Page 332 - Wind Energy Handbook

306 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

and the perturbation of rotor thrust by
þ
˜T ¼ 1 rÙ dC l uc(r)r dr (5:128)
2 dÆ
Þ
where the integral sign signifies that the integration is carried out over the whole
rotor. Hence the following expression for the variance of the rotor thrust is obtained:
2 þþ
2
o
ó ¼ 1 rÙ dC l ó 2 r (r 1 , r 2 ,0)c(r 1 )c(r 2 )r 1 r 2 dr 1 r 2 (5:129)
T 2 dÆ u u
2
o
o
where r (r 1 , r 2 , 0) is the normalized cross correlation function, k (r 1 , r 2 ,0)=ó for
u u u
o
points at radii r 1 and r 2 on the same or on different blades. k (r 1 , r 2 , 0) is given by
u
Equation (5.51), with Ùô replaced by the phase angle between the two blades on
which r 1 and r 2 measured. For a three-bladed, 40 m diameter rotor and an integral
length scale of 73.5 m, the reduction in the standard deviation of the stochastic rotor
thrust fluctuations is about 20 percent due to the lack of correlation of the wind
speed variations over the rotor. If the machine is rotating at 30 r.p.m. in an 8 m=s
wind and the turbulence intensity is 20 percent, the rotor thrust standard deviation
will be about 9 kN – i.e., 25 percent of the steady value
The derivation of the expression for the power spectrum of rotor thrust parallels
that for the power spectrum of blade root bending moment (Section 5.7.5), yielding:
2þþ
S T (n) ¼ 1 rÙ dC l S o (r 1 , r 2 , n)c(r 1 )c(r 2 )r 1 r 2 dr 1 dr 2 (5:130)
2 dÆ uJ,K
where S 0 (r 1 , r 2 , n) is the rotationally sampled cross spectrum for points at radii
uJ,K
r 1 and r 2 on blades J and K respectively. Note that on a machine with three blades,
A, B and C, S o uJ,K (r 1 , r 2 , n) is complex when J and K are different, but
S o (r 1 , r 2 , n) and S o (r 1 , r 2 , n) are complex conjugates, so the double integral in
uA,B uA,C
Equation (5.130) is still real. An example power spectrum of rotor thrust for a three
bladed machine is shown in Figure 5.44. It can be seen that there is some concentra-
tion of energy at the blade passing frequency of 1.5 Hz due to gust slicing, but that
the effect is not large. The concentration effect is significantly greater for two-bladed
machines (see Figure 5.45). This shows the power spectrum of rotor thrust for a
two-bladed machine with the same blade planform, but rotating 22.5 percent faster
to give comparable performance.
In addition to thrust fluctuations, longitudinal turbulence will also cause rotor
torque fluctuations and in-plane rotor loads due to differential loads on different
blades, both of which will result in tower sideways bending moments. The expression
1 2
for the in-plane component of aerodynamic lift per unit length, F Y (r) ¼ rW C l
2
c(r) sin ö,can be differentiatedwith respect to the wind fluctuation as follows:
dF Y 1 d 2 1 d
¼ rc(r) [W sin öC l ] ¼ rc(r) [WfU 1 (1 a) þ ugC l ]
du 2 du 2 du

1
ffi rc(r)WC l þ sin ö dC l
2 dÆ