Aircraft Equations of Motion - 1
 Robert Stengel, Aircraft Flight Dynamics, 
 MAE 331, 2012 " •  6 degrees of freedom" •  Angular kinematics" •  Euler angles" •  Rotation matrix" •  Angular momentum" •  Inertia matrix" 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 ! Lockheed F-104! •  Nonlinear equations of motion" –  Compute exact flight paths and motions" •  Simulate flight motions" •  Optimize flight paths" •  Predict performance" –  Provide basis for approximate solutions" •  Linear equations of motion" –  Simplify computation of flight paths and solutions" –  Define modes of motion" –  Provide basis for control system design and flying qualities analysis " What Use are the Equations of Motion?" dx(t ) dt = f x(t),u(t),w(t ),p(t),t [ ] dx(t ) dt = F x(t ) + G u(t) + L w(t) Translational Position Cartesian Frames of Reference" •  Two reference frames of interest" –  I: Inertial frame (fixed to inertial space)" –  B: Body frame (fixed to body)" Common convention (z up)" Aircraft convention (z down)" •  Translation" –  Relative linear positions of origins" •  Rotation" –  Orientation of the body frame with respect to the inertial frame" Measurement of Position in Alternative Frames - 1" •  Two reference frames of interest" –  I: Inertial frame (fixed to inertial space)" –  B: Body frame (fixed to body)" •  Differences in frame orientations must be taken into account in adding vector components " r = x y z ! " # # # $ % & & & r particle = r origin + Δr w.r.t . origin Inertial-axis view" Body-axis view" Euler Angles Measure the Orientation of One Frame with Respect to the Other" •  Conventional sequence of rotations from inertial to body frame" –  Each rotation is about a single axis" –  Right-hand rule " –  Yaw, then pitch, then roll" –  These are called Euler Angles " Yaw rotation ( ψ ) about z I " Pitch rotation ( θ ) about y 1 " Roll rotation ( ϕ ) about x 2 " •  Other sequences of 3 rotations can be chosen; however, once sequence is chosen, it must be retained " Effects of Rotation on Vector Transformation from Inertial to Body Frame of Reference" Yaw rotation ( ψ ) about z I – Intermediate Frame 1" Pitch rotation ( θ ) about y 1 – Intermediate Frame 2" Roll rotation ( ϕ ) about x 2 - Body Frame" x y z ! " # # # $ % & & & 1 = cos ψ sin ψ 0 −sin ψ cos ψ 0 0 0 1 ! " # # # $ % & & & x y z ! " # # # $ % & & & I = x I cos ψ + y I sin ψ −x I sin ψ + y I cos ψ z I ! " # # # $ % & & & ; r 1 = H I 1 r I x y z ! " # # # $ % & & & 2 = cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ ! " # # # $ % & & & x y z ! " # # # $ % & & & 1 ; r 2 = H 1 2 r 1 = H 1 2 H I 1 ! " $ % r I = H I 2 r I x y z ! " # # # $ % & & & B = 1 0 0 0 cos φ sin φ 0 −sin φ cos φ ! " # # # $ % & & & x y z ! " # # # $ % & & & 2 ; r B = H 2 B r 2 = H 2 B H 1 2 H I 1 ! " $ % r I = H I B r I The Rotation Matrix" H I B ( φ , θ , ψ ) = H 2 B ( φ )H 1 2 ( θ )H I 1 ( ψ ) •  The three-angle rotation matrix is the product of 3 single-angle rotation matrices: " = 1 0 0 0 cos φ sin φ 0 −sin φ cos φ # $ % % % & ' ( ( ( cos θ 0 −sin θ 0 1 0 sin θ 0 cos θ # $ % % % & ' ( ( ( cos ψ sin ψ 0 −sin ψ cos ψ 0 0 0 1 # $ % % % & ' ( ( ( = cos θ cos ψ cos θ sin ψ −sin θ −cos φ sin ψ + sin φ sin θ cos ψ cos φ cos ψ + sin φ sin θ sin ψ sin φ cos θ sin φ sin ψ + cos φ sin θ cos ψ −sin φ cos ψ + cos φ sin θ sin ψ cos φ cos θ # $ % % % & ' ( ( ( also called Direction Cosine Matrix (see supplement)" Properties of the Rotation Matrix" H I B ( φ , θ , ψ ) = H 2 B ( φ )H 1 2 ( θ )H I 1 ( ψ ) •  The rotation matrix produces an orthonormal transformation" –  Angles are preserved" –  Lengths are preserved" r I = r B ; s I = s B ∠(r I ,s I ) = ∠(r B ,s B ) = x deg r" s" Properties