Page 288 - Wind Energy Handbook

262 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

frequency for out-of-plane oscillations of 1.65 Hz, and so the relative effect of the
centrifugal loads is much reduced, even before allowance is made for the smaller
lever arms at which they act (Figure 5.24). In fact the increase in the first mode
frequency for in-plane oscillations due to centrifugal force is only 0.5 percent –
probably small enough to be ignored.

5.8.4 Aerodynamic and structural damping

Blade motion is generally resisted by two forms of viscous damping, aerodynamic
and structural, which are considered in turn.
An approximate expression for the aerodynamic damping per unit length in the
flapwise direction can be derived by a method analagous to that used in Section
5.7.5 to derive the linear relation

1
q ¼ rÙrc(r) dC l u (5:25)
2 dÆ
between blade load fluctuations per unit length, q, and fluctuations in the incident
wind, u. The wind-speed fluctuation, u, is simply replaced by the blade flapwise
x
velocity, _ x, giving
q dC l
1
^ c c a (r) ¼ ¼ rÙrc(r) (5:77)
x
_ x 2 dÆ
The rate of change of lift coefficient with angle of attack, dC l =dÆ, is constant and
equal to 2ð before the blade goes into stall, but can become negative post-stall,
leading to the risk of instability (see Section 7.1.9).
It can be seen that the aerodynamic damping per unit length, ^ c a (r), varies
c
spanwise as the product of radius and blade chord, and is therefore not very close
to being proportional to the mass per unit length, as is required to satisfy the
orthogonality condition. This will result in some aerodynamic coupling of modes,
which is not accounted for in normal modal analysis.
The aerodynamic damping ratio for the ith mode, defined as î ai ¼
Ð
2
c ai =2m i ø i ¼ R ^ c c a (r)ì (r)dr=2m i ø i , can be calculated using Equation (5.77) as fol-
0 i
lows:
ð R
2
1 rÙ dC l rc(r)ì (r)dr
2 dÆ i
î ai ¼ ð 0 (5:78)
R
2
2ø i m(r)ì (r)dr
i
0
In the case of fibreglass Blade TR described in Example 5.1, this yields values of 0.16
and 0.04 for the first and second modes respectively. These high values are a
consequence of the lightness of the blade in relation to its width in the vicinity of
the tip, an area which dominates the integrals thanks to the mode shape weighting.
The corresponding first mode logarithmic decrement (¼ 2ðî a ) is thus 1.0.
Structural damping can be considered as an internal resistance opposing the rate