Page 306 - Wind Energy Handbook

280 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

tion method (Section 5.8.5), the expressions for the displacement and velocity
increments at the tip of blade J are as follows:

6 _ € _ h €
f
f
˜Q J þ m 1 f f J0 þ 3f J0 þ c 1 3f J0 þ f f J0
h 2
˜ f J ¼ (5:115)
2
m 1 ø þ 3c 1 þ 6m 1
h h 2
3 h
_
_
f
˜ f J ¼ ˜ f 3f 0 € (5:116)
f
f f 0
h 2
The derivation of these expressions parallels that for the absolute values of
displacement and velocity at the end of the time step, given in Section 5.8.5.
Similar expressions obtain for the displacement and velocity increments at the
hub due to tower flexure.
(5) Solve Equations (5.108) and (5.114) for the accelerations at the end of the time
step.
(6) Solve the incremental equations of motion again – this time including the
changes in the coupled terms on the right hand side over the time step – to
obtain revised increments of displacement and velocity over the time step.

(7) Repeat Step 5 and Step 6 until the increments of displacement and velocity
converge.

(8) Repeat Steps 1–7 for the second and subsequent time steps.

If the analysis is being carried out to obtain the response to deterministic loads,
advantage may be taken of the fact that the behaviour of each blade mirrors that of
its neighbours with an appropriate phase difference. This means that the number of
equations of motion can be reduced to two, and the analysis iterated over a number
of revolutions until a steady-state response is achieved. For example, in the case of
a machine with three blades, A, B and C the values of blade B and blade C tip
velocities and accelerations, which are required on the right-hand side of Equation
(5.114), would be equated to the corresponding values for blade A occurring T=3
and 2T=3 earlier (T being the period of blade rotation).
Figure 5.32 shows the results from the application of the above procedure to the
derivation of blade tip and hub displacements in response to tower shadow
loading, considering only the blade and tower fundamental modes. The machine is
three bladed and the parameters chosen are, as far as the rotor is concerned,
generally the same as for the rigid tower example in Section 5.8.5 illustrated in
Figure 5.25. The tower natural frequency is 1.16 Hz, and the tower damping ratio
(which is dominated by the aerodynamic damping of the blades) is taken as 0.022.
It can be seen that the tower response is sinusoidal at blade passing frequency,
which is the forcing frequency. The amplitude is only about one fiftieth of the
maximum blade tip displacement of about 30 mm, reflecting the large generalized