of the Rotation Matrix" •  Inverse relationship; interchange sub- and superscripts" •  Because transformation is orthonormal," –  Inverse = transpose" –  Rotation matrix is always non-singular " r B = H I B r I ; r I = H I B ( ) −1 r B = H B I r B H B I = H I B ( ) −1 = H I B ( ) T = H 1 I H 2 1 H B 2 H B I H I B = H I B H B I = I Measurement of Position in Alternative Frames - 2" r particle I = r origin− B I + H B I Δr B Inertial-axis view" Body-axis view" r particle B = r origin− I B + H I B Δr I Angular Momentum Angular Momentum of a Particle ! •  Moment of linear momentum of differential particles that make up the body" –  (Differential masses) x components of the velocity that are perpendicular to the moment arms" •  Cross Product: Evaluation of a determinant with unit vectors (i, j, k) along axes, (x, y, z) and (v x , v y , v z ) projections on to axes" r × v = i j k x y z v x v y v z = yv z − zv y ( ) i + zv x − xv z ( ) j + xv y − yv x ( ) k dh = r × dmv ( ) = r × v m ( ) dm = r × v o + ω × r ( ) ( ) dm ω = ω x ω y ω z " # $ $ $ $ % & ' ' ' ' Cross-Product- Equivalent Matrix" r × v = i j k x y z v x v y v z = yv z − zv y ( ) i + zv x − xv z ( ) j + xv y − yv x ( ) k = yv z − zv y ( ) zv x − xv z ( ) xv y − yv x ( ) # $ % % % % % & ' ( ( ( ( ( =  rv = 0 −z y z 0 −x −y x 0 # $ % % % & ' ( ( ( v x v y v z # $ % % % % & ' ( ( ( ( Cross-product-equivalent matrix "  r = 0 −z y z 0 −x −y x 0 " # $ $ $ % & ' ' ' Cross product" Angular Momentum of the Aircraft" •  Integrate moment of linear momentum of differential particles over the body" h = r × v o + ω × r ( ) ( ) dm Body ∫ = r × v ( ) ρ (x, y, z)dx dy dz z min z max ∫ y min y max ∫ x min x max ∫ = h x h y h z % & ' ' ' ' ( ) * * * * ρ (x, y, z) = Density of the body h = r × v o ( ) dm Bo dy ∫ + r × ω × r ( ) ( ) dm Bo dy ∫ = 0 − r × r × ω ( ) ( ) dm Bo dy ∫ = − r × r ( ) dm × ω Bo dy ∫ ≡ −  r  r ( ) dmω Bo dy ∫ •  Choose the center of mass as the rotational center" Supermarine Spitfire! Location of the Center of Mass" r cm = 1 m r dm Body ∫ = r ρ (x, y,z)dx dy dz z min z max ∫ y min y max ∫ x min x max ∫ = x cm y cm z cm # $ % % % & ' ( ( ( The Inertia Matrix The Inertia Matrix" h = −  r  r ω dm Bo dy ∫ = −  r  r dm Bo dy ∫ ω = Iω •  Inertia matrix derives from equal effect of angular rate on all particles of the aircraft" I = −  r  r dm Bo dy ∫ = − 0 −z y z 0 −x −y x 0 # $ % % % & ' ( ( ( 0 −z y z 0 −x −y x 0 # $ % % % & ' ( ( ( dm Bo dy ∫ = (y 2 + z 2 ) −xy −xz −xy (x 2 + z 2 ) −yz −xz −yz (x 2 + y 2 ) # $ % % % % & ' ( ( ( ( dm Bo dy ∫ ω = ω x ω y ω z " # $ $ $ $ % & ' ' ' ' where" Moments and Products of Inertia" •  Inertia matrix" I = (y 2 + z 2 ) −xy −xz −xy (x 2 + z 2 ) −yz −xz −yz (x 2 + y 2 ) " # $ $ $ $ % & ' ' ' ' dm Body ∫ = I xx −I xy −I xz −I xy I yy −I yz −I xz −I yz I zz " # $ $ $ $ % & ' ' ' ' –  Moments of inertia on the diagonal" –  Products of inertia off the diagonal" I xx 0 0 0 I yy 0 0 0 I zz ! " # # # # $ % & & & & •  If products of inertia are zero, (x, y, z) are principal axes >" •  All rigid bodies have a set of principal axes" Ellipsoid of Inertia! € I xx x 2 + I yy y 2 + I zz z 2 = 1 Inertia Matrix of an Aircraft with Mirror Symmetry" I = (y 2 + z 2 ) 0 −xz 0 (x 2 + z 2 ) 0 −xz 0 (x 2 + y 2 ) " # $ $ $ $ % & ' ' ' ' dm Body ∫ = I xx 0 −I xz 0 I yy 0 −I xz 0 I zz " # $ $ $ $ % & ' ' ' ' •  Nose high/low product of inertia, I xz " North American XB-70! Nominal Configuration Tips folded, 50% fuel, W = 38,524 lb x cm @0.218 c I xx = 1.8 ×10 6 slug-ft 2 I yy = 19.9 ×10 6 slug-ft 2 I xx = 22.1×10 6 slug-ft 2 I xz = −0.88 ×10 6 slug-ft 2 Rate of Change of Angular Momentum Newtons 2 nd Law, Applied to Rotational Motion" •  In inertial frame, rate