Page 247 - Wind Energy Handbook
P. 247
STATIONARY BLADE LOADING 221
is the generalized mass, and ø 1 is the first mode natural frequency in radians=s. The
logarithmic decrement is obtained by multiplying the damping ratio by 2ð.
When the wind direction is angled to the blade so as to generate maximum lift,
the blade will be approaching stall, with the result that the aerodynamic damping is
effectively zero. In this situation tip deflections are limited only by the blade
structural damping. Structural damping is discussed in Section 5.8.4, and values for
typical blade materials given.
Root bending moment
The standard deviation of tip displacement in combination with the blade mode
shape yields an inertial loading distribution from which the standard deviation of
the resulting bending moment at any position along the blade may be calculated. In
particular, the standard deviation of the root bending moment may be expressed in
terms of the mean root bending moment as follows:
ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó M1 ó u ó x1
¼ 2 p ffiffiffiffiffiffi R u (n 1 ) K Sx (n 1 ):º M1 ¼ º M1 (5:7)
M U 2ä x 1
where
ð R
m(r)ì 1 (r)r dr ð R
º M1 ¼ 0 ð c(r)ì 1 (r)dr (5:8)
R 0
m 1 c(r)r dr
0
(See Section A5.5 in the Appendix for the derivation of the expression for º M1 .)
The standard deviation of the quasistatic root bending moment fluctuation, or
root bending moment background response is expressed in terms of the mean root
bending moment by
p ffiffiffiffiffiffiffiffiffiffiffi
ó MB ó u
¼ 2 K SMB (5:9)
M U
where K SMB is a size reduction factor to take account of lack of correlation of wind
fluctuations along the blade. As shown in Section A5.6 of the Appendix, K SMB is
usually only slightly less than unity because the blade length is small compared
with the integral length scale of longitudinal turbulence measured in the across
wind direction.
The variance of the total root bending moment fluctuations is equal to the sum of
the resonant and background response variances, i.e.,
ó 2 ¼ ó 2 þ ó 2
M M1 MB
Hence
