Page 334 - Wind Energy Handbook

308 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

so, approximately,

dF Y 1 dC l C l
¼ rÙ c(r)r þ sin ö (5:131a)
du 2 dÆ dC l =dÆ
Thus the standard deviation of rotor torque is approximately given by
þ
dC l 2 C l
1
ó Q ¼ rÙ ó u r c(r) þ sin ö dr (5:131b)
2 dÆ dC l =dÆ
(which parallels Equation (5.26)) provided the relationship between blade loading
and wind speed fluctuation remains linear and the turbulence length scale is large
compared with rotor diameter. Equation (5.131b) can be used to derive an expres-
sion for the variance of the rotor torque in the same way as for rotor thrust above.
At the top of the tower the stochastic M X (i.e., side-to-side) moment due to rotor
torque fluctuations is typically of the same order of magnitude as the stochastic M Y
(i.e., fore-aft) moment due to differential out-of-plane loads on the rotor, but at the
tower base the dominant effect of rotor thrust loading means that the stochastic
side-to-side moments are usually significantly less than the stochastic fore-aft
moments before the excitation of tower resonance is taken into account.

Analysis in the time domain

As noted in Section 5.7.5, there are situations, such as operation in stalled flow,
when the linear relationship between blade loading and wind speed fluctuations
required for analysis in the frequency domain does not apply. In these cases,
recourse must be made to analysis in the time domain using wind simulation
techniques such as described in Section 5.7.6.

5.12.5 Dynamic response to operational loads

The power spectrum of rotor thrust will usually contain some energy at the tower
natural frequency, leading to dynamic magnification of deflections, and hence of
tower bending moments. The power spectrum of hub deflection, S x1 (n), resulting
from the excitation of the tower first fore-aft flexural mode, is related to the power
spectrum of rotor thrust by
S T (n) 1
S x1 (n) ¼ 2 (5:132)
2
2
2 2
2 2
k 1 [(1 n =n ) þ 4î n =n ]
1
1
1
This relation is analagous to Equation (5.90), and derived in the same way.
The amplitude of tower base fore-aft moment at resonance in the first mode, M Y1 ,
can be derived from the corresponding amplitude of hub deflection, x H1 , as follows