of change of angular momentum = applied moment (or torque), M" dh dt = d Iω ( ) dt = dI dt ω + I d ω dt =  Iω + I  ω = M = m x m y m z " # $ $ $ $ % & ' ' ' ' •  Angular momentum and rate vectors are not necessarily aligned" h = Iω Angular Momentum and Rate" Rate of Change of Angular Momentum How Do We Get Rid of dI/dt in the Angular Momentum Equation?" •  Dynamic equation in a body-referenced frame" –  Inertial properties of a constant-mass, rigid body are unchanging in a body frame of reference" –  but a body-referenced frame is non-Newtonian or non-inertial" –  Therefore, dynamic equation must be modified for expression in a rotating frame" d Iω ( ) dt =  Iω + I  ω •  Chain Rule" and in an inertial frame"  I ≠ 0 Angular Momentum Expressed in Two Frames of Reference" •  Angular momentum and rate are vectors" –  Expressed in either the inertial or body frame" –  Two frames related algebraically by the rotation matrix" h B t ( ) = H I B t ( ) h I t ( ) ; h I t ( ) = H B I t ( ) h B t ( ) ω B t ( ) = H I B t ( ) ω I t ( ) ; ω I t ( ) = H B I t ( ) ω B t ( ) Vector Derivative Expressed in a Rotating Frame" •  Chain Rule" •  Consequently, the 2 nd term is"  h I = H B I  h B +  H B I h B Effect of ! body-frame rotation! Rate of change ! expressed in body frame! •  Alternatively"  h I = H B I  h B + ω I × h I = H B I  h B +  ω I h I  ω = 0 − ω z ω y ω z 0 − ω x − ω y ω x 0 # $ % % % % & ' ( ( ( ( " where the cross-product- equivalent matrix of angular rate is"  H B I h B =  ω I h I =  ω I H B I h B External Moment Causes Change in Angular Rate"  h B = H I B  h I +  H I B h I = H I B  h I − ω B × h B = H I B  h I −  ω B h B = H I B M I −  ω B I B ω B = M B −  ω B I B ω B "Positive rotation of Frame B w.r.t. Frame A is a negative rotation of Frame A w.r.t. Frame B" M I = m x m y m z ! " # # # # $ % & & & & I ; M B = H I B M I = m x m y m z ! " # # # # $ % & & & & B = L M N ! " # # # $ % & & & •  Moment = torque = force x moment arm" •  In the body frame of reference, the angular momentum change is" Rate of Change of Body-Referenced Angular Rate due to External Moment" •  For constant body-axis inertia matrix"  h B = H I B  h I +  H I B h I = H I B  h I − ω B × h B = H I B  h I −  ω B h B = H I B M I −  ω B I B ω B = M B −  ω B I B ω B •  In the body frame of reference, the angular momentum change is"  ω B = I B −1 M B −  ω B I B ω B ( ) •  Consequently, the differential equation for angular rate of change is"  h B = I B  ω B = M B −  ω B I B ω B Next Time: Aircraft Equations of Motion – 2   Reading Aircraft Dynamics,  Virtual Textbook, Parts 8,9  Supplemental Material Direction Cosine Matrix (also called Rotation Matrix)" H I B = cos δ 11 cos δ 21 cos δ 31 cos δ 12 cos δ 22 cos δ 32 cos δ 13 cos δ 23 cos δ 33 " # $ $ $ % & ' ' ' •  Cosines of angles between each I axis and each B axis" •  Projections of vector components" r B = H I B r I •  Moments and products of inertia tabulated for geometric shapes with uniform density" Moments and Products of Inertia" (Bedford & Fowler)" •  Construct aircraft moments and products of inertia from components using parallel-axis theorem" •  Model in Pro/ENGINEER, etc." . Aircraft Equations of Motion - 1 Robert Stengel, Aircraft Flight Dynamics, 
 MAE 3 31, 2 012 " •  6 degrees of freedom" •  Angular kinematics" • . of inertia, I xz " North American XB-70! Nominal Configuration Tips folded, 50% fuel, W = 38,524 lb x cm @0. 218 c I xx = 1. 8 10 6 slug-ft 2 I yy = 19 .9 10 6 slug-ft 2 I xx = 22 .1 10 6 . 22 .1 10 6 slug-ft 2 I xz = −0.88 10 6 slug-ft 2 Rate of Change of Angular Momentum Newtons 2 nd Law, Applied to Rotational Motion& quot; •  In inertial frame, rate of change of angular momentum