Page 435 - Wind Energy Handbook

BLADES 409

Substituting V for U 1 (1 a) for brevity, the in-plane damping coefficient is derived
as follows:

@F Y 1 @W @C l @C d
^ c c Y (r) ¼ ¼ rc [ C l V þ C d Ùr] þ W V þ Ùr C d (7:16)
2
y
y
y
y
@ _ y @ _ y @ _ y @ _ y
Noting that
@W Ùr
¼
@ _ y W
y
and

@C l @C l @Æ @C l @ö @C l V
¼ ¼ ¼
@ _ y @Æ @ _ y @Æ @ _ y @Æ W 2
y
y
y
this equation becomes:

2 2
2
Ùr V @C l 2Ù r þ V 2 @C d
1
^ c c Y (r) ¼ rc VC l þ þ C d V (7:17)
2 W Ùr @Æ Ùr @Æ
The ‘cross’ coefficients and the out-of-plane damping coefficient and are derived by
a similar procedure:

2 2
Ùr Ù r þ 2V 2 @C l @C d
1
^ c c YX (r) ¼ rc C l V þ VC d þ Ùr (7:18)
2 W Ùr @Æ @Æ

2 2
2
Ùr 2Ù r þ V 2 @C l V @C d
1
^ c c XY (r) ¼ rc þ C l V þ VC d (7:19)
2 W Ùr @Æ Ùr @Æ

2 2
Ùr @C l Ù r þ 2V 2 @C d
1
^ c c x (r) ¼ rc þVC l þ Ùr þ C d þ V (7:20)
2 W @Æ Ùr @Æ
It is apparent from inspection of the expressions for the two damping coefficients,
c
^ c c Y and ^ c X , that the choice of an aerofoil with a gentler stall – i.e., with a smaller lift
curve slope after stall onset – will increase the damping coefficient in both cases.
Note that the modal damping coefficient is dominated by the damping per unit
length over the outboard part of the blade, so it is important to select an aerofoil
with a gentle stall in this area only.
The choice of aerofoil also affects performance, so there is merit in expressing the
damping coefficients in terms of the power output in order to investigate possible
trade-offs between them. It transpires that the damping and ‘cross’ coefficients per
unit length can be formulated quite simply in terms of the power output per unit
length of blade, P9(r, V) ¼ Ùr( F Y ), and the blade thrust per unit length, F X ,as
follows: