Point-Mass Dynamics and Aerodynamic/Thrust Forces
 Robert Stengel, Aircraft Flight Dynamics, 
 MAE 331, 2012! •  Properties of the Atmosphere" •  Frames of reference" •  Velocity and momentum" •  Newtons laws" •  Introduction to Lift, Drag, and Thrust" •  Simplified longitudinal equations of motion" 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! The Atmosphere •  Air density and pressure decay exponentially with altitude" •  Air temperature and speed of sound are linear functions of altitude " Properties of the Lower Atmosphere! Wind: Motion of the Atmosphere" •  Zero wind at Earths surface = Inertially rotating air mass" •  Wind measured with respect to Earths rotating surface " Wind Velocity Profiles vary over Time! Typical Jetstream Velocity! •  Airspeed = Airplanes speed with respect to air mass" •  Inertial velocity = Wind velocity ± Airspeed " Air Density, Dynamic Pressure, and Mach Number" ρ = Air density, functionof height = ρ sealevel e β z = ρ sealevel e − β h ρ sealevel = 1.225 kg / m 3 ; β = 1/ 9,042 m V air = v x 2 + v y 2 + v z 2 ! " # $ air 1/2 = v T v ! " # $ air 1/2 = Airspeed Dynamic pressure = q = 1 2 ρ h ( ) V air 2 Mach number = V air a h ( ) ; a = speed of sound, m / s •  Airspeed must increase as altitude increases to maintain constant dynamic pressure" Contours of Constant Dynamic Pressure, " Weight = Lift = C L 1 2 ρ V air 2 S = C L qS •  In steady, cruising flight, " q Equations of Motion for a Point Mass Newtonian Frame of Reference" •  Newtonian (Inertial) Frame of Reference" –  Unaccelerated Cartesian frame whose origin is referenced to an inertial (non-moving) frame" –  Right-hand rule" –  Origin can translate at constant linear velocity" –  Frame cannot be rotating with respect to inertial origin" •  Translation changes the position of an object" r = x y z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ •  Position: 3 dimensions" •  What is a non-moving frame?" Velocity and Momentum " •  Velocity of a particle" v = dx dt =  x =  x  y  z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = v x v y v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ •  Linear momentum of a particle" p = mv = m v x v y v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ where m = mass of particle Newtons 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 dt mv ( ) = 0 ; mv t 1 = mv t 2 Newtons Laws of Motion: 
 Dynamics of a Particle " d dt mv ( ) = m dv dt = F ; F = f x f y f z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ •  Second Law" –  A particle of fixed mass acted upon by a force changes velocity with an acceleration proportional to and in the direction of the force, as observed in an inertial reference frame; " –  The ratio of force to acceleration is the mass of the particle: F = m a" ∴ dv dt = 1 m F = 1 m I 3 F = 1 / m 0 0 0 1 / m 0 0 0 1 / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ f x f y f z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Newtons Laws of Motion: 
 Dynamics of a Particle " •  Third Law" –  For every action, there is an equal and opposite reaction" Equations of Motion for a Point Mass: Position and Velocity " dv dt =  v =  v x  v y  v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = 1 m F = 1 / m 0 0 0 1 / m 0 0 0 1 / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ f x f y f z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ dr dt =  r =  x  y  z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = v = v x v y v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ F I = f x f y f z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ I = F gravity + F aerodynamics + F thrust ⎡ ⎣ ⎤ ⎦ I Rate of change of position! Rate of change of velocity! Vector of combined forces! Equations of Motion for a Point Mass "  ˙ x (t) = dx(t) dt = f[x(t),F] •  Written as a single equation" x ≡ r v " # $ % & ' = Position Velocity " # $ $ % & ' ' = x y z v x v y v z " # $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' •  With"  x  y  z  v x  v y  v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = v x v y v z f x / m f y / m f z / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ x y z v x v y v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ + 0 0 0 0 0 0 0 0 0 1 / m 0 0 0 1 / m 0 0 0 1 / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ f x f y f z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Dynamic equations are linear!  ˙ x (t) = dx(t) dt = f[x(t),F] Equations of Motion for a Point Mass ! Gravitational Force:
