Page 342 - Wind Energy Handbook

316 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

0.3

0.25 Square of frequency
Power spectral density, response function,
nSu(n)/su 2 = x1/100, for damping ratio
0.1417nT/(0.098 + of 0.1 and natural
nT ) 5/3 with T = 5 s frequency of 1 Hz
0.2
(stiffness, k, taken as
unity)
0.15

0.1

0.05

0
0.0001 0.001 0.01 0.1 1 10
Frequency (Hz, logarithmic scale)
Figure A5.1 Power Spectrum and Frequency Response Function

x
0:1417nL =U
nS u (n) ¼ ó 2 u u (A5:8)
x
(0:098 þ nL =U) 3 5
u
and is plotted as the non-dimensional power-spectral density function, R u (n) ¼
x
2
nS u (n)=ó , against a logarithmic frequency scale. The time scale, L =U, chosen is
u u
x
5 s, based on a mean wind speed, U,of 45 m=s and an integral length scale, L ,of
u
225 m.
In view of the fact that the resonant response usually occurs over a narrow band
of frequencies on the tail of the power spectrum, it is normal to treat it separately
from the quasistatic response at lower frequencies, and to ignore the variation in
nS u (n) on either side of the resonant frequency, n 1 (see, for example, Wyatt (1980)).
The variance of total tip displacement then becomes:
2
2
ó ¼ ó þ ó 2
x B x1
in which the variance of the first mode resonant response, ó x1 , is given by
" #
ð 2 ð
R 1
2
ó 2 ¼ C f rU ì 1 (r)c(r)dr S u (n 1 ) jH 1 (n)j dn (A5:9)
x1
0 0
2
and the resonant response of higher modes, ó , ó 2 x3 etc are ignored. The non-
x2
resonant response, ó B , is termed the background response, and can be derived from
simple static beam theory. Ð
2
It has been shown by Newland (1984) that 0 1 jH 1 (n)j dn reduces to
2
2
(ð =2ä)(n 1 =k ), where ä is the logarithmic decrement of damping. The logarithmic
1
decrement, ä,is2ð times the damping ratio, î 1 , defined as î 1 ¼ c 1 =2m 1 ø 1 . Hence
Equation (A5.9) becomes