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teetering rotor with built-in coning angle is treated separately in section 9.5.)
An exception to the general rule was the gyro-controlled ‘rigid’ rotor developed
by Lockheed, where the fundamental lag frequency was greater than 1Ω and the
design of such a system would have to include consideration of this type of instability.
However, the fundamental lag frequency of tail rotors is normally greater than 1Ω
(but significantly less than 2Ω in order to avoid excessive blade and hub vibratory
loading), and tail rotors must be designed to avoid this form of instability, which is
generally referred to as tail rotor ‘buzz’ and is usually of a mild ‘limit cycle’ nature.
Since the tendency to instability increases as the fundamental flap and lag frequencies
approach the same value, it is instructive to examine the factors which influence
these frequencies. It is usually the case that the elastic stiffness of the root region of
the blade is significantly less in the flatwise direction compared to the chordwise
direction. Hence, as blade pitch is increased, the lower flatwise stiffness reduces the
lag frequency from its value at zero pitch. Furthermore, due to the large value of
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
1.0 1.1 1.2 1.3 1.4 1.5 1.6
Flap frequency ratio, λ1
θ = 0.2
0.3
0.4
0.5
κ1 = λ1
Lag frequency ratio, κ1
Fig. 9.16 Stability boundaries for coupled flap–lag motion
Aeroelastic and aeromechanical behaviour 341
pitch–flap (δ3) coupling typically employed on tail rotors (in order to reduce flapping
in forward flight), the fundamental flapping frequency will be well above lΩ.
Consequently it is very probable that the flap and lag frequencies will coalesce at
high values of blade pitch.
Figure 9.16 clearly indicates the increasing area of instability as blade pitch is
increased.
9.8 Tail rotor pitch–flap–lag instability9
The analysis describing the tail rotor ‘buzz’ phenomenon referred to in section 9.7
does not require the inclusion of the torsion, or pitch, degree of freedom, and hence
may be considered to represent the case of a torsionally stiff blade and pitch control
circuit.
If this additional degree of freedom is included in the analysis, an additional
region of instability appears which represents the much more severe instability known
as tail rotor ‘bang’.
This form of instability is associated with the coalescence of the flap, lag and
torsion mode frequencies.
Figure 9.17 illustrates the position of the stability boundaries as a function of the
4
3
2
1
1.0 1.5 2.0
Lag mode frequency, Ω
Unstable
Stable
Torsion mode frequency, Ω
Fig. 9.17 Stability boundaries for coupled pitch–flap–lag motion
A
B
342 Bramwell’s Helicopter Dynamics
frequencies of the torsion and lag modes, for typical values of uncoupled flap frequency,
δ3 coupling, and pitch angle setting.
Region A of the figure represents the ‘buzz’ instability appropriate to high values
of torsional stiffness, and Region B represents the ‘bang’ instability.
One aspect which has to be given careful consideration is the placement of the
torsion mode frequency. Because the blades will experience different stiffnesses at
the pitch change track rod depending upon the phase relationship between the torsional
motion of each blade, a number of torsion mode frequencies will exist.
Each of these frequencies must be chosen so as to avoid the unstable region.
This is an example of a case where the behaviour of the tail rotor and control
system must be considered in combination in order to define the correct data for the
appropriate single blade analysis.
In practice, increasing the bending stiffness of the pitch change ‘spider’ arms, and
of the pitch change rod attached to the centre of the ‘spider’, has produced freedom
from both the ‘buzz’ and ‘bang’ instabilities.
9.9 Ground resonance
Any helicopter with a main rotor blade fundamental lag mode frequency less than 1Ω
is susceptible in principle to the instability known as ground resonance.
The degrees of freedom involved are the lead–lag motion of the rotor blades and
any fuselage mode containing motion of the rotor hub in the rotor plane.
If the blades lag in phase, the centre of gravity of the rotor system remains on the
axis of rotation, but out-of-phase oscillatory motion of the blades will cause the rotor
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Bramwell’s Helicopter Dynamics(168)
