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argument as that used for the flapping equation, we arrive at the mode shape equation
d
d
d
d
– d
d
d
d
– ( + 1) = 0
2
2
2
2
2 2 2 4
x
EI T
x
R
x
G
T
x
m RT 




ν Ω (7.52)
Y
α1
r
r1
Y
X
Y1
Ω2r1dm
∂
∂
2
1
2 d
Y
t
m
Fig. 7.16 Lagwise forces acting on blade
P
Structural dynamics of elastic blades 259
and the frequency equation
d2χ/dψ2 + ν 2χ = 0 (7.53)
The boundary conditions relating to eqn 7.52 are the same as those of flapping
motion:
(a) Hinged blade (including lag hinge offset)
At x = e, y = 0, d2y/dx2 = 0 (where y = Y/R)
At x = 1, d2y/dx2 = 0, d3y/dx3 = 0
(b) Hingeless blade
At x = 0, y = 0, dy/dx = 0
At x = 1, d2y/dx2 = 0, d3y/dx3 = 0
The lagging mode shape equation, eqn 7.52, is identical in form to the flapping
equation except that the frequency ratio ν appears as ν 2 + 1 in the mode equation.
This is explained physically by the fact that the centrifugal force field in the lagging
plane is radial instead of parallel as in the flapping plane, and the relationship between
the mode shape and the frequency is different.
The only exact solution of eqn 7.52 is T(x) = x for e = 0, which gives v2 + 1 = 1
or ν = 0. Thus, the first mode shape is a straight line from the hub associated with
zero lagging frequency. When the lag hinge offset is not zero there is no exact
solution to the lag equation but we would expect a close approximation to be a
straight line moving about the hinge point, i.e. like a rigid blade, with a frequency of
approximately √(3/2e)Ω, e being the non-dimensional lag hinge offset. As we mentioned
in Chapter 1, a typical value of e is about 0.05, in which case ν 1 = 0.274.
The calculation of the lag mode shapes and frequencies generally can be achieved
by the methods described for the flapping equation.
For hingeless blades the first mode shape is determined largely by the stiffness
near the root; in fact, the structural element at the root is designed to give suitable
flap and lag frequencies. Since the stiffening effect of the centrifugal field is much
less than in the flapping motion, the first lag frequency is much more sensitive to
changes of root stiffness. The lagging and flapping stiffnesses are ‘matched’ in such
a way as to minimize torsional moments when lagging and flapping motion occurs.
This point will be discussed in more detail in Chapter 9 on aeroelastic coupling, but
the typical first mode frequencies arising from such a choice of stiffnesses are usually
in the region of 0.55 Ω to 0.7 Ω, i.e. much higher than for the hinged blade.
When considering the higher modes, it will be appreciated that the lag stiffness
will be much greater than the flapping stiffness; typically, over most of the blade the
lagwise stiffness is about ten times greater. Calculations show that the shapes of the
lagging modes have the same general appearance as those of the flapping motion but,
whereas the first lag frequency is usually much smaller than the first flap frequency,
the reverse is true for all the corresponding higher modes since the curvature of the
blades invokes the much greater stiffness.
260 Bramwell’s Helicopter Dynamics
7.2.3 Torsional deflections
Consider a portion of the blade of span dr under the action of torsional moments. If
C is the torsional moment of inertia per unit length of span, the inertia moment due
to angular acceleration θ˙˙ is – Cθ˙˙ dr and that due to the ‘propeller moment’ section
1.10, is – CΩ2θ dr.
Then the elementary moment dL tending to twist the blade in the nose-up sense is
dL = – Cθ˙˙ dr – CΩ2θ dr (7.54)
Now consider the torques acting on the sides of the element and let us define the
positive value of the torque when its sense agrees with the positive direction of r. If
W is the torque on the element, Fig. 7.17, the value is –W on the left-hand side and
W + (∂W/∂r) dr on the right-hand side. For equilibrium we must have
–W + L d + + d = 0
r
r W
W
r
∂ r
∂
∂
∂
or
∂W/∂r + ∂L/∂r = 0 (7.55)
Now the relationship between the angle of twist θ and the torque on the element
is
　
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