Page 299 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 273
" # " #
ð R ð R
3
€
3
2
_
1
æ
æ
1
Iæ þ rÙ dC l r c(r)dr æ þ rÙ dC l r c(r)dr Ù(tan ä 3 )æ þ IÙ æ
2 dÆ R 2 dÆ R
ð R
1
¼ rÙ dC l u(r, t)c(r)rjrjdr (5:94)
2 dÆ R
assuming a frozen wake. By dividing through by the moment of inertia and writing
" ð R #
3
ç ¼ 1 r dC l r c(r)dr (5:95)
2 I dÆ R
this can be simplified to
ð R
_
€
2
æ
1
æ æ þ çÙæ þ (1 þ ç tan ä 3 )Ù æ ¼ r Ù dC l u(r, t)c(r)rjrj dr (5:96)
2 I dÆ R
ç is a measure of the ratio of aerodynamic to inertial forces acting on the blade, and
is one eighth of the Lock number.
Delta 3 coupling thus raises the natural frequency, ø n , of the teeter motion from
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
Ù to Ù 1 þ ç tan ä 3 . For a 40 m diameter rotor consisting of two TR blades
mounted on a teeter hinge set at a ä 3 angle of 308, ç ¼ 0:888 and tan ä 3 ¼ 0:577, so
the increase in natural frequency due to the ä 3 angle is 23 percent. The correspond-
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ing damping ratio, given by î ¼ (ç=2) 1 þ ç tan ä 3 is quite high at 0.36.
Teeter response to deterministic loads
The teeter response to deterministic loads can be found using the same step-by-step
integration procedure set out in Section 5.8.5. However, as the loadings due to wind
shear and yaw are both approximately sinusoidal, an estimate of the maximum
teeter angle for these cases may be obtained by using the standard solution for
forced oscillations. For a harmonically varying teeter moment, M T ¼ M T0 cos Ùt
due to wind shear, the teeter angle is given by

M TO cos (Ùt W)
æ ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5:97)
2 2
Iø n (1 (Ù=ø n ) ) þ (2îÙ=ø n ) 2
1
2
where W ¼ tan ((2îÙ=ø n )=(1 (Ù=ø n ) )) ¼ 908 ä 3 is the phase lag with respect
to the excitation.
For the two-bladed turbine described above, rotating at 30 r.p.m. in a wind with a
hub-height mean of 12 m=s and a shear exponent of 0.2, the teeter moment
amplitude, M o , is approximately 50 kNm (see Figure 5.11, which gives the blade
root bending moment variation with azimuth for a fixed hub machine, based on
2
momentum theory). Taking the rotor moment of inertia as 307 000 kg m for TR
blades, and ø n ¼ 1:23ð rad=s, the maximum teeter angle comes to 0:98 for ä 3 ¼ 308.
This increases by about 16 percent to 1:058 if the ä 3 angle is reduced to zero.