Page 350 - Wind Energy Handbook

324 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

A5.7 Peak Response

One of the key parameters required in blade design is the extreme value of the out-
of-plane bending moment. The 50 year return moment is defined as the expected
maximum moment occurring during the mean wind averaging period when the
mean takes the 50 year return value. Treating the moment as a Gaussian process,
Davenport (1964) has shown that the expected value of the maximum departure
from the mean is the standard deviation multiplied by the peak factor, g, where

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5772
g ¼ 2ln(íT) þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A5:42)
2ln(íT)

In this formula, í is the mean zero-upcrossing frequency of the root moment
fluctuations, and T is the mean wind speed averaging period. The variance of the
root bending moment is, in the same way as for the tip displacement, equal to the
sum of the variances of the background and resonant root bending moment
responses, i.e.,

ó 2 ¼ ó 2 þ ó 2 (A5:43)
M MB M1
Hence, from Equations (A5.39) and (A5.34), we obtain

ó 2 ð 2
ó 2 M ¼ ó 2 MB þ ó 2 M1 ¼ 4M 2 u K SMB þ R u (n 1 )K Sx (n 1 )º 2 M1 (A5:44)
U 2 2ä

Thus

0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
@ ó u ð 2 2 A
M max ¼ M þ gó M ¼ M 1 þ 2g K SMB þ R u (n 1 )K Sx (n 1 )º M1 (A5:45)
U 2ä

The mean zero up-crossing frequency of the root moment fluctuations, í, is defined
as

v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u ð 1
u 2
u n S M (n)dn
u
í ¼ u 0 ð 1 (A5:46)
t
S M (n)dn
0
where S M (n) is the power spectrum of the root moment fluctuations. If we separate
the power spectrum of the background response from the first mode resonant
response at frequency n 1 , then the above expression can be written