Launch Vehicle Dynamics

Home , Ariane (rocket family), Soviet rocketry

3/6/19

Seminar 4 Early Space Age Launch Vehicle Dynamics FRS 148, Princeton University Robert Stengel • Political Rains and First Fruit: The Cold War and Sputnik America Before Sputnik § Bashful Behemoth: Technology, the State, & the Birth of Deterrence § While Waiting for Technocracy: The ICBM & the First American Space Program Launch Vehicle Flight Dynamics

…the Heavens and the Earth, Ch 2 to 4 Understanding Space, Sec 9.1, 14.1 (pp. 535-542), 14.3

Political Rains and First Fruit: The Cold War and Sputnik

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German Vengeance (Aggregate) Weapons

A-3 A-4 (V-2) A-4b

V-1 “Buzz Bomb

http://en.wikipedia.org/w iki/Aggregate_(rocket_family) 3

V-2 (A-4) Rocket • Gyro Control • Thrust and aero steering • 6,084 built during WWII • 1000+ test flights • 3,225 launched in combat

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Peenemünde, Mittelwerk, Wernher von Braun, and the German Rocket Team

Helmut Gröttrup 5

German Vengeance Weapons and Space Launch Vehicles [never built]

A-11 A-12

A-12

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Sänger “Antipodal” Bomber Concept (1935-1941) Dr. Eugen Sänger (1905-1964), Austrian designer

Sub-Orbital “Skip” Trajectory

http://www.luft46.com/misc/sanger.html 7

Nuclear Weapons and the Balance of Power

Trinity Test Trinity Test

Hiroshima Explosion

“Enola Gay” B-29 “Fat Man” “Little Boy”

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Balance of Power in Europe Stalin’s 5-Yr Plan

Post-WWII Soviet Rocket Development

R-7

G Series: Principally interned German Engineers R Series: Principally Russian Engineers 10

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Sputnik 1 and the R-7 (October 4, 1957) R-7 (Semyorka) launch

Sergei Korolev vehicle (ICBM) (1907-1966)

Sputnik 1

Eight Characteristics of Soviet R&D

1) Need to borrow foreign 5) Scarcity of skilled technology and desire to workers, need to keep foster national creativity unskilled workers occupied 2) Lack of competitive stimulus 6) Tradition of pure and planning system that research, priority to inhibited innovation by science not technology production managers 7) Organizational distance 3) Inhibitions due to terror between R&D and (risk of failure) production 4) Te nsio ns between 8) Less concern for professionalization of R&D economics than technical and adherence to the party performance line

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Commonalities Between Post-War US and USSR § Continental superstates § Born of revolutions inspired by ideology of progress § Faith in the works of man § Patriotism rooted in § common ideas, values, and experience rather than § tradition, religion, or ethnicity … § or not? § Came of age internationally during the world wars and prevailed because of § geographical expanse § remoteness § unprecedented mobilization of technical resources § Emerged from isolationism to become superpowers

America Before Sputnik Bashful Behemoth: Technology, the State, and the Birth of Deterrence

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Buildup of Federal Support for R&D § Civil War spawned National Academy of Sciences, 1863 § American pattern § Diffuse entities for research § Army and navy arsenals the exception American assembly lines § Lack of European "command technology” § NACA established in 1915 [by President Woodrow Wilson] at Langley Research Center § 1920s § Veblen lobbies for federal support of R&D § Hoover lobbies for federal support of R&D § Great Depression, 1929 to 1939

Robert Goddard (1882-1945) • Physics Professor, Clark University, Worcester MA • “A Method of Reaching Extreme Altitudes”, 1919 • New York Timess editorial comment(1920):

"after the rocket quits our air … it will neither be accelerated nor maintained by the explosion of the charges it then might have left.”

“To claim that it would be is to deny a fundamental law of dynamics”

(The NYT Editors) expressed disbelief that Professor Goddard actually "does not know of the relation of action to reaction, and the need to have something better than a vacuum against which to react.” New York Times regretted its error on July 17, 1969, the day after Apollo 11 was launched to the Moon

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Robert Goddards Rockets, 1926-1941

US Rocket R&D in the 1930s American Rocket Society Guggenheim Aeronautical Laboratory, CIT Reaction Motors, Inc.

