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m′ [ ]
ah
a
C
q m
m
= – q
4
(1 – /2)
2
3
+ 3
2
+ –
16
(1 – /2)
+ ( ) 2 0 1 2 f
s
γ μ
θ λ μ
γ μ
(5.88)
Equation 5.87 displays the ‘Amer effect’8, i.e. the considerable and destabilising
change of the in-plane force during pitching, the terms in the square bracket of eqn
5.87 adding to those of tc, eqn 3.33. In hovering flight these extra terms reduce the
effect of the thrust tilt by about 25–30 per cent; in climbing flight (large λ) the
reduction is much greater.
5.4.3 The tailplane derivatives
The derivative (mu)T If MT is the pitching moment due to the tailplane, then, in the
notation of Chapter 4, section 4.2.2
MT VS l RCL
12
2
= – T T T ρ (5.89)
and
∂
∂
∂
∂
M
u
VS l R C V
C
L u
T L
T T
12
= – + T
ρ T
The tailplane lift coefficient can be expressed as
CLT a 0 = T(αT + θ – τ – ε)
from section 4.2.2. Then
V
C
u
a V
u
∂ L
∂
∂
∂
T = – T
ε
and, since ε = vi/V,
V
u V V
∂
∂
∂
∂
ε
= vi – vi
= ∂ i – i
∂
λ
μ
λ
μ
Flight dynamics and control 157
where vi is evaluated at the tailplane by the method discussed in Chapter 4.
Hence
∂
∂
∂
∂
M
u
VS l R CL a
T
T T
12
T
= – + i – i T ρ λ
μ
λ
μ
which in non-dimensional form is
( )T = – T + –
12
T
i i
T m V C a u L ′
μ λ
μ
λ
μ
∂
∂ (5.90)
The derivative (mw)T Differentiating eqn 5.89 with respect to w gives
∂
∂
∂
∂
M
w
V S l R
C
w
L T 12
2
= – ρ T T T
and
∂
∂
∂
∂
∂
∂
C
w
a
V
a
V w
LT = 1 – = 1 – T T i εα
λ
Hence
( )T = – 1 –
12
T T
m Va i
w w ′
μ ∂λ
∂ ˆ
(5.91)
Here again, the downwash term is evaluated at the tailplane.
The derivative (mq)T For a steady pitching rate q the change of incidence
at the tailplane is
ΔαT = lTRq /V
and the moment change is
Δ ρ M a VS l R q = – 12
T T T
2 2
Therefore
( )T = –
12
T T T
Mq ρaVS l2R2
In non-dimensional form,
( )T = –
12
mq ′ aTμVTlT (5.92)
The derivative (mw˙ )T This is the moment derivative arising from the time taken for
the changes of downwash to reach the tailplane, and may be calculated in the same
manner as for the fixed wing aircraft. It appears that for the small tailplanes typical
of most helicopters this derivative is of little importance. According to Bramwell9 the
derivative is
( )T = –
12
T T T
m aV l i
w˙ wˆ ′
∂
∂
λ
(5.93)
158 Bramwell’s Helicopter Dynamics
5.4.4 Summary of longitudinal derivatives
For convenience, all the longitudinal derivatives are collected together below:
x t
a t h
u = – c – –
1
D
c cD ∂
∂
∂
∂
∂
μ ∂
α
μ μ (5.41)
z
t
u = – ∂ c
∂μ (5.42)
x t
a
w
t
w
h
w w = – c – –
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Bramwell’s Helicopter Dynamics(81)
