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worsens at high speed. Autostabilisation is generally incorporated to bring about an
improvement. In mitigation of the inferior stability is the fact that the larger hub
moments allow much greater control power to be achieved.
As is well known, the longitudinal stability quartic of the fixed wing aircraft splits
up into two quadratics whose coefficients are related in a very simple way to the
coefficients of the quartic, leading to a simple physical interpretation of the motion.
Unfortunately this is not so for the helicopter. We have already seen that in hovering
flight the interaction of the pitching and horizontal motions leads to a characteristic
cubic equation, and this is further complicated by coupling with the vertical motion
as speed increases.
166 Bramwell’s Helicopter Dynamics
5.6 Lateral dynamic stability
5.6.1 The equations of motion
Writing the lateral equations of motion A.2.13, A.2.15 and A.2.17 in the same manner
as for the longitudinal equations, we have
( / ) W g – Y – Ypp + (W/g)Vr – Yrr – W cos c – W sin c v˙ vv φ τ ψ τ
= + YA1A1 Yθ tθ t (5.95)
– + – – – = + Lvv Ap˙ Lpp Er˙ Lrr LA1A1 Lθ tθ t (5.96)
– – – + – = + Nvv Ep˙ Npp Cr˙ Nrr NA1A1 Nθ tθ t (5.97)
The variables are defined as shown in Fig. 5.13; A1 and θ
t are the lateral cyclic and
tailrotor collective pitch angles respectively.
The non-dimensional derivatives are defined by:
yv = Yv/ρsAΩR, yp and yr are found to be negligibly small.
l v′ = Lv/ρsAΩR2 , n v′ = Nv/ρsAΩR2
lp′ = Lp/ρsAΩR3 , lr′ = Lr/ρsAΩR3
n′p = Np/ρsAΩR3, nr′ = Nr/ρsAΩR3
λim
0.8
0.6
0.4
0.3
0.35
0.2
0.1
0.2
0.3
0.35
0.2
0.3
0.1
–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
λ re
–0.2
–0.4
–0.6
μ = 0.35
Fig. 5.12 Stability roots of helicopter with hingeless blades
Hingeless
helicopter
Hingeless
helicopter
with
tailplane
Flight dynamics and control 167
lA′1 LA1 sA R = /ρ Ω2 3 lθ′ Lθ ρsA R t t = / Ω2 3
nA′1 NA1 sA R = /ρ Ω2 3, nθ′ Nθ ρsA R t t = / Ω2 3
yA1 YA1 sA R = /ρ Ω2 3, γθt θtρ = Y / sAΩ2R2
The non-dimensional moments and products of inertia are defined by
iA = A/mR2, ic = C/mR2, iE = E/mR2
and μ* and ˆt are the same as for the longitudinal case.
The non-dimensional forms of the equations of motion are then
d
d
– – cos +
d
d
c c – c sin c = 1 1 + t t
v
τ vv φ τ
ψ
ψ τ θ θ y w V
t
w yA A y
ˆ (5.98)
– +
d
d
–
d
d
–
d
d
–
d
d
= +
2
2
2
2 1 1 t t l l
i
i
p l lA l
E
A
vv r A
φ
τ
φτ
ψ
τ
ψτ
θ θ (5.99)
– –
d
d
–
d
d
+
d
d
–
d
d
= +
2
2
2
2 1 1 t n
i
i
E n n n A n
C
vv p r A t
φ
τ
φτ
ψ
τ
ψτ
θ θ (5.100)
where l v = μ*lv′/iA, lp = lp′/iA, … , nv = μ*nv′/iC, np = n′p/iC, … ,
5.6.2 Stick-fixed dynamic stability
Putting the control displacements A1 and θ
t to zero and assuming solutions of the
form
v = v0eλτ, φ = φ0eλτ, … , etc.
T
Y
V
p
βss
ψ
δb1
Tt
y
W
φ
Tt
v
Fig. 5.13 Nomenclature diagram for lateral stability
r
168 Bramwell’s Helicopter Dynamics
leads to the vanishing of the determinant
– – cos – sin
– – –( / ) –
– –( / ) – –
= 0
c c c c
2 2
2 2
λ τ λ τ
λ λ λ
λ λ λ λ
y w V w
l y l i i l
n i i n n
p E A r
E C p r
v
v
v
ˆ
and to the characteristic frequency equation
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