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feathering amplitude is B1 – a1s. Thus feathering and flapping can be interchanged
and either may be made to vanish by the appropriate choice of axis. The ability to
select an axis relative to which either the feathering or flapping vanishes is useful in
simplifying the analysis of rotor blade motion and for interpreting rotor behaviour.
The coning angle a0 and collective pitch θ0 play no part in the principle of equivalence.
Strictly speaking, the principle of equivalence fails if the flapping hinges are
offset, because the angle of the tip path plane will then no longer be the same as the
amplitude of blade flapping, as a sketch will easily show. However, the size of the
offset is usually so small that the equivalence idea can be generally applied. Offset
hinges, as will be seen later, make an important contribution to the moments on the
helicopter.
Another important feature of blade flapping motion can be deduced from the
flapping equation. Assuming ε to be negligible, the flapping equation (eqn 1.9) can
be written
d2β/dψ2 + β = MA/BΩ2
in which β is defined relative to a plane perpendicular to the shaft axis.
Now, assuming that higher harmonics can be neglected, steady blade flapping can
be expressed in the form
β = a0 – a1 cos ψ – b1 sin ψ
a1s B1
a b c
c′ b′ a′
Tip path plane
ψ = 90°
θ = B1 sin ψ
a b
b′
a′
–B1
ψ = 270°
Fig. 1.12 Equivalence of flapping and feathering
Basic mechanics of rotor systems and helicopter flight 15
and on substitution for β into the flapping equation above
MA = BΩ2a0 = constant
Thus, for first harmonic motion, the blade flaps in such a way as to maintain a
constant aerodynamic flapping moment. This does not necessarily mean that the
blade thrust is also constant, since, except in hovering flight, the blade loading
distribution varies with azimuth angle and the centre of pressure of the loading
moves along the blade.
1.6.3 Flapping motion due to pitching or rolling
An important hovering flight case for which the response of the rotor can be calculated
is pitching or rolling. Consider first the case of pitching at constant angular velocity
q. The equation of motion, eqn 1.3, with ε = 0, is
d2β /dt2 + Ω2β = MA/B – 2Ωq sin ψ (1.14)
Due to pitching and flapping, the velocity component normal to the blade at a
point distance r from the hub is r(q cos ψ – ˙β); with cos β = 1 and neglecting a very
small term in q, the chordwise velocity is Ωr. The corresponding change of incidence
Δα is therefore
Δα = (q cos ψ – ˙β )/Ω = ˆ q cos ψ – dβ/dψ
where ˆ q = q/Ω.
The contribution to the flapping moment of the flapping velocity ˙β has already
been considered in section 1.6.1; by a similar calculation the moment due to the
pitching velocity q is found to be
(MA)pitching = ρacΩ2R4 ˆ q cos ψ/8 (1.15)
Equation 1.14 now becomes
d
d
+
8
d
d
+ =
8
cos – 2 sin
2
2
β
ψ
γ βψ
β γqˆ ψ qˆ ψ (1.16)
Assuming a steady-state solution, β = a0 – a1 cos ψ – b1 sin ψ gives
a1= – 16qˆ/γ , b1= – qˆ (1.17)
Hence, when the shaft has a steady positive rate of pitch, the rotor disc tilts
forward by amount 16 ˆ q/γ and sideways (towards ψ = 270°) by amount ˆ q. The
longitudinal a1 tilt is due to the gyroscopic moment on the blade, and the lateral b1
tilt to the aerodynamic moment due to flapping. For typical values of γ, the lateral tilt
is roughly half the longitudinal tilt.
The same result can be obtained in a somewhat different way by focusing attention
on the rotor disc. If steady blade motion is assumed to occur, each blade behaves
identically and the rotor can be regarded as a rigid body rotating in space with
angular velocity components Ω about the shaft and q perpendicular to the shaft.
16 Bramwell’s Helicopter Dynamics
According to elementary gyroscopic theory, the rotor will experience a precessing
moment bCΩq tending to tilt it laterally towards ψ = 90°, bC being the moment of
inertia of all the blades in the plane of rotation. In addition, there is the aerodynamic
moment on the rotor due to its pitching rotation. Using eqn 1.15, we find that the total
moment for all the blades is bCρaΩ2R4 ˆ q/16 and is in the nose down sense. Now,
　
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