Aircraft Equations of Motion - 2
 Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2012 " •  Rotating frames of reference" •  Combined equations of motion" •  FLIGHT 6-DOF simulation program" Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! Euler Angle Rates Euler-Angle Rates and Body-Axis Rates" Body-axis angular rate vector (orthogonal)" ω B = ω x ω y ω z " # $ $ $ $ % & ' ' ' ' B = p q r " # $ $ $ % & ' ' ' Euler-angle rate vector" Form a non- orthogonal vector of Euler angles" Θ = φ θ ψ % & ' ' ' ( ) * * *  Θ =  φ  θ  ψ % & ' ' ' ( ) * * * ≠ ω x ω y ω z % & ' ' ' ' ( ) * * * * I Relationship Between Euler-Angle Rates and Body-Axis Rates" •  is measured in the Inertial Frame" •  is measured in Intermediate Frame #1" •  is measured in Intermediate Frame #2" •  Inverse transformation [(.) -1 ≠ (.) T ] " € ˙ φ € ˙ θ € ˙ ψ p q r ! " # # # $ % & & & = I 3  φ 0 0 ! " # # # $ % & & & + H 2 B 0  θ 0 ! " # # # $ % & & & + H 2 B H 1 2 0 0  ψ ! " # # # $ % & & & p q r ! " # # # $ % & & & = 1 0 −sin θ 0 cos φ sin φ cos θ 0 −sin φ cos φ cos θ ! " # # # $ % & & &  φ  θ  ψ ! " # # # $ % & & & = L I B  Θ  φ  θ  ψ $ % & & & ' ( ) ) ) = 1 sin φ tan θ cos φ tan θ 0 cos φ −sin φ 0 sin φ sec θ cos φ sec θ $ % & & & ' ( ) ) ) p q r $ % & & & ' ( ) ) ) = L B I ω B "Can the inversion become singular?" "What does this mean? " •  which is" Euler-Angle Rates and Body-Axis Rates" Avoiding the Singularity at θ = ±90°" !  Don’t use Euler angles as primary definition of angular attitude" !  Alternatives to Euler angles" -  Direction cosine (rotation) matrix" -  Quaternions" !  Propagation of rotation matrix (9 parameters)" -  From previous lecture"  H B I h B =  ω I H B I h B  H I B t ( ) = −  ω B t ( ) H I B t ( ) = − 0 −r t ( ) q t ( ) r t ( ) 0 −p t ( ) −q t ( ) p t ( ) 0 t ( ) # $ % % % % & ' ( ( ( ( B H I B t ( ) ; H I B 0 ( ) = H I B φ 0 , θ 0 , ψ 0 ( ) Consequently" Avoiding the Singularity at θ = ±90°" !  Propagation of quaternion vector" o  see Flight Dynamics for details" e 1 e 2 e 3 e 4 ! " # # # # # $ % & & & & & = Rotation angle, rad x-component of rotation axis y-component of rotation axis z-component of rotation axis ! " # # # # # $ % & & & & & !  Quaternion vector: single rotation from inertial to body frame (4 parameters)"  e t ( ) =  e 1 t ( )  e 2 t ( )  e 3 t ( )  e 4 t ( ) ! " # # # # # # $ % & & & & & & = 0 −r t ( ) −q t ( ) −p t ( ) r t ( ) 0 −p t ( ) q t ( ) q t ( ) p t ( ) 0 −r t ( ) p t ( ) −q t ( ) r t ( ) 0 ! " # # # # # # $ % & & & & & & e 1 t ( ) e 2 t ( ) e 3 t ( ) e 4 t ( ) ! " # # # # # # $ % & & & & & & = Q t ( ) e t ( ) ; e 0 ( ) = e φ 0 , θ 0 , ψ 0 ( ) Rigid-Body Equations of Motion Point-Mass Dynamics" •  Inertial rate of change of translational position" •  Body-axis rate of change of translational velocity" –  Identical to angular-momentum transformation"  r I = v I = H B I v B  v I = 1 m F I  v B = H I B  v I −  ω B v B = 1 m H I B F I −  ω B v B = 1 m F B −  ω B v B F B = X Y Z ! " # # # $ % & & & B = C X qS C Y qS C Z qS ! " # # # $ % & & & v B = u v w ! " # # # $ % & & &  r I t ( ) = H B I t ( ) v B t ( )  v B t ( ) = 1 m t ( ) F B t ( ) + H I B t ( ) g I −  ω B t ( ) v B t ( )  Θ I t ( ) = L B I t ( ) ω B t ( )  ω B t ( ) = I B −1 t ( ) M B t ( ) −  ω B t ( ) I B t ( ) ω B t ( ) # $ % & •  Rate of change of Translational Position " •  Rate of change of Angular