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these two moments must be in equilibrium with an aerodynamic moment produced
by a cyclic pitch variation in the tip path plane, and the rotor achieves this by
appropriate tilts a1 and b1 relative to the shaft. This cyclic pitch variation, by the
arguments of section 1.6.1, is easily seen to be – a1 sin ψ + b1 cos ψ. By comparing this
with eqn 1.11, the aerodynamic moment on one blade is seen to be – BΩ2γ (a1 sin ψ
– b1 cos ψ)/8. For all blades there would therfore be a steady moment bBΩ2γ b1/16
acting in the nose down sense and a moment bBΩ2γa1/16 in the direction ψ = 90°. For
these moments to be equal and opposite to those above, we must have, as before,
a1 = – 16 ˆ q/γ and b1 = – ˆ q
where we have taken B = C, since flapping and in-plane moments of inertia of the
blade are almost identical – typically, C ≈ 1.003B.
When the pitching rate q is not constant, eqn 1.3 becomes (e = 0)
d
d
+
8
d
d
+ =
8
( ) cos – 2 ( ) sin +
d
d
cos
2
2
β
ψ
γ βψ
β γˆψ ψ ˆψ ψ ψ ψ
ˆ
q q
q
(1.18)
According to the theory of differential equations there is a solution in the form
β = a0(ψ) – a1(ψ) cos ψ – b1(ψ) sin ψ (1.19)
where, as indicated, the flapping coefficients are no longer constants, as in the previous
cases, but functions of time or azimuth angle.
The case of sinusoidally varying pitching velocity, which is important in stability
investigations, has been analysed by Sissingh2 and Zbrozek3. Taking q = q0 sin vψ
and substituting eqn 1.19 into eqn 1.18 gives, after equating coefficients of sin ψ and
cos ψ,
γ
ψ
γ
ψ ψ
ψ 8
+ 2
d
d
–
8
d
d
–
d
d
1 = – 2 sin
1 1
2
1
a 2 0
a b b
qˆ v
γ
ψ ψ
γ
ψ ψ 8
d
d
+ 2
d
d
+
8
+ 2
d
d
1 = cos
2
1
2 1
1
0
a a
b
b
qˆ v v
The solutions for a1(ψ) and b1(ψ) are straightforward, but rather lengthy. Sissingh
has shown that the tip path plane oscillates relative to the shaft, performing a beat
motion out of phase with the shaft oscillation. Now, v is the ratio of the pitching
frequency to the rotational frequency of the shaft and in typical disturbed motion is
usually much less than 0.1. On this basis Zbrozek has shown that, to good
approximations, the lengthy expressions for a1 and b1 can be reduced to
a1 ≈ –16 ˆ q/γ + [(16/γ)2 – 1]d ˆ q/dψ
Basic mechanics of rotor systems and helicopter flight 17
To pilot’s control To pilot’s control
Fig. 1.13 Swash plate mechanism
b1 ≈ – ˆ q + (24 /γ)d ˆ q/dψ
Since a typical lateral or longitudinal stability oscillation is about 10 seconds, and
the period of the rotor is about 1
4 second (240 rev/min), v is about 0.025. With ˆ q =
ˆ q0 sin vψ, the second terms of a1 and b1 are quite small and by neglecting them
Zbrozek’s expressions for a1 and b1 become the same as for the steady case. Thus, in
disturbed motion, both a1 and b1 are proportional to q, and the rotor responds as if the
instantaneous values were steady. This is the justification for the ‘quasi-steady’ treatment
of rotor behaviour in which the rotor response is calculated as if the continuously
changing motion were a sequence of steady conditions. This assumption greatly
simplifies stability and control investigations. The ‘quasi-static’ behaviour of the
rotor might also have been expected from its response as a second order system. The
impressed motion considered above corresponds to forcing at a very low frequency
ratio, and it is well known that the response is almost the same as if the instantaneous
value of the forcing function were applied statically.
If the rolling case is considered, with constant angular velocity and ε = 0 as in the
pitching case, it can be shown that the equivalent to eqn 1.16 is
d
d
+
8
d
d
+ =
8
sin + 2 cos
2
2
β
ψ
γ βψ
β γpˆ ψ pˆ ψ (1.16a)
in which ˆp = p/Ω.
1.7 The cyclic and collective pitch control
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Bramwell’s Helicopter Dynamics(16)
