曝光臺 注意防騙 網曝天貓店富美金盛家居專營店坑蒙拐騙欺詐消費者

= – h1RδX + l1RδZ
Flight dynamics and control 149
where
h1 = h cos αs – l sin αs
≈ h – lαs
and
l1 = l cos αs + h sin αs
≈ l + hαs
αs being the incidence of the rotor hub axis in trimmed flight.
In addition to the moment of the forces there is the hub moment due to offset
hinges or hingeless blades, Msδa1, and the fuselage pitching moment δMf. Hence the
total moment increment is
δM = – h1RδX + l1RδZ + Msδa1 + δMf
from which
M hRX lRZ M
a
u
u = – 1 u + 1 u + s + (Mu)
1
f
∂
∂ (5.47)
M hRX lRZ M
a
w
w = – 1 w + 1 w + s + (Mw)
1
f
∂
∂ (5.48)
M hRX lRZ M
a
q
q = – 1 q + 1 q + s + (Mq )
1
f
∂
∂ (5.49)
which, in non-dimensional form, become
′ m hx lz C
a
u = – 1 u + 1 u + m + (mu)
1
s f
∂
∂μ (5.50)
′m hx lz C
a
w
w = – 1 w + 1 w + m + (mw)
1
s f
∂
∂ ˆ
(5.51)
′m hx lz C
a
q
q = – 1 q + 1 q + m + (mq )
1
s f
∂
∂ ˆ
(5.52)
hR
h1R
c.g.
δZ
δX
αs
l1R
Hub plane
Hub axis
lR
Fig. 5.6 Force components contributing to longitudinal moment
150 Bramwell’s Helicopter Dynamics
with
m
i
m m
i
m m
m
u i
B
u w
B
w q
q
B
= , =
*
, = –
μ* μ
′ ′
′
The moment derivatives can also be expressed in terms of the thrust and in-plane
forces. It can easily be verified that
δM = – (l – ha1s)RδT + hR(Tδa1 + δH) + Msδa1 + δMf
where a1s = a1 – B1, which is a small angle.
Then
′

 

 
m l ha t
h t
a h
C
a
u = – ( – 1s) + + + m + (mu)
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
μ μ μ ∂μ (5.53)
′

 

 
m l ha t
w
h t
a
w
h
w
C
a
w
w = – ( – 1s) + + + m + (mw)
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
ˆ ˆ ˆ ∂ ˆ
(5.54)
′

 

 
m l ha t
q
h t
a
q
h
q
C
a
q
q = – ( – 1s) + + + m + (mq )
c
c
1 c 1
f
D
s
∂
∂
∂
∂
∂
∂
∂
ˆ ˆ ˆ ∂ ˆ
(5.55)
5.4.1 The rotor force and flapping derivatives
To complete the calculation of the sets of force and moment derivatives above, we
need the basic rotor force and flapping derivatives ∂tc /∂μ, ∂a1/∂wˆ , … , etc. Now one
of the important variables in the expressions for tc, hc and a1 is the inflow ratio λ (or
λD), and it will be useful to find its derivatives first.
In order to do this, it will be assumed that the inflow remains constant for the
purpose of subsequent differentiation. In fact, the thrust coefficient, for example, has
an effect on the inflow via the lift deficiency function familiar in fixed wing aircraft
aerodynamic theory. This can be developed (see Johnson5) to provide a relationship
between local (elemental) momentum theory and local elemental lift which leads to
expressions for pitch and roll. These are dependent on the coefficients of an expression
for the local induced velocity coefficient which depends on radial position and
harmonically on azimuth angle. Pitt and Peters6, and Peters and Ha Quang7 have
formalised this ‘dynamic inflow’ approach and demonstrated its applicability in the
flight mechanics of manoeuvring flight.
However, the present development is aimed at a more elementary level, and thus
the inflow, as mentioned above, will be held constant whilst differentiating to obtain
　
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