Point-Mass Dynamics and Aerodynamic/Thrust Forces Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018 Learning Objectives • Properties of atmosphere • Frames of reference • Velocity and momentum • Newtons laws of motion • Airplane axes • Lift and drag • Simplified equations for longitudinal motion • Powerplants and thrust Reading: Flight Dynamics Introduction, 1-27 The Earth’s Atmosphere, 29-34 Kinematic Equations, 38-53 Forces and Moments, 59-65 Introduction to Thrust, 103-107 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html 1 http://www.princeton.edu/~stengel/FlightDynamics.html The Atmosphere 2 1 Properties of the Lower Atmosphere* Stratosphere Tropopause Troposphere • Density and pressure decay exponentially with altitude • Temperature and speed of sound are piecewise-linear functions of altitude 3 * 1976 US Standard Atmosphere Air Density, Dynamic Pressure, and Mach Number ρ = Air density, functionof height −βh βz = ρsealevele = ρsealevele 3 ρsealevel = 1.225kg/m ; β = 1/ 9,042m 1/2 1/2 V = !v2 + v2 + v2 # = !vT v# = Airspeed air " x y z $air " $air 1 2 Dynamic pressure = q = ρ (h)Vair ! "qbar" 2 V Mach number = air ; a = speed of sound, m/s a h ( ) 4 2 Contours of Constant Dynamic Pressure, q 1 • In steady, cruising flight, Weight = Lift = C ρV 2 S = C qS L 2 air L True airspeed must increase as altitude increases 5 to maintain constant dynamic pressure Wind: Motion of the Atmosphere Zero wind at Earths surface = Rotating air mass Wind measured with respect to Earths rotating surface Wind Velocity Profiles vary over Time Airspeed = Airplanes speed with respect to air mass Earth-relative velocity = Wind velocity True airspeed [vector] 6 3 Historical Factoids • Henri Pitot: Pitot tube (1732) Sir George Cayley • Sketched "modern" airplane configuration (1799) • Hand-launched glider (1804) • Applied aerodynamics (1809-1810) • Triplane glider carrying 10-yr-old boy (1849) • Monoplane glider carrying coachman • Benjamin Robins: Whirling (1853) arm "wind tunnel (1742) – Cayley's coachman had a steering oar with cruciform blades NASA SP-440, Wind Tunnels of NASA 7 Equations of Motion for a Particle (Point Mass) 8 4 Newtonian Frame of Reference § Newtonian (Inertial) Reference Frame § Unaccelerated Cartesian frame: origin referenced to inertial (non-moving) frame § Right-hand rule § Origin can translate at constant linear velocity § Frame cannot rotate with respect to inertial origin ⎡ x ⎤ § Position: 3 dimensions ⎢ ⎥ § What is a non-moving frame? r = ⎢ y ⎥ ⎣⎢ z ⎦⎥ Translation changes the position of an object 9 Velocity and Momentum § Velocity of a particle ⎡ ⎤ ⎡ x ⎤ vx dx ⎢ ⎥ ⎢ ⎥ v = = x = y = ⎢ vy ⎥ dt ⎢ ⎥ ⎢ z ⎥ ⎢ ⎥ ⎣ ⎦ vz ⎣⎢ ⎦⎥ § Linear momentum of a particle ⎡ v ⎤ ⎢ x ⎥ p = mv = m ⎢ vy ⎥ ⎢ v ⎥ ⎣⎢ z ⎦⎥ where m = mass of particle 10 5 Inertial Velocity Expressed in Polar Coordinates Polar Coordinates Projected on a Sphere γ : Vertical Flight Path Angle, rad or deg ξ : Horizontal Flight Path Angle (Heading Angle), rad or deg 11 Newtons Laws of Motion: Dynamics of a Particle § First Law: If no force acts on a particle, § it remains at rest or § continues to move in a straight line at constant velocity, as observed in an inertial reference frame § Momentum is conserved d (mv) = 0 ; mv = mv dt t1 t2 12 6 Newtons Laws