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+ + –
8
( ) + ( ) 1
2
1
2 g g
2
2 2
λ ν γ σ γ λ 2
r
ck
C k
k
k
B C k
252 A
+
8
( ) + 1
4
+ –
8
( ) = 0 1
2
2
1
2
1
2 g g
2
γν λν λ γ σ
C k
k
k
r
ck
B C k
A
(9.13)
328 Bramwell’s Helicopter Dynamics
Of the above constants, the only two which can readily be varied are the chordwise
blade c.g. position and the non-rotating torsional frequency. Divergence occurs when
the term independent of λ is equal to zero, i.e. when
ν γ σ
1 λ
2 g g
2
1
2 =
8
r ( )
ck
C k
(9.14)
For C(k) = 1, i.e. when unsteady aerodynamic effects are neglected, eqn 9.14
defines a straight line relating the torsional frequency to the chordwise c.g. location.
It is clear that the boundary exists for positive values of σg, i.e. a rearward location
of the c.g. relative to the 1
4 -chord point. The physical interpretation of divergence is
simply that, if the c.g. of the blade is located sufficiently far behind the 1
4 -chord
point, the component of centrifugal force about the blade-span axis, which arises
when the blade flaps, exerts a nose up moment which is greater than the torsional
restoring moment.
To find the flutter boundary we put λ = iω, the condition for undamped oscillations,
and equate the real and imaginary parts to zero. We then find
ν ω
ω
1
2 2
2
g
2
= +
– 1)( / )
4 ( )
( x k
C k
(9.15)
and
σ
γ
ω λ γ ω
ω g
2 2
g
2
2
2 2 2
2 = 2
( )
– +
( ) /64
– 1
k
k
ck
r C k
B C k
A
A
1
2 (9.16)
Equations 9.15 and 9.16 provide the relationship between ν1 and σg for which
undamped oscillations occur with ω2, the square of the flutter frequency ratio, as
parameter. Thus, if ν1 is given, eqn 9.15 can be solved for ω2, which can then be
inserted into eqn 9.16 to obtain σg.
A sketch of typical divergence and flutter boundaries as functions of σg and ν1
2 for
C(k) = 1 is shown in Fig. 9.6. The two curves correspond to the four roots of the
characteristic frequency quartic, eqn 9.14. If the torsional frequency is fixed at the
value corresponding to the point A of Fig. 9.6 and the chordwise c.g. position is
moved aft, the line AA′ is traced. The motion is completely stable until the intersection
with the divergence boundary is reached. At this point, A′, one of the roots of the
frequency quartic becomes zero, and just to the right of A′ the root becomes positive,
denoting the unstable divergence. The remaining roots between A′ and A″ are a
negative real root (stable) and a complex pair denoting a damped oscillation. To the
right of the point A″ the oscillation becomes undamped also. Moving along BB″ the
unstable flutter oscillation is met first, and then the divergence.
Analysis of the equations 9.13, 9.15 and 9.16 shows that the flutter and divergence
boundaries exist only for positive values of σg, which shows that pitch-flap motion
must be stable for σg < 0, i.e. when the centre of gravity of the blade is ahead of the
flexural axis. Since the torsional frequency must have a positive value, Fig. 9.6
shows that stable motion is possible even when the centre of gravity is behind the
Aeroelastic and aeromechanical behaviour 329
flexural axis. Inspection of eqns 9.15 and 9.16 shows that for flapping frequencies
greater than Ω, i.e. λ1 > 1, the flutter boundary is shifted to the left. This suggests that
offset or hingeless blades are rather less stable than centrally hinged blades.
The effect of taking unsteady aerodynamic coefficients (C(k) ≠ 1) into account is
quite small. The appropriate value of C(k) must be obtained iteratively, since the
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Bramwell’s Helicopter Dynamics(162)
