Aircraft Equations of Motion: Translation and Rotation Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives • What use are the equations of motion? • How is the angular orientation of the airplane described? • What is a cross-product-equivalent matrix? • What is angular momentum? • How are the inertial properties of the airplane described? • How is the rate of change of angular momentum calculated? Lockheed F-104 Reading: Flight Dynamics 155-161 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html 1 Review Questions § What characteristic(s) provide maximum gliding range? § Do gliding heavy airplanes fall out of the sky faster than light airplanes? § Are the factors for maximum gliding range and minimum sink rate the same? § How does the maximum climb rate vary with altitude? § What are “energy height” and “specific excess power”? § What is an “energy climb”? § How is the “maneuvering envelope” defined? § What factors determine the maximum steady turning rate? 2 1 Dynamic Systems Actuators Sensors Dynamic Process: Current state depends on Observation Process: Measurement may prior state contain error or be incomplete x = dynamic state y = output (error-free) u = input z = measurement w = exogenous disturbance n = measurement error p = parameter t or k = time or event index dx(t) y(t) = h[x(t),u(t)] = f [x(t),u(t),w(t),p(t),t] dt z(t) = y(t) + n(t) 3 Ordinary Differential Equations Fall Into 4 Categories dx(t) dx(t) = f [x(t),u(t),w(t),p(t),t] = F(t)x(t) + G(t)u(t) + L(t)w(t) dt dt dx(t) dx(t) = f [x(t),u(t),w(t)] = Fx(t) + Gu(t) + Lw(t) dt dt 4 2 What Use are the Equations of Motion? • Nonlinear equations of motion – Compute exact flight paths and dx(t) = f [x(t),u(t),w(t),p(t),t] motions dt • Simulate flight motions • Optimize flight paths dx(t) • Predict performance = Fx(t) + Gu(t) + Lw(t) – Provide basis for approximate dt 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 5 Examples of Airplane Dynamic System Models • Nonlinear, Time-Varying • Nonlinear, Time-Invariant – Large amplitude motions – Large amplitude motions – Negligible change in mass – Significant change in mass • Linear, Time-Varying • Linear, Time-Invariant – Small amplitude motions – Small amplitude motions – Perturbations from a dynamic flight path – Perturbations from an equilibrium flight path 6 3 Translational Position 7 Position of a Particle Projections of vector magnitude on three axes ⎡ ⎤ ⎡ x ⎤ cosδ x ⎢ ⎥ r ⎢ y ⎥ r cos = ⎢ ⎥ = ⎢ δ y ⎥ ⎢ z ⎥ ⎢ cosδ ⎥ ⎣ ⎦ ⎣⎢ z ⎦⎥ ⎡ cosδ ⎤ ⎢ x ⎥ ⎢ cosδ y ⎥ = Direction cosines ⎢ cosδ ⎥ ⎣⎢ z ⎦⎥ 8 4 Cartesian Frames of Reference • Translation • Two reference frames of interest – Relative linear positions of origins – I: Inertial frame (fixed to inertial space) • Rotation – B: Body frame (fixed to body) – Orientation of the body frame with respect to the inertial frame Common convention (z up) Aircraft convention (z down) 9 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) Inertial-axis view ⎡ x ⎤ r ⎢ y ⎥ = ⎢ ⎥ ⎣⎢ z ⎦⎥ rparticle = rorigin + Δrw.r.t. origin • Differences in frame orientations must be taken into account in adding vector components Body-axis view 10 5 Measurement of Position in Alternative Frames - 2 Inertial-axis view r r HI r particleI = origin−BI + BΔ B Body-axis view r r HB r particleB = origin−IB + I Δ I 11 Rotational Orientation 12 6 Direction Cosine Matrix ⎡ ⎤ cosδ11 cosδ 21 cosδ 31 B ⎢ ⎥ HI = ⎢ cosδ12 cosδ 22 cosδ 32 ⎥ ⎢ cosδ cosδ cosδ ⎥ ⎣ 13 23 33 ⎦ • Projections of unit vector components of one reference frame on another • Rotational orientation of one reference frame with respect to another B • Cosines of angles between rB = HI rI each I axis and each B axis 13 Properties of the Rotation Matrix B B ⎡ ⎤ ⎡ ⎤ cosδ11 cosδ 21 cosδ 31 h11 h12 h13 B ⎢ ⎥ ⎢ ⎥ HI = ⎢ cosδ12 cosδ 22 cosδ 32 ⎥ = ⎢ h21 h22 h23 ⎥ ⎢ ⎥ ⎢ ⎥ cosδ13 cosδ 23 cosδ 33 h31 h32 h33 ⎣ ⎦I ⎣ ⎦I B B rB = HI rI sB = HI sI Orthonormal transformation Angles between vectors are preserved r s Lengths are preserved rI = rB ; sI = sB ∠(rI ,sI ) = ∠(rB ,sB ) = x deg 14 7 Euler Angles • Body attitude measured with respect to inertial frame • Three-angle orientation expressed by sequence of three orthogonal single-angle rotations Inertial ⇒ Intermediate1 ⇒ Intermediate2 ⇒ Body • 24 (12) possible sequences of single-axis rotations • Aircraft convention: 3-2- 1, z positive down ψ : Yaw angle θ : Pitch angle φ : Roll angle 15 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 zI Pitch rotation (θ) about y1 Roll rotation (ϕ) about x2 Other sequences of 3 rotations can be chosen; however, once sequence is chosen, it must be retained 16 8 Reference Frame Rotation from Inertial to Body: Aircraft Convention (3-2-1) Yaw rotation (ψ) about zI axis ⎡ ⎤ ⎡ x ⎤ ⎡ cosψ sinψ 0 ⎤⎡ x ⎤ xI cosψ + yI sinψ ⎢ ⎥ ⎢ ⎥ 1 ⎢ y ⎥ ⎢ y ⎥ x sin y cos r = H r ⎢ ⎥ = ⎢ −sinψ cosψ 0 ⎥⎢ ⎥ = ⎢ − I ψ + I ψ ⎥ 1 I I ⎢ z ⎥ ⎢ 0 0 1 ⎥⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦1 ⎣ ⎦⎣ ⎦I ⎣ I ⎦ Pitch rotation (θ) about y1 axis ⎡ x ⎤ ⎡ cosθ 0 −sinθ ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 1 2 y = ⎢ ⎥ y r = H r = ⎡H H ⎤r = H r ⎢ ⎥ ⎢ 0 1 0 ⎥⎢ ⎥ 2 1 1 ⎣ 1 I ⎦ I I I ⎢ z ⎥ ⎢ sinθ 0 cosθ ⎥⎢ z ⎥ ⎣ ⎦2 ⎣ ⎦⎣ ⎦1 Roll rotation (ϕ) about x2 axis ⎡ x ⎤ ⎡ 1 0 0 ⎤⎡ x ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ B B 2 1 B y 0 cosφ sinφ y rB = H2 r2 = ⎣⎡H2 H1 HI ⎦⎤rI = HI rI ⎢ ⎥ = ⎢ ⎥⎢ ⎥ ⎢ z ⎥ ⎢ 0 −sinφ cosφ ⎥⎢ z ⎥ ⎣ ⎦B ⎣ ⎦2 ⎣ ⎦ 17 The Rotation Matrix The three-angle rotation matrix is the product of 3 single-angle rotation matrices: B B 2 1 HI (φ,θ,ψ) = H2 (φ)H1 (θ)HI (ψ) ⎡ 1 0 0 ⎤⎡ cosθ 0 −sinθ ⎤⎡ cosψ sinψ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = 0 cosφ sinφ ⎢ ⎥ ⎢ ⎥⎢ 0 1 0 ⎥⎢ −sinψ cosψ 0 ⎥ ⎢ 0 −sinφ cosφ ⎥ sinθ 0 cosθ ⎢ ⎥ ⎣ ⎦⎣⎢ ⎦⎥⎣ 0 0 1 ⎦ ⎡ cosθ cosψ cosθ sinψ −sinθ ⎤ ⎢ ⎥ = ⎢ − cosφ sinψ + sinφ sinθ cosψ cosφ cosψ + sinφ sinθ sinψ sinφ cosθ ⎥ ⎢ sinφ sinψ + cosφ sinθ cosψ −sinφ cosψ + cosφ sinθ sinψ cosφ cosθ ⎥ ⎣ ⎦ an expression of the Direction Cosine Matrix 18 9 Rotation Matrix Inverse Inverse