WIND ENERGY SYSTEMS - Cd3wd

Chapter 4—Wind Turbine Power 4–37
Figure 23: Simple torsional oscillator.
The solution to this equation is
θ(t) =A sin ωt + B cos ωt (44)
where A and B are constants to be determined from the initial conditions.
frequency of oscillation is given by
The radian
ω =
√
k T
I
rad/s (45)
We see that the frequency of oscillation is directly proportional to the torsional spring
constant and inversely proportional to the inertia of the disk. If the shaft is rotating rather
than fixed, then ω of Eq. 45 is the frequency of oscillation about the mean shaft speed.
Example
The turbine in the previous example suddenly loses the interconnection to the electrical grid, thus
allowing the high speed shaft to unwind. The inertia of the generator is 10.8 kg·m 2 , which is so much
smaller than the inertia of the turbine blades and gearbox that the generator acts like a torsional
oscillator with the other end of the high speed shaft fixed. Find an expression for θ as a function of
time.
From the previous example we can take θ(0+) = 0.0559 rad. The relative angular velocity dθ/dt
can not change instantaneously, so
dθ(0+)
dt
= dθ(0−)
dt
We can then insert these two initial conditions into Eq. 44 and evaluate the constants.
=0.
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001