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to similar assumptions and approximations. For a fully comprehensive text on helicopter
flight dynamics the reader is referred to Padfield3. The analysis begins with the
longitudinal motion.
5.2 The longitudinal equations of motion
The equations of motion of a rigid body referred to axes fixed in the body are derived
in Appendix A2. We found that the assumption of small disturbances enabled the
inertia terms to be linearised and the lateral and longitudinal dynamic terms to be
uncoupled. We have now to choose a suitable initial orientation of the axes in undisturbed
flight. Several axes systems were discussed in Chapter 1, but none of them appears
to have any particular advantage as an initial set for describing the dynamic stability.
At the same time, the rotor forces have no special relationship with the undisturbed
flow direction, as is the case with the conventional aeroplane, so the initial flight
direction (wind axes) offers no advantage in expressing the aerodynamic forces.
However, wind axes at least remove the terms qW0 and pW0 from the force equations
A.2.12 and A.2.13 and have the advantage of being thoroughly established in fixed
wing aircraft work.
Flight dynamics and control 141
We shall therefore use wind axes to describe the stability equations, i.e. in undisturbed
flight the x axis is directed parallel to the flight path, with the z axis pointing downwards
and the y axis pointing to starboard.
The linearised equations of longitudinal motion are therefore
(W/g)u˙ = –Wθ cos τc + ΔX (5.1)
(W/g)w˙ = –(W/g)Vθ˙ = – Wθ sin τc + ΔZ (5.2)
Bθ˙˙ = ΔM (5.3)
where ΔX, ΔZ, ΔM are the aerodynamic force and moment increments in disturbed
flight, Fig. 5.2, ˙ θ = , q and τc is the climb angle.
Since the disturbances in u, w, and q are supposed to be small, the force and
moment increments can be written as the first terms of a Taylor series, i.e. ΔX can be
written as
ΔX
X
u
u
X
w
w
X
q
q
X
B
B
X
= + + + +
1
1
0
0
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂θ
θ
= + + + + Xuu Xww Xqq XB1B1 Xθ 0θ0 (5.4)
where u, w, … , etc. are understood to be differential quantities. B1 and θ0 are the
cyclic and collective pitch control terms respectively. Similar expressions can be
written for ΔZ and ΔM. The terms Xu, Xw, … , Xq are called aerodynamic derivatives
in fixed wing aircraft work, but the term is less appropriate here because the force
and moment increments are due to rotor disc tilt as well as to changes in aerodynamic
forces. The derivatives are regarded as constants in disturbed motion. The variables
B1 and θ0 are due to pilot (or possibly autostabiliser) action and are specified functions
of time or of the other variables.
The stability eqns 5.1 to 5.3 can be written as
( / ) – – – + cos c = 1 1 + 0 0 Wgu Xuu Xww Xqq W XBB X ˙ θ τ θθ (5.5)
– + ( / ) – – – ( / ) + sin c = 1 1 + 0 0 Zuu Wg w Zww Zqq WgV W ZBB Z ˙ θ˙ θ τ θθ (5.6)
– – – + – = + Muu Mww Mw˙w˙ Bθ˙˙ Mqq MB1B1 Mθ 0θ0 (5.7)
V
ΔX
u θ
τc
w
Horizontal
ΔM
ΔZ W
Fig. 5.2 Longitudinal force components in longitudinal plane
142 Bramwell’s Helicopter Dynamics
Apart from the form of the control terms, the equations are identical to those of the
fixed wing aircraft. The term Mw˙w˙ allows for the effect of ‘downwash lag ’ on the
tailplane, if fitted, as in the fixed wing aircraft work.
5.2.1 Non-dimensionalisation of the equations
The fixed wing scheme of non-dimensionalisation can conveniently be used for the
helicopter, but the following reference quantities are more useful:
(i) the rotor blade radius R is the unit of length,
(ii) the rotor tip speed ΩR is the unit of speed,
(iii) the blade area sπR2 = sA is the reference area, where s = bc/πR is the rotor
solidity.
Let us define the following non-dimensional quantities:
uˆ = u/ΩR
wˆ = w/ΩR
qˆ = q/Ω
The non-dimensional aerodynamic unit of time is defined by
τ = t/tˆ
where tˆ = W/gρsAΩR.
Note that
ˆ ˆ
ˆ q q
t
d /d ; = 1
d
d ≠ θ τ θτΩ
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Bramwell’s Helicopter Dynamics(73)
