Page 349 - Wind Energy Handbook

ROOT BENDING MOMENT BACKGROUND RESPONSE 323

with ô set equal to zero. Measurements indicate that the normalized cross correla-
tion function decays exponentially, so it can be expressed as

r
r u (r r9) ¼ exp[ jr r9j=L ] (A5:38)
u
r
where L is the integral length scale for the longitudinal turbulence component
u
measured in the across wind direction along the blade, and is thus defined as
Ð
1
0 r u (r r)d(r r9). As the integral length scale for longitudinal turbulence meas-
z
ured vertically in the across wind direction (L ) is, if anything, less than that
u
y
measured horizontally (L ), it is conservative to treat it as being equal to that
u
y
r
measured horizontally, with the result that L can be taken as equal to L also.
u
u
x
y
Typically L is approximately equal to 30 percent of L , the integral length scale for
u
u
longitudinal turbulence measured in the along wind direction. Observing that
ð
R
1 2
M ¼ rU C f c(r)r dr
2
0
we can therefore write
p ffiffiffiffiffiffiffiffiffiffiffi
ó MB ó u
¼ 2 K SMB (A5:39)
M U
where K SMB , the size reduction factor for the root bending moment background
response, is defined as
ð ð
R R
x
exp[ jr r9j=0:3L ]c(r)c(r9)rr9 dr dr9
u
K SMB ¼ 0 0 ! (A5:40)
ð R 2
c(r)r dr
0
For a blade with a uniform chord, the integral is straightforward, giving

2 1 2 2 2 R
K SMB ¼ 4 þ exp( ö) þ where ö ¼ x (A5:41)
3ö ö 2 ö 4 ö 3 ö 4 0:3L u
x
As an example, K SMB comes to 0.927 for the case of R ¼ 20 m and L ¼ 230 m,
u
indicating that the effect of the lack of correlation of the wind fluctuations is rather
small.
For blades with a normal tapering chord, K SMB can be evaluated numerically. In
the case of a blade with a tip chord equal to 25 percent of the maximum chord,
K SMB is 0.924 for the same value of ö as before. It is seen that the taper has a
negligible effect on the end result.