Chapter 4 Rotating Blade Motion

1
Chapter 4Rotating Blade Motion

• Yanjie Li
• Harbin Institute Of Technology
• Shenzhen Graduate School

2
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

3
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
4.2 Types of Rotors
5
4.3 Equilibrium about the Flapping Hinge

• balance of aerodynamic, centrifugal forces
• flapping (conning) angle

Moment at the rotational axis by CF
6
Aerodynamic moment about the flap hinge
Coning angle for equilibrium
Equilibrium
For a parabolic lift, the center of lift is at ¾
radius
Ideal twist and uniform inflow produces linear
lift
7
4.4 Equilibrium about the Lead-Lag Hinge
Centrifugal Force on the blade element
component ? blade axis
Aerodynamic forces induced profile drag
Lag moment
8
From geometry

• which shows that centrifugal force acts at R (1
  e)/2

9
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

Centrifugal moment
Inertial moment
Aerodynamic moment
Mgt0, clockwise, reducing
10
Define mass moment of inertia about the flap hinge
11
For uniform inflow
Define Lock number
Flapping equation for e0
A more general form
where
Similar to a spring-mass-damper system
Undamped natural frequency
12
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
13
Assume uniform inflow, linearly twisted blades,
can be founded analytically
Substituting in Section 3.5

• In forward flight( ), periodic
  coefficients no analytical solution

14
The general flapping equation of motion cannot be
solved analytically for
Two options
Assume the solution for the blade flapping motion
to be given by the first harmonics only
We have
15
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

16
4.7 Dynamics of Blade Flapping with a Hinge Offset

• Hinge at eR
• Forces
• inertia
• centrifugal
• aerodynamic

Moment balance
Non-dimensional flap frequency
17
Analogy with a spring-mass-damper
system undamped natural frequency
Flapping equation
In hover, the flapping response to cyclic pitch
inputs is given
Phase lag will be less than
18
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

19
SwashplateRotating plate No-rotating plate
20
The movement of the swashplate result in changes
in blade pitch
21
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

22
Schematic of rotor reference axes and planes
23
4.10 Dynamics of a Lagging Blade with a Hinge
Offset
Offset eR
A wrong typo
24
Taking moments about the lag hinge
Moment of inertia about the lag hinge
Lag frequency with a hinge offset
Equation of motion about lead/lag hinge

• Centrifugal moment about the lag hinge is much
  smaller than in flapping
• Uncoupled natural frequency is much smaller

25
4.11 Coupled Flap-Lag Motion
coupled equation of motion
where
moment about flap hinge
26
coupled equation for motion
where
moment about lead/lag hinge
27
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
28
4.13 Other Types of Rotors

• Teetering rotor

Flapping motion
29
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

30

• 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

31
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
32
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
vertical force equilibrium
longitudinal force equilibrium
33
Lateral force equilibrium
Pitching moment about the hub
Rolling moment about the hub
Assume small angles
Torque
34
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
35
rotor side force
the rotor torque is given by
rotor rolling and pitching moments
36
additional equations for
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
37

• Thank You