Page 274 - Wind Energy Handbook

248 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

ð ð ð
2 R R 1
2 2
o
ó 2 M ¼ 1 2 rÙ dC l 0 0 0 S (r 1 , r 2 , n)dn c(r 1 )c(r 2 )r r dr 1 dr 2 (5:48)
u
1 2
dÆ
From this, it can be deduced that the power spectrum of the blade root bending
moment is
ð ð
2 R R
S M (n) ¼ 1 rÙ dC l o 2 2 (5:49)
u
2 dÆ 0 0 S (r 1 , r 2 , n)c(r 1 )c(r 2 )r r dr 1 dr 2
1 2
o
The derivation of the rotationally sampled cross spectrum, S (r 1 , r 2 , n), exactly
u
parallels the derivation of the rotationally sampled single point spectrum given
o
above, with the cross correlation function k (r 1 , r 2 , ô) between the longitudinal
u
wind fluctuations at points at radii r 1 and r 2 on the rotating blade replacing the
autocorrelation function in step 2. Here the expression for the separation distance,
s, given in Equation (5.36), is replaced by
2 2
2
2
2
s ¼ U ô þ r þ r 2r 1 r 2 cos Ùô (5:50)
1 2
The expression for the cross-correlation function thus becomes:
!1"
2ó 2 s=2 3 s s
o
k (r 1 , r 2 , ô) ¼ u þ
u 1 x K 1=3 x x
ˆ( ) 1:34L 1:34L 2(1:34L )
3 u u u
2 2 #
s r þ r 2r 1 r 2 cos Ùô
2
1
K 2=3 x 2 (5:51)
1:34L s
u
with s defined by Equation (5.50).
o
The form of the resulting normalized cross-correlation function, r (r 1 , r 2 , ô) ¼
u
2
o
k (r 1 , r 2 , ô)=ó , is illustrated in Figure 5.18 for the case considered in Example 5.2,
u u
taking r 1 ¼ 10 m and r 2 ¼ 20 m. In Figure 5.21, the rotationally sampled cross
spectrum for this case is compared with the rotationally sampled single point
spectra or ‘autospectra’ at these radii. It can be seen that the form of the cross
spectrum curve is similar to that of the autospectra with a pronounced peak at the
rotational frequency roughly midway between the peaks of the two autospectra. At
higher frequencies, however, the cross spectrum falls away much more rapidly.
The evaluation of the the power spectrum of the blade root bending moment is,
in practice, carried out using summations to approximate to the integrals in
Equation (5.49), as follows:

2 X X
o
2 2
S M (n) ¼ 1 rÙ dC l S (r j , r k , n)c(r j )c(r k )r r (˜r) 2 (5:52)
2 dÆ u j k
j k