WIND ENERGY SYSTEMS - Cd3wd

Chapter 3—Wind Measurements 3–21
free to rotate about the origin and the bearing friction is assumed to be negligible. The positive
direction for all angles, as well as dθ/dt and d 2 θ/dt 2 , is taken to be the counterclockwise
direction. The torque T is in the clockwise direction so the equation of motion can be written
as
I d2 θ
= −T = −Fr (19)
dt2 where I is the moment of inertia, r is the distance from the pivot point to the centroid of the
tail and F is the equivalent force of the wind acting at the centroid.
Figure 13: Wind vane angles
The force F on the tail is assumed to be an extension of Eq. 14 for the case of the wind
not perpendicular to the plate. The force is assumed to be proportional to the angle between
the vane and the apparent local wind, or β +∆β.
F = . cAρ u2
(β +∆β) (20)
2
where c is a constant depending on the aerodynamics of the vane, A is the area, ρ is the air
density, and u is the wind speed. The ∆β term is necessary because of the motion of the
vane. If dθ/dt is positive so the vane is rotating in a counterclockwise direction, the relative
motion of the vane with respect to the wind makes the wind appear to strike the vane at an
angle β +∆β. This is illustrated in Fig. 14 for the case β = 0. The vane centroid is rotating
in the counterclockwise direction at a speed rω, which when combined with the wind velocity
u yields an apparent wind velocity u’ atanangle∆β with respect to u. Thevelocityu is a
vector that shows both the speed and the direction of the wind. If rω is small compared with
u, as will normally be the case, then ∆β can be approximated by
Wind Energy Systems by Dr. Gary L. Johnson November 12, 2001