Page 302 - Wind Energy Handbook

276 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

the cross correlation function between the longitudinal wind fluctuations between
points at radii r 1 and r 2 on the rotating rotor and is given by the right-hand side of
Equation (5.51), with Ùô set equal to zero when r 1 and r 2 define points on the same
blade, and replaced by ð when r 1 and r 2 define points on different blades. Defining
o
o
2
r (r 1 , r 2 , 0) as the normalized cross correlation function, k (r 1 , r 2 ,0)=ó , Equation
u
u
u
(5.102) can be rewritten as:
ð ð
2 R R
ó 2 ¼ ó 2 1 rÙ dC l o (5:102a)
u
MT u 2 dÆ R R r (r 1 , r 2 ,0)c(r 1 )c(r 2 ):r 1 r 2 jr 1 jjr 2 j dr 1 dr 2
The corresponding expression for the standard deviation of the mean of the two-
blade root bending moments is:
ð ð
2 R R
1 2
ó 2 ¼ ó 1 rÙ dC l o 2 2 (5:103)
u
1 2
M 4 u 2 dÆ R R r (r 1 , r 2 ,0)c(r 1 )c(r 2 ):r r dr 1 dr 2
By inspection of the integrals, it is easily shown that:
ð ð
2 R R
o
1 2 þ ó 2 ¼ ó 2 1 rÙ dC l r (r 1 , r 2 ,0)c(r 1 )c(r 2 ):r r dr 1 dr 2 ¼ ó 2 (5:104)
ó
2 2
4 MT M u 2 dÆ 0 0 u 1 2 M
where ó M is the standard deviation of root bending moment for a rigidly mounted
blade. Thus, if the rotor is allowed to teeter, the standard deviation of the blade root
bending moment will drop from ó M to ó where ó M is given by the equation
M
above. The extent of the reduction is driven primarily by the ratio of rotor diameter
to the integral length scale of the wind turbulence. For a two-bladed rotor with TR
blades and an integral length scale of 73.5 m, the reduction is 11 percent.

5.8.9 Tower coupling

In the preceding sections, consideration of the dynamic behaviour of the blade has
been based on the assumption that the nacelle is fixed in space, i.e., that the tower is
rigid. In practice, of course, no tower is completely rigid, so fluctuating loads on the
rotor will result in fore–aft flexure of the tower, which, in turn, will affect the blade
dynamics. This section explores the effect the coupling of the blade and tower
motions has on blade response.
The application of standard modal analysis techniques to the dynamic behaviour
of the system comprising the tower and rotating rotor treated as a single entity is
complicated by the system’s continually changing geometry, which means that the
mode shapes and frequencies of the structure taken as a whole would have to be re-
evaluated at each succeeding rotor azimuth position.
An alternative approach is to base the analysis on the mode shapes and
frequencies of the different elements of the structure considered separately, with
the displacements arising from each set of modes superposed. Thus the tower
modes are calculated on the basis of a completely rigid rotor, and the blade modes