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flutter frequency ω must be known before C(k) can be estimated. The flutter boundary
is modified as sketched in Fig. 9.7.
Stammers4 has shown that forward flight has a stabilising influence on flutter
which tends to occur at half-integer frequencies.
9.4 Main rotor stall flutter
This is a single degree of freedom flutter problem involving torsional oscillation of
the blade. It normally takes the form of a limit cycle oscillation giving rise to high
B
B″
B′
A A′
Flutter
Unstable
Divergence
Stable
σg
A″
Fig. 9.6 Stability boundaries for torsional motion
C(k) ≠ 1
C(k) = 1
σg
Fig. 9.7 Stability boundaries for torsional motion (C(k) ≠ 1)
ν1
2
ν1
2
330 Bramwell’s Helicopter Dynamics
oscillatory loads in the blade pitch control circuit, and occurs over a stalled region of
the retreating blade side of the rotor disc. This region must cover a sufficient proportion
of the total disc area for the instability to allow one or more cycles of torsional
oscillation before the blade moves into a stable (non-stalled) region of the rotor disc.
Consider an aerofoil oscillating periodically in pitch in an airstream of velocity V.
The pitching moment coefficient can be expressed as
C
M
V c
m = a a pt a pt b pt
( )
= + sin + sin 2 + + cos + 12
2 2 0 1 2 1
α
ρ
… …
where α is the instantaneous incidence, p is the angular frequency, and c is the chord.
The work done in one cycle is
W = ∫ M dα
If the oscillations are harmonic, the incidence can be written as
α = α0 + α1 sin pt
where α0 is the mean incidence and α1 is the amplitude of the oscillations.
We then find that
W Vcp b pt t
p
= cos d 12
2 2
0
2 /
1 1
ρ α 2
π ∫
= 12
2 2
πρV c α1b1
The work is therefore proportional to the coefficient b1; if b1 is negative, work is
dissipated and the damping is positive. Thus, the damping depends on the out-ofphase
component of the pitching moment.
The basic mechanism of the instability is negative damping in pitch due to
aerodynamic pitching moment hysteresis caused by the periodic shedding of intense
vorticity at a blade angle of attack instantaneously greater than the static stalling
angle5.
Hovering rotor tests of these oscillations have been described by Ham and Young6.
Explanation of the phenomenon has already been alluded to in Chapter 6. We saw
there that at high incidence a suction peak occurs over the rear part of the aerofoil,
resulting in a large nose down pitching moment, and this gives rise to a substantial
out-of-phase moment of negative damping, particularly at the comparatively low
values of the reduced frequency typical of helicopter operation. The variation of
damping with incidence and reduced frequency is graphically displayed in the threedimensional
diagram taken from Carta’s paper, Fig. 9.8. The ‘hollow’ indicates the
region in which negative damping occurs.
Now, as we saw in Chapter 6, the aerofoil undergoes a large variation of incidence,
and therefore of rate of change of incidence, at high tip speed ratios. It will be assumed
that the rotor blade responds to the changing conditions at its fundamental torsional
frequency ωθ, which, in association with the local chordwise velocity ΩR(x + μ sin ψ),
Aeroelastic and aeromechanical behaviour 331
enables us to define the instantaneous value of the reduced frequency k, i.e. we assume
that the appropriate value of k is given by
k(x, ψ) = ωθb/ΩR (x + μ sin ψ)
where b is the semi-chord. Thus, for a given radial position, the variations of incidence
and reduced frequency trace out a closed path on the surface as shown in Fig. 9.8. In
the range, or ranges, of azimuth angle which lie in the region of negative damping,
torsional flutter can be expected to occur. It has been assumed that the damping at
any azimuth angle is the same as if the incidence and reduced frequency were fixed,
i.e. that the damping corresponding to given values of α and k applies instantaneously
at the given azimuth angle. For the blade as a whole we can define a ‘weighted-mean’
damping by integrating the local two-dimensional values in association with the first
torsional mode shape Q1(x) to obtain
ζmean ζα
0
1
1
= ∫ Q2 (x) ( , k) dx
where the reduced frequency to be used is that defined above.
　
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