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mode shape and throw doubt on the validity of eqn 7.106.
To avoid this difficulty, another expression for the root moment can be derived.
References to eqn. 7.104, the root moment can be written as
M t r
F
r
r mt Z Z r
R R
(0, ) = d – + d
0
2
0
2
∫ Ω∫  2
   
∂
∂
∂
∂ψ
(7.107)
Writing, as before
Z R S x
n
= n( )n( )
=1
Σ ∞
φ ψ
eqn 9.110 becomes
M t r
F
r
r R MxS x
R
n
n
(0, ) = d – n n
d
d
+ d
0
2 3
=1
2
2
0
1 ∫ Ω  ∫
 

 
Σ ∞
∂
∂
φ
ψ
φ
Equation 7.106 has been used by Bramwell2 to obtain stability derivatives of a
hingeless rotor.
Defining a root moment coefficient by
Mc = M/ρbcΩ2R4
282 Bramwell’s Helicopter Dynamics
eqn 7.106 can be written in non-dimensional form as
M
a
c b
1
2
1
= 1
( – 1)
( )
λ
γ φ ψ (7.108)
where γ
ρ
1
0
1
1
=
d
acR
∫ mx S x
which is equivalent to the lock number for the hinged rigid blade.
Since, for considerations of performance and stability and control, we may consider
only the 1st harmonic response, the flapping of the hingeless blade can be written in
the form
ϕ1(ψ) = β = a0 – a1 cos ψ – b1 sin ψ
when β , a0, a1, b, are analogous to the rigid blade flapping angle and flapping
coefficients. Then the pitching and rolling moment coefficients Cm and Cl of a hingeless
rotor with b blades are
C
a a
m z
1
2
1
1
=
(λ – 1)
γ (7.109)
C
a b
l =
( – 1)
2
1
2
1
1
λ
γ (7.110)
The flapping coefficients a0, a1, b1 can be calculated by a similar analysis to that for
the hinged rigid blade. The equations, although quite straight forward, are rather
cumbersome because unlike those of the centrally hinged blade, the integrals involving
the mode shapes do not lend to simple fractions and the equations do not give the
coefficients explicitly. The equations have been given by Bramwell21 and Curtiss and
Shupe20. The calculations show that the amplitude of flapping is only a little less than
that if the blade were hinged.
Returning to the determination of the hub moment, and since we are interested
only in the pitching and rolling moments on the helicopter, only the first harmonic
motion need be considered, in which case, for the nth mode
φn = (a0 )n – (a1)n cos ψ – (b1)n sin ψ
so that
d
d
+ = ( )
2
2
φ
ψ
φ ω
n
n a n
hence, substituting in the modified version of eqn 7.110, we have
M r
F
r
r R mxS x
n
(0, ) = d + 2 3 ( )n nd
=1
0
0
1
ψ ∂
∂
φ Ω Σ
∞ ∫
Structural dynamics of elastic blades 283
When all the blades are taken together to form the total pitching and rolling
moments, the last term on the right-hand side vanishes, so that the hub moment
reduces to
M r
F
r
r
R
(0, ψ ) = ∂ d
∂ ∂ ∫ (7.111)
In this equation, given by Simons and by Curtiss and Shupe, no restriction need be
placed on the loading, although, in practice, the loading will depend on the number
of modes chosen to represent the blade motion. According to Curtiss and Shupe, only
one mode is necessary over a well range of conditions, but for high values of μ two
modes ought to be used for better accuracy. On the other hand, if the general equation
in terms of blade flapping is used, eqn 7.106, we have to include that number of
modes which, in superposition, approximate to the true loading shape. In general,
this is longer than that for eqn 7.111.
Let us now consider modelling the hingeless rotor by defining an equivalent
hinged rigid blade system.
Although Young derived eqn 7.106 for the particular case of a cyclic pitch application
in hovering flight, instead of proceeding with the calculations in terms of mode
shape, as has been discussed above, he replaced the hingeless blade with a rigid blade
　
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