Chapter 4
Rotating Blade Motion
Yanjie Li
Harbin Institute Of Technology
Shenzhen Graduate School
Outline
•
Blade motions
•
Types of rotors
•
Equilibrium about the flapping hinge
•
Equilibrium about the lead

lag hinge
•
Equation of motion for a flapping blade
•
Dynamics of blade flapping with a hinge offset
•
Blade feathering and the swashplate
•
Dynamics of a lagging blade with a hinge offset
•
Coupled flap

lag motion and pitch

flap motion
•
Other types of rotors
•
Rotor trim
4.1 Rotating Blade Motion
3 blade motions
•
flapping
–
balance asymmetries in
forward flight
•
lead

lag
–
balance Coriolis forces
•
feathering
–
change pitch
–
change
collective thrust
–
cyclic: pitch, roll control
4.2 Types of Rotors
4.3 Equilibrium about the Flapping Hinge
•
balance of aerodynamic, centrifugal forces
–
flapping (conning) angle
Moment at the
rotational axis by CF
Centrifugal Force (CF)
Aerodynamic moment about the flap hinge:
Equilibrium
Coning angle for
equilibrium
For a parabolic lift, the center of lift is at ¾ radius
Ideal twist and uniform inflow produces linear lift
4.4 Equilibrium about the Lead

Lag Hinge
Centrifugal Force on the blade element
component
⊥
blade axis
Lag moment
Aerodynamic forces = induced + profile drag =
From geometry:
which shows that centrifugal force acts at
R
(1 +
e
)/2
4.5
Equation of Motion for Flapping Blade
In hovering flight, coning angle is a constant
In forward flight, coning angle varies in a periodic manner with azimuth
M>0,
clockwise,
reducing
Centrifugal moment:
Inertial moment:
Aerodynamic
moment:
Define mass moment of inertia about the flap hinge
For uniform inflow
y
U
T
Define Lock number
Flapping equation
for e=0
A more general form:
where
Similar to a spring

mass

damper system
Undamped natural frequency
1
If no aerodynamic forces the flapping motion reduces to
The rotor can take up arbitrary orientation
In forward flight, the blade flapping motion can be represented as infinite Fourier series
Fourier coefficient
Assume: uniform inflow, linearly twisted blades, can be founded analytically
M
Substituting in Section 3.5
P
T
U
U
,
In forward flight( ), periodic coefficients; no analytical solution
0
The general flapping equation of motion cannot be solved analytically for
0
Two
options:
Assume the solution for the blade flapping motion to be given by the first harmonics only:
We have
Notice by setting
There is an equivalence between pitching motion and flapping motion
If cyclic pitch motion is assumed to be
the flapping response
flapping response lags the blade pitch (aerodynamic) inputs by 90
°
4.7
Dynamics of Blade Flapping with a Hinge Offset
Hinge at eR
Forces
inertia
centrifugal
aerodynamic
Moment balance
Mass moment
of inertia
Non

dimensional flap frequency
Analogy with a spring

mass

damper system:
undamped natural frequency
rev
/
1
Flapping equation
In hover, the flapping response to cyclic pitch inputs is given
Phase lag will be less than
0
90
4.8 Blade Feathering and the Swashplate
Blade pitch
where
Blade

pitch motion comes from two sources:
control input
Elastic deformation (twist) of the blade and control system
Swashplate=Rotating plate + No

rotating plate
The movement of the swashplate result in changes in blade pitch
4.9 Review of Rotor Reference Axes
Several physical plane can be used to describe the equations of motion of the rotor
blade
. Each has advantages over others for certain types of analysis.
Hub Plane (HP)
Perpendicular to the rotor shaft
An observer can see both flapping and feathering
Complicated, but linked to a physical part of the aircraft; often used for blade
dynamic and flight dynamic analyses
No Feathering Plane (NFP) :
An observer cannot see the variation in cyclic pitch, i.e.
still see a cyclic variation in blade flapping angle; used for performance analyses
Tip Path Plane (TPP)
cannot see the variation in flapping, i.e.
used for aerodynamic analyses
Control Plane (CP)
represents the commanded cyclic pitch plane; swashplate plane
Schematic of rotor reference axes and planes
4.10
Dynamics of a Lagging Blade with a Hinge Offset
Offset = eR
A wrong typo
Taking moments about the lag hinge:
Moment of inertia about the lag hinge
Equation of motion about lead/lag hinge
Lag frequency with a hinge offset
Centrifugal moment about the lag
hinge is much smaller than in
flapping
Uncoupled natural frequency
is much smaller
4.11 Coupled Flap

Lag Motion
moment about flap hinge:
coupled equation of motion
where
moment about lead/lag hinge
coupled equation for motion
where
4.12 Coupled Pitch

Flap Motion
Pitch

flap coupling using a hinge to reduce cyclic flapping
Used to avoid a lead

lag hinge, save weight
Achieved by placing the pitch link/pitch horn connection to lie off the flap hinge
axis
Flapping by , pitch angle is reduced by
Eq. 4.39
Where uniform inflow has been assumed. Flapping frequency is increased to
Coning angle becomes
4.13 Other Types of Rotors
Teetering rotor
Flapping motion
4.13.2 Semi

Rigid or Hingeless Rotors
•
Flap and lag hinges are replaced by flexures
•
If feathering is also replaced: bearingless
•
Equivalent spring stiffness at an equivalent hinge offset e
•
is the pre

cone angle,
•
nonrotating flapping frequency
Natural flapping frequency
where we assumed . If , the frequency reduces to that for an
articulated rotor
Equivalent hinge offset and flap stiffness can be found by looking at the
slope at a point at 75% of the radius
effective spring stiffness
4.14 Introduction to Rotor Trim
•
Trim
–
calculation of rotor control settings, rotor disk orientation(pitch,
flap) & overall helicopter orientation for the prescribed flight
conditions
•
Controls
–
Collective pitch
•
increases all pitch angles change thrust
–
Lateral & Longitudinal cyclic pitch
•
Lateral ( ) tilts rotor disk left & right
•
Longitudinal ( ) tilts rotor disk forward & aft
–
Yaw
•
use tail rotor thrust
cross coupling is possible,
flight control system can minimize cross

coupling effects
4.14.1 Equations for Free

Flight Trim
Moments can be written in terms of the contribution from different parts
where hub plane (HP) is used as reference and flight path angle is
Assume: No sideslip (fuselage side force ) ;no contribution from horizontal
and vertical tails
vertical force equilibrium
longitudinal force equilibrium
Lateral force equilibrium
Pitching moment about the hub
Rolling moment about the hub
Torque
Assume small angles
Thrust = average blade lift number of blades
Complexity of the expression of , this should be evaluated numerically
Assume ; ;
rotor torque, side force, drag force & moments can be computed similarly
rotor drag force
rotor side force
the rotor torque is given by
rotor rolling and pitching moments
additional equations for
s
'
The vehicle equilibrium equations, along with the inflow equations, can be written as
Where X is the vector of rotor trim unknowns, defined as
Nonlinear equations

solved numerically
Section 4.14.2 introduce a typical trim solution procedure
Thank You
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