relationship: interchange sub- and superscripts B rB = HI rI −1 r = HB r = HI r I ( I ) B B B Because transformation is orthonormal Inverse = transpose Rotation matrix is always non-singular B −1 B T I ⎣⎡HI (φ,θ,ψ )⎦⎤ = ⎣⎡HI (φ,θ,ψ )⎦⎤ = HB (ψ ,θ,φ) I B −1 B T I 1 2 HB = (HI ) = (HI ) = H1H2HB I B B I 19 HB HI = HI HB = I Checklist q What are direction cosines? q What are Euler angles? q What rotation sequence is used to describe airplane attitude? q What are properties of the rotation matrix? 20 10 Angular Momentum 21 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 " ω x % dh = (r × dm v) = (r × v m )dm $ ' ω = $ ω y ' $ ' = ⎣⎡r × (vo + ω × r)⎦⎤dm ω #$ z &' • Cross Product: Evaluation of a determinant with unit vectors (i, j, k) along axes, (x, y, z) and (vx, vy, vz) projections on to axes i j k x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k vx vy vz 22 11 Cross-Product- Equivalent Matrix i j k Cross product x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k vx vy vz # yv − zv & # & % ( z y ) ( # 0 −z y & vx % ( % ( % ( = zv − xv = rv = z 0 x vy % ( x z ) ( % − ( % ( % ( % ( % ( −y x 0 vz (xvy − yvx ) $ ' $% '( $% '( " 0 −z y % Cross-product-equivalent $ ' matrix r = $ z 0 −x ' $ −y x 0 ' # & 23 Angular Momentum of the Aircraft • Integrate moment of linear momentum of differential particles over the body ⎡ ⎤ hx xmax ymax zmax ⎢ ⎥ h = ⎡r × v + ω × r ⎤dm = r × v ρ(x, y,z)dx dydz = h ∫ ⎣ ( o )⎦ ∫ ∫ ∫ ( ) ⎢ y ⎥ Body xmin ymin zmin ⎢ h ⎥ ⎣⎢ z ⎦⎥ ρ(x, y,z) = Density of the body • Choose the center of mass as the rotational center Supermarine Spitfire h = r × v dm + r × ω × r dm ∫ ( o ) ∫ ⎣⎡ ( )⎦⎤ Body Body = 0 − ∫ ⎣⎡r × (r × ω)⎦⎤dm Body = − (r × r)dm × ω ≡ − (r!r!)dmω ∫ ∫ 24 Body Body 12 Location of the Center of Mass # & xcm 1 xmax ymax zmax % ( r = r dm = rρ(x, y,z)dx dydz = % y ( cm m ∫ ∫ ∫ ∫ cm Body xmin ymin zmin % z ( $ cm ' 25 The Inertia Matrix 26 13 The Inertia Matrix " ω % $ x ' h = − r! r! ω dm = − r! r! dm ω = ω ω = ω ∫ ∫ I $ y ' $ ' Body Body ω #$ z &' where ⎡ 0 −z y ⎤⎡ 0 −z y ⎤ ⎢ ⎥⎢ ⎥ I = − ∫ r! r! dm = − ∫ ⎢ z 0 −x ⎥⎢ z 0 −x ⎥ dm Body Body ⎢ −y x 0 ⎥⎢ −y x 0 ⎥ ⎣ ⎦⎣ ⎦ ⎡ (y2 + z2 ) −xy −xz ⎤ ⎢ ⎥ = ∫ ⎢ −xy (x2 + z2 ) −yz ⎥dm Body ⎢ 2 2 ⎥ ⎢ −xz −yz (x + y ) ⎥ ⎣ ⎦ Inertia matrix derives from equal effect of angular rate on all particles of the aircraft 27 Moments and Products of Inertia 2 2 ⎡ ⎤ ⎡ (y + z ) −xy −xz ⎤ I xx −I xy −I xz ⎢ ⎥ ⎢ ⎥ 2 2 − − I = ∫ ⎢ −xy (x + z ) −yz ⎥dm = ⎢ I xy I yy I yz ⎥ Body ⎢ 2 2 ⎥ ⎢ ⎥ −xz −yz (x + y ) − xz − yz zz ⎣⎢ ⎦⎥ ⎣⎢ I I I ⎦⎥ Inertia matrix • Moments of inertia on the diagonal • Products of inertia off the diagonal ⎡ ⎤ • If products of inertia are zero, (x, y, z) I xx 0 0 are principal axes ---> ⎢ ⎥ 0 0 • All rigid bodies have a set of principal ⎢ I yy ⎥ axes ⎢ 0 0 ⎥ ⎣⎢ I zz ⎦⎥ Ellipsoid of Inertia 2 2 2 I xx x + I yy y + I zzz = 1 28 14 Inertia Matrix of an Aircraft with Mirror Symmetry ⎡ 2 2 ⎤ ⎡ ⎤ (y + z ) 0 −xz I xx 0 −I xz ⎢ ⎥ ⎢ ⎥ 2 2 0 0 I = ∫ ⎢ 0 (x + z ) 0 ⎥dm = ⎢ I yy ⎥ Body