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(9.32)
The interpretation of eqns 9.31 and 9.32 is that the centre of gravity of the rotor
whirls round the hub with a displacement consisting of two modes, one with frequency
(κ + 1)Ω and amplitude rgξ0σ1/2b and the other with frequency (κ – 1)Ω and amplitude
rgξ0σ-1/2b. The motion corresponding to the upper frequency (κ + 1)Ω is in the same
direction as the rotor rotation and is described as a ‘progressive’ motion; the motion
corresponding to the lower frequency (κ – 1)Ω is in the same or opposite direction to
the rotor rotation according to whether κ – 1 is positive or negative. When negative,
the motion is described as ‘regressive’.
The whirling centre of gravity produces periodic inertia forces which excite motion
of the whole airframe on its undercarriage (‘chassis’ mode). If one of the frequencies
of the oscillating inertia forces coincides with a chassis frequency, the potential for
the occurrence of ground resonance exists.
This coincidence of frequencies can be represented in diagrammatic form, Fig.
9.19, which applies to an articulated rotor with no lag hinge restraint. The comparable
diagram for a rotor with hinge restraint, or utilising an elastic hub element is shown
in Fig. 9.20.
For a blade without hinge restraint, i.e. without a drag-hinge spring or elastic
element, the lag frequency will always be less than the rotor speed and, therefore,
κ – 1 will always be negative. In Figs 9.19 and 9.20 the horizontal line represents the
(constant) chassis frequency ωc.
Aeroelastic and aeromechanical behaviour 345
In Fig. 9.20 the point A represents the blade frequency when the rotor hub is
stationary and is assumed here to be higher than the chassis frequency.
As the rotor speed increases, the branch corresponding to the whirl frequency
(κ – 1)Ω intersects the chassis line at B and again at C when (κ – 1) < 0.
If the chassis frequency were higher than the frequency of the non-rotating blade
there would have been an intersection with the (κ + 1)Ω branch.
The corresponding intersection for the case of the articulated rotor is point D on
Fig. 9.19. It will be shown later that this intersection and the one corresponding to the
point B cannot lead to instability. Ground resonance, if it occurs, is associated only
with intersection C.
Let us now derive the equations of motion of the chassis–rotor system. It will be
assumed that the chassis motion is confined to a single degree of freedom in the plane
of the rotor – say, the lateral direction. The extension to two degrees of chassis
motion is quite straightforward, but the analysis becomes rather too involved for
simple results to be obtained. In practice, however, the chassis frequencies in the two
directions are often far enough apart for the single-degree-of-freedom analysis to be
applied with reasonable approximation to either direction separately.
D
C
(κ + 1)Ω, progressive
(κ – 1)Ω, regressive
ωc
Rotor speed Ω
Blade frequency, κΩ
Fig. 9.19 Uncoupled chassis and rotor frequencies – articulated rotor, no hinge restraint
(κ – 1)Ω, regressive
C
B
ωc
Rotor speed Ω
Blade frequency, κΩ
(κ + 1)Ω, progressive
A
Fig. 9.20 Uncoupled chassis and rotor frequencies – rotor with drag hinge spring or elastic element
346 Bramwell’s Helicopter Dynamics
The equations of blade lagging motion, relative to an unaccelerated hub, have
already been obtained in Chapters 1 and 7. We must now include the inertia moment
acting on the blade due to lateral oscillations of the chassis mode. Referring to Fig.
9.21 the inertia force on a blade element due to hub acceleration is ˙y˙ dm in the
negative y direction. Hence, the inertia lagging moment Ni about the real or virtual
hinge is
N ry m
eR
R
i = – ∫ ˙˙ cos (ψk + ξk)d
For small lagging angle ξk, and neglecting the product ˙y˙ξ k , we have, approximately,
N y k r m
eR
R
i = – ˙˙ cos ψ ∫ d
= – Mbrg ˙y˙ cos ψk (9.33)
where Mb is the mass of the blade. The equation of motion of the kth blade can then
be written
ξ˙˙k + 2κ δξ˙k + κ2 2ξ˙k = – M r ˙y˙ cos ψk /I
Ω Ω b g
= – ( ˙y˙/lb ) cos ψ k (9.34)
where δ is the damping coefficient, which may include both aerodynamic and artificial
　
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