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φ = 1.25[cos(3π /2)t – 9 cos(π /2)t + 8]°
The quantities d2φ /dτ2 and dφ /dτ are therefore known and inserted in eqn 5.178 to
obtain the lateral control A1 directly.
It can be seen that for the helicopter with zero offset hinges the manoeuvre is only
about one quarter completed when the stick begins to move in the opposite direction,
and is barely half completed before considerable opposite stick has to be applied to
check the rolling motion and settle into the required bank angle. The hingeless
helicopter, on the other hand, requires very little opposite stick to achieve the same
manoeuvre. The amount of stick movement required is not only much less but is
smoother than that of the helicopter with zero offset. This response would be regarded
as very satisfactory, and is one of the benefits of the hingeless rotor.
0 1 2 3 4 5
t seconds
15
10
5
Rate of roll, deg/s
Hingeless e = 0.04
e = 0
Fig. 5.25 Rate of roll in response to sudden application of lateral control
Hingeless
e = 0 Roll angle
demanded
20°
10°
1 2
t seconds
Angle of bank
Lateral cyclic pitch
6°
5°
4°
3°
2°
1°
0°
–1°
–2°
–3°
–4°
Fig. 5.26 Lateral cyclic pitch to achieve given roll manoeuvre
190 Bramwell’s Helicopter Dynamics
5.9.5 Response to vertical gusts
In this section we shall describe briefly the response of a helicopter to a sharp vertical
gust. A thorough investigation taking into account the time taken for the rotor to enter
the gust and the subsequent blade bending and flapping response would be very
lengthy. We shall assume that the rotor takes a negligible time to enter the gust
completely and, as in the section on forward flight response, we shall assume that
changes of forward speed are negligible, at least for the important initial response.
We shall also neglect the flapping motion of the blades.
The effect of meeting an up-gust, as far as the calculation of the blade forces and
moments is concerned, is identical to the effect of a downward velocity of the complete
helicopter; i.e. if the gust has velocity wg, the vertical force due to the gust alone is
Zwwg and the pitching moment is Mwwg.
The non-dimensional equations of motion corresponding to level flight are therefore
d
d
– –
d
d
= g
w
z w z w w w τ μ θτ
(5.179)
–
d
d
– + d
d
–
d
d
=
d
d
+
2
2
g
m g
w
m w m m
w
w˙τ w q w˙ mww
θ
τ
θ
τ τ (5.180)
The increment of normal acceleration ng is given, as before, by
ng
g
w
w
=
d
d
–
d
c d
μ θ
τ τ


or
n
z
w
= – w (w + w)
c
g (5.181)
from eqn 5.179.
By comparing eqn 5.181 with eqn 5.174, the Laplace transformation of the response
to a sharp-edged gust is
n
z w
w p
z
p
p B p C
w
= – 1 + w
+
+ +
g
c
1
2
1c 1
Γ
′ ′

 

 
(5.182)
Equation 5.182 has been solved for the cases of the helicopter with 4 per cent
flapping hinge offset and one with a hingeless rotor, both with and without a tailplane,
and the results for μ = 0.3 are shown in Figs 5.27 and 5.28.
Both tailless helicopters are unstable, as has been shown before, and we see that,
apart from a slight initial drop, the normal acceleration increases indefinitely, the
hingeless helicopter diverging more rapidly due to the larger, positive, value of mw.
When a tailplane is fitted, the response is improved and the normal acceleration of
the helicopter with offset hinges subsides quite quickly. The hingeless helicopter at
this forward speed is just about neutrally stable, and the normal acceleration remains
practically constant. At higher forward speeds, the hingeless helicopter becomes
unstable and the normal acceleration diverges.
Flight dynamics and control 191
2
1
0
Normal acceleration (g units)
1 2 3
e = 0.04
Hingeless
t seconds
Fig. 5.27 Longitudinal acceleration in response to sharp-edged vertical gust for a tailless helicopter
　
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