of Motion: Dynamics of a Particle § Second Law: Particle of fixed mass acted upon by a force § changes velocity with acceleration proportional to and in direction of force, as observed in inertial frame § Mass is ratio of force to acceleration of particle: ⎡ ⎤ fx d dv ⎢ ⎥ F = ma (mv) = m = F ; F = ⎢ fy ⎥ dt dt ⎢ f ⎥ ⎣⎢ z ⎦⎥ ⎡ ⎤ ⎡ 1 / m 0 0 ⎤ fx dv 1 1 ⎢ ⎥ ⎢ ⎥ ∴ = F = I3F = 0 1 / m 0 ⎢ fy ⎥ dt m m ⎢ ⎥ ⎣⎢ 0 0 1 / m ⎦⎥ ⎢ f ⎥ ⎣⎢ z ⎦⎥ 13 Newtons Laws of Motion: Dynamics of a Particle § Third Law § For every action, there is an equal and opposite reaction Force on Rocket = Force on Exhaust Gasses 14 7 Equations of Motion for a Particle: Position and Velocity Force vector ⎡ f ⎤ ⎢ x ⎥ F = f = ⎡F + F + F ⎤ I ⎢ y ⎥ ⎣ gravity aerodynamics thrust ⎦I ⎢ ⎥ fz ⎣⎢ ⎦⎥I Rate of change of velocity ⎡ ⎤ ⎡ ⎤ v!x ⎡ 1/ m 0 0 ⎤ fx dv ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ = v! = v! = F = 0 1/ m 0 f dt ⎢ y ⎥ m ⎢ ⎥⎢ y ⎥ ⎢ v ⎥ ⎣⎢ 0 0 1/ m ⎦⎥⎢ f ⎥ ⎢ !z ⎥ ⎢ z ⎥ ⎣ ⎦I ⎣ ⎦I ⎡ ⎤ ⎡ x! ⎤ vx Rate of change dr ⎢ ⎥ ⎢ ⎥ of position = r! = y! = v = v dt ⎢ ⎥ ⎢ y ⎥ ⎢ z! ⎥ ⎢ v ⎥ ⎣ ⎦I ⎢ z ⎥ 15 ⎣ ⎦I Integration for Velocity with Constant Force Differential equation ⎡ ⎤ ⎡ ⎤ fx m ax dv t 1 ⎢ ⎥ ⎢ ⎥ ( ) f m = v! (t) = F = ⎢ y ⎥ = ⎢ ay ⎥ dt m ⎢ ⎥ f m ⎢ a ⎥ ⎢ z ⎥ ⎢ z ⎥ ⎣ ⎦ ⎣ ⎦ Integral T dv(t) T 1 T v(T ) = dt + v(0) = Fdt + v(0) = adt + v(0) ∫0 dt ∫0 m ∫0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ vx (T ) fx m vx (0) ax vx (0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ vy (T ) ⎥ = ⎢ fy m ⎥dt + ⎢ vy (0) ⎥ = ⎢ ay ⎥dt + ⎢ vy (0) ⎥ ∫0 ∫0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v T f m v 0 a v 0 ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ⎦⎥ ⎣⎢ z ( ) ⎦⎥ ⎣⎢ z ⎦⎥ ⎣⎢ z ( ) ⎦⎥ 16 8 Integration for Position with Varying Velocity Differential equation ⎡ ⎤ ⎡ ⎤ x!(t) vx (t) dr(t) ⎢ ⎥ ⎢ ⎥ = r!(t) = v(t) = ⎢ y!(t) ⎥ = ⎢ vy (t) ⎥ dt ⎢ ⎥ ⎢ ⎥ z! t v t ⎣⎢ ( ) ⎦⎥ ⎢ z ( ) ⎥ ⎣ ⎦ Integral T dr(t) T r(T ) = dt + r(0) = v(t)dt + r(0) ∫0 dt ∫0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x(T ) vx (t) x(0) ⎢ ⎥ T ⎢ ⎥ ⎢ ⎥ ⎢ y(T ) ⎥ = ⎢ vy (t) ⎥dt + ⎢ y(0) ⎥ ∫0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z T v t z 0 ⎣⎢ ( ) ⎦⎥ ⎢ z ( ) ⎥ ⎣⎢ ( ) ⎦⎥ ⎣ ⎦ 17 Gravitational Force: Flat-Earth Approximation § Approximation § g is gravitational § Flat earth reference assumed to acceleration be inertial frame, e.g., § mg is gravitational force § North, East, Down § Range, Crossrange, Altitude (–) § Independent of position § z measured down ⎡ 0 ⎤ ⎢ ⎥ F = F = mg = m 0 ( gravity )I ( gravity )E E ⎢ ⎥ ⎢ go ⎥ ⎣ ⎦E g ! 9.807 m/s2 at earth's surface o 18 9 Flight Path Dynamics, Constant Gravity, No Aerodynamics Integral vx (T ) = vx v 0 v 0 x ( ) = x0 T v 0 = v z ( ) z0 vz (T ) = vz − gdt = vz − gT 0 ∫ 0 0 x(0) = x0 x T x v T ( ) = 0 + x0 z(0) = z0 T Differential equation 2 z(T ) = z0 + vz T − gt dt = z0 + vz T − gT 2 0 ∫ 0 0 v!x (t) = 0 v!z (t) = −g (z positive up) x!(t) = vx (t) z!