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US R&D Through 1948

§ 1941: R&D turnaround § Military needs § Federally funded labs § "Arsenal of democracy" § Vannevar Bush: Civilian extension of Office of Scientific Research and Development § Henry Stimson and James Forrestal (Princeton, Forrestal campus), Research Board for National Security § R&D without politics – good idea? § 1944: Henry Smyth (Princeton) report on peaceful use of atomic energy § 1945: Atomic Energy Commission § 1946: Canadians crack Russian atomic spy ring § 1948: Military R&D: 62% of federal total 19

Post-War Developments § 1943: Marshall: Deterrence, long-range bomber atomic strike capability § 1947: George Kennan's Foreign Affairs article, "The Sources of Soviet Conduct”: containment § Truman's Board of Budget director: James Webb § 1945-49: US sought a counterweight to Soviet's conventional military might: Strategic Air Command § Cruise missiles (e.g., Northrop Snark) vs ballistic missiles § USAF: responsibility for strategic ballistic missiles § Army: responsibility for tactical ballistic missiles

B-52 Snark B-47

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Significance of Map Projections Cylindrical Projection: Mercator Map

Azimuthal Projection: Polar Map

Project Bumper (V-2/WAC Corporal) 1948-1950 •White Sands, NM •Cape Canaveral, FL nd •2 stages 2 •8 flights, 4 failures Stage •Engineering development •High-altitude photography •Atmospheric temperature profile •Cosmic radiation

V-2

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Dawn of the Missile Age, 1956

Hydrogen Bomb First Test First Deliverable Device Tested USSR Aug 1953 Nov 1955 US Nov 1952 May 1956

Contest for influence in the “3rd World”

While Waiting for Technocracy: The ICBM and the First American Space Program

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Precursors to US ICBMs, 1946-1948 § Convair Project MX-774 § Swiveling engines § Separable nose cone § Outer skin was propellant-tank wall § RAND study of satellite launch Convair RAND World- MX-774 circling Spaceship

Post-World War II Science Fact and Fiction Catalyzed human imagination

Chesley Bonestell (1888-1986)

Willy Ley (1906-1969) Wernher von Braun (1912-1977)26

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Hydrogen Bombs, 1952-1953 Soviet Joe-4, 1953 US Ivy MIKE, 1952

Military-Industrial Complex § UN debates § Baruch Plan (international control) § Gromyko's response (unilateral US disarmament) § How would ban on nuclear weapons have affected military power balance? § Roles and missions contested within American military -- inseparable from the Cold War (Daniel Yergin) § Navy, ground power, air power, ... § Tradition, size, power, sick aviation industry, conversion of factories § Importance of aviation industry to US security § Truman's recognition (1946) § Continued growth of aviation R&D, including rockets and missiles

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Resistance to ICBMs and Satellites § Invasion of South Korea by North Korea, June 1950 § Bureaucratic resistance to ICBMs § Guidance systems, MIT Instrumentation Lab § Blunt body re-entry vehicle proposed, NACA Langley MiDAS § Reports of Soviet rocketry supported need for a crash ICBM program § Atlas given highest priority, June 1954 § RAND studied utility of satellites for strategic and meteorological reconnaissance,1950 § Political problems with over-flight, even in orbit. § How would the Soviets respond? § This worried Eisenhower § Policy must be charted in advance § WS-117L: Missile Defense Alert System (MiDAS) 29

Launch Vehicle Flight Dynamics

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Launch Phases and Loading Issues-1 • Liftoff – Reverberation from the ground – Random vibrations – Thrust transients • Winds and Transonic Aerodynamics – High-altitude jet stream – Buffeting • Staging – High sustained acceleration – Thrust transients 31

Launch Phases and Loading Issues-2

• Heat shield separation – Mechanical and pyrotechnic transients • Spin stabilization – Tangential and centripetal acceleration – Steady-state rotation • Separation – Pyrotechnic transients

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Equations of Motion for a Point Mass

⎡ x˙ ⎤ ⎡v x⎤ dr ⎢ ⎥ ⎢ ⎥ = r˙ = v = y˙ = v dt ⎢ ⎥ ⎢ y⎥ ⎣⎢ z˙ ⎦⎥ ⎣⎢v z ⎦⎥

⎡1 /m 0 0 ⎤⎡ f x⎤ dv 1 ⎢ ⎥⎢ ⎥ = v˙ = F = 0 1/m 0 f dt m ⎢ ⎥⎢ y⎥ ⎣⎢ 0 0 1/m⎦⎥⎣⎢ f z⎦⎥

