Page 344 - Wind Energy Handbook

318 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES
ð
1
k u (r, r9, ô) ¼ 1 S uu (r, r9, n) exp(i2ðnô)dn
2
1
giving
" ð # ð
1 T 1
k u (r, r9,0) ¼ u(r, t)u(r9, t)dt ¼ S uu (r, r9, n)dn for ô ¼ 0 (A5:15)
T 0 0
Hence

ð ð ð
R R 1
2
ó Q1 ¼ (rUC f ) 2 S uu (r, r9, n)dn c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9 (A5:16)
0 0 0
N
The normalized cross spectrum is defined as S (r, r9, n) ¼ S uu (r, r9, n)=S u (n), and
uu
like S uu (r, r9, n), is in general a complex quantity, because of phase differences
between the wind speed fluctuations at different heights. As only in-phase wind
speed fluctuations will affect the response, we consider only the real part of the
normalized cross spectrum, known as the normalized co-spectrum, and denoted by
N
ł (r, r9, n). Substituting in Equation (A5.16), we obtain:
uu
ð ð ð
R R 1
N
2
ó Q1 ¼ (rUC f ) 2 S u (n)ł (r, r9, n)dn c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9 (A5:17)
uu
0 0 0
From this, it can be deduced that the power spectrum of the generalized load with
respect to the first mode is
ð ð
R R
N
S Q1 (n) ¼ (rUC f ) 2 S u (n)ł (r, r9, n)c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9 (A5:18)
uu
0 0
Note that the power spectrum for the along wind turbulence shows some variation
with height, and so should strictly be written S u (n, z) instead of S u (n). However,
the variation along the length of a vertical blade is small, and is ignored here.
As for the initial case when wind loadings along the blade were assumed to be
perfectly correlated, the power spectrum for first mode tip displacement is equal to
the product of the power spectrum of the generalized load (with respect to the first
mode) and the square of the frequency response function, i.e.,

S 1x (n) ¼ S Q1 (n)jH 1 (n)j 2 (A5:19)

As before, S Q1 (n) is assumed constant over the narrow band of frequencies
straddling the resonant frequency, and the standard deviation of resonant tip
response becomes:
ð 1 2
2
ó 2 ¼ S Q1 (n 1 ) jH 1 (n)j dn ¼ S Q1 (n 1 ) ð n 1 (A5:20)
x1 2
0 2ä k 1