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λ(A2λ4 + B2λ3 + C2λ2 + D2λ + E2) = 0 (5.101)
where
A2 iE iAiC
= 1 – 2 / (5.102)
B 2 y iEiAiC N
2
= – v (1 – / ) – 2 (5.103)
C yN P l i i nV E C 2 2 2 = + + ( / ) + v v v V ˆ ˆ (5.104)
D2 = yvP2 + lvQ2 – nvR2 (5.105)
E2 = lvS2 – nvT2 (5.106)
and
N2 = lp + nr + (iE/iC)lr + (iE/iA)np (5.107)
P2 = lpnr – lrnp (5.108)
Q2 = npVˆ – wc cos τc (5.109)
R2 = lpVˆ – wc sin τc (5.110)
S2 = npwc sin τc + nrwc cos τc (5.111)
T2 = lpwc sin τc + lrwc cos τc (5.112)
the suffix 2 denoting the lateral coefficients.
The zero root of eqn 5.101 implies that the aircraft has no preference for a particular
heading.
5.6.3 The lateral stability derivatives
Referring to Fig. 5.13 the side force ΔY in disturbed lateral motion will be
ΔY = Tδb1 + δTt + δYf
where δTt and δYf are the incremental tailrotor and fuselage forces respectively. We
assume that in lateral motion the main rotor thrust remains constant and that, apart
from rolling motion, the force in the plane of the rotor disc remains very small.
In general, the helicopter longitudinal axis will be inclined to the wind axes, Fig.
5.14, and the effective tailrotor height ht′R and the rearward distance lt′R from the
c.g. are related to the datum distances by
Flight dynamics and control 169
ht′R = (ht cos αs – lt sin αs )R (5.113)
≈ (ht – ltαs)R
and
lt′R = (lt cos αs + ht sin αs )R (5.114)
≈ ltR
The rolling moment will consist of the moment of the main rotor thrust and side
force, the hub moment due to rotor tilt, and the moment of the tailrotor thrust, i.e.
ΔL = h1RTδb1 + h1RδY + Msδb1 + ht′RδTt
It will be assumed that the contribution of the fuselage to the rolling moment is
negligible.
The yawing moment contributions will arise from changes in the tailrotor thrust
and the fuselage and fin. Thus
ΔN = –lt′RδTt + δNf
The sideslip, rolling, and yawing disturbances all give rise to axial velocity
components at the tailrotor, and the associated thrust changes can be calculated from
the relations already obtained from the main rotor. Thus, if we denote any of these
velocity components by w, we have
∂Tt/∂ v = –∂ T/∂w
In the rolling motion, w ≡ – ht′Rp, and
∂
∂
∂
∂
T
p
h R
T
w
t
= – t′
and for yawing, since w ≡ lt′Rr,
hR h1R
V l1R
htR
ltR lR
lt′R ht′R
Fig. 5.14 Moment arms for calculating lateral moments
αs
170 Bramwell’s Helicopter Dynamics
∂
∂
T ∂
r
l R
T
w
t
= t
d ′
Then
L hR T
b Y
h R
T
w
M
b
v v v v
= 1 + – +
1
t s
∂ 1
∂
∂
∂
∂
∂
∂
∂
′ (5.115)
L hR T
b
p
Y
p
h R
T
w
M
b
p = 1 + – + p
1
t
2 2
s
∂ 1
∂
∂
∂
∂
∂
∂
∂
′ (5.116)
L hlR
T
r = t t w
′ ′ 2
∂
∂ (5.117)
N lR T
w
v = t′ + (Nv)f
∂
∂ (5.118)
N hlR
T
w
p = tt + (Np)
2
′ ′ f
∂
∂ (5.119)
N l R
T
w
r = – t + (Nr)
2 2
′ f
∂
∂ (5.120)
To calculate ∂b1/∂v we recall that, when a rotor is placed in a stream of air of
velocity V, the rotor tilts backwards with angle a1 and sideways with angle b1 with
reference to the plane of no-feathering (Chapter 3). The resultant tilt is therefore
( + ) 1
2
1
a b2 1/2 at angle ψ0 = tan–1 (b1/a1) towards the advancing blade, ψ0 being
measured from the rearmost position of the blade. Then, when a small side wind
blows, the relative wind appears to come from a new direction, making sideslip angle
βss = v /(V cos αnf) to the original direction, in the no-feathering plane. Thus, the
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Bramwell’s Helicopter Dynamics(87)
