Page 316 - Wind Energy Handbook

290 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

(2) A probability density function for the load cycle ranges can be derived
empirically, based on the spectral properties of the power spectrum of the
stochastic and periodic components of loading combined.

The second approach is considered in the next Section.

5.9.3 Fatigue prediction in the frequency domain

The probability density function (p.d.f.) of peaks of a narrow band, Gaussian
process are given by the well-known Rayleigh distribution. As each peak is
associated with a trough of similar magnitude, the p.d.f. of cycle ranges is Rayleigh
likewise.
Wind turbine blade loading cannot be considered as narrow band, despite the
concentration of energy at the rotational frequency by ‘gust slicing’ (Section 5.7.5),
and neither can it be considered as Gaussian because of the presence of periodic
components. Dirlik (1985) produced an empirical p.d.f. of cycle ranges applicable to
both wide and narrow band Gaussian processes, in terms of basic spectral proper-
ties determined from the power spectrum. This was done by considering 70 power
spectra of various shapes, computing their rainflow cycle range distributions and
fitting a general expression for the cycle range p.d.f. in terms of the first, second and
fourth spectral moments. Dirlik’s expression for the cycle range p.d.f. is:

2
D 1 Z=Q D 2 Z ( Z =2R ) ( Z =2)
2
2
e þ e þ D 3 Ze
Q R 2
p(S) ¼ p ffiffiffiffiffiffiffi (5:117)
2 m o
where
2
2
p ffiffiffiffiffiffiffi 2(x m ª ) (1 ª D 1 þ D )
Z ¼ S= m o , D 1 ¼ , D 2 ¼ 1 , D 3 ¼ 1 D 1 D 2
1 þ ª 2 1 R
r ffiffiffiffiffiffiffi
2
1:25(ª D 3 D 2 R) ª x m D 1 m 1 m 2 m 2
Q ¼ , R ¼ 2 , x m ¼ , ª ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffi ,
(1 ª D 1 þ D )
D 1 m 0 m 4 m 0 m 4
1
ð
1
i
m i ¼ n S ó (n)dn S ó (n) is the power spectrum of stress,
0
and S is the cycle stress range:
Although the Dirlik cycle range p.d.f. was not intended to apply to signals
containing periodic components, several investigations (Hoskin et al. (1989), Mor-
gan and Tindal (1990), Bishop et al. (1991)) have been carried out to determine its
validity for wind turbine fatigue damage calculations, using monitored data for
flapwise bending from the MS1 wind turbine on Orkney. Cycle range p.d.f.s were
calculated from power spectra of monitored strains using the Dirlik formula and
fatigue damage rates derived from these p.d.f.s compared with damage rates