Newton's Laws As Applied to "Rocket Science"

Section 3 Rocket Science Review 101: Ballistic Equations of Motion

Newton's Laws as Applied to "Rocket Science" ... its not just a job ... its an adventure

MAE 6530, Propulsion Systems II 1 Real World Launch Analysis

Orbital designs a unique mission trajectory for each Pegasus flight to maximize payload performance while complying with the satellite and 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. • 6-DOF simulations costs Appropriate data for mass properties, aerodynamics, A LOT! To run and are typically and motor ballistics are input. POST then selects values Not used for Trajectory for the optimization parameters that target the desired design! orbit with specified constraints on key parameters such as angle of attack, dynamic loading, payload thermal, and ground track. • We are going to start with A simple 2+-D Ballistic code that After POST calculates optimum launch trajectory, a works well for sounding-rocket Pegasus-specific six degree of Freedom simulation mission profile development program used to verify Trajectory acceptability with realistic attitude dynamics, including separation analysis on all stages. MAE 6530, Propulsion Systems II 2 Newtonian Dynamics • General 3-DOF Equations of Motion

MAE 6530, Propulsion Systems II 3 RS 101: Summary External Forces Acting on Rocket

•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) MAE 6530, Propulsion Systems II 4 External Forces Acting on Rocket (2) •Lift – acts perpendicular to flight path (non-conservative) •Drag – acts along flight path (non-conservative)

Define L D C = ift , C = rag L 1 D 1 ⋅ ρ ⋅V 2 ⋅ A ⋅ ρ ⋅V 2 ⋅ A 2 ref 2 ref

CL = lift coefficient, CD = drag coefficient 1 ⋅ ρ ⋅V 2 = "dynamic pressure"(q) 2

Aref = "reference area"− > typically maximum frontal area ρ = "local air density"− > function of altitude

MAE 6530, Propulsion Systems II 5 External Forces Acting on Rocket (3) •Gravitational Force

MAE 6530, Propulsion Systems II 6 Perifocal Coordinate System

Maps spherical earth onto flat earth

! ⎡ ⎤ ⎡ ⎤ ir local vertical ⎢ ! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ iν ⎥ = local horizontal(along flight path) ⎢ ! ⎥ ⎢ ⎥ i ⎢ ι ⎥ ⎢ local horizontal( perpendicular to flight path) ⎥ ⎣ ⎦ ⎣ ⎦ MAE 6530, Propulsion Systems II 7 Perifocal Coordinate System Sub-orbital Image

Sub-Orbital Launch: Spherical Earth Sub-Orbital Launch: Flat Earth

~ Symmetric trajectory Vr V

MAE 6530, Propulsion Systems II 8 Ballistic versus Non -Ballistic Trajectories

• Non-ballistic trajectories sustain significantly non -zero angles of attack

… lift is a factor in resulting trajectory α ≈ 0 … so is induced drag ⎡V ⎤ → θ = γ = tan−1 r ballistic ⎢V ⎥ • Ballistic rocket trims trajectory at α ≈ 0 ⎣ υ ⎦ … lift is a negligible factor

MAE 6530, Propulsion Systems II 9 Example of Ballistic Trajectory

~ Symmetric trajectory

• Ballistic Trajectories Offer minimum drag profiles α ≈ 0 ( Induced drag primarily due to Alpha-dither, assume small)

