Page 335 - Wind Energy Handbook

TOWER LOADING 309

ð
H
ð m(z)ì(z)z dz
H
2 2 0
M Y1 ¼ ø x H1 m(z)ì(z)z dz ¼ ø x H1 m T1 H ð H (5:133)
1
1
0 2
H m(z)ì (z)dz
0
The quotient on the right-hand side is close to unity because of the dominance of
2
the tower head mass, so, substituting k 1 for ø m T1 , the equation reduces to
1
M Y1 ¼ x H1 k 1 H, which applies at any exciting frequency. Hence the power spectrum
for the tower base fore-aft bending moment due to rotor thrust loading is given by
1
2
S My1 (n) ¼ S T (n)H (5:134)
2
[(1 n =n ) þ 4î n =n ]
2
2 2
2 2
1 1
The aerodynamic damping is almost entirely provided by the rotor, the damping
ratio for the first tower mode being approximately
ð R
1 rÙ dC l
2 dÆ rc(r)dr
î a1 ¼ N 0 (5:135)
2m T1 ø 1
where N is the number of blades (see Section 5.8.4). The overall damping ratio is
obtained by adding this to the structural damping ratio for the tower (see Table
5.4), and is generally low compared to the blade first mode damping because of the
large tower head mass. The effect of a low damping ratio is illustrated by the power
spectrum of fore-aft tower bending moment shown in Figure 5.44, which has a very
high peak at the tower natural frequency of 1.16 Hz, despite this frequency being
somewhat removed from the blade passing frequency of 1.5 Hz. The damping ratio
is calculated as 0.022, consisting of 0.019 due to aerodynamic damping and 0.03 due
to structural damping (for a welded steel tower).
In the example shown in Figure 5.44, the tower dynamic response increases the
standard deviation of the tower base fore-aft bending moment by 15 percent.
However, the effect of tower dynamic response results in a larger increase of 25
percent in the case of the two-bladed machine featured in Figure 5.45 despite the
reduction in tower natural frequency in proportion to blade-passing frequency.
It is important to note that the rotor provides negligible aerodynamic damping in
the side-to-side direction, so that effectively the only damping present is the
structural damping. This means that, even though the side-to-side loadings are
small in relation to the fore-aft loads, the side-to-side tower moment fluctuations
can sometimes approach the fore-aft ones in magnitude.

5.12.6 Fatigue loads and stresses

The tower moments at height z are related to the hub-height loads as follows:

M Y (z, t) ¼ F X (H, t):(H z) þ M Y (H, t) M X (z, t) ¼ F Y (H, t):(H z) þ M X (H, t)
M Z (z, t) ¼ M Z (H, t) (5:136)