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Substituting eqn 5.136 in 5.134 and equating coefficients of sin ψ and cos ψ gives
A1′′ + (2/TfΩ)A1′ + 2B1′ + (2/TfΩ)B1 = 2clqˆ (5.137)
and
2A1′ + (2/TfΩ)A1 – B1′′ – (2/TfΩ)B1′ = 0 (5.138)
where the dashes denote differentiation with respect to ψ.
The free motion corresponding to eqns 5.137 and 5.138 is found to consist of a
high frequency ‘nutation’ mode, of no practical interest, and a significant low frequency
mode. An approximation to this latter mode can be obtained by ignoring the A1 terms,
giving
dB1/dψ + (1/TfΩ)B1 = clqˆ (5.139)
A similar relationship exists between the lateral cyclic pitch A1 and the rate of
roll p.
In terms of aerodynamic time, eqn 5.139 can be written as
d /d + ( / ) = d /d 1 f 1 B tTB clτ ˆ θτ (5.140)
Then considering hovering flight, the equations of stick fixed longitudinal motion
are
du/dτ – xuu + wcθ = 0
– + d2/d 2 – d /d – = 0
1 1 muu θ τ mqθ τ mB B
cldθ/dτ – dB1/dτ – (tˆ/Tf)B1 = 0
The characteristic equation of this motion can be written as
( + / f ){hovering cubic} – 1 ( – ) = 0 λtˆ T clmBλλ xu (5.141)
where the ‘hovering cubic’ is the uncontrolled characteristic equation, eqn 5.94. The
control derivative mB1 is the moment coefficient for unit cyclic pitch change; and
therefore, for our example helicopter, mB1 = –0.0214 (zero offset).
The quartic, eqn 5.141, has been solved for a large range of values of the following
time T t f /ˆ and the two linkage ratios c c l l = and = 1. 12
The roots indicate a heavily
damped oscillation of short period and a lightly damped oscillation of long period,
which can be regarded as the original undamped stick-fixed motion modified by the
presence of the bar. The roots of this latter oscillation are shown in Fig. 5.18. It can
be seen that the amount of stabilisation provided by the bar is rather limited.
178 Bramwell’s Helicopter Dynamics
The Hiller stabilising bar is similar to the Bell bar, except that small aerofoils on
the bar provide aerodynamic damping in place of the viscous damping. A further
difference is that the pilot controls the bar directly, which acts as a servo control
between the stick and the rotor blades. By their nature, Bell and Hiller stabilising bars
are well adapted to the two-bladed teetering rotors which are a characteristic of these
helicopters.
5.8.2 Automatic flight control systems
Some automatic flight control systems (AFCS) are specifically designed to deal with
the basic instability inherent in conventional helicopters. The latter implies continuous
pilot activity on the controls in order to fly uncomplicated manoeuvres or even
simply straight and level, which can be tiring over long periods. The stability
augmentation system (SAS) described briefly in (a) below is designed to address this
problem. The automatic stabilisation system (ASE) described in (b) below is aimed
at maintaining a desired manoeuvre attitude and therefore normally operates over a
shorter time period. More detailed descriptions of these and other types of AFCS
fitted to helicopters may be found in McLean12 and Pallett and Coyle13. Generally,
these systems are fitted to the larger and more expensive machines, the application of
mechanical gyro devices such as the Bell and Hiller bars being confined to the
smaller helicopters. The latter tend to confer only a limited improvement of stability
and can add considerably to the drag of the rotor head, whilst the former offers a
more flexible means of stabilisation and is accommodated completely within the
airframe.
S
No bar
Tf/tˆ=0
cl =1
–0.2
3
2
cl =1/2
2
34
–0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3
–0.6
–0.4
0.2
0.4
0.8
0.6
4
r
1
Fig. 5.18 Effect of Bell stabilising bar on lateral stability roots
Tf/tˆ=1
Flight dynamics and control 179
(a) SAS (stability augmentation system)
The SAS is designed to maintain the helicopter at the datum to which it has been
trimmed. It uses a simple feedback control in which a rate gyro senses pitch rate, for
example, which, on integration, provides a correcting input at the swash plate (if this
is the means of rotor control). Within the feedback loop, however, there is a so-called
‘leaky integrator’ path parallel with the output from the rate gyro, the effect of these
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Bramwell’s Helicopter Dynamics(91)
