Page 343 - Wind Energy Handbook

EFFECT OF ACROSS-WIND TURBULENCE DISTRIBUTION 317

" ð R # 2 2
ó 2 ¼ C f rU ì 1 (r)c(r)dr S u (n 1 ) ð n 1 (A5:10)
x1 2
0 2ä k 1
For comparison, the first mode component, x 1 , of the steady response is obtained by
1
2
setting ø ¼ 0 and q 0 (r) ¼ rU C f c(r) in Equation (A5.3), yielding
2
ð
1 R
1 2
x 1 ¼ rU C f ì 1 (r)c(r)dr (A5:11)
2
k 1 0
Hence the ratio of the standard deviation of the first mode resonant response to the
first mode component of the steady response is
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð n 1 S u (n 1 ) ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó x1 ó u ó u
¼ 2 p ffiffiffiffiffiffi 2 ¼ 2 p ffiffiffiffiffiffi R u (n 1 ) (A5:12)
U 2ä ó U 2ä
x 1 u
Note that towards the upper tail of the power spectrum of along wind turbulence,
p
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
where n 1 is likely to be located, R u (n 1 ) tends to 0:1417=(nL u =U) 3.
x
A5.4 Effect of Across-wind Turbulence Distribution on
Resonant Displacement Response

In the foregoing treatment, the wind was assumed to be perfectly correlated along
the blade. The implications of removing this simplifying assumption will now be
examined.
The fluctuating load on the blade, q(r, t), becomes C f rUu(u, r)c(r) per unit length,
and the generalized fluctuating load with respect to the first mode becomes
ð R ð R
Q 1 (t) ¼ ì 1 (r)q(r, t)dr ¼ C f rU u(r, t)c(r)ì 1 (r)dr (A5:13)
0 0

The standard deviation, ó Q ,of Q(t) is given by
ð ð " ð # " ð #
1 T 2 1 T R R
2
ó Q1 ¼ Q (t)dt ¼ (rUC f ) 2 u(r, t)c(r)ì 1 (r)dr u(r9, t)c(r9)ì 1 (r9)dr9 dt
1
T 0 T 0 0 0
" #
ð ð ð T
R R
¼ (rUC f ) 2 1 u(r, t)u(r9, t)dt c(r)c(r9)ì 1 (r)ì 1 (r9)dr dr9 (A5:14)
0 0 T 0
Now the expression within the square brackets is the cross correlation function,
k u (r, r9, ô) ¼ Efu(r, t)u(r9, t þ ô)g with ô set equal to zero. The cross correlation
function is related to the cross spectrum, S uu (r, r9, n), as follows: