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eqn 1.2 becomes
d2β/dψ2 + (1 + ε)β = MA/BΩ2 (1.9)
This equation is valid for any case of steady rectilinear flight including hovering.
The problem is to express MA as a function of ψ and then to solve the equation. We
now consider some simple but important examples in hovering flight.
1.6.1 Disturbed flapping motion at constant blade pitch angle
We suppose that the blades are set at a constant blade pitch angle relative to the shaft
and that the rotor is rotating steadily with angular velocity Ω. Since we are interested
only in the character of the disturbed motion, the aerodynamic moment corresponding
to the constant pitch angle will be ignored and attention will be concentrated on the
aerodynamic moments arising from disturbed flapping motion.
Now, when a blade flaps with angular velocity ˙β , there is a relative downwash of
velocity r˙ β at a point on the blade distance r from the hinge. Assuming cos β = 1, the
chordwise component of wind velocity is Ω(r + eR), so that the local change of
incidence Δα due to flapping is
Δα β β ψ
=
–
( + )
=
– d /d
+
r
r eR
x
x e
˙
Ω
where x = r/R.
Assuming a constant lift slope a for the blade section, the lift on an element of
blade is
dL = – 12
ρacΩ2R3(x + e)x(dβ/dψ)dx
The moment of this lift about the flapping hinge is rdL and the total aerodynamic
moment, assuming the blade chord c to be constant, is
M rL ac R x x e x
R e
A
0
(1– )
12
2 4
0
(1– e)
= ∫ d = – ρΩ ∫ 2( + )(dβ/dψ )d
giving
MA/BΩ2 = – (γ /8)(1 – e)3(1 + e/3)dβ /dψ
where γ = ρacR4/B is called Lock’s inertia number.
Writing n for (1 – e)3(1 + e/3), the flapping equation becomes
d2β/dψ2 + (nγ/8)dβ /dψ + (1 + ε)β = 0 (1.10)
Equation 1.10 is the equation of damped harmonic motion with a natural undamped
frequency Ω√(1 + ε). If ε is zero (no flapping hinge offset), the natural undamped
frequency is exactly equal to the shaft frequency. Normally ε is about 0.06, giving an
undamped flapping frequency about 3 per cent higher than the shaft frequency.
Taking a typical value of γ of 6 gives a value for nγ /8 of about 0.7. This means that
the damping of the motion is about 35 per cent of critical, or that the time-constant
12 Bramwell’s Helicopter Dynamics
in terms of the azimuth angle is about 90° or 1
4 of a revolution. Thus, the flapping
motion is very heavily damped. It has already been remarked that the centrifugal
moment acts like a spring, and we now see that flapping produces an aerodynamic
moment proportional to flapping rate, i.e. in hovering flight the blade behaves like a
mass–spring–dashpot system. In forward flight the damping is more complicated and
includes a periodic component, but the notion of the blade as a second order system
is often a useful one in a physical interpretation of blade motion.
1.6.2 Flapping motion due to cyclic feathering
Suppose that, in addition to a constant (collective) pitch angle θ0, the blade pitch is
veried in a sinusoidal manner relative to the hub plane. The blade pitch θ can then be
expressed as
θ = θ0 – A1 cos ψ – B1 sin ψ (1.11)
To simplify the calculations we will take e = 0, since the small values of flapping
hinge offset normally employed have little effect on the flapping motion.
In calculating the flapping moment MA, the induced velocity, or rotor downwash,
to be discussed in Chapter 2, will be ignored. By a similar analysis to that above, the
flapping moment is easily found to be given by
MA/BΩ2 = γ (θ0 – A1 cos ψ – B1 sin ψ)/8
Substituting in eqn 1.9 leads to the steady-state solution
β = γθ0/8 – A1 sin ψ + B1 cos ψ (1.12)
The term γθ0/8 represents a constant flapping angle and corresponds to a motion
in which the blade traces out a shallow cone, and for this reason the angle is called
the coning angle. If the induced velocity had been included, the coning angle would
have been reduced somewhat. For our present purpose the exact calculation of the
coning angle is unimportant. The terms –A1 sin ψ + B1 cos ψ represent a tilt of the
axis of the cone away from the shaft axis. Since ψ is usually measured from the
rearmost position of the blade, i.e. along the axis of the rear fuselage, a positive value
　
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