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arrangement is shown in Fig. 1.1.
2 Bramwell’s Helicopter Dynamics
In this figure, the flapping and lag hinges intersect, i.e. the hinges are at the same
distance from the rotor shaft, but this need not necessarily be the case in a particular
design. Neither are the hinges always absolutely mutually perpendicular.
Consider the arrangement shown in Fig. 1.2. Let OX be taken parallel to the bladespan
axis and OZ perpendicular to the plane of the rotor hub. Let OP represent either
the flapping hinge axis or the lag hinge axis. The flapping hinge is referred to as the
δ-hinge and the lag hinge as the α-hinge. We then define:
– the angle between OZ and the projection of OP onto the plane OYZ as δ1 or α1,
– the angle between OZ and the projection of OP onto the plane OXZ as δ2 or α2,
– the angle between OY and the projection of OP onto the plane OXY as δ3 or α3.
These are the definitions in common use in industry. The most important angles in
practice are α2, which leads to pitch resulting from lagging of the blade, and δ3,
which couples pitch and flap, as follows.
Lag hinge
Flapping hinge
Pitch change
(or feathering) hinge
Flapping
Lagging
Feathering
Fig. 1.1 Typical hinge arrangement
α1, δ1
Z
α2, δ2
α3, δ3
Y
O
P
Fig. 1.2 Blade hinge angles
X
Basic mechanics of rotor systems and helicopter flight 3
Referring to Fig. 1.3, when δ3 is positive, positive blade flapping causes the blade
pitch angle to be reduced. It will also be appreciated that, if the drag hinge is mounted
outboard of the flapping hinge, movement about the lag hinge produces a δ3 effect.
If the blade moves through angle ξ0 and flaps through angle β relative to the hub
plane, the change of pitch angle Δθ due to flapping is found to be
Δθ = –tan β tan(δ3 – ξ0)
or for small angles
Δθ = –β tan(δ3 – ξ0)
so Δθ is proportional to β.
The dynamic coupling of blade motions will be dealt with in more detail in
Chapter 9.
The blades of two-bladed rotors are usually mounted as a single unit on a ‘seesaw’
or ‘teetering’ hinge, Fig. 1.4(a). No lag hinges are fitted, but the Coriolis root
δ3
Fig. 1.3 The δ3-hinge
(a)
β0
Flapping
(b)
Fig. 1.4 (a) Teetering or see-saw rotor. (b) Underslung rotor, showing radial components of velocity on
upwards flapping blade
4 Bramwell’s Helicopter Dynamics
bending moments may be greatly reduced by ‘underslinging’ the rotor Fig. 1.4(b). It
can be seen from the figure that, when the rotor flaps, the radial components of
velocity of points on the upwards flapping blade below the hinge line are positive
while those above are negative. Thus the corresponding Coriolis forces are of opposite
sign and, by proper choice of the hinge height, the moment at the blade root can be
reduced to second order magnitude. This assumes that a certain amount of pre-cone
or blade flap, β0, is initially built in.
Although, as stated earlier, the adoption of blade hinges was an important step in
the evolution of the helicopter, several problems are posed by the presence of hinges
and the dampers which are also fitted to restrain the lagging motion. Not only do the
bearings operate under very high centrifugal loads, requiring frequent servicing and
maintenance, but when the number of blades is large the hub becomes very bulky and
may contribute a large proportion of the total drag. Figure 1.5(a) shows a diagrammatic
view of the Westland Wessex hub, on which, as may be observed, the flapping and
lag hinges intersect. Figure 1.5(b) is a photograph of the same rotor hub, showing
also the swash plate mechanism that enables the cyclic and collective pitch control
(discussed in section 1.7).
Fig. 1.5 (a) Diagrammatic view of Westland Wessex hub
Lag hinge
Feathering
bearing
assembly
Flapping hinge
Basic mechanics of rotor systems and helicopter flight 5
More recently, improvements in blade design and construction enabled rotors to
be developed which dispensed with the flapping and lagging hinges. These ‘hingeless’,
or less accurately termed ‘semi-rigid’, rotors have blades which are connected to the
shaft in cantilever fashion but which have flexible elements near to the root, allowing
the flapping and lagging freedoms. Such a design is shown in Fig. 1.6(a) which is that
of the Wesland Lynx helicopter. In this case, the flexible element is close in to the
　
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