MAE 6530, Propulsion Systems II 10 Collected Ballistic Equations

⎡V ⎤ ⎡ 2 2 ⎤ γ = tan−1 r V F V ⎢V ⎥ ⎢ ν µ ⎡ thrust ρ ∞ ⎤ ⎥ ⎣ ν ⎦ ⎡ • ⎤ − + − sin(γ ) ⎢ 2 ⎢ ⎥ ⎥ M ⎢ V r ⎥ r r ⎣ m 2β ⎦ β = CD ⋅ Aref ⎢ • ⎥ ⎢ ⎥ 2 • ⎢ VrVν ⎡ Fthrust ρV∞ ⎤ ⎥ ⎢ V ν ⎥ X = f [X,Fthrust ] ⎢ − + ⎢ − ⎥cos(γ ) ⎥ ⎢ ⎥ r ⎣ m 2β ⎦ ⎢ ⎥ ⎢ ⎥ ⎡ V ⎤ ⎢ ⎥ ⎢ ⎥ r = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Vν ⎥ ⎢ • ⎥ ⎢ Vr ⎥ ! ⎢ ⎥ r ⎢ ⎥ X = ⎢ r ⎥ ⎢ ⎥ V ⎢ ν ⎥ ⎢ • ⎥ ⎢ ν ⎥ ⎢ M ⎥ ⎢ ν ⎥ ⎢ r ⎥ ⎢ ⎥ • ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ Fthrust m ⎢ − ⎥ ⎡ V" ⎤ ⎣⎢ ⎦⎥ r ⎢ g0 Isp ⎥ ⎢ ⎥ ⎢ V" ⎥ ⎣ ⎦α =0 ν !" ⎢ ⎥ ( X ) = ⎢ r" ⎥ ⎢ ν ⎥ ⎢ −m" ⎥ ⎢ motor ⎥ ⎣ ⎦ MAE 6530, Propulsion Systems II 11 Ballistic 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. … • β = m / (CD ⋅ Aref ) … 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 Low Ballistic Coefficients Dissipate More Energy Due to drag

MAE 6530, Propulsion Systems II 13 Numerical Integration of the 2-D Launch Equations of Motion

•Simple Second Order predictor/corrector works well for Small-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

MAE 6530, Propulsion Systems II 14 Trapezoidal Rule Predictor/Corrector Algorithm

“trapezoidal rule” MAE 6530, Propulsion Systems II 15 Fixed Earth for Sounding Rocket Launch Approximation • Ignore effects of Earth’s rotation

• Vinertial=Vground

• ginertial=gground

• Accurate for Short Duration, Non-orbital flights

MAE 6530, Propulsion Systems II 16 Ground Launch: Down Range Calculation • Integrated trajectory gives

• Inertial Downrange Horizontal Motion

υ2 ⎛ rk +1 + rk ⎞ Δx = r ⋅ dυ ≈ ⋅(υk +1 − υk ) ∫ ⎝⎜ 2 ⎠⎟ υ1

Recursive Formula for Horizontal Motion

⎛ rk +1 + rk ⎞ xk +1 = xk + ⋅(υk +1 − υk ) ⎝⎜ 2 ⎠⎟

MAE 6530, Propulsion Systems II 17 Velocity Off of the Rail

Vrail

Fdrag Fthrust

Ffric

1839 MAE 6530, Propulsion Systems II Exit Conditions off of Rail = Initial Conditions for Free Flight Simulation Vrail

Altitude : h = r − Rearth Fdrag Fthrust

Ffric

MAE 6530, Propulsion Systems II 19 Rail Simulation Block Diagram

Initial ⎡Vrail ⎤ ⎡ 0 ⎤ ⎢ ⎥ ⎢ ⎥ do Parameters and Conditions zrail = 0 ⎢ ⎥ ⎢ ⎥ → zrail ≤ lrail ⎢ ⎥ ⎢ ⎥ Constants ⎣ m ⎦0 ⎣M initial ⎦ while t = 0 C h l D0 launch rail A C ⎡zrail ⋅ cosγ rail + hlaunch ⎤ ref γ launch Frail h ⎢ ⎥ ⎡ ⎤ Rearth + h F M M ⎢ ⎥ ⎢ ⎥ thrust initial prop r ⎢ µ ⎥ 1976 Standard ⎢ ⎥ = − I g µ ⎢g ⎥ ⎢ r2 ⎥ h → → ρ(h) sp 0 (r) ⎢ ⎥ Atmosphere ⎢ ⎥ m ⎣ β ⎦ ⎢ ⎥ Rearth ⎢ ⎥ ⎣ CD ⋅ Aref ⎦ V ⎡ F ρ ⎤ ⎡ rail ⎤ ⎢ thrust − g ⋅ sinγ + C ⋅ cosγ − (h) ⋅V 2 ⎥  (r) ( rail Frail rail ) rail ⎢ ⎥ ⎡Vrail ⎤ ⎢ m 2 ⋅ β ⎥ z ⎢ ⎥ ⎢ ⎥ rail z = V ⎢ ⎥ ⎢ rail ⎥ ⎢ rail ⎥ ⎢ m ⎥ ⎣⎢ m ⎦⎥ ⎢ ⎛ F ⎞ ⎥ ∫ ⎣ ⎦t thrust ⎢ −⎜ ⎟ ⎥ ⎢ ⎝ g0 ⋅ Isp ⎠ ⎥ ⎣ ⎦ t + Δt

