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3
sin + 2 sin + 4
3
0 – 1.46 cos
2 2 1/2
i
γ θ μ ψμ ψ λ νλ ψ 1
+ 2μ sinλ –ψ 0.69μ iν1λ/2 sin 2ψ
(3.48)
Equation 3.48 is a linear equation with periodic coefficients and there is no known
solution in closed form. Moreover, as discussed in section 3.7, it is valid only for the
advancing region 0 < ψ < 180°, since in the reverse flow area the lift and flapping
moment are incorrectly evaluated. For the speeds typical of present day helicopters,
however, this results in negligible error, as was stated earlier.
The free motion of the blade is found by putting the terms on the right-hand side
of eqn 3.48 equal to zero. The flapping equation is then
d
d
+
8
1 + 4
3
sin
d
d
+ + + 4
3
cos + sin 2 = 0
2
2
2 β
ψ
γ μ ψ βψ
( ) ε γ ( μ ψ μ ψ) β
1
8
(3.49)
Considerable attention has been given to this equation, since its solution answers
the important question of the stability of the flapping motion. In hovering flight,
μ = 0, as we have seen already, the equation reduces to one with constant coefficients,
eqn 1.10, and it was found that the corresponding motion is heavily damped. It is
reasonable to expect that the damping would remain high for low values of μ
but further investigation is required to find the effect of the periodic terms for high
values of μ. As we shall see shortly, the denominator of one of the flapping coefficients
of steady motion is the term 1 – μ2/2, indicating that infinite flapping amplitudes
might be expected to occur at μ = √2.
Several attempts to solve eqn 3.49 analytically have been made, notably by Glauert
and Shone,10 Bennett,11 Horvay,12 Shutler and Jones,13 and Lowis.14 Of these, only
Lowis has attempted to take the reverse flow area into account. The others indicate
that the flapping motion appears to be stable for μ < 1 but their results are not really
valid for values of μ greater than about 0.7 because of the neglect of the reverse-flow
region. The last three authors make use of Floquet’s theorem, which states that an
equation of the type eqn 3.49 has a solution of the form
β = α1eν ψ 1(ψ) + α2eν ψ 2(ψ)
1P 2P (3.50)
where α1, α2, ν1, ν2 are constants and P1(ψ), P2(ψ) are periodic functions of period
2π. Hovering flight is a special case of eqn 3.50 with solution
β = e νψ [α1 sin (1 – ν)ψ + α cos (1 – ν)ψ]
2 1/2
2
– ˆ ˆ ˆ2 1/2
where νˆ = γ /16.
The stability of the motion is determined by the values of ν1 and ν2, which may not
be real, and the investigations mentioned have been directed to finding their values.
106 Bramwell’s Helicopter Dynamics
An exact analytical determination is not possible, and ν1 and ν2 must be evaluated
numerically for a range of flight parameters. Lowis found an approximate method of
taking the reverse flow region into account which amounted merely to changing the
sing of γ in eqn 3.49 over a range of ψ in the retreating region depending on the value
of μ. His results showed that flapping instability occurs for μ in the range 2.2 to 2.8,
depending on the inertia number γ.
Another method of dealing with eqn 3.49 is to use computational methods. The
reversed-flow area and periodic coefficients can be easily and exactly taken into
account. Of course, the output gives the flapping response to a given set of conditions,
which does not provide as much information as the values of ν1 and ν2. Nevertheless,
by suitably choosing the initial conditions, the response can give an adequate picture
of the flapping behaviour and stability. Investigations of this kind have been made by
Wilde and Bramwell15 and Sissingh16. The results obtained by Wilde and Bramwell
for γ = 6 and ε = 0 are shown in Fig. 3.24; they agree closely with the results of Lowis
and Sissingh.
3.12 The flapping coefficients
In the previous section the free blade motion was discussed; we now wish to find the
forced blade motion, that is, the steady motion in forward flight corresponding to a
given collective pitch angle, tip speed ratio, and inflow ratio – these conditions
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Bramwell’s Helicopter Dynamics(56)
