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systems’, J. Mech. Eng. Sci., 19(6), 1977.
15. Done, G. T. S., ‘Relative energy concepts in rotating system dynamics’, 2nd Int. Conf. on
Vibrations in Rotating Machinery, Cambridge, UK, 2–4 Sept. 1980.
Structural dynamics of elastic blades 289
16. Wilkinson, R. and Shilladay, J. D., ‘A comparison of the azimuthwise integration methods
used in the rotor performance computer programs’, Westland Helicopters Res. Memo. 55,
1969.
17. Hohenemser, K. H., ‘Hingeless rotorcraft flight dynamics’, AGAR Dograph 197, 1974.
18. Young, M. I., ‘A simplified theory of hingeless rotors with application to tandem helicopters’,
Proc. 18th Annual natn. Forum Am. Helicopter Soc., 1962.
19. Simons, I. A., ‘Some thoughts on a stability and control research programme with special
reference to hingeless rotor helicopter’, Westland Helicopters Res. Memo. 79, 1970.
20. Curtiss, H. C., jnr and Shupe, N. K., ‘A stability and control theory for hingeless rotors’, 27th
Annual natn. Forum Am. Helicopter Soc., Preprint 541, 1971.
21. Bramwell, A. R. S., ‘A method for calculating the stability and control derivatives of helicopters
with hingeless rotors’, Res. Memo. City Univ. Lond. Aero. 69/4, 1969.
22. Bramwell, A. R. S., ‘Further note on the calculation of the hub moments of a hingeless
helicopter rotor’, Res. Memo. City Univ. Lond. Aero. 71/2, 1971.
8
Rotor induced vibration
8.1 Introduction
We have seen in Chapter 3 that the aerodynamic loads on a helicopter rotor blade
vary considerably as it moves round the rotor disc, and in steady flight these loads are
periodic. The rotor forces and moments causing fuselage vibration are transmitted
from the blades to the rotor hub and then by means of the main rotor drive shaft into
the main rotor gearbox bearings and hence into the gearbox casing, and finally into
the fuselage at the gearbox attachment points.
Again, as we saw in Chapter 3, these loads arise from the aerodynamic forces on
the rotor blades, together with the inertia forces produced by the flapping and lagging
motions of the blade.
In addition to the main rotor loads, tail rotor force fluctuations may also be of
concern but in the vast majority of cases the main rotor forcing is the prime cause of
unwanted vibration.
The control of vibration is important for four main reasons:
(i) to improve crew efficiency, and hence safety of operation;
(ii) to improve the comfort of passengers;
(iii) to improve the reliability of avionic and mechanical equipment;
(iv) to improve the fatigue lives of airframe structural components.
Hence it is very important to control vibration throughout the design, development
and in-service stages of a helicopter project.
It may be appreciated from section 3.11 that the generation of oscillatory aerodynamic
loads at frequencies which are integral multiples of the rotational speed is fundamental
to the edgewise operation of the rotor in forward flight, and hence forced vibrations
of the helicopter cannot be entirely eliminated. Therefore the efforts of the design and
development organisations must be devoted to the minimisation of the vibratory
loads and to the minimisation of the fuselage response.
Subsequent sections of this chapter will indicate how this is achieved, although it
Rotor induced vibration 291
should be recognised that it is not yet possible to predict accurately the vibration
level of a helicopter in a specific flight condition. However, much valuable information
can be generated which can lead to an acceptable standard of vibration.
8.2 The exciting forces
The hub forces and moments from each blade can be resolved into force components
X, Y, Z, and moment components L, M, N relative to fixed axes in the helicopter,
Fig. 8.1. The axes are taken to conform with standard stability notation, i.e. with
the Z axis pointing downwards along the hub axis and the Y axis pointing to starboard.
The moment N can be disregarded since it forms part of the rotor torque.
In Chapter 1, section 1.11, we defined the rotating reaction forces along and
perpendicular to the hub axes as R1, R2, R3, Fig. 8.2.
If ψ is the azimuth angle of the zeroth (reference) blade, the azimuth angle of the
kth blade is ψk = ψ + 2πk/b. The corresponding components in the fixed X, Y, Z
directions are given by
Xk = – R1k cos ψk + R2k sin ψk
M
y
z
x
L
N
Fig. 8.1 Force components resolved along helicopter body axes
X
Y
Z
ψ
R1
R2
R3
Fig. 8.2 Reactions at blade hinge
292 Bramwell’s Helicopter Dynamics
　
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