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∂
∂
t
w
z
a
w as
c = – =
2
ˆ 8 +
μ
μ (5.79)
∂
∂
a
w as
1
2
2 =
16
ˆ (1 – /2)(8 + )
μ
μ μ
(5.80)
and
∂
∂
h
w
a
as
c
2
0
2
D
2
D =
4
8 +
(1 – 9 /2)/6 +
ˆ 1 – /2
μ
μ
θ μ λ
μ
(5.81)
5.4.2 The q-derivatives
In the equation for the thrust coefficient,
t a
c 0
= 2
4
2
3
[θ(1 + 3μ/2) + λ] (3.33)
all the terms are independent of the pitch rate q; therefore
Flight dynamics and control 155
∂ ∂ t q zq c / ˆ = – = 0
Now a rate of pitch applied to the rotor shaft gives rise to extra aerodynamic and
inertia terms in the flapping equation. According to eqn 1.16 we must add the terms
γqˆ cos ψ/8 – 2qˆ sin ψ to the right-hand side of eqn 3.48. Then we find, from
examination of the various harmonic terms, that the longitudinal flapping Δa1 due to
the rate of pitch is
Δ γ μ
a
q
1 2 = – 16
1 – /2
⋅
ˆ
or ∂
∂
a
q
1
2 = – 16 1
ˆ γ 1 – μ /2
⋅ (5.82)
We find that the aerodynamic incidence changes represented by the term 2 ˆ q sin ψ
cause sideways flapping given by
∂
∂
b
q
1
2 = – 1
ˆ (1 + μ /2)
(5.83)
Also, from eqn 3.51, it follows that
∂a0/∂qˆ = 0 (5.84)
Due to the pitching motion, the normal velocity at the rotor blade becomes modified to
UP = ΩR(λD – μa0 cos ψ + xqˆ cos ψ)
Using this expression in the calculation of the in-plane force HD, we find for its
coefficient
hc a a0b1 a0 a q a b q
2
D 0 1 D 0 D 1 = 1
4
–
4
1
3
– 12
+ 1
3
– 12
+ + 1
8 μδ μ λ μθ λ μ ˆ ˆ 
 
 
(5.85)
or, in terms of λ (referred to the no-feathering axis),
hc a a0b1 a0 a q a a a b q
2
0 1 1
2
0
2
D 0 1 1 = 1
4
–
4
1
3
–12
+ 1
3
– 12– 12
+ + + 1
8 μδ  μ ˆ λ μ μθ λ μ θ μˆ
 

 
(5.86)
In these expressions for hcD we have not cancelled the first two terms in the
brackets, as in eqn 3.39. Then differentiating eqn 5.86, using eqns 5.83 and 5.84, and
remembering that ∂λ/∂ ˆ q = 0, gives
∂
∂
∂
∂
∂
∂
∂
∂
h
q
a a
a
a
q
a
a
q
a
q
c 0
2 0
1
1
1 2
0
D = – 1
4
– /3
(1 + /2)
+ 1
3
– 12
– +
ˆ μ ˆ ˆ ˆ
 λ μ μθ
 
+ 1
8
–
8(1 + /2) 1 2 μ μ
μ
b
qˆ 
 
Typically ˆ q is very small compared with a0, a1, b1 and the final term in this
156 Bramwell’s Helicopter Dynamics
expression may be neglected. Assuming that ˆ q does not alter the induced velocity
unduly, then the same assumptions that followed eqn 3.38 may be made, leading to
∂
∂
∂
∂
h
q
a a
a
q
c a a
1
2
0
1
2
0
2
D =
4
12
+ – –
ˆ ˆ12 (1 + /2) λ μ μ θ μ
μ [ ]
Again, the final term involving μ2 can be shown to be small compared with the
bracketed term and may be neglected, leading to
∂
∂
h
q
a
a c
2 1
2
0
D = –
4
(1 – /2)
1
2
+ –
ˆ γ μ
λ μ μ θ 
 
 
(5.87)
Since ∂tc /∂qˆ is zero, eqn 5.55 becomes
　
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