Page 433 - Wind Energy Handbook

BLADES 407

sive design stress has to be reduced significantly to guard against buckling, with
the result that blade flexibility is reduced. For example, in the case of the 19.5 m TR
blade, the compressive design stress is reduced to about 90 MPa, resulting in a tip
deflection of about 2 m under extreme loading, and a strength-to-stiffness ratio less
than that for wood/epoxy. In this connection, it is noteworthy that Hancock,
Sonderby and Schubert (1997) records a 130 percent proof load test on a 31.2 m
birch/epoxy blade which resulted in a tip deflection of 3.4 m.

7.1.9 Blade resonance

One of the most important objectives of blade design is the avoidance of resonant
oscillations, which, in a mild form, exacerbate fatigue damage and in an extreme
form can lead to rapid failure. The excitation of blade resonance can be minimized
by maximizing the damping and ensuring that the blade flapwise and edgewise
natural frequencies are well separated from the exciting frequencies – i.e. the
rotational frequency and its harmonics, particularly the blade passing frequency –
and from the frequencies of other vibration modes with which there is an identifi-
able risk of coupled oscillations.

Vibrations in stall

On stall-regulated machines, the lift curve slope dC l =dÆ, goes negative when a
section of the blade goes into stall, resulting in local negative aerodynamic damping
of blade motion in the lift direction. If the overall aerodynamic damping for a
particular mode shape is negative, and exceeds the modal structural damping in
magnitude, then divergent oscillations can develop from any initial disturbance,
regardless of the relationship between the mode natural frequency and exciting
frequencies. The first mode in each direction is most susceptible to such behaviour
because the structural damping increases with frequency while the aerodynamic
damping diminishes. If conditions favouring first-mode oscillations are to be
avoided, the factors affecting the aerodynamic damping of both edgewise and
flapwise oscillations need to be understood, so these are explored below.
Consider a turbine operating in steady conditions in a perpendicular airflow. If a
blade cross section at radius r experiences out-of-plane and in-plane perturbations
x
y
with velocities _ x in the downwind direction and _ y in the direction opposite to that
of blade rotation (assumed clockwise), then the relative velocity triangle is as in
Figure 7.16(a). The lift and drag forces per unit length on a blade element, L and D,
can be resolved into out-of-plane and in-plane forces F X and F Y (see Figure 7.16(b)),
leading to Equations (5.18) and (5.19). Ignoring the small rotational induction factor,
which is very small, these may be rewritten as:

1
y
x
F Y ¼ W[ C l (U 1 (1 a) _ x) þ C d (Ùr _ y)] rc (7:12)
2
1
F X ¼ W[C l (Ùr _ y) þ C d (U 1 (1 a) _ x)] rc (7:13)
y
x
2