WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–38
θ(0+) = 0.0559 = A sin 0 + B cos 0
dθ(0+)
= 0 = Aω cos 0 + Bω(− sin 0)
dt
From these two equations we observe that B = 0.0559 and A =0.
The relative angular velocity about the mean angular velocity is given by Eq. 45.
The expression for θ is then
ω =
√
20, 400
10.8
=43.46 rad/s
θ =0.0559 cos43.46t rad
The generator will oscillate with respect to the gearbox at the rate of 43.46 rad/s or 6.92 Hz. The
amplitude is not large but the torque reversal twice per cycle would probably produce audible noise.
An actual wind turbine drive train is quite complicated[7]. The rotor itself is not a perfect
rigid body, but is able to flex back and forth in the plane of rotation. The rotor can therefore
be modeled as a stiff shaft supplying power to the rotor hub. We have the various inertias
of the rotor blades, hub, gearbox, generator, and shafts. There is damping caused by the
wind, the oil in the gearbox, and various nonlinear elements. This damping causes any
oscillations to die out if they are not being continually reinforced. A reasonably complete
model[7] may have six inertial masses separated by five shafts and described by five second
order differential equations. These equations are all coupled so the solution process requires
a computer program. The full solution contains a number of oscillation frequencies, some of
which may be heavily damped and others rather lightly damped. If the system is pulsed at
the lightly damped oscillation frequencies, serious damage can occur.
The most important source of a pulsation in the driving function is that of the rotor blades
passing by the tower each revolution. If we had only one blade on the rotor, we would have
one pulse per revolution. If the rotor were spinning at 40 r/min, the pulsation frequency seen
by the shaft would be 40 pulses per minute, or 40/60 pulses per second, 0.667 Hz. A rotor
with two perfectly identical blades will have the lowest pulsation frequency equal to two pulses
per revolution, or 1.33 Hz for a 40 r/min rotor. In practice, the two blades are not identical,
so both 0.667 and 1.33 Hz would be available to drive oscillations near those frequencies.
These driving frequencies are normally referred to as 1P and 2P. Oscillation frequencies near
a multiple of the driving frequencies can also be excited, especially 4P, 6P, 8P, and so on.
Example
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001