Page 246 - Wind Energy Handbook

220 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

5.6.3 Dynamic response
Tip displacement

Wind fluctuations at frequencies close to the first flapwise mode blade natural
frequency excite resonant blade oscillations and result in additional, inertial load-
ings over and above the quasistatic loads that would be experienced by a comple-
tely rigid blade. As the oscillations result from fluctuations of the wind speed about
the mean value, the standard deviation of resonant tip displacement can be ex-
pressed in terms of the wind turbulence intensity and the normalized power
2
spectral density at the resonant frequency, R u (n 1 ) ¼ n:S u (n 1 )=ó , as follows:
u
ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó x1 ó u
¼ 2 p ffiffiffiffiffiffi R u (n 1 ) K Sx (n 1 ) (5:6)
x 1 U 2ä
Here x 1 is the first mode component of the steady tip displacement, U is the mean
wind speed (usually averaged over 10 min), ä is the logarithmic decrement of
damping and K Sx (n 1 ) is a size reduction factor, which results from the lack of
correlation of the wind along the blade at the relevant frequency. Note that the
2
1
2
1
2
2
dynamic pressure, 1 2 rU ¼ r(U þ u) ¼ r(U þ 2Uu þ u ), is linearized to
2
2
1
2 rU(U þ 2u) to simplify the result. See Sections A5.2–4 in the Appendix for the
derivations of Equation (5.6) and the expression for K Sx (n 1 ).
Damping
It is evident from Equation (5.6) that a key determinant of resonant tip response is
the value of damping present. Generally the damping consists of two components,
aerodynamic and structural. In the case of a vibrating blade flat on to the wind, the
1
2
fluctuating aerodynamic force per unit length is given by r(U _ x) C d :c(r)
x
2
1 2
x
2 rU C d :c(r) ffi rU _ xxC d :c(r), where _ x is the blade flatwise velocity, C d the drag coeffi-
cient and c(r) the local blade chord. Hence the aerodynamic damping per unit
c
length, ^ c a (r), is rUC d c(r), and the first mode aerodynamic damping ratio,
ð R
î a1 ¼ c a1 =2m 1 ø 1 ¼ 2
^ c c a (r)ì (r)dr=2m 1 ø 1
1
0
is given by
ð
R
2
î a1 ¼ rUC d ì (r)c(r)dr=2m 1 ø 1
1
0
Here ì 1 (r) is the first mode shape,
ð R
2
m 1 ¼ m(r)ì (r)dr
1
0