Equations of Motion for a Point Mass Velocity and position dynamics expressed in a single equation ⎡ x ⎤ ⎢ ⎥ ⎢ y ⎥ dx(t) ⎡r ⎤ ⎢ z ⎥ x˙ (t) f[x(t),F] x ≡ ⎢ ⎥ = ⎢ ⎥ = = v v dt ⎣ ⎦ ⎢ x⎥ ⎢v y⎥ ⎢ ⎥ ⎣v z ⎦

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Combined Equations of Motion for a Point Mass

⎡ x˙ ⎤ ⎡ vx ⎤ ⎡0 0 0 1 0 0⎤⎡ x ⎤ ⎡ 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ y˙ v 0 0 0 0 1 0 y 0 0 0 ⎢ ⎥ ⎢ y ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎡ f x⎤ ⎢ z˙ ⎥ ⎢ vz ⎥ ⎢0 0 0 0 0 1⎥⎢ z ⎥ ⎢ 0 0 0 ⎥⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥ + f y v˙ f /m ⎢0 0 0 0 0 0⎥ v ⎢1 /m 0 0 ⎥⎢ ⎥ ⎢ x⎥ ⎢ x ⎥ ⎢ ⎥⎢ x⎥ ⎢ ⎥⎢ f ⎥ ⎣ z⎦I ⎢v ˙ y⎥ ⎢ f y /m⎥ ⎢0 0 0 0 0 0⎥⎢ v y⎥ ⎢ 0 1/m 0 ⎥ ⎢v ˙ ⎥ ⎢ f /m⎥ ⎢ ⎥⎢ v ⎥ ⎢ ⎥ ⎣ z ⎦I ⎣ z ⎦I ⎣0 0 0 0 0 0⎦⎣ z⎦I ⎣ 0 0 1/m⎦

⎡ f x⎤ With ⎢ ⎥ F = f = F + F + F I ⎢ y⎥ [ gravity aerodynamics thrust ]I ⎢ f ⎥ ⎣ z⎦I 35

Math Models of Gravity

• Flat-earth approximation ⎡ 0 ⎤ – g is gravitational ⎢ ⎥ 2 mg m 0 ; g 9.807 m /s acceleration f = ⎢ ⎥ o = – mg is gravitational force ⎣⎢g o⎦⎥ – Independent of position ⎡g x⎤ • Round, rotating earth ⎢ ⎥ g = g = g [non − rotating frame] – Inverse-square gravitation r ⎢ y⎥ gravity ⎢ g ⎥ – Centripetal acceleration ⎣ z ⎦ – Non-linear function of gr = ggravity + grotation [rotating frame] position ⎡ x⎤ ⎡ x⎤ 5 3 2 −µ ⎢ ⎥ 2 ⎢ ⎥ 2 2 2 1/ 2 – μ = 3.986 x 10 km /s = y − Ω y ; r = x + y + z 3 ⎢ ⎥ ⎢ ⎥ [ ] –5 r – Ω = 7.29 x 10 rad/s ⎣⎢ z⎦⎥ ⎣⎢0 ⎦⎥ 36

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Equations of Motion with Round-Earth Gravity Model (Inertial, Non-Rotating Frame)

⎡ x˙ ⎤ ⎡ 0 0 0 1 0 0⎤⎡ x ⎤ ⎡ 0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ y˙ 0 0 0 0 1 0 y 0 0 0 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎛ f x ⎞ ⎛ f x ⎞ ⎢ z˙ ⎥ ⎢ 0 0 0 0 0 1⎥⎢ z ⎥ ⎢ 0 0 0 ⎥⎢ ⎜ ⎟ ⎜ ⎟ ⎥ ⎢ ⎥ = ⎢ 3 ⎥⎢ ⎥ + ⎢ ⎥ f y + f y v˙ −µ /r 0 0 0 0 0 v 1/m 0 0 ⎢⎜ ⎟ ⎜ ⎟ ⎥ ⎢ x⎥ ⎢ ⎥⎢ x⎥ ⎢ ⎥⎢ ⎜ ⎟ ⎜ ⎟ ⎥ f z f z 3 ⎣⎝ ⎠ aero ⎝ ⎠ thrust ⎦I ⎢v ˙ y⎥ ⎢ 0 −µ /r 0 0 0 0⎥⎢ v y⎥ ⎢ 0 1/m 0 ⎥ ⎢v ˙ ⎥ ⎢ 3 ⎥⎢ v ⎥ ⎢ ⎥ ⎣ z ⎦E ⎣ 0 0 −µ /r 0 0 0⎦⎣ z⎦I ⎣ 0 0 1/m⎦ Position of the vehicle (in spherical coordinates) R : Earth's radius ⎡ x⎤ ⎡cos Lcosλ⎤ ⎢ ⎥ ⎢ ⎥ h : Altitude (height) r y cosLsin R h = ⎢ ⎥ = ⎢ λ⎥( + ) L : Latitude ⎣⎢ z⎦⎥ ⎣⎢ sinL ⎦⎥ λ : Longitude 37

