WIND ENERGY SYSTEMS - Cd3wd

Chapter 3—Wind Measurements 3–24
These coefficients can be evaluated for given vane parameters and a particular wind speed
and inserted in Eq. 29 to give an expression for θ as a function of time.
When the roots are complex conjugates a slightly different general solution is normally
used. Equation 29 is valid whether the roots are real or not, but another form of the solution
yields more insight into the physical phenomena being observed. This alternative form is
where
θ = γ 1 + Be −ωnξt sin(ω d t + σ) rad (32)
ω d = ω n
√1 − ξ 2 rad/s (33)
This form of the solution is a sinusoidal function of time with a phase angle σ andanexponentially
decaying amplitude, a so- called damped sinusoid. When the initial conditions for θ
and dθ/dt are substituted into this expression, we get two equations involving the unknown
quantities B and σ.
0 = γ 1 + B sin σ
0 = Bω d cos σ − Bω n ξ sin σ (34)
Solving for B and σ yields
B = − γ 1
sin σ
σ = tan −1 ω √
d
1 − ξ
ω n ξ =tan−1 ξ
(35)
When ξ is zero, the sinusoid of Eq. 32 is not damped but oscillates at a frequency ω d =
ω n . We therefore call ω n the natural frequency of the system. The quantity ξ is called the
damping ratio since it contributes to the decay or damping of the sinusoid. When ξ is less
than unity, we have oscillation, and this is referred to as the underdamped case. When ξ is
greater than unity, we no longer have oscillation but rather a monotonic change of angle from
the initial to the final position. This is called the overdamped case. Critical damping occurs
for ξ = 1. For most direction vanes used in meteorological applications, ξ is well below 1,
giving a damped oscillatory response.
Wind Energy Systems by Dr. Gary L. Johnson November 12, 2001