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equation
Aλ6 + Bλ5 + Cλ4 + Dλ3 + Eλ2 + Fλ + G = 0
where
A = 1
B = 2ν
C = 4 + 2χ + ν2 + F(kA+ kB)
D = 2ν(2 + χ) + (kA+ kB) (K+ Fν)
E = χ2 + ν2 + (kA+ kB) [F (4 + χ) + Kν] + kAkB F2
F = (kA+ kB) (χK + 2Fν) + 2FKkAkB
G = kAkB (K2 + 4F2 )
Taking as typical values
ν = 0.836, χ = 0.245, F = 1.08, K = 1.146
kΑ = 0.102, kΒ = 0.0204,
the roots of the sextic are found to be
λ1,2 = – 0.408 ± 2.03i
λ3,4 = – 0.215 ± 0.246
λ5 = – 0.356, λ6 = – 0.07
If the complex roots are substituted back into the equations of motion and the
356 Bramwell’s Helicopter Dynamics
ratios of the variables are examined, as was done in Chapter 5, it is found that the first
complex pair corresponds to a tilt of the rotor disc whose axis moves in the same
direction as the rotor with a frequency just over twice that of the rotor; the second
pair of roots corresponds to a rotor tilt which is a retrograde, or backward, motion,
with a frequency about a quarter that of the rotor. It is this latter mode of motion
whose frequency is likely to lie close to the lagging frequency of the blade.
Let us suppose that the shaft is fixed, i.e. that the pitching and rolling motion is
now absent. This assumption can be expressed mathematically by making kA and kB
infinitely large. The sextic is found to degenerate into a quartic which, with the other
constants having the same values as before, has the roots
λ1,2 = – 0.42 ± 2.03i
λ3,4 = – 0.42 ± 0.21i
It can be seen that the fast, forward, motion is hardly affected by constraining the
angular motion of the fuselage and that only the damping of the slow, backward,
mode is changed significantly. It can easily be shown15 that the motion described
above is exactly the same as that of a damped spring-restrained gyroscope whose
polar moment of inertia is the same as that of the rotor, and the spring stiffness is the
same as the hub moment due to the rotor tilt.
The motion of the helicopter consists, therefore, of a fast rotor precession, which
is practically independent of the motion of the fuselage, and a slow precession in
which the fuselage motion, mainly rolling, has some influence on the damping and
frequency. Because the fuselage moves considerably in this latter mode, and appears
to rock under the rotor, it is often referred to as the ‘pendulum’ mode, but this is an
incorrect description since, as we have seen, the mode is still present even when the
fuselage is fixed. The ‘slow gyroscopic mode’ is suggested as a more apt description.
The two real roots of the sextic correspond to the damping in pitch and roll, as
already discussed in Chapter 5.
We now have to investigate the interaction of this blade flapping and fuselage
motion with the lagging motion of the blades. Due to the rolling and pitching of the
fuselage, the associated acceleration of the hub contributes inertia terms to the lagging
equations as in the case of ground resonance. However, blade flapping also occurs in
this motion, and the corresponding Coriolis inertia terms must be included. Also, the
inertia of the lagging blades causes further pitching and rolling moments on the
helicopter.
The Coriolis terms due to blade flapping are proportional to β dβ/dψ, which, for
small perturbations, can be linearised to a0 dβ/dψ, a0 being assumed constant at the
value for steady hovering flight. There is also a lagwise aerodynamic force due to the
change of direction of the local flow when the blade flaps, and this is clearly proportional
to the local induced velocity and to the flapping rate dβ/dψ. Thus, both the coning
angle and the induced velocity are parameters of the motion.
If the co-ordinate of a point of the lagging blade is expressed as
Y = RT1 (x) ξ(ψ)
Aeroelastic and aeromechanical behaviour 357
the following non-dimensional quantities may be defined15, as
ˆH
bg R
W
mT x x J
mS T S x x
mT x
=
2
( )d ; =
(d /d )d
0 d
1
1
1 1 1 ∫ ∫
∫
0
1
0
1
1
2
L aJ H
mT x
mT x
= 2 ; =
d
d
0
1
1
2
∫
∫
0
1
0
1
and in addition to eqns 9.50 and 9.51 we have the roll, pitch, and lagging equations
in the form
hH
i
k b
p
A
A
ˆ ˆ
+ –
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Bramwell’s Helicopter Dynamics(176)
