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can be interpreted, in the case of two opposing blades, as the couple due to the
displaced centrifugal force vectors, and these are parallel to the tip path plane which
is the plane of steady rotation, Fig. 1.20. For small offsets, the inclination of the tip
path plane is approximately equal to the flapping angle.
26 Bramwell’s Helicopter Dynamics
1.12 Rotor forces and choice of axes
In the previous section, the rotor force components were expressed in terms of a set
of axes fixed in the helicopter. As explained there, such a formulation is necessary to
study the forces and moments on the whole helicopter. However, it is more usual, and
natural, when considering the rotor as a lifting device, to regard it as producing a
thrust, defined along some convenient direction, together with small components of
force in the other two perpendicular directions. For this purpose, three axes systems
are in common use, as follows. The question as to which axis is the most useful
depends upon the problem being considered, and will become apparent in the
applications dealt with later, although some indications are given below.
1.12.1 The no-feathering or control axis
As explained in section 1.6.2, this is the axis normal to the plane of the swash plate.
By definition, no cyclic feathering occurs relative to this axis, the blade pitch being
the constant value supplied by the collective pitch application. Since the pitch angle
is constant, the only other blade motion contribution to the local blade incidence is
that due to the flapping. The no-feathering axis is often used to express the blade
flapping, especially when blade aerodynamic forces are being established, since
constant blade pitch at a section eases the mathematical development. The rotor
aerodynamics and dynamics established in Chapter 3 are expressed using this axis
system.
1.12.2 The tip path plane or disc axis
The tip path plane axis is the axis perpendicular to the plane through the blade tips
and, for zero offset flapping hinges, it is therefore the axis of no flapping. The
definition applies only to first harmonic motion since, when there are higher harmonics,
the blades no longer trace out a plane. Of the higher harmonics, only the odd values
affect the tilt of the disc, and these are usually extremely small compared with the
first harmonics, as explained in section 1.6.2. Now, although there is no first harmonic
flapping relative to the tip path plane, there will be cyclic feathering and the amount
of feathering is exactly equal to the flapping relative to the no-feathering axis. Thus,
in this case, the blade incidence will be determined from the collective pitch and the
apparent feathering motion in the tip path plane.
Fig. 1.20 Centrifugal force couple on tilted rotor with offset hinges
Basic mechanics of rotor systems and helicopter flight 27
When the flapping hinges are offset, the tip path plane axis is no longer the axis
of no flapping, as can be easily seen from a diagram like Fig. 1.20 with exaggerated
hinge offset. Strictly speaking, both feathering and flapping occur relative to the tip
path plane but, provided the offset is small, as it usually is, the error in assuming that
there is no flapping is negligible.
1.12.3 The shaft or hub plane axis
This axis is usually less convenient for calculating the rotor forces, as the blade
incidence must be expressed in terms of both feathering and flapping. It is, nevertheless,
a useful axis for dealing with hingeless rotors, since blade flapping relative to the hub
is of prime importance. Indeed, the blade mechanics developed so far in the current
chapter have been with reference to the shaft axis.
1.13 The rotor disc
In all the three cases discussed above, it is usual to call the force component along the
axis, whichever of the above it is, the thrust T. The component perpendicular to this
axis and pointing rearward is called the H force, and the third component, pointing
sideways to starboard, is the Y force. Usually the Y force is very small and attention
is mainly focused on the thrust and the H force, i.e. the longitudinal force components.
Calculations and measurements show that the resultant rotor force is almost
perpendicular to the tip path plane, usually pointing backwards slightly. It is for this
reason that the tip path plane axis is useful since the resultant force is almost exactly
equal to the thrust; the H force can be regarded as a kind of rotor drag.
Let us denote the thrust and the H force relative to the no-feathering axis by T and
H respectively, and use the subscript D to denote the tip path plane (disc) axis and s
to denote the shaft axis. Now the flapping and feathering angles are usually small –
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