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C S
n n
n b + i =
e (e – 1)
e – 1
i 2 i
2 i/
ψ π
π
If n is not an integer, C + iS can be written
C S
n n n n
n b n b n b + i =
e e (e – e )
e (e – e )
i i i –i
i / i –i
ψ π π π
π π/ π/
=
sin
sin ( / )
cos +
– 1
+ i sin +
– 1
π
π ψ π ψ π n
n b
n
b
b
n
b
b
giving
C
n
n b
n b
b
=
sin
sin ( / )
cos + – 1
π
π ψ π
(A.3.1)
and
S
n
n b
n b
b
=
sin
sin ( / )
sin + – 1
π
π ψ π
(A.3.2)
If n is an integer but not a multiple of the number of blades b, we see that C +
iS = 0. If n is a multiple of b
C + iS = 0/0
By L’Hospital’s theorem,
C S
n
n
n
n
n b
n b
+ i = e
d(e – 1)/d
d(e – 1)/d
i
2 i
2 i/
= multiple of
ψ
π
π
=
e [2 i e ]
2 i e
i 2 i
2 i/
b n n
n b
ψ π
π
π
π
= b einψ
giving
C = b cos nψ and S = b sin nψ (A.3.3)
Appendices 367
Thus
C
n
n b
n b
b
= n
sin
sin ( / )
cos + – 1 , if is not an integer
π
π ψ π
= 0, if n is not a multiple of b
= b cos nψ, if n is a multiple of b
S
n
n b
n b
b
= n
sin
sin ( / )
sin + – 1 , if is not an integer
π
π ψ π
= 0, if n is not a multiple of b
= b sin nψ, if n is a multiple of b
The Coleman co-ordinates
In ground and air resonance problems (Chapter 9) there are equations of the form
˙x˙k + 2Ωkx˙k + Ω2xk = p(t) sin ψk + q(t) cos ψk + … (A.3.4)
where p(t) and q(t) are functions of time.
The equation can be taken to represent a variable quantity xk which is measured
with respect to the rotating kth blade. We wish to find the total effect of all the blades.
To do this we define new co-ordinates u and v, say, such that
u b x
k=
b
= – (2/ ) k cos k
0
–1
Σ ψ (A.3.5)
v b x
k=
b
= – (2/ ) k sin k
0
–1
Σ ψ (A.3.6)
where, as in the previous section, ψk takes the values ψ, ψ + 2π/b, …, ψ + 2π(b – 1)/b.
Differentiating eqns A.3.5 and A.3.6, we easily find, remembering dψ/dt = Ω,
Σ Ω Σ Ω
k=
b
k k k=
b
x b u xk k b u
0
–1
0
–1
˙ sin ψ = ( – v˙)/2; ˙ cos ψ = – (v + ˙)/2
Σ Ω Ω
k=
b
xk k b u
0
–1
˙˙ sin ψ = – (v˙˙ – 2 ˙ – 2v)/2
Σ Ω Ω
k=
b
xk k bu u
0
–1
˙˙ cos ψ = – (˙˙ + 2 v˙ – 2 )/2
Also, we can show that
Σ Σ k=
b
k k=
b
k b
0
–1
2
0
–1
sin ψ = cos2 ψ = /2
and Σ k=
b
0 k k
–1
sin ψ cos ψ = 0
368 Bramwell’s Helicopter Dynamics
Then, multiplying eqn A.3.4 by cos ψk, summing over the blades, and using the
above relationships, we get
u˙ ˙ + 2Ωku˙ + 2Ωv˙ + 2Ω2kv = – q(t)
Similarly, performing the same procedure with sin ψk,
v˙ ˙ + 2Ωkv˙ – 2Ωu˙ – 2Ω2ku = – p(t)
The transformations represented by eqns A.3.5 and A.3.6, called the Coleman
transformations, have removed the periodic terms from eqn A.3.4. They effectively
resolve a rotating quantity into components along axes fixed in the helicopter body.
A.4 The frequency response of a second order system
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Bramwell’s Helicopter Dynamics(181)
