Page 248 - Wind Energy Handbook

222 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð 2
ó M ó u 2
¼ 2 K SMB þ R u (n 1 )K Sx (n 1 )º M1 (5:10)
M U 2ä
The design extreme root bending moment is typically calculated as that due to the
50 year return, 10 min mean wind speed plus the number of standard deviations of
the root bending moment fluctuations corresponding to the likely peak excursion in
a 10 min period. Thus

M max ¼ M þ g:ó M (5:11)

where g is known as the peak factor, and depends on the number of cycles of root
bending moment fluctuations in 10 min, according to the formula
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:577
g ¼ 2 ln(600í) þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5:12)
2 ln(600ı)

Here, í, is the mean zero-upcrossing frequency (i.e., the number of times per second
the moment fluctuation changes from negative to positive) of the root bending
moment fluctuations, which will be intermediate between that of the quasistatic
wind loading and the blade natural frequency, n 1 (see Section A5.7 of the Appen-
dix). (Note that, as g varies relatively slowly with frequency, it is a reasonable
approximation to set g at an upper limit of 3.9, which corresponds to a frequency of
about 1.9 Hz, as is suggested in DS 472.)
Substituting Equation (5.10) into Equation (5.11) yields
2 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2
ó M 4 ó u ð 2 5
M max ¼ M 1 þ g ¼ M 1 þ g 2 K SMB þ R u (n 1 )K Sx (n 1 )º M1 (5:13)
M U 2ä
The expression in square brackets corresponds to the formula for the impact factor
j in Annex B of DS 472:
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó u
j ¼ 1 þ 2g k b þ k r (5:14)
U
It is often necessary to express the maximum moment in terms of the quasistatic
moment due to the 50 year return gust speed, U e50 . In order to do this, we equate
the latter quantity to the quasistatic component of Equation (5.13), obtaining

ð
R p ffiffiffiffiffiffiffiffiffiffiffi
1
C f : rU 2 c(r)r dr ¼ M 1 þ g 0 :2 ó u K SMB (5:15)
2 e50
0 U
Here the peak factor, g takes a lower value, g 0 , corresponding to the lower
frequency of the quasistatic root bending moment fluctuations. Equation (5.15) can
then be combined with Equation (5.13) to yield