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the latter case. Let us write the equations in the non-dimensional form originally used
by Coleman and Stempin. They are
d
d
+ d
d
+ ( + ) – (1 – )
d
d
– = 0
2
2 A A
θ
ψ
θψ
θ Ι βψ
HD I Kθ′ HDβ (9.17)
Ω
β
2β0
θ
Fig. 9.11 Teetering rotor with combined flapping and feathering
2θ0
334 Bramwell’s Helicopter Dynamics
d
d
+
d
d
+ + (1 – ) d
d
+ = 0
2
2 A
β
ψ
βψ
β Ι θψ
HB IB HBθ (9.18)
In the above equations, HB and HD represent damping of the pitching and flapping
motion; IA and IB are defined by
Ι A
C B
A
H
A
= ′ – ′ + 1
′
′
′
and
Ι B
C A
B
H
B
= ′ – ′ – 5
′
′
′
and we have approximately
1 – = 2 + 0
I B 2 1
A
H
A β A′
′
and
1 – = – 0
I 2 5
H
B 2β B′
′
where the terms in H1′ and H5′ represent aerodynamic damping. It can be seen that,
since B/A is usually a very large ratio, 1 – IA is strongly dependent on coning angle
whereas 1 – IB varies little. Kθ′ is the stiffness of the pitch control. Coleman and
Stempin state that HD depends strongly on coning angle but that HB is practically
unaffected.
The equations of motion 9.17 and 9.18 are similar in form to those of section 9.3
(eqns 9.11 and 9.12) except for the presence here of the terms in dβ /dψ and dθ/dψ.
(Since the motion has been referred to principal axes through the centre of gravity of
the blade, the term in β˙˙ is absent.) Coleman and Stempin’s calculations show that
these extra terms are destabilising and that the instability depends mainly on the
coning angle and to a lesser extent on the pitch setting angle θ0. An important result
of their investigations is that instability can occur even when the centre of mass of the
blade is ahead of the 1
4 -chord point. A sketch of the variation of the stability boundaries
with control stiffness, coning angle, and chordwise c.g. location is given in Fig. 9.12.
When instability occurs, the blade tips trace out a wavy or ‘weaving’ path which
gives rise to the name of the phenomenon.
9.6 Tail rotor pitch–flap (‘umbrella mode’) instability9
This type of instability depends on the existence of some form of coupling between
blade pitching and flapping motions, and is very similar to the main rotor blade
pitch–flap flutter problem described in section 9.3.
In the case of the tail rotor, the coupling usually arises from the large δ3 coupling
introduced to reduce tail rotor blade flapping and stresses in forward flight. This
Aeroelastic and aeromechanical behaviour 335
coupling is normally of the order of one degree of reduction in blade pitch per degree
of (‘upward’) blade flapping.
This form of instability is usually experienced as a limit cycle oscillation with all
blades moving in phase. Hence the description ‘umbrella mode’ applied to this type
of motion.
The stiffness and damping of the pitch control circuit between the blades and the
pitch control actuator have a fundamental influence on this phenomenon.
In general, stable solutions can be found with both high and low pitching mode
frequencies, in the absence of pitch circuit damping.
Where the possibility of instability arises, the onset of this is difficult to predict
due to the presence of backlash and friction damping effects in the pitch control
circuit, and in fact the occurrence in practice tends to be somewhat erratic. Disturbances
below a certain threshold will subside, but beyond this divergence will occur, rapidly
reaching a constant level.
The frequency of the oscillation is at or close to the fundamental flapping frequency
of the tail rotor blade which is typically 1.2Ω. The reason that it is significantly
greater than 1Ω is primarily due to the additional aerodynamic stiffness arising from
the δ3 coupling.
Figure 9.13 indicates a typical stability boundary as a function of pitch control
circuit stiffness and damping.
9.7 Main and tail rotor flap–lag instability
We have already seen in Chapter 1 that blade flapping produces large Coriolis moments
　
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