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The torsional damping defined in this way has been calculated by Carta for a case
in which μ = 0.17 and CT /σ = 0.111, and the result is shown in Fig. 9.9. We see that
negative damping occurs in the region 225° to 10°, and in this region we can expect
stall flutter to occur. To test the validity of this assumption, the torsional stress and
pitch link loads for the same flight case, Fig. 9.10, have been examined. It can be
seen that large values of torsional stress and pitch link loads occur in this region,
demonstrating that the calculation of the damping gives a good indication of the
possibility of stall flutter.
The high pitch link loads which occur in stall flutter may be a serious obstacle to
increased flight speeds. According to Ham7, the most profitable way of reducing
these loads is to use aerofoil sections which have high dynamic stall angles, so that
they can operate in the retreating blade region below the stall angle as much as
possible.
0 4 8 12 16 20 24 28 32
Unstable
region
1.2
0.8
0.4
0
0.4
0.3
0.2
0.1
0
Reduced frequency, k
Aerodynamic
damping
0.5
Incidence, α
Fig. 9.8 Contours of blade torsional damping
332 Bramwell’s Helicopter Dynamics
9.5 Main rotor blade weaving
Blade weaving is the name given to a form of instability which can affect two-bladed
‘teetering’ rotors, i.e. rotors with two blades rigidly connected together at their root
ends with a built-in non-zero coning angle, and suspended on a central flapping or
‘teetering’ hinge. The phenomenon involves the same basic coupling mechanism as
the main rotor pitch–flap flutter problem described in section 9.3, but since both
blades move as a single entity, extra terms arise in addition to those of eqns 9.11 and
9.12.
Consider the two-bladed teetering rotor of Fig. 9.11. The blades are joined at angle
2β0 (twice the built-in coning angle) and are feathered so that the steady pitch angle
between the blades is 2θ0. The two blades are then imagined to move as a rigid body
except that torsional flexibility of the control system will allow some mutual feathering.
The latter motion will have a negligible effect on the moments of inertia of the whole
0 50 100 150 200 250 300 350
80
60
40
20
0
ω 0 /Ω 6
Aerodynamic damping moment
Azimuth angle ψ
Fig. 9.9 Aerodynamic torsional damping as a function of azimuth
Pitch link load
Torsion strain
at 90% radius
Torsion strain
at 46% radius
60° 180° 300° 60°
120° 240° 360°
ψ
Fig. 9.10 Variation of pitch link loads with azimuth angle
Nm
Aeroelastic and aeromechanical behaviour 333
rotor system, so the assumption of complete rigidity is justified. If the lines of the
centre of mass intersect at the hub, Euler’s equations, eqns A.1.11 to A.1.13 can be
applied to the rotor as a whole. Coleman and Stempin8, who first investigated this
motion, have shown that, if A, B, C are the principal moments of inertia of the
individual blades, the corresponding moments of inertia of the rotor with coning
angle β0 and collective pitch setting θ0 are
A′ = 2A cos2β0 + 2B sin2θ0 sin2β0 + 2C cos2θ0 sin2β0 ≈ 2(A + Bβ 0
2 )
B′ = 2B cos2θ0 + 2C sin2θ0 ≈ 2B
C′ = 2A sin2β0 + 2B sin2θ0 cos2β0 + 2C cos2θ0 cos2β0 ≈ 2(C – Bβ 0
2 )
assuming A + B = C for the individual blades.
As might be expected, the flapping and lagging moments of inertia are very little
different from those of the original blades, but the pitching moment of inertia is
greatly increased by coning angle. The angular velocity components of the rotor are
the same as those of the previous section, and the linearised Euler equations are
found to be
θ˙˙ + ′ – ′ 2θ + + – β˙ = A
′
′ ′ ′
′
′
C B
A
A B C
A
L
A Ω Ω
β˙˙ + ′ – ′ 2β – + – θ˙ = A
′
′ ′ ′
′ ′
C A
B
A B C
B
M
B Ω Ω
The important difference between these equations and those of the pitch–flap
coupling of section 9.3 is that, since the rotor can no longer be regarded as a lamina,
A′ + B′ – C′ ≠ 0 so that the coefficients of Ω˙ θ and Ω˙ β do not vanish as they did in
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