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difficult to make estimates of the pitching moment. Also, because of the problem of
correctly representing the downwash, it is difficult to obtain reliable wind tunnel
measurements of the pitching moment. The contribution from a tailplane, however,
can be estimated comparatively accurately, although it depends on the fuselage incidence,
which is not known in advance. A method for calculating the tailplane moment will
be given later in the chapter.
120 Bramwell’s Helicopter Dynamics
For the purpose of illustrating the calculation of the trim, we assume Cmf = 0.
From the data given we find also that
t h w h Cm
cD c s = = 0.0214 and = 0.0274
The two quantities above represent the control moment contributions, expressed
in non-dimensional form, referred to in section 1.14, i.e. the thrust moment and the
centrifugal couple due to the flapping hinge offset, which has been taken as e = 0.04.
It can be seen that this typical value of flapping hinge offset results in a total control
moment which is more than double that due to the thrust alone.
Equation 4.14 provides the longitudinal cyclic pitch angle B1 to trim for different
c.g. positions and to calculate the fuselage attitude θ, we can write eqn 3.50 in terms
of the disc axes as
(D/W) cos τc + HD/W + TDa1/W = B1 – θ
or, in non-dimensional form, as
θ = 1 – 1 – c / c – μd /
12
2
D 0 c B a h w w
A negative sign for θ indicates that the fuselage is in a nose-down attitude. Except
at low speeds, by far the largest term in the equation for θ is the fuselage drag term
12
2
μ d0/wc.
Values of B1 and θ for three different values of f (c.g. position) are shown in Table 4.1.
The variation of longitudinal trim parameters with tip speed ratio in the range 0 to
0.35 are shown in Figs 4.2 to 4.6. The effects of hinge offset and c.g. position are
shown in Figs 4.5 and 4.6 for B1 and fuselage attitude θ.
The variation of λ with μ, Fig. 4.2, displays the reduction of the thrust velocity
part of the induced velocity at the higher speeds, shown by the broken line, and then
the gradually increasing total inflow due to the forward tilt of the rotor disc required
to overcome the helicopter drag (full line).
Table 4.1 Values of cyclic pitch B1 and fuselage pitch attitude θ for different c.g. positions
f 0 0.01 0.02
C.g. position On shaft 7.9 cm fwd 15.8 cm fwd
B1° 6.32 5.31 4.31
θ° –7.45 –8.45 –9.45
– 0.08
– 0.06
– 0.04
– 0.02
0 0.1 μ 0.2 0.3
λ
λi
λ
Fig. 4.2 Inflow ratio in trimmed flight
Trim and performance in axial and forward flight 121
0 0.1 0.2 0.3
μ
14°
12°
10°
8°
6°
4°
2°
θ0
Fig. 4.3 Collective pitch variation in trimmed flight
10°
8°
6°
4°
2°
0 0.1 0.2 0.3
μ
a1
a0
bi
Fig. 4.4 Flapping angles in trimmed flight
12°
10°
8°
6°
4°
2°
0
–2°
–4°
e = 0
e = 0.04
f = 0
0.01
0.02
0.2
μ 0.3
Long-cyclic pitch, B1
Fig. 4.5 Longitudinal pitch angle to trim
In order that the thrust should be kept constant the collective pitch angle θ0, as
follows from eqn 3.33, must vary in almost exactly the same way as λ, and this can
also be seen in Fig. 4.3.
As might be expected, at constant rotor thrust the coning angle a0 is practically
constant over the entire speed range, Fig. 4.4, while the longitudinal flapping angle
a1 varies roughly linearly with speed.
The variation of longitudinal cyclic pitch B1 required for trim is shown in Fig. 4.5
for three c.g. positions and for the flapping offsets represented by e = 0 and e = 0.04.
122 Bramwell’s Helicopter Dynamics
Referring to eqn 4.14, we see that if Cmf = 0 and if the c.g. is on the shaft (f = 0),
B1 differs from a1 only by the small term in hcD , and that the difference becomes
smaller the larger the hinge offset or hub couple. Our previous calculations show
that, at μ = 0.3, the difference between B1 and a1 is only 0.36° for the offset e = 0.04;
for zero offset, e = 0 = Cms and the difference is 0.79°.
With zero hinge offset, the effect of moving the c.g. is merely to change the
amount of cyclic pitch to trim by the ratio of c.g. distances f /h, as explained in section
1.14. According to eqn 4.14, with h = 0.25 a forward c.g. displacement of 7.9 cm
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