View
238
Download
0
Category
Preview:
Citation preview
7/28/2019 coaxial rotor
1/43
Unmanned Aircraft Design,Modeling and Control
Modeling of Coaxial Helicoptersw a ocus on c ass c ro or ynam cs eory
7/28/2019 coaxial rotor
2/43
7/28/2019 coaxial rotor
3/43
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
7/28/2019 coaxial rotor
4/43
7/28/2019 coaxial rotor
5/43
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
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 5
Flow Region
Helicopter rotor in forward flight:
Lift loss on retreating blade limitsmaximum forward flight velocity
To some extent compensated on coaxialrotor
7/28/2019 coaxial rotor
6/43
Part 1 Coaxial Helicopter Acuation Systems
Full-Scale:
Dual swashplate
Miniature-Scale
ary ng
Single swashplate &
stabilizer bar
No collective
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 6
Note the fundamental difference of theacutation system in a full-scale and small-scale (toy) coaxial helicopter
7/28/2019 coaxial rotor
7/43
Part 1 ASL Coaxial Helicopter History
The CoaX Family
AIRobots CX
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 7
7/28/2019 coaxial rotor
8/43
7/28/2019 coaxial rotor
9/43
Part 2 Model Overview
Lower Rotor Block Diagram
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 9
Collective and cyclic pitch swashplate
7/28/2019 coaxial rotor
10/43
Part 1 Rotor Degrees of Freedom
Feathering (Pitch)
Flapping
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 10
Lead-lag degree of freedom not asrelevant for free-flight dynamics
as flapping and feathering
7/28/2019 coaxial rotor
11/43
Part 1 Rotor Degrees of Freedom
Feathering (Pitch)
Flapping
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 11
Rotor azimuthPeriodic functionsof rotor azimuth
Collective pitchLongitudinal cyclic Lateral cyclic
Rotor coning Longitudinal flapping Lateral flapping
7/28/2019 coaxial rotor
12/43
Part 2 Rigid Body Dynamics
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 12
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
7/28/2019 coaxial rotor
13/43
Part 2 Rigid Body Dynamics
Momentum
Conservation
Angular Momentum
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 13
Conservation
Velocities and accelerations derived fromkinematics of rigid body
7/28/2019 coaxial rotor
14/43
Part 2 External Forces and Moments
ThrustGravity Hub Body Drag
orceorce orce orce
Thrust Tilt
Moment
Flapping
Moment
Hub Force
Moment
Rotor
Torque
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 14
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
7/28/2019 coaxial rotor
15/43
7/28/2019 coaxial rotor
16/43
7/28/2019 coaxial rotor
17/43
7/28/2019 coaxial rotor
18/43
Part 2 Blade Aerodynamics: Blade Forces
Rotor Torque
Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 18
7/28/2019 coaxial rotor
19/43
Part 2 Blade Aerodynamics: Blade Forces
Rotor Torque
Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 19
Usually neglectable
7/28/2019 coaxial rotor
20/43
Part 2 Blade Aerodynamics: Blade Forces
Rotor Torque
Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 20
To compute the lift and drag forces we
need models for the lift and dragcoefficients AND the inflow velocity"U"
7/28/2019 coaxial rotor
21/43
7/28/2019 coaxial rotor
22/43
7/28/2019 coaxial rotor
23/43
7/28/2019 coaxial rotor
24/43
Part 2 Blade Aerodynamics: Blade Lift and Drag
Rotor Torque
Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 24
Introduce ourcoefficient models
7/28/2019 coaxial rotor
25/43
Part 2 Blade Aerodynamics: Inflow Velocities (1)
Tangential
Inflow
Inflow
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 25
Because on a helicopter rotor itsmuch larger than the perpendicularinflow
Rotorcraft body
7/28/2019 coaxial rotor
26/43
Part 2 Blade Aerodynamics: Inflow Velocities (2)
Rotor
VelocityInflow due
to Pitch Motion
Longitudinal Inflow
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 26
Rotorcraft bodypitch and roll rateswith respect to
hub-wind frame {B'}
Rotorcraft bodypitch and roll rates
with respect to hubframe {H}
7/28/2019 coaxial rotor
27/43
Part 2 Blade Aerodynamics: Inflow Velocities (3)
Induced
Velocity
Roll rate
about hub y-axis
Longitudinal Inflow
due to Flapping
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 27
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)
7/28/2019 coaxial rotor
28/43
Part 2 Rotor Forces & Torques (Connecting the Pieces)
Average over rotor azimuth
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 28
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
7/28/2019 coaxial rotor
29/43
Simplified torque
7/28/2019 coaxial rotor
30/43
Part 2 Rotor Torque
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 30
Simplified torquecoefficient forhovering rotor
7/28/2019 coaxial rotor
31/43
Part 2 Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 31
We do the same
for thrust
Tip-path plane normal defines direction of thrustvector.
7/28/2019 coaxial rotor
32/43
Part 2 Rotor Thrust
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 32
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
7/28/2019 coaxial rotor
33/43
Part 2 Rotor Flapping Moment
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 33
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
7/28/2019 coaxial rotor
34/43
Part 2 Blade Flapping: Rotor Hub Designs
Teetering
Hingeless
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 34
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
7/28/2019 coaxial rotor
35/43
Part 2 Blade Flapping: Linear Flap Spring Model
Virtual Hinge
with Spring
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 35
tua ge toalso capture thehingeless rotor
Part 2 Blade Flapping: Flapping Dynamics (1)
7/28/2019 coaxial rotor
36/43
Part 2 Blade Flapping: Flapping Dynamics (1)
Derivation Procedure
Formulate Angular Momentum
Conservation Law for Blade Body
Extract Flap Dynamics
Introduce Steady-State Solution
Solve for Flap Coefficients
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 36
Part 2 Blade Flapping: Flapping Dynamics (2)Look at the bladeforces & moments
7/28/2019 coaxial rotor
37/43
Part 2 Blade Flapping: Flapping Dynamics (2)
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 37
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
7/28/2019 coaxial rotor
38/43
7/28/2019 coaxial rotor
39/43
Part 2 Blade Flapping: Steady State Flapping
7/28/2019 coaxial rotor
40/43
Part 2 Blade Flapping: Steady State Flapping
Steady-State
Fla in Res onseForcing Terms
Flapping Behavior Dominated by and
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 40
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
7/28/2019 coaxial rotor
41/43
Part 2 Rotor Flapping Moment
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 41
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
7/28/2019 coaxial rotor
42/43
Part 2 Extension to Full Coaxial Rotor System
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 42
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
7/28/2019 coaxial rotor
43/43
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)
Unmanned Aircraft Design, Modeling and Control - Rotorcraft 43
Many more
but e.g. also in forward flight requiresadvanced numerical methods.
This would go far beyond the scope ofthis lecture...
Recommended