Position " •  Rate of change of Translational Velocity " •  Rate of change of Angular Velocity " r I = x y z ! " # # # $ % & & & I Θ I = φ θ ψ % & ' ' ' ( ) * * * I v B = u v w ! " # # # $ % & & & B ω B = p q r " # $ $ $ % & ' ' ' B •  Translational Position " •  Angular Position " •  Translational Velocity" •  Angular Velocity " Rigid-Body Equations of Motion" (Euler Angles)" Aircraft Characteristics Expressed in Body Frame of Reference" I B = I xx −I xy −I xz −I xy I yy −I yz −I xz −I yz I zz " # $ $ $ $ % & ' ' ' ' B F B = X aero + X thrust Y aero + Y thrust Z aero + Z thrust ! " # # # $ % & & & B = C X aero + C X thrust C Y aero + C Y thrust C Z aero + C Z thrust ! " # # # # $ % & & & & B 1 2 ρ V 2 S = C X C Y C Z ! " # # # $ % & & & B q S Aerodynamic and thrust force " Aerodynamic and thrust moment " Inertia matrix " Reference Lengths b = wing span c = mean aerodynamic chord M B = L aero + L thrust M aero + M thrust N aero + N thrust ! " # # # $ % & & & B = C l aero + C l thrust ( ) b C m aero + C m thrust ( ) c C n aero + C n thrust ( ) b ! " # # # # # $ % & & & & & B 1 2 ρ V 2 S = C l b C m c C n b ! " # # # $ % & & & B q S Rigid-Body Equations of Motion: Position"  x I = cos θ cos ψ ( ) u + −cos φ sin ψ + sin φ sin θ cos ψ ( ) v + sin φ sin ψ + cos φ sin θ cos ψ ( ) w  y I = cos θ sin ψ ( ) u + cos φ cos ψ + sin φ sin θ sin ψ ( ) v + −sin φ cos ψ + cos φ sin θ sin ψ ( ) w  z I = −sin θ ( ) u + sin φ cos θ ( ) v + cos φ cos θ ( ) w  φ = p + qsin φ + r cos φ ( ) tan θ  θ = q cos φ − r sin φ  ψ = qsin φ + r cos φ ( ) sec θ •  Rate of change of Translational Position " •  Rate of change of Angular Position " Rigid-Body Equations of Motion: Rate"  u = X / m − gsin θ + rv − qw  v =Y / m + gsin φ cos θ − ru + pw  w = Z / m + g cos φ cos θ + qu − pv  p = I zz L + I xz N − I xz I yy − I xx − I zz ( ) p + I xz 2 + I zz I zz − I yy ( ) " # $ % r { } q ( ) I xx I zz − I xz 2 ( )  q = 1 I yy M − I xx − I zz ( ) pr − I xz p 2 − r 2 ( ) " # $ %  r = I xz L + I xx N − I xz I yy − I xx − I zz ( ) r + I xz 2 + I xx I xx − I yy ( ) " # $ % p { } q ( ) I xx I zz − I xz 2 ( ) •  Rate of change of Translational Velocity " •  Rate of change of Angular Velocity " Mirror symmetry, I xz ≠ 0# FLIGHT -  Computer Program to Solve the 6-DOF Equations of Motion http://www.princeton.edu/~stengel/FlightDynamics.html! FLIGHT - MATLAB Program" http://www.princeton.edu/~stengel/FlightDynamics.html! FLIGHT - MATLAB Program" Examples from FLIGHT Longitudinal Transient Response to Initial Pitch Rate" Bizjet, M = 0.3, Altitude = 3,052 m! •  For a symmetric aircraft, longitudinal perturbations do not induce lateral- directional motions " Transient Response to Initial Roll Rate" Lateral-Directional Response" Longitudinal Response" Bizjet, M = 0.3, Altitude = 3,052 m! •  For a symmetric aircraft, lateral- directional perturbations do induce longitudinal motions " Transient Response to Initial Yaw Rate" Lateral-Directional Response" Longitudinal Response" Bizjet, M = 0.3, Altitude = 3,052 m! Crossplot of Transient Response to Initial Yaw Rate" Bizjet, M = 0.3, Altitude = 3,052 m! Longitudinal-Lateral-Directional Coupling" Alternative Reference Frames Velocity Orientation in an Inertial Frame of Reference" Polar Coordinates" Projected on a Sphere" Body Orientation with Respect to an Inertial Frame" Relationship of