曝光台 注意防骗
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of the hingeless blade.
In order to investigate the second of the benefits of the hingeless rotor listed
above, a method is needed for calculating the moments transmitted to the hub due to
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
S1(x), λ1 = 1.092
S2(x), λ2 = 3.157
S3(x), λ3 = 5.57
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 7.31 Typical flapping mode shapes for hingeless blade
280 Bramwell’s Helicopter Dynamics
the deflection of the blades. The first analysis for such calculations was made by
Young18 who produced a very simple expression for the hub moment. Fig. 7.32
refers.
Young calculated the moment at the blade root M(0, t) of the aerodynamic, centrifugal,
and inertia loadings as
(0, ) = – – d
0
2
2
2 M t F
r
m Z m Z
t
r r
∫R Ω
∂
∂
∂
∂
(7.104)
By means of the blade bending equation
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2 – + =
r
EI
Z
r r
G
Z
r
m Z
t
F
r
the terms ∂F/∂r – m∂ 2Z/∂t 2 can be eliminated from eqn 7.104 giving
M t
r
EI
Z
r r
G
Z
r
m Z r r
R
(0, ) = – – d
0
2
2
2
2
2 ∫
Ω
∂
∂
∂
∂
∂
∂
∂
∂
Writing the blade deflection as
Z R S r t
n
= n( )n( )
=1
Σ ∞
φ
the root moment can be written
M t R
r
EI
S
r r
G
S
r
m S r r
R
n
n
n n
(0, ) = d n
d
d
d
– d
d
d
d
– d
0 =1
2
2
2
2
2 ∫ Σ
∞
Ω
φ
But the blade mode shapes satisfy the equation
d
d
d
d
– d
d
d
d
– = 0
2
2
2
2
2
r
EI
S
r r
G
S
r
n n m S
n n
λ Ω
Z
Ω
0
dF
Ω2r dm
Z
r X
Fig. 7.32 Flapwise forces acting on blade
∂ Ζ
∂
2
2 d
t
m
Structural dynamics of elastic blades 281
so that the root moment finally becomes
M t R mrS r
n
n n
R
(0, ) = 2 ( – 1) n d
=1
2
0
Ω Σ
∞
φ λ ∫ (7.105)
Since, in the case of the rotor with a flapping hinge we only need to use the first rigid
blade mode for considerations of performance and stability and central, let us assume
that we need only consider the first bending mode of the hingeless rotor for the same
purposes.
Then if only the first mode is considered, the moment is
M(0, t) = ( – 1) R3 (t ) mxSi dx
1
0
1
λ2 φ ∫ (7.106)
This is a slightly more general form of the equation given by Young, who calculated
only the increment due to a cyclic pitch change in hovering flight.
It has been shown by Simons19 and by Curtiss and Shupe20 that the derivation of
eqn 7.106 implies that the aerodynamic loading distribution has the same shape as
the first bending mode, i.e. that
∂
∂
F
x
∝ S1(x) ⋅ m(x)
This implicit assumption may not be seriously in error at low values of μ, but one
might expect that at high values the loading might depart considerably from the first
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Bramwell’s Helicopter Dynamics(141)
