Page 323 - Wind Energy Handbook

HUB AND LOW-SPEED SHAFT LOADING 297

o
tion function, k (r 1 , r 2 , 0) will be increased when r 1 and r 2 relate to different blades,
u
because of the reduced separation between the two blade elements resulting from
the 1208 degree angle between the blades. Equation (5.119) can be rewritten in terms
o
o
2
of the normalized cross correlation function, r (r 1 , r 2 ,0) ¼ k (r 1 , r 2 ,0)=ó ,as
u u u
follows:
ð ð p ffiffiffi p ffiffiffi
2 R R 3 3
o
ó 2 ¼ ó 2 1 rÙ dC l r (r 1 , r 2 ,0)c(r 1 )c(r 2 )
Mzs u 2 dÆ R R u 2 r 1 2 r 2 jr 1 jjr 2 j dr 1 dr 2
(5:119a)
In the case of 40 m diameter turbines with TR blades operating in wind with a
turbulence length scale of 73.5 m, the standard deviation of shaft moment due to
stochastic loading for a three-bladed machine is 82 percent of that for a two-bladed,
ffiffiffi
p
fixed hub machine rotating at the same speed. This ratio would rise to 3=2 if the
effect on the cross correlation function of the 1208 blade spacing were ignored. It is
worth noting that, for a three-bladed machine, the standard deviation of the shaft
moment M YS due to stochastic loading is the same as that of M ZS .
By analogy with the derivation of the shaft moment above, the standard deviation
of the hub ‘dishing’ moment for a three-bladed machine due to stochastic loading is
given by:

2ð R ð R p ffiffiffi p ffiffiffi
o
2
1 2
ó 2 Mh ¼ ó u 1 2 rÙ dC l R R r (r 1 , r 2 ,0)c(r 1 )c(r 2 ) 2 3 r 2 1 2 3 r dr 1 dr 2 (5:120)
2
u
dÆ
4
where the integrations are carried out over two blades only, and the cross correla-
tion function is modified as before to account for the 1208 angle between the blades.
It can be shown that
3 2
1 2 þ ó 2 ¼ ó (5:121)
ó
4 MZS Mh 4 My1
5.10.4 Gravity loading

An important component of shaft loading is the cyclic cantilever bending moment
due to rotor weight, which usually has a dominant effect on shaft fatigue design. As
an illustration, a rotor consisting of three TR blades, each weighing 2 tonnes, and a
5 tonne hub cantilevered 0.85 m beyond the shaft main bearing, will produce a
maximum shaft gravity moment of about 90 kNm. This compares with a shaft
moment range due to wind shear of 70 kNm for a hub height wind of 12 m=s and a
shear exponent of 0.2, and a shaft moment standard deviation of 50 kNm due to
turbulence, taking a turbulence intensity of 20 percent and the same hub-height
mean wind speed. Note that the shaft moment due to wind shear relieves that due
to gravity, so it would be wise to adopt a smaller wind shear exponent for shaft
fatigue calculations.