Effect of Launch Site on Launch Velocity

• Launch site and azimuth – Earths rotation adds up to 465 m/s to final inertial velocity – Function of launch latitude and azimuth angles

ΔVlaunch ≈ ΩRcos L cosβ β :Launch azimuth angle (rotating frame, from East) 38

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Properties of the Lower Atmosphere

• Air density and pressure decay exponentially with altitude • Air temperature and speed of sound are linear functions of altitude

US Standard Atmosphere, 1976 39

Lower Atmosphere Rotates With The Earth

• Zero wind at Earths surface = Inertially rotating air mass Jet Stream (typical) • Wind measured with respect to Earths rotating surface • Jet stream magnitude typically peaks at 10-15- km altitude

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Aerodynamic Forces

⎡ ⎤ ⎡ Drag ⎤ CD ⎢ ⎥ ⎢ ⎥ 1 Side Force = C ρV 2S ⎢ ⎥ ⎢ Y ⎥ 2 ⎢ Lift ⎥ ⎢ C ⎥ ⎣ ⎦ ⎣ L ⎦ • V = air-relative velocity = velocity w.r.t. air mass • Drag measured opposite to the air-relative velocity vector • Lift and side force are perpendicular to the velocity vector 41

Aerodynamic Force Parameters ρ = air density, functionof height, h −βh = ρsealevele 3 ρsealevel = 1.225 kg / m ; β = 1/ 9,042 m

2 2 2 1/2 T 1/2 V = ⎣⎡vx + vy + vz ⎦⎤ = ⎣⎡v v⎦⎤ , m s 1 q = ρV 2 = Dynamic pressure, N m2 2 2 S = referencearea, m ⎡ C ⎤ ⎢ D ⎥ ⎢ CY ⎥ = non - dimensionalaerodynamiccoefficients ⎢ C ⎥ ⎣ L ⎦ 42

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Aerodynamic Drag 1 Drag = C ρV 2S D 2

• Drag components sum to produce total drag – Parallel to airstream – Skin friction – Base pressure differential

– Forebody pressure differential (M > 1) 43

Flat-Earth 2-D Equations of Motion for a Point Mass Restrict motions to a vertical plane (i.e., motions in y direction = 0)

⎡ x˙ ⎤ ⎡ vx ⎤ ⎡ 0 0 1 0⎤⎡ x ⎤ ⎡ 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ z˙ vz 0 0 0 1 z 0 0 ⎡ fx⎤ ⎢ ⎥ = ⎢ ⎥ = ⎢ ⎥⎢ ⎥ +⎢ ⎥⎢ ⎥ ⎢ v˙x ⎥ ⎢ fx / m⎥ ⎢ 0 0 0 0⎥⎢ vx⎥ ⎢1 / m 0 ⎥⎣ fz⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ v˙z ⎦ ⎣ fz / m⎦ ⎣ 0 0 0 0⎦⎣ vz⎦ ⎣ 0 1/ m⎦

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Flat-Earth 2-D Equations of Motion for a Point Mass Transform velocity from Cartesian to polar coordinates

⎡ 2 2 ⎤ x! + z! ⎡ x! ⎤ ⎡ r! ⎤ ⎡ V cosγ ⎤ ⎡ V ⎤ ⎢ ⎥ ⎡ Velocity ⎤ " = ⎢ ⎥ ⎢ ⎥ = ⎢ h! ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ! ⎥ γ −1 ⎛ ⎞ Flight path angle ⎣ −z! ⎦ ⎣ h ⎦ ⎢ V sinγ ⎥ ⎣⎢ ⎦⎥ ⎢ sin ⎜ ⎟ ⎥ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎢ ⎝ V ⎠ ⎥ ⎣ ⎦

Flat-Earth Model • Ignore round, rotating Earth effects (!) • i.e., assume that flat-Earth-relative frame is inertial

⎡ 2 2 ⎤ x! + z! ⎡ x! ⎤ ⎡ V cosγ ⎤ ⎡ V ⎤ ⎢ ⎥ ⎡ Velocity ⎤ = = ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ −1 ⎛ z! ⎞ ⎥ z! −V sinγ ⎢ γ ⎥ −sin ⎢ Flight path angle ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ ⎣ ⎦ ⎢ ⎝⎜ V ⎠⎟ ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ • Adequate model for investigating early phase of launch 46

