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ρe Degree of elastic coupling = κβ/κβB = κξ/κξB
ρm Fuselage mass ratio = M2/M1 (DAVI)
σ Solidity based on local radius = bc/π r
σg Distance of c.g. of blade elemental strip behind flexural
axis in terms of c
σ1, σ–1 Functions of lag frequency (ground resonance)
τ Period of one rotor revolution
τ Non-dimensional aerodynamic unit of time = t/tˆ
τc Climb angle
τdes Angle of descent
Ω Rotor or blade angular velocity
Ω Angular velocity vector = ω1i + ω2 j + ω3k
ω Total wake swirl velocity
ω Circular frequency
Normalised excitation frequency
ωb Component of ω due to bound circulation
ωn Natural frequency
ωnr Natural frequency of non-rotating blade
ωt
Component of ω due to trailing vortices
ωβ Uncoupled rotating flap natural frequency
ωξ Uncoupled rotating lag natural frequency
ωθ Rotating torsional natural frequency
ω0 Non-rotating torsional natural frequency
ψ Azimuthal angular position of blade, or general angular
coordinate
xxvi Notation
ψ Imaginary part of velocity potential
ψ Yaw displacement of helicopter (from steady state)
ψn Angle between adjacent blades
ψw Wake azimuth relative to blade
ζ Non-dimensional damping factor = c/ccrit
ζmean Weighted mean damping (stall flutter)
ζi
(t) ith generalised coordinate for blade torsion
Suffices
The following suffices refer to:
A Aerodynamic
A, B, D, E Inertia moments and products
D Rotor disc (tip-path plane)
D Drag
L Lift
M Moment
N Normal force
P Perpendicular
P Profile drag
T Tip of blade
T Thrust
T Tailplane
T Tangential
b Bound vorticity
c Climbing
c Coefficient
c Chassis
e Effective
f Fuselage
g Blade c.g., or c.g. of system of particles
h On matrices indicates row is used to correspond to hinge
i Induced
l, u Lower and upper surfaces
kg C.g. of kth blade relative to hub (ground resonance)
nf No feathering
nr Non-rotating
p Pressure
r Radial direction
r Root of blade
r Rotor
rg Rotor c.g. (ground resonance)
s Shaft
Notation xxvii
s Due to centrifugal force at blade hinge
s Setting angle of tailplane
s Shed part of wake
ss Sideslip
t Trailing vortices
t Tail rotor
w Wake
z In z direction
β Flap
ξ Lag
θ Pitch
ψ In tangential direction
0 Generally modulus, or amplitude of
∞ At infinity
1
Basic mechanics of rotor systems
and helicopter flight
1.1 Introduction
In this chapter we shall discuss some of the fundamental mechanisms of rotor systems
from both the mechanical system and the kinematic motion and dynamics points of
view. A brief description of the rotor hinge system leads on to a study of the blade
motion and rotor forces and moments. Only the simplest aerodynamic assumptions
are made in order to obtain an elementary appreciation of the rotor characteristics. It
is fortunate that, in spite of the considerable flexibility of rotor blades, much of
helicopter theory can be effected by regarding the blade as rigid, with obvious
simplifications in the analysis. Analyses that involve more detail in both aerodynamics
and blade properties are made in later chapters. The simple rotor system analysis in
this chapter allows finally the whole helicopter trimmed flight equilibrium equations
to be derived.
1.2 The rotor hinge system
The development of the autogyro and, later, the helicopter owes much to the introduction
of hinges about which the blades are free to move. The use of hinges was first
suggested by Renard in 1904 as a means of relieving the large bending stresses at the
blade root and of eliminating the rolling moment which arises in forward flight, but
the first successful practical application was due to Cierva in the early 1920s. The
most important of these hinges is the flapping hinge which allows the blade to flap,
i.e. to move in a plane containing the blade and the shaft. Now a blade which is free
to flap experiences large Coriolis moments in the plane of rotation and a further
hinge – called the drag or lag hinge – is provided to relieve these moments. Lastly,
the blade can be feathered about a third axis, usually parallel to the blade span, to
enable the blade pitch angle to be changed. A diagrammatic view of a typical hinge
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Bramwell’s Helicopter Dynamics(9)