(t) = vz (t) 19 Flight Path with Constant Gravity and No Aerodynamics 20 10 Aerodynamic Force on an Airplane Earth-Reference Frame Body-Axis Frame Wind-Axis Frame ⎡ ⎤ ⎡ X ⎤ CX ⎢ ⎥ 1 F = ⎢ Y ⎥ = C ρV 2 S I ⎢ ⎥ ⎢ Y ⎥ air 2 ⎡ ⎤ Z ⎢ ⎥ CX ⎡ C ⎤ ⎣⎢ ⎦⎥E CZ D ⎣ ⎦E ⎢ ⎥ ⎢ ⎥ FB = ⎢ CY ⎥ q S FV = ⎢ CY ⎥q S ⎡ ⎤ ⎢ ⎥ CX C Z ⎢ C ⎥ ⎢ ⎥ ⎣ ⎦B ⎣ L ⎦ = ⎢ CY ⎥ q S ⎢ ⎥ Aligned with the CZ Aligned with and ⎣ ⎦E aircraft axes (e.g., perpendicular to the Referenced to the Earth, not Nose, Wing Tip, velocity vector the aircraft (e.g., North, East, Down) (Forward, Sideward, Down) Down) 21 Non-Dimensional Aerodynamic Coefficients Body-Axis Frame Wind-Axis Frame ⎡ C ⎤ ⎡ axial force coefficient ⎤ ⎡ C ⎤ ⎡ drag coefficient ⎤ ⎢ X ⎥ ⎢ ⎥ ⎢ D ⎥ ⎢ ⎥ ⎢ CY ⎥ = ⎢ side force coefficient ⎥ ⎢ CY ⎥= ⎢ side force coefficient ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ CZ normal force coefficient CL lift coefficient ⎣ ⎦B ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ • Functions of flight condition, control settings, and disturbances, e.g., CL = CL(δ, M, δE) • Non-dimensional coefficients allow application of sub-scale model wind tunnel data to full-scale airplane 22 11 Longitudinal Variables γ = θ − α (with wingtips level) u(t) :axial velocity • along vehicle centerline w(t) :normal velocity • perpendicular to centerline V (t) :velocity magnitude • along net direction of flight α (t) :angle of attack • angle between centerline and direction of flight γ (t) :flight path angle • angle between direction of flight and local horizontal • angle between centerline and local horizontal θ(t) :pitch angle 23 Lateral-Directional Variables ξ = ψ + β (with wingtips level) β(t) :sideslip angle • angle between centerline and direction of flight ψ (t) :yaw angle • angle between centerline and local horizontal • angle between direction of flight and compass reference ξ (t) :heading angle (e.g., north) φ t :roll angle • angle between true vertical and body z axis ( ) 24 12 Introduction to Lift and Drag 25 Lift and Drag are Referenced to Velocity Vector 1 % ∂C (1 Lift = C ρV 2 S ≈ C + L α ρV 2 S L 2 air &' L0 ∂α )*2 air • Lift components sum to produce total lift – Perpendicular to velocity vector – Pressure differential between upper and lower surfaces 1 1 Drag = C ρV 2 S ≈ $C +εC 2 & ρV 2 S D 2 air % D0 L '2 air • Drag components sum to produce total drag – Parallel and opposed to velocity vector – Skin friction, pressure differentials 26 13 2-D Aerodynamic Lift 1 1 % ∂C ( L•iftFast= C flowρV over2 S ≈ topC + +slowC flow+C over bottomρV 2 S ≈ = CMean+ flowL α +qS L air ( Lwing L fuselage Ltail ) air ' L0 * Circulation2 2 & ∂α ) • Upper/lower speed difference proportional to angle of attack • Stagnation points at leading and trailing edges • Kutta condition: Aft stagnation point at sharp trailing edge Streamlines Streaklines Instantaneous tangent to velocity vector Locus of particles passing through points Dye injected at fixed points http://www.diam.unige.it/~irro/profilo_e.html 27 3-D Lift Inward-Outward Flow Tip Vortices Inward flow over upper surface Outward flow over lower surface Bound vorticity of wing produces tip vortices 28 14 2-D vs. 3-D Lift Identical Chord Sections Infinite vs. Finite Span Finite aspect ratio reduces lift slope What is aspect ratio? Talay, NASA-SP-367 29 Aerodynamic Drag 1 1 Drag = C ρV 2 S ≈ C + C + C ρV 2 S ≈ ⎡C + εC 2 ⎤qS D 2 air ( Dp Dw Di ) 2 air ⎣ D0 L ⎦ • Drag components – Parasite drag (friction, interference, base pressure differential) – Wave drag (shock-induced pressure differential) – Induced drag (drag due to lift generation) • In steady, subsonic flight – Parasite (form) drag increases as V2 – Induced