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Simplified Launch Trajectory Equations of Motion • Gravity-turn, flat earth, vertical plane – Thrust aligned with velocity vector (α = 0) – Lift = 0 – Round, rotating earth effects neglected Thrust − Drag + m(t)g sinγ (t) V!(t) = [ ] m(t)

⎡⎛ 1 2 ⎞ ⎤ = ⎢⎜Thrust − CD ρ(h)V (t)⎟ m(t) − g sinγ (t)⎥ ⎣⎝ 2 ⎠ ⎦ γ!(t) = −gcosγ (t) /V(t) V = velocity

h!(t) = −z!(t) = V(t)sinγ (t) γ = flight path angle r!(t) = x!(t) = V(t)cosγ (t) h = height (altitude) r = range 47

Gravity-Turn Flight Path • For vertical launch, – trajectory is vertical unless – vehicle is pitched over via thrust-vector control • Following pitch-over, – if thrust is aligned with the velocity vector, – the result is called a gravity turn, with α = 0 • Gravity-turn flight path depends on 3 initial conditions – Initial pitch-over angle (from vertical launch) – Velocity at pitch-over – Acceleration profile determined by thrust-to-weight ratio, T/W 48

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Typical Velocity Loss due to Drag During Launch

• Aerodynamic effects on launch Thrust-to- Velocity Loss, vehicle are most important Weight Ratio m/s below ~50-km altitude 2 336 3 474 • Maintain angle of attack and 4 581 sideslip angle near zero to minimize side force and lift • Typical velocity loss due to drag for vertical launch – Constant thrust-to-weight ratio 2 – CDS/m = 0.0002 m /kg – Final altitude above 80 km 49

Effects of Gravity and Drag on the Velocity Vector

Thrust/Weight = T/W = 2 ⎡ 1 ⎤ Thrust = 1960 Thrust − C S ρ(h)V 2 (t) + mg sinγ(t) ⎣⎢ D 2 ⎦⎥ CD = 0.2 V˙ (t) = m S = 0.1 ˙ (t) gcos (t)/V(t) Mass = 100 γ = − γ

• Significant reduction in velocity magnitude

• Strong curvature of the flight path 50

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Typical Launch Trajectories Thrust/Weight = 2 • Max dynamic pressure, Mach = 1, max drag, and max jet stream magnitude occur at similar altitudes • Aerodynamic effects negligible above ~50- km

Typical Launch Trajectories Thrust/Weight = 4 • Max dynamic pressure, Mach = 1, max drag, and max jet stream magnitude occur at similar altitudes • Aerodynamic effects negligible above ~50- km

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Gravity and Drag Effects during Single-Stage Orbital Thrust/Weight = T/W = 2 Launch

• Launch trajectory using flat-earth model • Red line: velocity due to rocket alone • Several km/s lost • With higher T/W to gravity and drag – Shorter time to orbit – Increased loss due to drag 53 – Decreased loss due to gravity

Typical Ariane 4 Launch Profile

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Mass-Ratio Effect on Final Load Factor • Thrust-to-weight ratio = load factor

Thrust Thrust Final Load = n (load factor) = Factor Weight mgo Initial Load Mass Mass Thrust Thrust Factor Ratio = 2 Ratio = 5 n = ; n = 1.3 2.6 6.5 initial m g final m g 2 4 10 initial o final o 3 6 15 • If thrust is constant n m final = initial = µ ninital m final • If thrust is reduced, limit load factor can be enforced 55

Expendable vs. Reusable Launch Vehicles

• Expendable Vehicle • Reusable Vehicle – Low cost per vehicle – High initial cost – New vehicle for each – High structural ratio launch – Maintenance and – Low structural ratio repair – Continued production – Non-reusable parts – Launch preparation and supplies – Upgrade in production – Launch preparation – Return to launch site

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Reusable Space Vehicles § Space Transportation § SpaceX Falcon Heavy System § First Stage Recovery § Space Shuttle Vehicle § Heat-shield Recovery § Solid-rocket Booster Recovery

Evolution of the Soyuz Booster from the R-7

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Big Dumb Boosters,” c. 1963 Objective: 450,000 kg to low earth orbit Douglas 1-1/2 Stage General Dynamics, Martin, and Douglas Concepts

Plug nozzle 1-1/2 stage, fully recoverable Nozzle = Reentry Heat Shield Recovery at sea Fully recoverable Ducted rocket 59

