coaxial rotor

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Unmanned Aircraft Design,Modeling and Control

Modeling of Coaxial Helicoptersw a ocus on c ass c ro or ynam cs eory


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Part 1 Goals of the Course

Know how to derive models for the most relevant effects

Gain capability to read into more advanced topics

Unmanned Aircraft Design, Modeling and Control - Rotorcraft 3


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Part 1 Coaxial Helicopters

All thrust forces used for lifting

Advantage in forward flight

Hig er comp exity rotor u

Higher maintenance cost

Retreating

BladeAdvancing

Blade

Reverse


Flow Region

Helicopter rotor in forward flight:

Lift loss on retreating blade limitsmaximum forward flight velocity

To some extent compensated on coaxialrotor


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Part 1 Coaxial Helicopter Acuation Systems

Full-Scale:

Dual swashplate

Miniature-Scale

ary ng

Single swashplate &

stabilizer bar

No collective


Note the fundamental difference of theacutation system in a full-scale and small-scale (toy) coaxial helicopter


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Part 1 ASL Coaxial Helicopter History

The CoaX Family

AIRobots CX



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Part 2 Model Overview

Lower Rotor Block Diagram


Collective and cyclic pitch swashplate


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Part 1 Rotor Degrees of Freedom

Feathering (Pitch)

Flapping


Lead-lag degree of freedom not asrelevant for free-flight dynamics

as flapping and feathering


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Part 1 Rotor Degrees of Freedom

Feathering (Pitch)

Flapping


Rotor azimuthPeriodic functionsof rotor azimuth

Collective pitchLongitudinal cyclic Lateral cyclic

Rotor coning Longitudinal flapping Lateral flapping


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Part 2 Rigid Body Dynamics


General concept of modelling:

1.) Model rigidbody dynamics2.) Attach external forces & moments to bodydynamics (e.g. rotor forces & moments)3.) Model forces & moments in detail


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Part 2 Rigid Body Dynamics

Momentum

Conservation

Angular Momentum


Conservation

Velocities and accelerations derived fromkinematics of rigid body


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Part 2 External Forces and Moments

ThrustGravity Hub Body Drag

orceorce orce orce

Thrust Tilt

Moment

Flapping

Moment

Hub Force

Moment

Rotor

Torque


Body drag and hub

forces/momentshave NOT beendiscussed...

Next steps:

Derive models for rotor thrust, torque and

flapping moments.

To do so we want to first find expressionsfor the thrust and torque as well as for the

the flapping coefficients


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Part 2 Blade Aerodynamics: Blade Forces

Rotor Torque

Rotor Thrust



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Rotor Torque

Rotor Thrust


Usually neglectable


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Rotor Torque

Rotor Thrust


To compute the lift and drag forces we

need models for the lift and dragcoefficients AND the inflow velocity"U"


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Part 2 Blade Aerodynamics: Blade Lift and Drag

Rotor Torque

Rotor Thrust


Introduce ourcoefficient models


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Part 2 Blade Aerodynamics: Inflow Velocities (1)

Tangential

Inflow

Inflow


Because on a helicopter rotor itsmuch larger than the perpendicularinflow

Rotorcraft body


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Rotor

VelocityInflow due

to Pitch Motion

Longitudinal Inflow


Rotorcraft bodypitch and roll rateswith respect to

hub-wind frame {B'}

Rotorcraft bodypitch and roll rates

with respect to hubframe {H}


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Induced

Velocity

Roll rate

about hub y-axis

Longitudinal Inflow

due to Flapping


m Descent

Rate

B a e F ap

Velocity

In case of the lower coaxialrotor, w' may also be useful toaccount for the downwash ofthe upper rotor (only near hoverthough)


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Part 2 Rotor Forces & Torques (Connecting the Pieces)

Average over rotor azimuth


um over num er o a es

For this course wefocus on the mostrelevant inflowcomponents

Introduce inflowmodels to lift anddrag increments

Calculate main forces

Integrate overblade radius

Compute the average overrotor azimuth


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Simplified torque


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Part 2 Rotor Torque


Simplified torquecoefficient forhovering rotor


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Part 2 Rotor Thrust


We do the same

for thrust

Tip-path plane normal defines direction of thrustvector.


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Part 2 Rotor Thrust


Note:the flapping coefficients are defined with respectto the hub-wind frame {B'} and NOT the bodyframe {B}

l i

Having defined thrust and torque weinvestigate a more complexphenomenon: blade flapping


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Part 2 Rotor Flapping Moment


The flapping moment is most relevant

for the pitch/roll motion of therotorcraft

phenomenon: blade flapping...

Spring stiffness

coefficient

P t 2 Bl d Fl i R t H b D iThe hub-design fundamentallyaffects the flapping behavior


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Part 2 Blade Flapping: Rotor Hub Designs

Teetering

Hingeless


affects the flapping behavior.

All of these designs can be

captured with ONE mathematicalmodel only.

P t 2 Bl d Fl i Li Fl S i M d lVirtual hinge to


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Part 2 Blade Flapping: Linear Flap Spring Model

Virtual Hinge

with Spring


tua ge toalso capture thehingeless rotor

Part 2 Blade Flapping: Flapping Dynamics (1)


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Derivation Procedure

Formulate Angular Momentum

Conservation Law for Blade Body

Extract Flap Dynamics

Introduce Steady-State Solution

Solve for Flap Coefficients


Part 2 Blade Flapping: Flapping Dynamics (2)Look at the bladeforces & moments


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o ces & o e tsaffecting flapping

Only y-component of flapping dynamics relevant

Moment due to

aerodynamic bladeforces

Moment due togravity (neglected)

Flap spring moment


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Part 2 Blade Flapping: Steady State Flapping


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Part 2 Blade Flapping: Steady State Flapping

Steady-State

Fla in Res onseForcing Terms

Flapping Behavior Dominated by and


We introduce the steady-state flapping model to find expressionsfor steady state flapping

We can derive a system of equations for the flappingcoefficients

only blade

feathering can beactively controlled

Matrices A heavily depend onflapping frequency ratio and Locknumber

Looking at thedetails of thissystem allowsinsight into the

basic flap behaviorof a rotor (e.g. flapphase-lag etc.)

Part 2 Rotor Flapping Moment For a simple low-frequency rotor model we can directly


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Part 2 Rotor Flapping Moment


For a simple low frequency rotor model we can directlyintroduce the steady-state coefficients into our flapmoments.

A better flapping model would account for the flappingdynamics (and not only for the steady-state response)

Part 2 Extension to Full Coaxial Rotor System To derive a model of a coaxiald h i l


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Part 2 Extension to Full Coaxial Rotor System


rotor we extend the single rotortheory

Part 2 Conclusion

How the downwash of the upper rotoraffects the lower rotor not only in hover

b t l i f d fli ht i


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Part 2 Conclusion

Modeling the dynamics of helicopters is difficult

Helico ters are vibrations ke t to ether b differential e uations

J. Watkinson

Coaxial Rotor Interaction has not been treated

Modeling in Forward-Flight & Axial Descent very difficult

u c g

Principles of Helicopter Aerodynamics, G. Leishman

Helicopter Performance, Stability and Control, R. Prouty

Helicopter Flight Dynamics, G. Padfield (pdf @ ethbib)


Many more

but e.g. also in forward flight requiresadvanced numerical methods.

This would go far beyond the scope ofthis lecture...

coaxial rotor

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