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1
1 1
B m z m z
m z m z
w u u w
μ B w w B
Now, when calculating the control derivatives, we found that zB1 zw = – μ , so that
d
d
=
–
( + )
1
1
B m z m z
z m m
w u u w
μ w B μ w
=
( + )
1
c 1
E
zwt mB μmw
from eqns 5.21, 5.25, and 5.27, with τc = 0.
Thus, the rate of change of cyclic pitch to trim with forward speed is proportional
to the constant term in the stability quartic, i.e. it is a measure of the static stability
of the helicopter. The result is analogous to that of the fixed wing aircraft. Now, in
the theory of the static stability of subsonic aircraft it is usual to regard mu as zero,
so that the static stability is entirely determined by the sign of mw, which is proportional
to ∂Cm/∂CL. But ∂Cm/∂CL is proportional to the distance of the c.g. from the neutral
point, and for this reason – ∂Cm/∂CL is called the ‘static margin’. Thus the static
stability of the fixed wing aircraft is related simply to a quantity which has a clear
physical meaning. Unfortunately, as we have seen, in helicopter work there is no
such simple parameter which directly controls the static and dynamic stability. Both
types of stability are affected by a number of quantities over which the designer has
only limited powers of variation. At low speeds mw is small, and the static stability
is dominated by mu which, although positive, is responsible for the helicopter’s
characteristic divergent oscillation. At high speeds the static stability is often improved,
because mu has changed very little and mw, although being much larger and positive
194 Bramwell’s Helicopter Dynamics
(i.e. in the unstable sense), is usually associated with a positive zu. When a tailplane
is fitted to improve the dynamic stability, at the high speeds for which zu is usually
positive, the combination zumw is such as to decrease the value of E1. Thus the
desirability of a positive static stability is not as straight/forward for the helicopter as
it is with a fixed wing aircraft.
The coefficient C1. Consider a helicopter performing a pull-up manoeuvre at constant
speed in a vertical plane, Fig. 5.30. It will be assumed that the flight path is circular
and the helicopter is in trim. The pitching moment will be a function of the vertical
velocity w, the rate of pitch q, and the cyclic pitch B1; i.e. in the trimmed manoeuvre
we have Cm(w, q, B1) = 0 with w and q being dependent on ng, the excess normal
acceleration. Therefore
d
d
=
d
d
+
d
d
+
d
d
= 0
1
C 1
n
C
w
w
n
C
q
q
n
C
B
B
n
m ∂ m m m
∂
∂
∂
∂
∂
or
d
d
= –
( / )d /d + ( / )d/d
/
1
1
B
n
C w w n C q q n
C B
m m
m
∂ ∂ ∂ ∂
∂ ∂ (5.183)
where, in eqn 5.183, w and q have been non-dimensionalised (section 5.2). Then, if
ΔT is the increase of thrust in the manoeuvre, we have
ng
T
W g
=
/
Δ
or
n
T w w R
W
z
w
= w w
( / )
= –
c
∂ ∂ Ω
so that
d
d
w = – c
n
w
zw
Now
q
ng
V
= Ω
therefore
d
d
= =
*
q c
n
g
V
w
Ω μμ
Substituting in eqn 5.183 and expressing in derivative form gives
Fig. 5.30 Helicopter in vertical pull-up
V
q
Flight dynamics and control 195
d
d
=
/ – / * 1 c c
1
B
n
m w z m w
m
w w q
B
′ ′
′
μμ
=
–
1
μ
μ
m zm
z m
w w q
w B
∝ C1′
Hence, as with the fixed wing aircraft, the rate of change of control angle to trim
with normal acceleration is proportional to C1′, i.e. the value of C1 when the forward
velocity terms have been neglected. Now we have seen that the normal acceleration
response to either a longitudinal control input or a vertical gust depends on the value
of C1′; if it is positive the motion is stable (since the coefficient B1′c is always
positive), and if it is negative the response is divergent. Thus the slope of the control
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Bramwell’s Helicopter Dynamics(99)
