WIND ENERGY SYSTEMS - Cd3wd

Chapter 3—Wind Measurements 3–23
These roots are
s 2 +2ξω n s + ω 2 n = 0 (27)
s 1 = −ω n ξ + ω n
√ξ 2 − 1
s 2 = −ω n ξ − ω n
√ξ 2 − 1 (28)
We see that the character of the solution changes as ξ passes through unity. For ξ greater
than unity, the roots are real and distinct, but when ξ is less than unity, the roots are complex
conjugates. We shall first consider the solution for real and distinct roots.
Suppose that the vane is at rest with both θ and γ equal to zero, when the wind direction
changes suddenly to some angle γ 1 . We want to find an expression for θ which describes the
motion of the vane. Our initial conditions just after the initial change in wind direction are
θ(0+) = 0 and ω(0+) = 0, because of the inertia of the vane. Since ω = dθ/dt, weseethat
dθ/dt is zero just after the wind direction change. After a sufficiently long period of time
the vane will again be aligned with the wind, or θ(∞) =γ 1 . The latter value would be the
particular solution for this case and would have to be added to Eq. 26 to get the general
solution for the inhomogeneous differential equation, Eq. 23.
θ = A 1 e s1t + A 2 e s2t + γ 1 (29)
Substituting the initial conditions for θ and dθ/dt in this equation yields
0 = A 1 + A 2 + γ 1
0 = A 1 s 1 + A 2 s 2 (30)
Solving these two equations for the coefficients A 1 and A 2 yields
A 1 =
γ 1
s 1 /s 2 − 1
A 2 =
γ 1
s 2 /s 1 − 1
(31)
Wind Energy Systems by Dr. Gary L. Johnson November 12, 2001