Newtons Laws as Applied to Rocket Science its not just a job its an adventure 1 RS 102 Summary 2 Vertically accelerating rocket Neglected aerodynamic drag How High will my Rocket go ID: 760516
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Rocket Science 103:Ballistic Equations of Motion
Newton's Laws asApplied to "Rocket Science"... its not just a job ... its an adventure
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Slide2RS 102: Summary
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Vertically accelerating rocket
Neglected aerodynamic drag
Slide3How High will my Rocket go?
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Slide4Forces Acting on Rocket
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Lift – acts perpendicular to flight path (non-conservative)Drag – acts along flight path (non-conservative)Thrust – acts along longitudinal axis of rocket (non-conservative)Gravity – acts downward (conservative)
Slide5Forces Acting on Rocket (2)
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Lift – acts perpendicular to flight path (non-conservative)Drag – acts along flight path (non-conservative)
Define
Slide6Friction (Drag) Losses (2)
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For constant
C
D, M
“Ballistic Coefficient”
drag
Slide7Ballistic Coefficient
• When effects of lift are negligible aerodynamic effects can be incorporated into a single parameter …. Ballistic Coefficient …. • b is a measure of a projectile's ability to coast. …… M is the projectile's mass and … CDAref is the drag form factor. • At any given velocity and air density, the deceleration of a rocket from drag is inversely proportional to b
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Low Ballistic
Coefficients DissipateMore Energy Due to drag
Slide8Real World Launch Analysis
Trajectory Design OptimizationOrbital designs a unique mission trajectory foreach Pegasus flight to maximize payloadperformance while complying with the satelliteand launch vehicle constraints. Using the 3-Degree of Freedom Program for Optimization of Simulation Trajectories(POST), a desired orbit is specified and a set of optimization parameters and constraints are designated. Appropriate data for mass properties, aerodynamics, and motor ballistics are input.POST then selects values for the optimization parameters that target the desired orbit with specified constraints on key parameters such as angle of attack, dynamic loading, payload thermal, and ground track. After POST has been used to determine the optimum launch trajectory, a Pegasus-specific six degree of Freedom simulation program is used to verify Trajectory acceptability with realistic attitude dynamics, including separation analysis on all stages.
• 6-DOF
simulations costs
A LOT! To run and are typicallyNot used for Trajectory design!• We are going to start withA simple 2+-D code that works Well for mission profile development
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Slide9Perifocal Coordinate SystemSub-orbital Image
n
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Sub-Orbital Launch: Spherical Earth
Sub-Orbital Launch: Flat Earth
~ Symmetric
trajectory
Slide10Perifocal Coordinate System
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Maps spherical
earth onto flat earth
Slide11Newtonian Dynamics
• Must resort to Newton’s laws to describe these orbits
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Slide12Velocity Vector
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Slide13Acceleration Vector
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Slide14Acceleration Vector (cont’d)
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Slide15Newton’s Second Law
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Slide16Newton’s Second Law
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Slide17Gravitational (conservative) Forces
• Assume spherical earth .. Always acts in i
r direction
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Slide18Gravitational Forces (2)
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Sea Level Acceleration of gravity at 21 deg. latitude
Slide19Vehicle Mass
Initial mass of vehicle
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Slide20Non-Conservative Forces
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Slide21Aerodynamic Forces
“Dynamic Pressure”
A
ref
… reference
Area … planformOr diameter based
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Slide22Collected Equations
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Slide23Ballistic versus Non -Ballistic Trajectories
• Non-ballistic trajectories sustain
significantly non -zero angles of attack … lift is a factor in resulting trajectory … so is induced drag• Ballistic rocket trims trajectory at … lift is a negligible factor
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Slide24Non-Ballistic “Gravity Turn”
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Yup this is real
! Made to minimize
Gravity losses
Slide25More “Gravity Turn”
Space Shuttle
Launch (STS 115 – Atlantis) as seen from ISS“definitelyNot ballistic”
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Slide26Example of Ballistic Trajectory
~ Symmetric
trajectory
• Ballistic Trajectories
Offer minimum drag profiles( No induced drag)
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Slide27Collected Equations, Ballistic Trajectory
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Slide28Numerical Analysis of the 2-D Launch Equations of Motion
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Slide29Integrated Equations of Motion
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Slide30Numerical Approximation ofthe Integral
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Slide31Numerical Approximation ofthe Integral (cont’d)
“Trapezoidal rule”
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Slide32Numerical Approximation ofthe Integral (cont’d)
“Trapezoidal rule”
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Slide33Numerical Approximation ofthe Integral (cont’d)
“trapezoidal rule”
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Slide34Predictor/Corrector Algorithm
“trapezoidal rule”
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Slide35Higher Order Integrators
•Simple Second Order predictor/corrector works well forSmall-to-moderate step sizes … but at larger step sizes can be come unstable• Good to have a higher order integration scheme in our bag of tools • 4th Order Runge-Kutta method is one most commonly used• we’ll only use the second order integrators for this simulation
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Slide36Summary Slides on E.O.M.
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Slide37Collected Equations, Ballistic Trajectory
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This what we are going to program
Slide38Predictor/Corrector Algorithm
“trapezoidal rule”
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Slide39Fixed Earth Approximation
• Ignore effects of rotation
• Vinertial=Vground• ginertial=gground• Accurate for Short Duration lower altitude flights
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Slide40Ground Launch: Down Range Calculation
• Integrated trajectory gives
• Inertial Downrange
Recursive Formula
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Slide41Velocity Off of the Rail
V
rail
F
thrust
F
fric
F
drag
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Slide42Velocity Off of the Rail… Initial Conditions for simulation
V
rail
F
thrust
F
fric
F
drag
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Slide43Rail Simulation Block Diagram
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Initial
Conditions
Parameters and
Constants
Slide44Ballistic Simulation Initial Conditions
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Final
conditions
Slide45Ballistic Simulation Block Diagram
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Initial
Conditions
Parameters and
Constants
Slide46Design Friday: 2009 “Pike” Ballistic Simulation
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… Build functional block diagram of the simulation, Identify key elements
and computational blocks
Rail Launch ..
Single degree of freedom model for rail dynamics, rail launch angle
assumed to be constant, assume constant thrust, deplete mass during burn
Initial conditions:…
Use rail exit conditions for calculating initial conditions for
Ballistic simulation
Motor Burn phase …
Assume constant thrust, constant drag coefficient,
ballistic equations of motion, deplete mass during burn
Coast phase …
zero thrust (following depletion of propellant mass), constant mass,
Constant drag coefficient, ballistic equations of motion
Use trapezoidal rule for integration, Use 1976 US standard atmosphere to calculate
Density, pressure, temperature, etc .. As function of vehicle altitude
Slide47Design Friday: 2009 “Pike” Ballistic Simulation (2)
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Slide48Design Friday: 2009 “Pike” Ballistic Simulation (3)
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Check simulation results for CD0=0 (b >> 1) .. Using analytical expressions for Vtburn, htburn , hapogee
… Plot Achieved apogee (with drag active) as a function of initial launch rail angle for nominal motor impulse… Plot achieved apogee as function motor impulse forI = {1000, …. 5000 Nt-sec) at 85 deg launch angle
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Questions??