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 Newtons 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 Newtons 2 nd Law, Applied to Rotational Motion& quot; • In inertial frame, rate of change of angular momentum