 Flat-Earth Approximation" •  g is gravitational acceleration" •  mg is gravitational force! •  Independent of position! •  z measured down" F gravity ( ) I = F gravity ( ) E = mg f = m 0 0 g o ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ •  Approximation" –  Flat earth reference is an inertial frame, e.g.," •  North, East, Down" •  Range, Crossrange, Altitude (–)" g o  9.807 m / s 2 at earth's surface Aerodynamic Force" F I = X Y Z ! " # # # $ % & & & I = C X C Y C Z ! " # # # $ % & & & I 1 2 ρ V air 2 S = C X C Y C Z ! " # # # $ % & & & I q S •  Referenced to the Earth not the aircraft" Inertial Frame" Body-Axis Frame" Velocity-Axis Frame" F B = C X C Y C Z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ B q S F V = C D C Y C L ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ q S •  Aligned with the aircraft axes" •  Aligned with and perpendicular to the direction of motion" Non-Dimensional Aerodynamic Coefficients" Body-Axis Frame" Velocity-Axis Frame" C X C Y C Z ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ B = axial force coefficient side force coefficient normal force coefficient ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ C D C Y C L ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = drag coefficient side force coefficient lift coefficient ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ •  Functions of flight condition, control settings, and disturbances, e.g., C L = C L (δ, M, δE)" •  Non-dimensional coefficients allow application of sub-scale model wind tunnel data to full-scale airplane" u(t) :axial velocity w(t) :normal velocity V t ( ) : velocity magnitude α t ( ) : angle of attack γ t ( ) : flight path angle θ (t) : pitch angle •  along vehicle centerline! •  perpendicular to centerline! •  along net direction of flight! •  angle between centerline and direction of flight! •  angle between direction of flight and local horizontal! •  angle between centerline and local horizontal! Longitudinal Variables" γ = θ − α (with wingtips level) Lateral-Directional Variables" β (t) : sideslip angle ψ (t) : yaw angle ξ t ( ) : heading angle φ t ( ) : roll angle •  angle between centerline and direction of flight! •  angle between centerline and local horizontal! •  angle between direction of flight and compass reference (e.g., north)! •  angle between true vertical and body z axis! ξ = ψ + β (with wingtips level) Introduction to Lift and Drag Lift and Drag are Oriented to the Velocity Vector" •  Drag components sum to produce total drag" –  Skin friction" –  Base pressure differential" –  Shock-induced pressure differential (M > 1)" •  Lift components sum to produce total lift" –  Pressure differential between upper and lower surfaces" –  Wing" –  Fuselage" –  Horizontal tail" Lift = C L 1 2 ρ V air 2 S ≈ C L 0 + ∂ C L ∂α α % & ' ( ) * 1 2 ρ V air 2 S Drag = C D 1 2 ρ V air 2 S ≈ C D 0 + ε C L 2 $ % & ' 1 2 ρ V air 2 S Aerodynamic Lift" •  Fast flow over top + slow flow over bottom = Mean flow + Circulation" •  Speed difference proportional to angle of attack" •  Kutta condition (stagnation points at leading and trailing edges)" Chord Section! Streamlines! Lift = C L 1 2 ρ V air 2 S ≈ C L wing + C L fuselage + C L tail ( ) 1 2 ρ V air 2 S ≈ C L 0 + ∂ C L ∂α α % & ' ( ) * qS 2D vs. 3D Lift" •  Inward flow over upper surface" •  Outward flow over lower surface" •  Bound vorticity of wing produces tip vortices" Inward-Outward Flow! Tip Vortices! 2D vs. 3D Lift" Identical Chord Sections! Infinite vs. Finite Span! •  Finite aspect ratio reduces lift slope" What is aspect ratio?! Aerodynamic Drag" •  Drag components" –  Parasite drag (friction, interference, base pressure differential)" –  Induced drag (drag due to lift generation)" –  Wave drag (shock-induced pressure differential)" •  In steady, subsonic flight" –  Parasite (form) drag increases as V 2 " –  Induced drag proportional to 1/V 2 " –  Total drag minimized at one particular airspeed" Drag = C D 1 2 ρ V air 2 S ≈ C D p + C D i + C D w ( ) 1 2 ρ V air 2 S ≈ C D 0 + ε C L 2 $ % & ' qS 2-D Equations of Motion 2-D Equations of 
