Page 348 - Wind Energy Handbook

322 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES
Ð R
2
where m 1 ¼ m(r)ì (r)dr is the generalized mass with respect to the first
0 1
mode, and the exponential expression within the double integral allows for
the lack of correlation of wind fluctuations along the blade. Substituting
Ð
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
( R c(r)ì 1 (r)dr): K Sx (n 1 ) for the square root of the double integral, using Equation
0
A5.25, leads to
ð
R !
ó M1 ó u ð p ffiffiffiffiffiffiffiffiffiffiffiffi 0 m(r)ì 1 (r)r dr ð R p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 2 p ffiffiffiffiffiffi R u (n) ð c(r)ì 1 (r)dr : K Sx (n 1 ) (A5:32)
M U 2ä R 0
m 1 c(r)r dr
0
Defining the ratio of the integrals,
ð
R !
m(r)ì 1 (r)r dr ð R ó M1
0 M
ð R c(r)ì 1 (r)dr ¼ ó x1 (A5:33)
c(r)r dr 0
m 1
x 1
0
as º M1 we obtain
ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó M1 ó u
¼ 2 p ffiffiffiffiffiffi R u (n 1 ) K Sx (n 1 )º M1 (A5:34)
M U 2ä

A5.6 Root Bending Moment Background Response

The root bending moment background response can be expressed in terms of the
standard deviation of the root bending moment excluding resonant effects. If the
wind is perfectly correlated along the blade, this is given by

ð
R
ó MB ¼ C f rUó u c(r)r dr (A5:35)
0
However, if the lack of correlation of wind fluctuations along the blade is taken into
account,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð ð
R R
ó MB ¼ C f rUó u r u (r r9)c(r)c(r9)rr9 dr dr9 (A5:36)
0 0
where r u (r r9) is the normalized cross correlation function between simultaneous
wind-speed fluctuations at two different blade radii, and is defined as
1
r u (r r9) ¼ Efu(r, t)u(r9, t þ ô)g (A5:37)
ó 2 u