Alternative Lunar Landers and Launch Vehicles

Saturn 1 Saturn 5 Nova (Saturn 8)

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NASA, SpaceX, and Blue Origin Launch Vehicles

Specific Energy Contributed in Boost Phase

• Total E n e rg y = Kinetic + Potential Energy (relative to flat earth) mV 2 E = + mgh 2 • Specific Total Energy = Energy per unit weight = Energy Height (km) V 2 ES = + h 2g 62

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Specific Energy Contributed in Boost Phase

• Specific Energy contributed by first stage of launch vehicle – Less remaining drag loss (typical) – Plus Earths rotation speed (typical)

Earth- Remaining Mach RelativeVelocity, Drag Loss, Earth Rotation Specific Kinetic Total Specific Percent Altitude, km Number km/s km/s Speed, km/s Energy, km Energy, km of Goal Scout 1st-Stage Burnout 22 4 1.2 0.05 0.4 123.42 145.42 3.93% Subsonic Horizontal Launch 12 0.8 0.235 0.15 0.4 12.05 24.05 0.65% Supersonic Horizontal Launch 25 3 0.93 0.04 0.4 85.57 110.57 2.99% Scramjet Horizontal Launch 50 12 3.6 0 0.4 829.19 879.19 23.74% Target Orbit 300 25 7.4 0.4 3403.34 3703.34

Aerodynamic Lift Force 1 ∂C 1 Lift = C ρV 2S ≈ L α ρV 2S L 2 ∂α 2

• Perpendicular to airstream • Angle between x axis and airstream = angle of attack, α • Lift components integrate over length to produce net lift – Increase in cross-sectional area – Tail fins 64 • For symmetric vehicle, lift = 0 if α = 0

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Normal Force about equal to Lift 1 ∂C 1 Normal Force = C ρV 2S ≈ N α ρV 2S N 2 ∂α 2

• Perpendicular to body centerline • For small angle of attack, normal force is approximately the same as lift 65

Aerodynamic Pitching Moment

1 ∂C 1 Pitching Moment = C ρV 2Sr ≈ m α ρV 2Sr m 2 ∂α 2 r = Reference Length

• Pitching moment components • ... plus pitching moment integrate over length to produce due to thrust vectoring net pitching moment for control – Increase in cross-sectional area 66 – Tail fins

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Pitching Moment Distribution Causes Large Bending Effects

Aerodynamic and thrust-vectoring effects bend the vehicle Trajectory shaped to reduce structural loads 67

Angular Attitude Perturbations

• Pitch-angle perturbation, Δθ, is about the same as angle-of-attack perturbation, Δα Net Pitching Moment Δθ˙˙ ≈ Δα˙˙ = Pitching Moment of Inertia

M y + M y 1 ⎡∂ M y ∂ M y ⎤ Δα!! = aero thrust ≈ ⎢ net Δα! + net Δα ⎥ I I ∂α! ∂α 68 yy yy ⎣⎢ ⎦⎥

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Attitude Stability

1 ⎡∂ M y ∂ M y ⎤ Δα!! = ⎢ net Δα! + net Δα ⎥ I ∂α! ∂α yy ⎣⎢ ⎦⎥ • Attitude perturbations are stable if ∂M ∂M ynet < 0, ynet < 0 ∂α˙ ∂α

• Non-oscillatory divergence • Oscillatory divergence if if

∂M y ∂M net > 0 Dynamic Instability ynet > 0 Static Instability ∂α˙ ∂α Thrust-vector feedback control normally required

to provide static and dynamic stability 69

Jet Stream Profiles • Launch vehicle must be able to fly through strong wind profiles • Design profiles assume 95th-99th-percentile worst winds and wind shear

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Typical Thrust-Vector Angle Requirements

71 NASA TN D-4662, 1968

Trajectory shaped to reduce structural loads

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Next Time: Antecedents to the Apollo Program Rocket Propulsion and Staging

Supplemental Material

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Pre-World War II Rocket Societies

US R&D in the 1930s

§ 1935-1936: Universities spent about $50M for research; $6M came from the government. § Total federal R&D: about $70M, plus $50M for social sciences and statistics. § Industry spent $100M. Total: about $264M. § Aloofness toward public finance of R&D, but US was not anti-tech § Hoover and Roosevelt were big boosters of research, engineering, and federal intervention. § Goddard, LOX, pressurized feed, integration with gasoline fuel (like kerosene, or RP-1). 83 patents