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1
2
i c i
4 =
2
ˆ (1 – /2)(1 + ( /4) / + )
μ
μ λ v
(5.75)
∂
∂
a
w as
1
2
2
16
ˆ (1 – /2)(8 + ) ≈ μ
μ μ
(for μ > 0.08) (5.80)
∂
∂
h
w
a
a t
a c
i c i
4
12
1 0
D
2
D =
4(1 + ( /4) / + )
– +
ˆ λ 1 – /2
μθ μλ
v μ
(5.77)
∂
∂
h
w
a
as
c
2
0
2
D
2
D =
4
8 +
(1 – 9 /2)/6 +
1 – /2
(for > 0.08)
ˆ
μ
μ
θ μ λ
μ
μ (5.81)
∂
∂
t
q
c = 0,
160 Bramwell’s Helicopter Dynamics
–1.2
–1.0
–0.8
–0.6
–0.4
–0.2
zw
xw
xu
zu
0.1 μ
0.2
0.02
0.01
0
–0.01
–0.02
0.1 0.2 0.3 μ
mw′ , f = 0
mu′, f = 0.02
mu′, f = 0
mw′ , f = 0.02
mq′ /10
Fig. 5.8 Typical variation of helicopter derivatives with speed
∂
∂
a
q
1
2 = – 16 1
ˆ γ 1 – μ /2
⋅ (5.82)
∂
∂
∂
∂
h
q
a a
a
q
c 12
1
2
0
D = 1
4
+ –
ˆ ˆ [λ μ μ θ] (5.87)
5.5 The longitudinal stability characteristics
The longitudinal derivatives for the example helicopter of Chapter 4 have been
calculated and are shown in Fig. 5.8. But, before discussing the stability over the
whole speed range, let us examine the hovering case since, although the analysis is
very much simplified by the absence of a number of derivatives, the stability
characteristics are typical of most of the flight range.
The stability derivatives for the hovering case (c.g. on shaft axis) are
xu = – 0.032 xw = 0 xq = 0
zu = 0 zw = – 0.52 zq = 0
mu′ = 0.016 mw′ = 0 mq′ = – 0.099
0 0.3
Flight dynamics and control 161
The relative density parameter μ* = 47.6 and ˆt = 1.82 seconds.
Taking iB = 0.11 as the non-dimensional longitudinal moment of inertia gives
mu = 6.8, mw = 0, mq = – 0.90
In hovering flight ˆ V = = 0 c τ and, since zu = 0 also, we see at once from eqn 5.15
that the vertical motion is uncoupled from the pitching and fore-and-aft motion, and
eqn 5.15 gives
λ – zw = 0
or λ = – 0.52
indicating a heavily damped subsidence. The other modes of motion are determined
from
–
– –
= 0 c
2
λ
λ λ
x w
m m
u
u q
or λ3 – (xu + mq)λ2 + xumqλ + muwc = 0 (5.94)
This is the characteristic equation for hovering flight; it could also have been
derived from the characteristic quartic, of course, which would have given the root
λ = zw. Inserting the numerical values above gives
λ3 + 0.93λ3 + 0.029λ + 0.58 = 0
which has the solution
λ1 = – 1.26, λ2,3 = 0.165 ± 0.65i
The real root represents a heavily damped subsidence, whose amplitude is halved
in one second, and the complex roots represent a divergent oscillation with a period
of 17.5 seconds and whose amplitude doubles in 7.1 seconds.
The motions corresponding to the roots of the characteristic cubic, eqn 5.94,
involve attitude and speed changes only; the vertical motion in hovering flight, as we
have seen, is independent of these two degrees of freedom. To get some idea of the
nature of these modes of motion we substitute the values of λ back into the equations
of motion from which the roots originated, i.e. eqns 5.14 and 5.16, with w absent and
τc = 0, giving
(λ – xu) u0 + wcθ0 = 0
–muu0 + (λ2 – mqλ)θ0 = 0
Since the characteristic equation expresses the consistency of these equations,
either can be used to find the ratio u0/θ0 for the root in question. Then from eqn 5.16
we have, using the numerical values of the derivatives,
u0/θ0 = (λ2 + 0.9λ)/6.8
Taking the real root λ = – 1.26 gives
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Bramwell’s Helicopter Dynamics(83)
