Point-Mass Dynamics and Aerodynamic/Thrust Forces

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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

The Atmosphere

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)

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

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

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

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 ⎢ ⎥ ⎢ ⎥ = v! (t) = F = ⎢ fy m ⎥ = ay dt m ⎢ ⎥ ⎢ f m ⎥ ⎢ ⎥ ⎢ z ⎥ az ⎣ ⎦ ⎣⎢ ⎦⎥ 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) ⎥ ⎢ vz (t) ⎥ ⎣ ⎦ ⎣ ⎦ 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 = v t ( ) z ( )

Flight Path with Constant Gravity and No Aerodynamics

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

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

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

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 drag(due to lift) proportional to 1/V2 – Total drag minimized at one particular airspeed

15 2-D Equations of Motion with Aerodynamics and Thrust

2-D Equations of Motion for a Point Mass • Motions restricted to vertical plane • Inertial frame, wind = 0 • z positive down, flat-earth assumption • Point-mass location coincides with aircraft’s center of mass

⎡ ⎤ ⎡ x! ⎤ vx ⎢ ⎥ 1 2 2 ⎢ ⎥ q = ρ (z)(vx + vz ) z! ⎢ vz ⎥ 2 ⎢ ⎥ = ⎢ v! ⎥ ⎢ ⎥ x fx / m ⎢ ⎥ ⎢ ⎥ C = Thrust coefficient v! T ⎢ z ⎥ ⎢ fz / m ⎥ ⎣ ⎦ ⎣ ⎦ θ = Pitch angle ⎡ 0 0 1 0 ⎤⎡ x ⎤ ⎡ 0 0 ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ C cosθ + C qS ⎤ z ( T XI ) ⎢ 0 0 0 1 ⎥⎢ ⎥ ⎢ 0 0 ⎥⎢ ⎥ = v + ⎢ ⎥ ⎢ 0 0 0 0 ⎥⎢ x ⎥ ⎢ 1/ m 0 ⎥ −C sinθ + C qS + mg ⎢ ⎥ ⎢ ( T ZI ) o ⎥ ⎢ 0 0 0 0 ⎥ v ⎢ 0 1/ m ⎥⎣ ⎦ ⎣ ⎦⎣⎢ z ⎦⎥ ⎣ ⎦ 32

16 Transform Velocity from Cartesian to Polar Coordinates

Inertial axes -> wind axes and back

⎡ 2 2 ⎤ ⎡ v2 v2 ⎤ ⎡ ⎤ ⎡ ⎤ x + z ⎢ x + z ⎥ ⎡ x ⎤ vx V cosγ ⎡ V ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ ⇒ ⎢ ⎥ = ⎢ z ⎥ = ⎢ v ⎥ z v −V sinγ γ −1 ⎛ ⎞ −1 ⎛ z ⎞ ⎣ ⎦ ⎣⎢ z ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎢ −sin ⎜ ⎟ ⎥ ⎢ −sin ⎜ ⎟ ⎥ ⎢ ⎝ V ⎠ ⎥ ⎢ ⎝ V ⎠ ⎥ ⎣ ⎦ ⎣ ⎦ Rates of change of velocity and flight path angle

⎡ d 2 2 ⎤ ⎢ vx + vz ⎥ ⎡ V! ⎤ dt ⎢ ⎥ = ⎢ ⎥ γ! ⎢ d −1 ⎛ vz ⎞ ⎥ ⎣⎢ ⎦⎥ − sin ⎢ dt ⎝⎜ V ⎠⎟ ⎥ ⎣ ⎦ 33

Longitudinal Point- Mass Equations of Motion x :range z : –height (altitude) x!(t) = vx = V(t)cosγ (t) V :velocity z!(t) = vz = −V(t)sinγ (t) γ :flight path angle

1 2 (CT cosα − CD ) ρ(z)V (t)S − mgo sinγ (t) V!(t) = 2 m

1 2 (CT sinα + CL ) ρ(z)V (t)S − mgo cosγ (t) γ!(t) = 2 mV(t)

17 Steady, Level (i.e., Cruising) Flight

In steady, level flight with cos α ~ 1, sin α ~ 0 Thrust = Drag, Lift = Weight

Vcruise = x!(t) = vx

0 = z!(t) = vz

1 2 (CT − CD ) ρ(z)VcruiseS 0 = 2 m

1 2 CL ρ(z)VcruiseS − mg(z) 0 = 2 mV cruise 35

Introduction to Aeronautical Propulsion

18 Internal Combustion Reciprocating Engine Linear motion of pistons converted to rotary motion to drive propeller

Turbojet Engines (1930s) Thrust produced directly by exhaust gas

Axial-flow Turbojet (von Ohain, Germany)