Inertial Axes to Body Axes" •  Transformation is independent of velocity vector" •  Represented by" –  Euler angles" –  Rotation matrix " v x v y v z ! " # # # # $ % & & & & = H B I u v w ! " # # # $ % & & & u v w ! " # # # $ % & & & = H I B v x v y v z ! " # # # # $ % & & & & Velocity-Vector Components of an Aircraft" V , ξ , γ V , β , α Velocity Orientation with Respect to the Body Frame" Polar Coordinates" Projected on a Sphere" •  No reference to the body frame" •  Bank angle, μ, is roll angle about the velocity vector " V ξ γ # $ % % % & ' ( ( ( = v x 2 + v y 2 + v z 2 sin −1 v y / v x 2 + v y 2 ( ) 1/ 2 # $ & ' sin −1 −v z / V ( ) # $ % % % % % & ' ( ( ( ( ( v x v y v z ! " # # # # $ % & & & & I = V cos γ cos ξ V cos γ sin ξ −V sin γ ! " # # # $ % & & & Relationship of Inertial Axes to Velocity Axes" Relationship of Body Axes to Wind Axes" •  No reference to the inertial frame " u v w ! " # # # $ % & & & = V cos α cos β V sin β V sin α cos β ! " # # # $ % & & & V β α # $ % % % & ' ( ( ( = u 2 + v 2 + w 2 sin −1 v / V ( ) tan −1 w / u ( ) # $ % % % % & ' ( ( ( ( Angles Projected on the Unit Sphere" € α : angle of attack β : sideslip angle γ : vertical flight path angle ξ : horizontal flight path angle ψ : yaw angle θ : pitch angle φ : roll angle (about body x − axis) µ : bank angle (about velocity vector) •  Origin is airplanes center of mass" Alternative Frames of Reference" •  Orthonormal transformations connect all reference frames" Next Time: Linearization and Modes of Motion   Reading Flight Dynamics, 234-242, 255-266, 274-297, 321-330  Virtual Textbook, Part 10  Supplemental Material  r I = H B I v B  v B = 1 m F B + H I B g I −  ω B v B  ω B = I B −1 M B −  ω B I B ω B ( ) •  Rate of change of Translational Position " •  Rate of change of Rotation Matrix " •  Rate of change of Translational Velocity " •  Rate of change of Angular Velocity " r I = x y z ! " # # # $ % & & & I Θ I = fcn H I B ( ) v B = u v w ! " # # # $ % & & & B ω B = p q r " # $ $ $ % & ' ' ' B •  Translational Position " •  Angular Position " •  Translational Velocity" •  Angular Velocity " Rigid-Body Equations of Motion" (Attitude from Rotation Matrix)"  H I B = −  ω B H I B  r I = H B I v B  v B = 1 m F B + H I B g I −  ω B v B  ω B = I B −1 M B −  ω B I B ω B ( ) •  Rate of change of Translational Position " •  Rate of change of Rotation Matrix " •  Rate of change of Translational Velocity " •  Rate of change of Angular Velocity " r I = x y z ! " # # # $ % & & & I Θ I = fcn H I B e ( ) " # $ % v B = u v w ! " # # # $ % & & & B ω B = p q r " # $ $ $ % & ' ' ' B •  Translational Position " •  Angular Position " •  Translational Velocity" •  Angular Velocity " Rigid-Body Equations of Motion" (Attitude from Quaternion Vector)"  e = Qe . Aircraft Equations of Motion - 2 Robert Stengel, Aircraft Flight Dynamics, MAE 331, 20 12 " •  Rotating frames of reference" •  Combined equations of motion& quot; •  FLIGHT 6-DOF. reference frames" Next Time: Linearization and Modes of Motion   Reading Flight Dynamics, 23 4 -2 42, 25 5 -2 66, 27 4 -2 97, 32 1-3 30  Virtual Textbook, Part 10  Supplemental Material  r I =. 0# FLIGHT -  Computer Program to Solve the 6-DOF Equations of Motion http://www.princeton.edu/~stengel/FlightDynamics.html! FLIGHT - MATLAB Program" http://www.princeton.edu/~stengel/FlightDynamics.html! FLIGHT