Rotating Blade Motion

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16 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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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