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The helicopter longitudinal relative density parameter μ* is defined by
μ* = W/gρsAR = Ωtˆ
and the non-dimensional moment of inertia iB is defined by
iB = B/(WR2/g)
Finally we define the non-dimensional derivatives as
xu = Xu /ρsAΩR, xw = Xw/ρsAΩR, xq′ = Xq/ρsAΩR2
zu = Zu /ρsAΩR, zw = Zw/ρsAΩR, zq′ = Zq/ρsAΩR2
mu′ = Mu/ρsAΩR2 , mw′ = Mw/ρsAΩR2 , mq′ = Mq/ρsAΩR3
xB1 XB1 sA R = /ρ Ω2 2, mw′˙ = Mw˙/ρsAR2 mB′1 MB1 sA R = /ρ Ω2 3
xθ Xθ ρsA R 0 0 = / Ω2 2, zB1 ZB1 sA R = /ρ Ω2 2, mθ′ Mθ ρsA R 0 0 = / Ω2 3
zθ Zθ ρsA R 0 0 = / Ω2 2,
Flight dynamics and control 143
Then dividing the force eqns 5.5 and 5.6 by ρsAΩ2R2 and the moment eqn 5.7 by
ρsAΩ2R3 we have the non-dimensional form of the stability equations as
d
d
– – –
*
d
d
+ c cos c = 1 1 + 0 0
ˆ
u ˆ ˆ
x u x w
x
u w w x B x
q
τ μ B
θτ
θ τ θθ
′
(5.8)
– +
d
d
– – +
*
d
d
+ c sin c = 1 1 + 0 0 z u
w
z w V
z
u w w z B z
q
B ˆ
ˆ
ˆ ˆ
τ μ
θτ
θ τ θθ
′
(5.9)
–
*
–
*
–
d
d
+ d
d
– d
2
2
μ μ
τ
θ
τ
θ i
m u
i
m w
m
i
w m
B i
u
B
w
w
B
q
B
′ ′ ′ ′
ˆ ˆ
˙ ˆ
=
*
+
*
1 1 0 0
μi μ θ θ
m B
i
m
B
B
B
′ ′ (5.10)
in which wc = W/ρsAΩ2R2.
The above system of non-dimensionalisation, based on the original work of Bryant
and Gates4, was intended to display the mass and inertia parameters represented by
μ* and iB. However, in practice, the slight advantage does not justify the somewhat
unwieldy notation and we propose here to write mu for μ*mu′/iB, xq = xq′/μ* etc. It
was for this reason that a ‘dash’ was applied to some of the non-dimensional derivatives
above, indicating that the final forms for these derivatives had yet to be defined.
Since, also, the non-dimensional variables uˆ and wˆ appear only in combination
with the non-dimensional derivatives, which are written in lower case letters, there
should be no ambiguity if uˆ and wˆ are written simply as u and w. Then the final nondimensional
form of the equations can be written as
d
d
– – –
d
d
+ c cos c = 1 1 + 0 0
u
τ xuu xww xq w xB B x
θτ
θ τ θθ (5.11)
– +
d
d
– – ( + )
d
d
+ c sin c = 1 1 + 0 0 z u
w
u τ zww V zq w zB B z
θτ
θ τ θθ
ˆ (5.12)
– – –
d
d
+ d
d
–
d
d
= +
2
2 1 1 0 0 m u m w m
w
u w w˙ τ mq mBB m
θ
τ
θτ
θ θ (5.13)
5.3 Longitudinal dynamic stability
To study the longitudinal dynamic stability, the controls are assumed fixed. It is
worth noting here that in comparing the stability of a helicopter with that of a fixed
wing aircraft, the concept of static stability is not as meaningful because of the
relative lack of importance of the c.g. position in determining the longitudinal behaviour
of the helicopter. Neither is the concept of stick-free stability, due to the inherent
instability with respect to incidence at low speeds (section 5.1.2), and lack of a
natural force feedback to the pilot’s controls that relates to longitudinal control (as in
the case for fixed wing aircraft).
144 Bramwell’s Helicopter Dynamics
Thus we put B1 = θ0 = 0, remembering that, in the equations, B1 and θ0 are
variations from the trim values. We also find that zq is always zero and that xq is
negligibly small.
Equations 5.11, 5.12, and 5.13 are linear differential equations with constant
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