MAE 6530, Propulsion Systems II 20 Ballistic Simulation Initial Conditions Final Initial → Conditions rail Conditions ballistic simulation saimulation ⎡ ⎤ V ⋅sinγ ⎢ rail rail ⎥ t = t ⎡ ⎤ 0 ballistic final rail Vr ⎢ ⎥ ⎢ ⎥ Vν ⋅cosγ rail ⎢ V ⎥ ⎢ ⎥ ⎢ ν ⎥ ⎢ ⎥ ⎢ ⎥ x0 = lrail ⋅ cosγ rail ≈ 0 ⎢ ⎥ R + h +l ⋅sinγ ballistic ⎢ earth launch rail rail ⎥ ⎢ r ⎥ = ⎢ ⎥ ⎢ ⎥ ⎛ l cos ⎞ ⎢ ⎥ ⎢ rail ⋅ γ rail ⎥ ν ⎢ ⎜ ⎟ ⎥ ⎢ ⎥ ⎝ Rearth + hlaunch +lrail ⋅sinγ rail ⎠ ⎢ ⎥ ⎢ ⎥ ⎢ m ⎥ ⎢ ⎥ ⎣ ⎦initial ⎢ ⎥ conditions mrail ⎣⎢ ⎦⎥ Final conditions

MAE 6530, Propulsion Systems II 21 Ballistic Simulation Block Diagram

⎡Vr ⎤ ⎢V ⎥ Initial ⎢ υ ⎥ ⎢ r ⎥ t do 0ballistic Parameters and Conditions ⎢ ⎥ → h ≤ hlaunch ⎢ ν ⎥ while Constants ⎢ m ⎥ ⎢ ⎥ x ⎣ ⎦0 C h l D0 launch rail

⎡ r − Rearth ⎤ h ⎢ ⎥ A γ C ⎡ ⎤ ⎛ m > (m0 − mprop ) → Fnom ⎞ ref launch Frail ⎢ ⎥ ⎢ ⎥ Fthrust ⎢⎜ ⎟ ⎥ ⎢ ⎥ ⎝ m ≤ (m0 − mprop ) → 0 ⎠ ⎢ ⎥ ⎢ ⎥ Fthrust M initial M prop ⎢ µ ⎥ ⎢ ⎥ − 1976 Standard ⎢ ⎥ = ⎢ r2 ⎥ g(r) ⎢ ⎥ h → → ρ(h) I g µ ⎢ ⎥ m sp 0 ⎢ ⎥ ⎢ ⎥ Atmosphere ⎢ C ⋅ A ⎥ ⎢ β ⎥ D ref ⎢ ⎥ ⎢ ⎥ R ⎢ ⎥ earth ⎢ γ ⎥ −1 ⎡Vr ⎤ ⎣ ⎦ ⎢ tan ⎢ ⎥ ⎥ ⎣⎢ ⎣Vυ ⎦ ⎦⎥ ⎡Vr ⎤ ⎢V ⎥ ⎢ υ ⎥ ⎢ r ⎥ t + Δt ⎢ ⎥ ∫ ⎢ ν ⎥ ⎢ m ⎥ ⎢ ⎥ ⎣ x ⎦t MAE 6530, Propulsion Systems II 22 Questions??

MAE 6530, Propulsion Systems II 23