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shown in Fig. 7.18.
It is usual that the first mode frequency is several times the rotor frequency and
may be as high as 6Ω or even more.
7.2.4 Intermodal coupling
In the previous sections we have examined the flapping and lagging and torsional
mode shapes and frequencies on the assumptions that the motions are independent of
one another. In general, however, a certain amount of elastic and inertial coupling
exists when there is built-in twist and if the elastic and mass axes do not coincide.
The complete coupled equations have been given by Houbolt and Brooks11 and, more
recently, by Sobey12.
Structural dynamics of elastic blades 263
Aerodynamic and inertia coupling of the first-mode motions relating to classical
blade flutter will be considered in Chapter 9.
To give an indication of how the torsional motion of the blade modifies the
flapping equation, consider a blade with zero built-in twist. Referring to Figs 7.19
and 7.20 we see that twist introduces the following flapping moments and shears.
(i) A component GeA sin θ of the moment GeA, GeA being the moment of the
blade tension about the elastic axis when the latter is displaced from the
section centroid.
(ii) The moment of the centrifugal force due to the displacement of the centre of
gravity of the blade section relative to the elastic axis. The moment is easily
seen to be equal to
r
R
∫ 2mr(e )dr
Ω 1 1θ1 1.
(iii) The change of shear force dS due to the inertia loading – meθ˙˙ dx resulting
from the twisting motion.
1
0
–1
Q1(x)
Q2(x)
Q3(x)
r/R 1
Fig. 7.18 Torsional mode shapes of a uniform blade
Elastic axis
Centroid
c.g.
θ
eA
e
Elastic axis
Centroid
c.g.
Fig. 7.19 Blade section geometry
Fig. 7.20 Blade section forces and moments
GeA
(Z˙˙ + eθ˙˙) dm
θ
264 Bramwell’s Helicopter Dynamics
Then, if M′ is the sum of the moments due to the effects of twist, we have, for
small θ,
∂
∂
∂
∂
∂
∂
θ ∂
∂
θ θ
2
2
2
2 A
= = ( ) + ( 2 ) – M′
r
S
r r
Ge
r
Ω rme me˙˙
Adding these terms to the uncoupled flap-bending equation, eqn 7.6 we get
∂
∂
∂
∂
θ ∂
∂
∂
∂
∂
∂
θ θ
2
2
2
2 A
– – – ( 2 ) + ( + ) = 0
r
EI
Z
r
Ge
r
G
Z
r r
rme m Z e
Ω ˙˙ ˙˙
This agrees with the equation of flapwise bending given by Houbolt and Brooks
for the special case of zero built-in twist. By similar arguments we could derive the
coupled lag and torsional equations of motion.
When the blade also has built-in twist, there is elastic coupling between the flapping
and lagging motion. The full equations are given by Houbolt and Brooks, but a
simple model used by Ormiston and Hodges13 to investigate flap-Jag instability will
serve here to give a simple illustration of the effects of elastic coupling on the
flapping and lagging motion. The flexibility of the blade is represented in the diagram
of Fig. 7.21. Part of the flexibility is contained in the hub springs, which have
flapping and lagging stiffnesses of κβH κξH and respectively. The remaining part of
the blade flexibility lies just outboard of the feathering hinge so that the associated
spring system, with stiffnesses κ β B and κ ξB , rotates when blade pitch is applied.
It is clear that, when pitch is applied (i.e. θ ≠ 0), flapping motion, i.e displacements
of the blade perpendicular to the plane of rotation, causes moments in the plane of
rotation, and vice versa. It is not difficult to show that for flap and lag displacements
β and ξ the elastic moments Me and Le in the flap and lag planes are given by
Me e
= – [ + ( – ) sin 2 ] – e
2
( – ) sin 2
β κ ρ κ κ θ
ξρ
Δβ ξ β Δκξ κβ θ (7.68)
Ω
Kξ H Kβ H
Kξ B
˙θ
Fig. 7.21 Representation of stiffness coupling
Kβ B
Structural dynamics of elastic blades 265
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Bramwell’s Helicopter Dynamics(132)
