Page 287 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 261

(7) calculate new deflected profile resulting from this bending moment distribu-
tion;
(8) calculate revised estimate of natural frequency from:
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Trial tip deflection
ø9 j ¼ ø j
Tip deflection calculated for new deflected profile

(9) repeat steps (2)–(8) with revised mode shape and frequency until calculated
mode shape converges.

It is important to note that the lateral loads and deflections of a centrifugally
loaded beam do not conform to Betti’s Law, so, as a consequence, the mode shapes
are not orthogonal. It is for this reason that the ‘purification’ stage has been included
in each cycle of iteration. When convergence of the calculated mode shape has
occurred, it will be found that it differs significantly from the ‘purified’ mode shape
input into each iteration, indicating that a true solution has not been obtained. It is
then necessary to use a trial and error approach to modify the magnitudes of the
‘purifying’ corrections applied until the output mode shape and input ‘purified’
mode shape match. A few further iterations will be required until the natural
frequency settles down.
A quick estimate of the first mode frequency of a rotating blade can be derived
using the Southwell formula reported by Putter and Manor (1978) as follows
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ø 1 ¼ ø 2 1,0 þ ö 1 Ù 2 (5:75)

in which ø 1,0 is the corresponding frequency for the non-rotating blade. The value
of ö 1 depends on the blade mass and stiffness distribution, and Madsen et al. (1984)
suggest the value 1.73 for wind-turbine blade out-of-plane oscillations. In the case
of Blade TR rotating at 30 r.p.m., this yields a percentage increase in first mode
frequency due to centrifugal stiffening of 7.7 percent compared to the correct value
of 8.1 percent. Typically, centrifugal stiffening results in an increase of the first
mode frequency for out-of-plane oscillations of between 5 percent and 10 percent.
For higher modes, the magnitude of the centrifugal forces is less in proportion to
the lateral inertia forces, so the percentage increase in frequency due to centrifugal
stiffening becomes progressively less.
The procedure for deriving the blade first mode shape and frequency in the case
of in-plane oscillations is the same as that described above for out-of-plane
vibrations, except that the formula for the bending moment distribution due to the
centrifugal forces has to be modified to:
ð R
2 r
M X:CF (r ) ¼ m(r)Ù r ì(r) ì(r ) dr (5:76)
r r
The first mode frequency for in-plane oscillations of Blade TR in the absence of
centrifugal force is 3.13 Hz. This is approximately double the corresponding