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ΔM = ∂M = M (k + k )
∂
∂
θ ∂
Δθ
θ
ξξ ββ
The change in flapping angle β, if ξ is at sufficiently low frequency, can be
expressed as (change of lift moment)/(centrifugal moment) or,
β
ξξ ββ θ =
( + ) /
2
k k M
B
∂ ∂
Ω
But in steady motion, with lift moment ΔM, the coning angle β0 is given by
β θ θ
0
0
2 =
∂M/∂
BΩ
so that
β = (β0 /θ0) (kξξ + kββ)
or β
β θ ξ
β θ
ξ
β
=
( )
1 – ( / )
0 0
0 0
/ k
k
(9.8)
As a result of the flapping motion, the Coriolis moment N causing lagging is, for
small disturbances,
N C C
k
k
= 2 = 2
0 1 – ( / )
0 0 0
Ωβ β Ω
β
θ
ξ
β θ
ξ
β
˙
˙
0
2
⋅
from eqn 9.8. Since this moment is proportional to ˙ ξ , it may be regarded as a viscous
damping moment. Then, including the damping of the lag damper and the drag
moment, represented by F˙ξ, the total damping will be positive if
F
k
k
C ˙ξ
ξ
β θ β
β
θ +
2
1 – ( / )
> 0
0 0 0
⋅ 0
2
Ω
as given by eqn 9.7.
For the helicopter with hinged blades, the criterion can be expressed in terms of
the α2 and δ3 hinges as
324 Bramwell’s Helicopter Dynamics
F C ˙ξ
α
β θ δ
β
θ +
2 tan
1 – ( / )tan
2 > 0
0 0 3 0
⋅ 0
2
Ω
Since F˙ξ represents the lag damping, and is therefore positive, instability can
occur only if the second term is a sufficiently large negative number. The sign of this
term can be regarded as depending only on α2, since δ3 would have to be unusually
large and positive to change the sign of the denominator. Thus, instability is possible
when α2 is negative, i.e. when the blade pitch increases as the blade moves forward
in lagging motion.
9.3 Main rotor pitch–flap flutter
Essential for this form of instability is a mechanism coupling blade pitch and flap.
The type of coupling normally encountered is due to adverse offsets of the chordwise
centre of gravity of the blade section from the blade feathering axis which is normally
coincident with the 1
4 -chord point. This instability is very similar in nature to the
fixed wing bending torsion flutter problem, and has been avoided on almost all
helicopter rotor blades by mass balancing the blade so that the chordwise position of
the centre of gravity of each spanwise element is forward of the blade section 1
4 -
chord point. This is usually accomplished by the use of non-load-carrying mass
balance weights along the full length of the blade.
In the region of the blade root end attachment, local reinforcing usually has the
effect of moving the centre of gravity well aft of the 1
4 -chord point. However, analysis
indicates that for a blade which is reasonably stiff in flatwise bending and torsion, the
product of inertia of the blade about axes coincident with the flapping and feathering
hinge lines is the significant parameter. Therefore, the effect of an aft movement of
the centre of gravity in the root region can be counteracted by the addition of a
relatively small mass positioned near the leading edge of the blade close to the tip.
For the case where complete decoupling is not achieved by mass balancing, then
the blade flatwise and torsional stiffnesses, and the effective torsional stiffness of the
pitch control circuit at the blade root, are important parameters.
The presence of blade flutter would normally be detected by the presence of high
oscillatory loads in the blade pitch control circuit.
Since on a conventional rotor blade the amount of mass balance material is of the
order of 9 to 12 per cent of total blade mass, then considerable weight saving could
be achieved if it were possible to relax the mass balance requirements. However, a
reduction in blade mass implies an increase in blade coning angle for a given value
of lift, and this can lead to problems involving increased lag plane loading, and more
severe damping requirements in lag to suppress certain types of instability. A reduction
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Bramwell’s Helicopter Dynamics(159)
