Modeling of coaxial helicopters - eth

Unmanned Aircraft Design,Modeling and ControlModeling of Coaxial Helicopters(with a focus on classic rotor dynamics theory)Christoph Hürzeler

Course Contents— Part 1 - Introduction:— Goals of the Course— Overview of Coaxial Helicopters— General Modeling Methodologies— Levels of Modeling— Part 2 – Rotorcraft Modeling— Modeling Introduction— Modeling Main Aerodynamic Forces— Modeling Blade Flapping— The Coaxial RotorUnmanned Aircraft Design, Modeling and Control - Rotorcraft 2

Part 1 – Goals of the Course— Know how to derive models for the most relevant effects— Better understand why and how helicopters work— Gain capability to read into more advanced topicsUnmanned Aircraft Design, Modeling and Control - Rotorcraft 3

Part 1 – Modeling Levels & Methodologies— Blackbox— Identify input-output behavior— No understanding of underlying system— Greybox— Based on laws of physics— Identify model parameters— Whitebox— Purely based on laws of physics— All parameters predictedwww.johnson-aeronautics.comwww.continuum-dynamics.comUnmanned Aircraft Design, Modeling and Control - Rotorcraft 4

Part 1 – Coaxial Helicopters— All thrust forces used for lifting— No exposed tail rotor— Advantage in forward flight— Higher complexity rotor hub— Higher maintenance costAdvancingBladeRetreatingBladeReverseFlow RegionUnmanned Aircraft Design, Modeling and Control - Rotorcraft 5

Part 1 – Coaxial Helicopter Acuation Systems— Full-Scale:— Fixed RPM— Dual swashplate— Miniature-Scale— Varying RPM— Single swashplate &stabilizer bar— No collectiveUnmanned Aircraft Design, Modeling and Control - Rotorcraft 6

Part 1 – ASL Coaxial Helicopter History— The CoaX Family— muFly Prototypes— AIRobots CXUnmanned Aircraft Design, Modeling and Control - Rotorcraft 7

Part 1 – AIRobots CX Actuation SystemUnmanned Aircraft Design, Modeling and Control - Rotorcraft 8

Part 2 – Model Overview— Lower Rotor Block DiagramUnmanned Aircraft Design, Modeling and Control - Rotorcraft 9

Part 1 – Rotor Degrees of Freedom— Feathering („Pitch“)— FlappingUnmanned Aircraft Design, Modeling and Control - Rotorcraft 10

Part 1 – Rotor Degrees of Freedom— Feathering („Pitch“)— FlappingUnmanned Aircraft Design, Modeling and Control - Rotorcraft 11

Part 2 – Rigid Body DynamicsUnmanned Aircraft Design, Modeling and Control - Rotorcraft 12

Part 2 – Rigid Body DynamicsMomentumConservationAngular MomentumConservationUnmanned Aircraft Design, Modeling and Control - Rotorcraft 13

Part 2 – External Forces and MomentsGravity Thrust Hub Body DragForce ForceForce ForceThrust TiltMomentFlappingMomentHub ForceMomentRotorTorqueUnmanned Aircraft Design, Modeling and Control - Rotorcraft 14

Part 2 – Coordinate Frame Definition— {W}: World frame with origin O— {B}: Rigid body frame with origin S— {B‘}: Hub-wind frame with origin S‘— {H}: Rotor hub frame with origin N— {K}: Rotor flap frame with origin E— G: Rotor blade center of gravityUnmanned Aircraft Design, Modeling and Control - Rotorcraft 15

Part 2 – Special Case of the Hub-Wind Frame— {W}: World frame with origin O— {B}: Rigid body frame with origin S— {B‘}: Hub-wind frame with origin S‘— {H}: Rotor hub frame with origin N— {K}: Rotor flap frame with origin E— G: Rotor blade center of gravityUnmanned Aircraft Design, Modeling and Control - Rotorcraft 16

Part 2 – Blade Aerodynamics: Blade Element TheoryUnmanned Aircraft Design, Modeling and Control - Rotorcraft 17

Part 2 – Blade Aerodynamics: Blade ForcesRotor TorqueRotor ThrustUnmanned Aircraft Design, Modeling and Control - Rotorcraft 18

Part 2 – Blade Aerodynamics: Blade ForcesProfile DragInduced DragRotor TorqueRotor ThrustUnmanned Aircraft Design, Modeling and Control - Rotorcraft 19