Centrifugal-flow Turbojet (Whittle, UK)

19 Birth of the Jet Airplane

Heinkel He. 178 (1939) Gloster Meteor (1943)

Gloster E/28/39 (1941)

Bell P-59A (1942)

Messerschitt Me 262 (1942)

Turbojet + Afterburner (1950s)

• Dual rotation rates, N1 and N2, typical

Fuel added to exhaust Additional air may be introduced 40

20 Turboprop Engines (1940s)

Exhaust gas drives propeller to produce thrust

High Bypass Ratio Turbofan

Propfan Engine

Aft-fan Engine Geared Turbofan Engine

21 Ramjet and Scramjet

Ramjet (1940s) Scramjet (1950s)

Talos X-51 Waverider

X-43 Hyper-X

Electric Propulsion

22 Electric Aircraft

• Solar cells/batteries • Fuel cells • Batteries

Hybrid-Electric Aircraft • Turbine engines • Generators • Thrust • Drag reduction

23 Thrust and Specific Impulse

Thrust m/s Isp = " Specific Impulse, Units = 2 = sec m! go m/s m! ≡ Mass flow rate of on - board propellant g Gravitational acceleration at earth's surface o ≡

Thrust and Thrust Coefficient

1 Thrust ≡ C ρV 2S T 2 • Non-dimensional thrust coefficient, CT – CT 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 48

24 Sensitivity of Thrust to Airspeed

1 Nominal Thrust = T ≡ C ρV 2S N TN 2 N

(.)N = Nominal (or reference) value

Turbojet thrust is ~independent of airspeed over wide range

Power Assuming thrust is aligned with airspeed vector 1 Power = P = Thrust × Velocity ≡ C ρV 3S T 2 Propeller–driven power is ~independent of (subsonic) airspeed over a wide range (reciprocating or turbine engine, with constant RPM or variable-pitch prop)

25 Next Time: Low-Speed Aerodynamics

Reading: Flight Dynamics Aerodynamic Coefficients, 65-84

Supplementary Material

26 MATLAB Scripts for Flat-Earth Trajectory, No Aerodynamics

Analytical Solution Numerical Solution Calling Routine g = 9.8; t = 0:0.1:40; tspan = 40; xo = [10;100;0;0]; vx0 = 10; [t1,x1] = ode45('FlatEarth',tspan,xo); vz0 = 100; x0 = 0; Equations of Motion z0 = 0; function xdot = FlatEarth(t,x) vx1 = vx0; % x(1) = vx vz1 = vz0 – g*t; % x(2) = vz x1 = x0 + vx0*t; % x(3) = x z1 = z0 + vz0*t - 0.5*g*t.* t; % x(4) = z g = 9.8; xdot(1) = 0; xdot(2) = -g; xdot(3) = x(1); xdot(4) = x(2); xdot = xdot'; end 53

27 Early Reciprocating Engines

• Rotary Engine: – Air-cooled – Crankshaft fixed Sopwith Triplane – Cylinders turn with propeller – On/off control: No throttle

• V-8 Engine: – Water-cooled SPAD S.VII – Crankshaft turns with propeller

Reciprocating Engines

• Rotary • In-Line

• Radial

• V-12 • Opposed

28 Turbo-compound Reciprocating Engine

• Exhaust gas drives the turbo-compressor • Napier Nomad II shown (1949)

Turbofan Engine (1960s)

• Dual or triple rotation rates

29 Jet Engine Nacelles

Propeller-Driven Aircraft of the 1950s

Reciprocating Engines Turboprop Engines

Vickers Viscount Douglas DC-6

Bristol Britannia

Douglas DC-7

Lockheed Electra Lockheed Starliner (Constellation) 1649

30 Pulsejet

Flapper-valved motor (1940s) Dynajet Red Head (1950s)

Pulse Detonation Engine V-1 Motor on Long EZ (1981)

http://airplanesandrockets.com/motors/dynajet-engine.htm 61

Jet Transports of the 2000s

Airbus A380 Boeing 787

Embraer 195 Boeing 747-8F

31 SR-71: P&W J58 Variable-Cycle Engine (Late 1950s)

Hybrid Turbojet/Ramjet