^ c YX þ ^ c XY ) sin Ł cos Ł þ ^ c X sin Ł (7:25)
c
c
c
c
Y
Substitution of Equations (7.21)–(7.24) in Equation (7.25) yields:

2

^ c c ¼ cos Ł 1 2P9 þ V @P9 þ cos Ł sin Ł 1 @P9 þ 2F X V @F X
Y 2 2
Ù r @V Ùr @V @V

2
þ sin Ł @F X (7:26)
@V