Page 297 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 271

1
2
2
S My1 (n) ¼ (k 1 R÷ M1 ) S x1 (n) ¼ (R÷ M1 ) S Q1 (n) (5:91c)
2
2 2
2 2
2
[(1 n =n ) þ 4î n =n ]
1 1 1
For Blade TR, the ratio ÷ M1 takes a value of 1.4.
5.8.7 Response to simulated loads
The blade dynamic response to time varying loading derived from wind simulation
(Section 5.7.6) can be obtained by a step-by-step dynamic analysis such as that
described for use with deterministic loads in Section 5.8.5. The procedure is
essentially the same, except that it is more important to select realistic values for the
initial blade tip displacement, velocity and acceleration, unless the results from the
first few rotation cycles are to be discarded.

5.8.8 Teeter motion

When the rotor is rigidly mounted on the shaft, out-of-plane aerodynamic loads on
the blades result in fluctuating bending moments in the low speed shaft additional
to those due to gravity. In the case of two bladed machines, the transfer of blade
out-of-plane aerodynamic moments to the shaft can be eliminated and blade root
bending moments reduced by mounting the rotor on a hinge with its axis perpendi-
cular to both the low speed shaft and the axis of the rotor. This allows the rotor to
teeter to and fro in response to differential aerodynamic loads on each blade.
The restoring moment is generated by the lateral components of the centrifugal
force acting on each blade element (see Figure 5.29). It is given by
ð R
2
2
M R ¼ r:m(r)Ù r:æ dr ¼ IÙ æ (5:92)
0
where æ is the teeter angle and I is the rotor moment of inertia about its centre. The
€
2
æ
equation of motion for free teeter oscillations is thus Iæ þ IÙ æ ¼ 0 (omitting the
aerodynamic damping term for the moment), indicating that the natural frequency
of the teeter motion with the teeter hinge perpendicular to the rotor axis is equal to
the rotational frequency. Since both the deterministic and stochastic components of
the exciting moment are dominated by this frequency, it is clear that the system
operates at resonance, with aerodynamic damping alone controlling the magnitude
of the teeter excursion.
The magnitude of teeter excursions would clearly be reduced if the teeter natural
frequency were moved away from the rotational frequency. This can be done by
rotating the teeter hinge axis relative to the rotor in the plane of rotation, as
illustrated in Figure 5.29, so that teeter motion results in a change of blade pitch –
positive in one blade and negative in the other – known as Delta 3 coupling.
Consider the case of blade A slicing through a gust. The increased thrust on the
blade will cause it to move in the downwind direction, by rotating about the teeter