Environmental Engineering Reference
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where
and
N
= (
M
- 1)/2 if
M
is odd and
N
= (
M
- 2)/2 if
M
is even.
Note that the coeffi cients in (30) depend on time through
ψ
m
=
ψ
+
m
Δψ
ψ
. This time dependency
is particular. For the usual case of a three-bladed rotor,
M
M
1
⎧
cos(
k
ψ=
),
k
iM
1
⎧
sin(
k
ψ =
),
k
iM
∑
∑
cos(
k
ψ=
)
,
sin(
k
ψ=
)
⎨
⎨
m
m
M
0
M
0
⎩
⎩
m
=
1
m
=
1
so the
coeffi cients will only contain the harmonics that are multiples of the num-
ber of blades, i.e.
k
= 3, 6, 9, … Thus by taking as example the non-dimensional fl ap
defl ection at the blade tip,
v
will be the cone angle of the tip-path plane while
v
v
,1
and
will give the tilt and yaw angles of the rotor. The rotational transformation
is not restricted to the DOF. The same transform is applied to the dynamic equations,
an operation equivalent to considering each equation as a dependent variable to
which eqn (30) applies.
In practical terms, the above analysis is carried out as follows. Consider the full
set of non-linear aeroelastic equations. The fi rst step is to construct a periodic solu-
tion. To this end, the non-linear equations are integrated in time until a periodic
response (with respect to the rotor speed) is reached. If the conditions are close to
instability the time domain calculations provide a response that contains signifi -
cant components in all the basic frequencies of the system. In such a case, all fre-
quencies besides the rotational frequency 1/rev and its basic multiple
M
/rev are
fi ltered by means of Fourier transformation.
The next step is to linearize the problem. Based on the periodic solution obtained,
the system is reformulated in perturbed form. To this system, the rotational transfor-
mation is applied on both the DOF and the equations. The end result of this proce-
dure is a dynamic system with constant coeffi cients and therefore the standard
eigenvalue analysis can provide directly its stability characterization.
The passage from the rotating to the non-rotating system affects the eigenfrequen-
cies. For a simple system, the modes in the non-rotating system will be equal to those
in the rotating system +/- the rotational speed. The modes produced in this way are
called
progressive
and
regressive
. However in the case of a complicated system as that
of a complete wind turbine, the regressive and progressive modes will be coupled with
the non-rotating parts of the system and so their identifi cation is more diffi cult [ 25 ].
Note that the perturbed equations are general and apply to both linear and non-
linear contexts. In fact, non-linear responses can be obtained by iteratively solving
the linearized set of equations until perturbations are eliminated. Therefore this
kind of formulation can be also used for non-linear stability identifi cation. Typi-
cally non-linear damping computations are based on either the assessment of the
aerodynamic work [26] or signal processing of the transient responses [27]. The
assessment of the aerodynamic work is applied at the level of the isolated blade, as
a means of validating linear analysis results. In such a method the work done by
the aerodynamic loads acting on the isolated blade is calculated for the blade
undergoing a harmonic motion according to the shape and the frequency of the
specifi c aero-elastic mode considered. It has been shown that the aerodynamic
work is directly related to the damping of the mode considered [26]. So, it is
v
,1
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