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˙
+ + 4
3
i
2C
a
D θλ ξ
˙
Ω (9.24)
The constant terms on the right-hand sides of eqns 9.23 and 9.24 give the steady
state flapping and lagging angles β0 and ξ0. Since we are concerned only with
perturbations from the steady state, these terms can be omitted; but, since the last two
terms of eqns 9.19 and 9.20 representing the Coriolis acceleration are products, they
must be written in first order form as 2BΩβ0
˙ ξ
and –
2CΩβ0
˙β
.
Finally, if it is assumed that the flapping and lagging moments of inertia, B and C,
are equal, the equations of disturbed motion become
d
d
+
d
d
+ +
d
d
= 0
2
1
2 β
ψ
γ βψ
λ β ξψ
2 8 ξ
C˙ (9.25)
Fβ˙ Fξ˙
βψ
ξ
ψ
ξψ
κ ξ
d
d
+
d
d
+
d
d
+ = 0
2
1
2
2 (9.26)
Aeroelastic and aeromechanical behaviour 339
where
C˙ξ = 2β – γ 2θ – 4λ
3
0 8 i
F˙β = γ θ – 8λ β
3
8 i – 2 0
F k
C
a
D
ξ˙ ξ˙
= γ + θλ
2
+ 4
3
8 i
and where k˙ξ is the non-dimensional artificial lag damping, if any. The characteristic
equation of this motion is
( 2 + /8 + )( + + ) – = 0
1
2 2
1
λ γλ λ λ λ κ2 λ2 ξ ξβ F˙ C˙F˙
which is of the form
Aλ4 + Bλ3 + Cλ2+ Dλ + E = 0
where
A = 1
B = γ /8 + F˙ξ
C = λ κ γ1 ξ ξ β
2
1
+ 2 + F˙/8 – C˙F˙
D = γκ λ 1 ξ
2
1
/8 + F˙ 2
E = λ κ 1
2
1
2
To find the neutral stability boundaries we equate Routh’s discriminant to zero,
i.e. we put
R = BCD – D2 – B2E = 0
It has been shown by Ormiston and Hodges10 that this expression can be put into
the form
( – 4 /3) =
2( – 1)(2 – )
2
+ +
64 ( – )
(1 + )( + ) i
2 1
4
1
2
1
2
1
2
1
2
2
1
2
1
2 θ λ
λ
λ λ
α λ κ
ξ γ ακ αλ
C
a
D k
˙
(9.27)
where
α = k˙ξ + 2CD/a + 4θλi/3
Equation 9.27 includes artificial damping omitted by Ormiston and Hodges.
The relationship, eqn 9.27 shows clearly that instability, if it occurs at all, does so
only if 1 < λ1
2 < 2, i.e. when the flapping frequency is between Ω and Ω√2. Further,
for a given value of λ1, the lowest possible value of θ for instability occurs when κ1
= λ1, i.e. when the lagging and flapping frequencies are identical, in which case the
corresponding value of θ is given by
340 Bramwell’s Helicopter Dynamics
( – 4 /3) =
(2 / )
2( – 1)(2 – ) i
2 1
4
1
2
1
2 θ λ
λ
λ λ
CD akξ˙
The absolute minimum value of collective pitch occurs when λ1 = κ1 = √(4/3) =
1.153, from which we easily find that
θ = 4λi/3 + 2√(2CD/a + k˙ξ) (9.28)
The stability boundaries for a case in which k˙ξ = 0, taken from Ormiston and
Hodge’s paper, is shown if Fig. 9.16. It is clear that, unless the lag frequency is higher
than about 0.95Ω, instability would not be expected to occur.
The vast majority of rotor systems, both articulated and hingeless, with more than
two blades have fundamental lag frequencies significantly below this value and are
therefore not likely to be susceptible to this form of instability. (The case of the twobladed
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Bramwell’s Helicopter Dynamics(167)
