曝光台 注意防骗
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n
φn
ψ (7.88)
from the orthogonal relations eqn 7.82
The strain energy UB due to bending is10
U EI
Z
r
B r
R
= 12
d
0
2
2
2 ∫
∂
∂
= 1
2
d
0
1 2
2
2
R
EI
z
x
∫ x
∂
∂
(7.89)
= 1
2
d
d
d
d
d
=1 =1 0
1 2
2
2
R 2
EI
S
x
S
x
x
m n
m n
Σ Σ m n
∞ ∞ φ φ ∫ (7.90)
The potential energy UG due to the centrifugal stiffening effect arises from the centrifugal
force G acting on a mass element doing negative work when the radius of its point
of application decreases as a result of blade bending (the ‘shortening effect’). Then
UG Gr r Z r r
R
= { – [1 – ( / ) ] } d
0
2 1/2 ∫ ∂ ∂
= 12
d
0
1 2
R G
z
x
∫ x
∂
∂ (7.91)
for small deflections. Substituting for ∂z/∂x we have
U R G
S
x
S
x
G x
m n
m n
m n = 12
d
d
d
d
d
=1 =1 0
1
Σ Σ
∞ ∞ φ φ ∫
The total strain and potential energy is
U U U
R
EI
S
x
S
x
R G
S
x
S
x
B G x
m n
m n
= + = 1 m n m n
2
d
d
d
d
+
d
d
d
d
d
=1 =1 0
1 2
2
2
2
Σ Σ 2
∞ ∞ ∫
φ φ
272 Bramwell’s Helicopter Dynamics
= 12
2 2 3 d
=1 =1
2
0
1
λn m n φ
ΩR Σ Σ n mSmSn x
∞ ∞ ∫ (7.92)
from eqn 7.81, so that
U n R f n
n
n = 12
2 2 3 ( )
=1
2 λ φ Ω Σ
∞
(7.93)
on using the orthogonal properties again.
The external work done by the elementary force (∂F/∂r)dr in an arbitrary displacement
is
δ δ ∂
∂
( ) = d ( )δφ
=1
W R
F
x
x S x
n
n n Σ
∞
and for the whole blade
δ δφ ∂
∂ W R
F
x
S x
n
= n nd
=1 0
1
Σ ∞
∫
so that
∂
∂φ
∂
∂
W
R
F
x
S x
n
= n d
0
1 ∫ (7.94)
Note that since only blade bending is being considered, the Coriolis force, which acts
in the lead–lag sense, does not contribute to this work form.
Now, Lagrange’s equations for small displacements are
d
d
– + =
t
T T U W
n n n n
∂
∂φ
∂
∂φ
∂
∂φ
∂
∂φ
or, with ψ = Ωt,
d
d
– + = ψ
∂
∂φ
∂
∂φ
∂
∂φ
∂
∂φ
T T U W
n′ n n n
(7.95)
Then, using eqns 7.88, 7.93, and 7.94, Lagrange’s equations give
d
d
+ = 1
( )
( )d
2
2
2
2 2
0
φ 1
ψ
λ φ ∂
∂
n
n n n R f n
F
x
S x x
Ω ∫ as before (7.96)
The response equations of the blade in lagging motion are of the same form as for
the flapping motion, i.e. they take the form
d
d
+ = 1
( )
( )d
2
2
2
2 2
0
χ 1
ψ
ν χ
∂
∂
n
n
y
n R g n
F
x
T x x
Ω ∫
where
g(n) = mTn (x)dx
0
1
2 ∫
Structural dynamics of elastic blades 273
is the generalised inertia for the nth lag bending mode, and Fg is the lagwise external
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Bramwell’s Helicopter Dynamics(137)
