Page 341 - Wind Energy Handbook

RESONANT DISPLACEMENT RESPONSE IGNORING WIND VARIATIONS 315

A5.3 Resonant Displacement Response Ignoring Wind
Variations along the Blade

A5.3.1 Linearization of wind loading

For a fluctuating wind speed U(t) ¼ U þ u(t), the wind load per unit length on the
2
1
2
2
1
blade is C f rU (t)c(r) ¼ C f r[U þ 2Uu(t) þ u (t)]c(r), where C f is the lift or drag
2 2
coefficient, as appropriate, and c(r) is the local blade chord dimension. In order to
permit a linear treatment, the third term in the square brackets, which will normally
be small compared to the first two, is ignored, so that the fluctuating load q(r, t)
becomes C f rUu(t)c(r).
A5.3.2 First mode displacement response

Setting q(r, t) ¼ C f rUu(t)c(r), the first mode tip displacement response to a sinusoi-
dal wind fluctuation of frequency n (¼ ø=2ð) and amplitude u o (n) given by Equa-
tion (A5.3) becomes
ð R
f 1 (t) ¼ ì 1 (r)C f rUc(r)dru o (n)jH 1 (n)j cos(2ðnt þ ö )
1
0
ð
R
¼ C f rU ì 1 (r)c(r)dru o (n)jH 1 (n)j cos(2ðnt þ ö ) (A5:5)
1
0
Hence power spectrum of first mode tip displacement is
" #
ð R 2
S 1x (n) ¼ C f rU ì 1 (r)c(r)dr S u (n)jH 1 (n)j 2 (A5:6)
0
where S u (n) is the power spectrum for the along wind turbulence. Thus the
standard deviation of the first mode tip displacement is given by
ð R ð 1
2
ó 2 ¼ [C f rU ì 1 (r)c(r)dr] 2 S u (n)jH 1 (n)j dn (A5:7)
1x
0 0

A5.3.3 Background and resonant response

Normally the bulk of the turbulent energy in the wind is at frequencies well below
the frequency of the first out-of-plane blade mode. This is illustrated in Figure A5.1,
where a typical power spectrum for wind turbulence is compared with the square,
2
jH 1 (n)j , of an example frequency response function for a 1 Hz resonant frequency.
The power spectrum is that due to Kaimal (and adopted in Eurocode 1, 1997):