Environmental Engineering Reference
In-Depth Information
Taking as example the ONERA model, loads are expressed by in total four circu-
lation parameters:
2L for the lift which correspond to the attached and sepa-
rated contributions respectively,
Γ
1L ,
Γ
2M for the pitching moment,
each satisfying a second order differential equation. This nice feature of the model
allows including the blade aerodynamics into the system as extra dynamic equations.
The spanwise piecewise constant distributions of the four circulation parameters
become new DOF and are treated in the same way as any other. The combined set
of the element DOF defi ne the so-called aeroelastic element [21]. Consequently
linearization with respect to a reference state can be extended to also include blade
aerodynamics. The complete linearized system is the basis for linear stability anal-
ysis which in this case provides the coupled aeroelastic eigenmodes and eigenfre-
quencies. There is however a signifi cant complication: the coeffi cients of the
dynamic system are no longer constant but time varying.
Γ
2D for the drag and
Γ
6 Rotor stability analysis
Linear stability analysis of fi rst order systems
with varying coeffi -
cients is still possible using Floquet's theory provided that the system is periodic
with period T : A ( t - T ) = A ( t ), b ( t - T ) = b ( t ) [ 22 , 23 ]. The solution obtained
takes a form similar to that given in eqn (26):
yAyb
=+
t
y=
(, )
tt
y
+ ℜ
(, ) d
ttt
b
(28 )
00
t
0
where
denotes the state transition matrix of the system with respect to
initial conditions defi ned at t 0 . The diffi culty in applying eqn (28) is linked to the
construction of the transition matrix. For a system involving a large number of
DOF, for each one of them the equations must be integrated over one period. So
depending on the size of the system this task can become exceedingly expensive.
Fortunately, for rotors equipped with identical blades and rotating at constant
speed
(, )
tt
0
stability analysis of the rotor system as a whole, is simplifi ed signifi cantly
by applying the Coleman transformation [23, 24]. Let
Ω
t denote the azimuth
position. Then for an M -bladed rotor, the expressions of any quantity defi ned on
the blades v ( m ) (
ψ
=
Ω
ψ
) will be the same except for an azimuth shift m
Δ ψ
,
Δ
ψ
= 2
π
/ M .
By introducing the following new variables:
M
M
1
1
()
m
()
m
v
()
ψ=
v
()
ψ
v
()
ψ=
v
()cos(
ψ
k
ψ
)
0
ck
,
m
M
M
m
=
1
m
=
1
M
M
1
1
( 29)
()
m
m
()
m
v
()
ψ=
v
( (1)
ψ−
v
()
ψ=
v
()sin(
ψ
k
ψ
)
M
/2
s k
,
m
M
M
m
=
1
m
=
1
it follows that:
N
(
)
()
m
m
v
()
ψ= +
v
v
cos(
k
ψ +
)
v
sin(
k
ψ +
)
v
(1)
( 30)
0
ck
,
m
sk
,
m
M
/ 2
k
=
1
if M even
 
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