 Motion for a Point Mass"  x  z  v x  v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = v x v z f x / m f z / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x z v x v z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ + 0 0 0 0 1 / m 0 0 1 / m ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ f x f z ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ •  Restrict motions to a vertical plane (i.e., motions in y direction = 0)" •  Assume point mass location coincides with center of mass" Transform Velocity from Cartesian to Polar Coordinates"  x  z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = v x v z ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = V cos γ −V sin γ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⇒ V γ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =  x 2 +  z 2 −sin −1  z V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = v x 2 + v z 2 −sin −1 v z V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥  V  γ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = d dt v x 2 + v z 2 −sin −1 v z V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = d dt v x 2 + v z 2 − d dt sin −1 v z V ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ •  Inertial axes -> wind axes and back" •  Rates of change of velocity and flight path angle" Longitudinal Point-Mass Equations of Motion"  r(t) =  x(t) = v x = V (t)cos γ (t)  h(t) = −  z(t) = −v z = V (t)sin γ (t)  V (t) = Thrust − Drag − mg h ( ) sin γ (t) m = C T − C D ( ) 1 2 ρ h ( ) V 2 (t)S − mg h ( ) sin γ (t) m  γ (t) = Lift − mg h ( ) cos γ (t) mV(t) = C L 1 2 ρ h ( ) V 2 (t)S − mg h ( ) cos γ (t) mV(t) r = range h = height (altitude) V = velocity γ = flight path angle •  Equations of motion, assuming mass is fixed, thrust is aligned with the velocity vector, and windspeed = 0" •  In steady, level flight" •  Thrust = Drag" •  Lift = Weight" Introduction to Propulsion Reciprocating (Internal Combustion) Engine (1860s)" •  Linear motion of pistons converted to rotary motion to drive propeller" Single Cylinder" Turbo-Charger (1920s)" •  Increases pressure of incoming air" •  Thrust produced directly by exhaust gas" Axial-flow Turbojet (von Ohain, Germany)! Centrifugal-flow Turbojet (Whittle, UK)! Turbojet Engines (1930s)" Turboprop Engines (1940s)" •  Exhaust gas drives a propeller to produce thrust" •  Typically uses a centrifugal-flow compressor" Turbojet + Afterburner (1950s)" •  Fuel added to exhaust" •  Additional air may be introduced" •  Dual rotation rates, N1 and N2, typical" Turbofan Engine (1960s)" •  Dual or triple rotation rates" High Bypass Ratio Turbofan" Propfan Engine! Aft-fan Engine! Ramjet and Scramjet" Ramjet (1940s)" Scramjet (1950s)" Talos! X-43! Hyper-X! Thrust and Thrust Coefficient" Thrust ≡ C T 1 2 ρ V 2 S •  Non-dimensional thrust coefficient, C T! –  C T is a function of power/throttle setting, fuel flow rate, blade angle, Mach number, " •  Reference area, S, may be aircraft wing area, propeller disk area, or jet exhaust area" I sp = Thrust  m g o  Specific Impulse, Units = m/s m/s 2 = sec  m ≡ Mass flow rate of on − board propellant g o ≡ Gravitational acceleration at earth's surface Thrust and Specific Impulse" [...]...Sensitivity of Thrust to Airspeed " 1 2 ρVN S 2 (.)N = Nominal ( or reference) value Nominal Thrust = TN ≡ CTN •  Turbojet thrust is independent of airspeed over a wide range" Power " •  Assuming thrust is aligned with airspeed vector" Power = P = Thrust × Velocity ≡ CT •  If power is independent of velocity (= constant)" ∂P ∂C 1 3 2 = 0 = T ρVN3S + CTN ρVN S ∂V ∂V 2 2 ∂ CT = −3CTN / VN ∂V •  If thrust is... Nomad II shown (1949)" •  •  V-12" Jet Engine Nacelles " Opposed" Pulsejet " Flapper-valved motor (1940s)" Dynajet Red Head (1950s)" Fighter Aircraft and Engines " Lockheed P-38! Convair/GD F-102! V-1 Motor! Pulse Detonation Engine" on Long EZ (1981)" http://airplanesandrockets.com/motors/dynajet-engine.htm! SR-71: P&W J58 Variable-Cycle Engine (Late 1950s)" Hybrid Turbojet/ Ramjet " Allison V-1710! Turbocharged... ∂V 2 2 ∂ CT = −3CTN / VN ∂V •  If thrust is independent of velocity (= constant)" ∂T ∂C 1 2 = 0 = T ρVN S + CTN ρVN S ∂V ∂V 2 ∂ CT = −CTN / VN ∂V Next Time: Aviation History Reading Airplane Stability and Control, Ch 1 ! Virtual Textbook, Part 3 1 3 ρV S 2 •  Velocity-independent power is typical of propellerdriven propulsion (reciprocating or turbine engine, with constant RPM or variable-pitch prop)" . Point-Mass Dynamics and Aerodynamic/ Thrust Forces Robert Stengel, Aircraft Flight Dynamics, 
 MAE 331, 2012! •  Properties of the Atmosphere" •  Frames of reference" •  Velocity and. Scramjet" Ramjet (1940s)" Scramjet (1950s)" Talos! X-43! Hyper-X! Thrust and Thrust Coefficient" Thrust ≡ C T 1 2 ρ V 2 S •  Non-dimensional thrust coefficient, C T! –  C T is a function of. (1950s)" V-1 Motor! Pulse Detonation Engine" on Long EZ (1981)" http://airplanesandrockets.com/motors/dynajet-engine.htm! Fighter Aircraft and Engines" Lockheed P-38! Allison V-1710! Turbocharged