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so that for given values of l/h and Mf, the cyclic pitch to trim is roughly that required
to eliminate the backward flapping of the rotor. This is a convenient way of interpreting
the cyclic pitch to trim and shows the advantage of using the tip path plane as a
reference plane in this case. With Mf = 0, Fig. 1.23 can be redrawn as Fig. 1.24.
1.15 The attitude of the helicopter
Since T ≈ W, eqn 1.44 can be written
(D/W) cos τc + H/W = B1 – θ (1.50)
Eliminating B1 by means of eqn 1.46 gives
θ τ = –
cos
– +
– + +
+
c f s 1
s
D
W
HW
M WlR HhR M a
WhR M
(1.51)
If Ms is negligible,
θ = – (D/W) cos τc – l/h + Mf/WhR
Thus, for a given c.g. position and supposing Mf to be constant, the helicopter
fuselage attitude is directly proportional to the drag and, hence, the square of the speed.
Another important conclusion is that, unlike a fixed wing aircraft, the attitude depends
very little on the angle of climb for, since the angle of climb is contained only in
cos τc even quite steep climbs have little effect on (D/W) cos τc and therefore on θ.
1.16 Lateral control to trim
Referring to Fig. 1.25, resolving horizontally, with T ≈ W and ignoring the sideways
pointing Y force,
W(A1 + b1 + φ) + Tt = 0 (1.52)
where Tt is the tailrotor thrust.
Taking moments about O, and for small angles of bank,
WfR + WhR(A1 + b1) + Ms(A1 + b1) + TthtR = 0 (1.53)
where fR is the lateral displacement of the c.g. and htR is the tailrotor height.
Solving eqn 1.53 for A1 gives
A b
WfR T h R
1 1 WhR M
t t
s
= – –
+
+
(1.54)
which is the lateral cyclic pitch to trim.
Basic mechanics of rotor systems and helicopter flight 31
Eliminating A1 from eqn 1.54 gives the trimmed bank attitude:
φ = – +
+
+
t t t
s
T
W
Wf R T h R
WhR M
(1.55)
If Ms = 0 (no offset hinge) and ht = h, which is usually approximately true, then
φ ≈ f/h
which means that the c.g. lies vertically below the rotor, for the rotor thrust vector
T
A1
Tt
φ
W
Fig. 1.26 Helicopter lateral attitude
Fig. 1.25 Forces and moments in lateral plane
T = TD
Resultant
c.g.
a1
B1
N.F.A.
tan–1(HD/TD)
HD
Fig. 1.24 Rotor force components in tip path (disc) plane
b1
A1
Tt
htR
hR
fR
W
φ
O
N.F.A.
32 Bramwell’s Helicopter Dynamics
must be tilted relative to the vertical to balance the tailrotor side force and tilted away
from the c.g. to balance the tailrotor moment, Fig. 1.26.
References
1. Stewart, W., ‘Higher harmonics of flapping on the helicopter rotor’, Aeronautical Research
Council CP 121, 1952.
2. Sissingh, G. J., ‘The frequency response of the ordinary rotor blade, the Hiller servo-blade and
the Young-Bell stabilizer’, Aeronautical Research Council R&M 2860, 1950.
3. Zbrozek, J. K., ‘The simple harmonic motion of a helicopter rotor with hinged blades’, Aeronautical
Research Council R&M 2813, 1949.
4. Correspondence in Aircraft Engineering, September and November 1955.
2
Rotor aerodynamics in
axial flight
2.1 Introduction
One of the most important aerodynamic problems of the helicopter is the determination
of the loading of the rotor blades. For this purpose it is essential to know the local
components of airflow at any station along the blade, and this in turn requires a
knowledge of the air velocity induced by the lift of the blades. In developing the
analysis, reference is made to well known and fundamental theorems, laws and
equations in aerodynamics, which may be found in standard texts, such as Houghton
and Carpenter1.
An element of a rotor blade can be regarded as an elementary aerofoil and, in
accordance with the Kutta–Zhukowsky theorem, there is a bound vortex of circulation
Γ about the aerofoil which, in general, varies along the span. Now Helmholtz’s
theorem implies that a vortex cannot terminate in the interior of a fluid, and the
vortex bounding the element continues as free vortex lines springing from the trailing
edge of the element, Fig. 2.1(a). These free vortex lines are called trailing vortices.
If Γ is the strength of the bound vortex of an element, and if Γ + dΓ is the vortex
strength of the neighbouring element, the neighbouring trailing vortices are in the
(b)
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