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 f

orward 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

yUT

Define Lock number

Flapping equation for e=0

A more general form:

where

Similar to a spring-mass-damper systemUndamped 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

PT UU ,

In forward flight( ), periodic coefficients; no analytical solution0

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 090

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, fla

p) & 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