Part 2 – Blade Aerodynamics: Blade ForcesRotor TorqueRotor ThrustUnmanned Aircraft Design, Modeling and Control - Rotorcraft 20

Part 2 – Blade Aerodynamics: Angle of AttackUnmanned Aircraft Design, Modeling and Control - Rotorcraft 21

Part 2 – Blade Aerodynamics: Lift and Drag Coefficients (1)Unmanned Aircraft Design, Modeling and Control - Rotorcraft 22

Part 2 – Blade Aerodynamics: Lift and Drag Coefficients (2)— Lift & Drag Polars for NACA 4412Unmanned Aircraft Design, Modeling and Control - Rotorcraft 23

Part 2 – Blade Aerodynamics: Blade Lift and DragRotor TorqueRotor ThrustUnmanned Aircraft Design, Modeling and Control - Rotorcraft 24

Part 2 – Blade Aerodynamics: Inflow Velocities (1)PerpendicularInflowTangentialInflowUnmanned Aircraft Design, Modeling and Control - Rotorcraft 25

Part 2 – Blade Aerodynamics: Inflow Velocities (2)RotorVelocityInflow dueto Pitch MotionLongitudinal InflowUnmanned Aircraft Design, Modeling and Control - Rotorcraft 26

Part 2 – Blade Aerodynamics: Inflow Velocities (3)InducedVelocityClimb/DescentRateRoll rateabout hub y-axisBlade FlapVelocityLongitudinal Inflowdue to FlappingUnmanned Aircraft Design, Modeling and Control - Rotorcraft 27

Part 2 – Rotor Forces & Torques (Connecting the Pieces)— Integrate over blade radius— Average over rotor azimuth— Sum over number of bladesUnmanned Aircraft Design, Modeling and Control - Rotorcraft 28

Part 2 – Rotor TorqueInduced DragProfile DragUnmanned Aircraft Design, Modeling and Control - Rotorcraft 29

Part 2 – Rotor TorqueUnmanned Aircraft Design, Modeling and Control - Rotorcraft 30

Part 2 – Rotor ThrustUnmanned Aircraft Design, Modeling and Control - Rotorcraft 32

Part 2 – Rotor Flapping MomentUnmanned Aircraft Design, Modeling and Control - Rotorcraft 33

Part 2 – Blade Flapping: Rotor Hub Designs— Fully Articulated— Teetering— HingelessUnmanned Aircraft Design, Modeling and Control - Rotorcraft 34

Part 2 – Blade Flapping: Linear Flap Spring ModelVirtual Hingewith SpringUnmanned Aircraft Design, Modeling and Control - Rotorcraft 35

Part 2 – Blade Flapping: Flapping Dynamics (1)— Derivation Procedure— Dettach Blade from Hub— Formulate Angular MomentumConservation Law for Blade Body— Extract Flap Dynamics— Introduce Steady-State Solution— Solve for Flap CoefficientsUnmanned Aircraft Design, Modeling and Control - Rotorcraft 36

Part 2 – Blade Flapping: Flapping Dynamics (2)Unmanned Aircraft Design, Modeling and Control - Rotorcraft 37

Part 2 – Blade Flapping: Flapping Dynamics (3)Flapping = Damped OscilatorUnmanned Aircraft Design, Modeling and Control - Rotorcraft 38

Part 2 – Blade Flapping: Flapping Dynamics (3)Flapping Frequency RatioLock NumberUnmanned Aircraft Design, Modeling and Control - Rotorcraft 39

Part 2 – Blade Flapping: Steady State FlappingSteady-StateFlapping ResponseForcing TermsFlapping Behavior Dominated by andUnmanned Aircraft Design, Modeling and Control - Rotorcraft 40

Part 2 – Rotor Flapping MomentUnmanned Aircraft Design, Modeling and Control - Rotorcraft 41

Part 2 – Extension to Full Coaxial Rotor SystemUnmanned Aircraft Design, Modeling and Control - Rotorcraft 42

Part 2 – Conclusion— Modeling the dynamics of helicopters is difficult— “Helicopters are vibrations kept together by differential equations”J. Watkinson— Coaxial Rotor Interaction has not been treated— Modeling in Forward-Flight & Axial Descent very difficult— (especially for the coaxial)— Further recommended reading— “Principles of Helicopter Aerodynamics”, G. Leishman— “Helicopter Performance, Stability and Control”, R. Prouty— “Helicopter Flight Dynamics”, G. Padfield (pdf @ ethbib)— Many more …Unmanned Aircraft Design, Modeling and Control - Rotorcraft 43