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16 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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Chapter 4

Yanjie Li

Harbin Institute Of Technology

Outline

Types of rotors

-
lag hinge

Equation of motion for a flapping blade

Dynamics of blade flapping with a hinge offset

Dynamics of a lagging blade with a hinge offset

Coupled flap
-
lag motion and pitch
-
flap motion

Other types of rotors

Rotor trim

flapping

balance asymmetries in
forward flight

-
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

-
Lag Hinge

Centrifugal Force on the blade element

component

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

where

-
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
. 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

Lag frequency with a hinge offset

hinge is much smaller than in
flapping

Uncoupled natural frequency
is much smaller

4.11 Coupled Flap
-
Lag Motion

coupled equation of motion

where

coupled equation for motion

where

4.12 Coupled Pitch
-
Flap Motion

Pitch
-
flap coupling using a hinge to reduce cyclic flapping

-
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

-
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

Torque

Assume small angles

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

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