The Unsteady Aerodynamics of Turbine Yaw and Over-Speed Protection - Small Wind Turbines: Analysis, Design, and Application - page 150

the apparent wind speed and angle of attack on the tail fin moving with angular
velocity h under the action of aerodynamic force F. Note that h is not necessarily
the yaw rate as the co-ordinate system used in Fig. 8.4 is an inertial one only in
specific cases such as the wind tunnel test of a delta tail fin shown in Fig. 8.5 .
All angles are assumed to be small, the drag is neglected, and the tail fin's lift is
linear with slope K. Taking moments about the yaw axis and assuming no fric-
tional moment in the yaw bearing results in the following second order linear
differential equation:

d 2 h

dt 2 þ 2fx n dh

dt þ x n h ¼ x n u

ð 8 : 4 Þ

where u is the wind direction. The natural frequency x n and damping ratio f are
given by ( 8.5a ,

b ) respectively:

r

qrAK
2I

x n ¼ U wake ð 8 : 5a Þ

and

r

qr 3 AK
8I

f ¼ ð 8 : 5b Þ

where q is the air density, A the tail fin area, and I the inertia about the yaw axis.
It is important to note that I has contributions from the tail boom, nacelle, and
blades. These ''pseudo-static'' relationships ( 8.5a , b ) were originally derived for
wind vanes used to measure wind direction, e.g. Weiringa [ 9 ] and Kristensen [ 10 ],

Fig. 8.5 Wind tunnel testing
of a 1/4 scale model of the
tail fin in Fig. 6.1 . The verti-
cal shaft has an encoder to
measure angle. From Wright
[ 21 ]