Back to 6DOF Spaceshot Flight & Structural Simulator overview

Spaceshot Sizing Documentation

Data Collection

Data collection consists of:
  1. Generating rocket configurations
  2. Effect of altitude
  3. Configuration assumptions (Fin geometry, nosecone geometry, protuberances, skin roughness, etc)
  4. Finless data
Aerodynamic data is then packed into a binary file. Then, aerodynamic coefficients for any rocket at any altitude and mach number and AoA can be interpolated or extrapolated from this data.

Rocket configurations

RASAero II is used to generate the aerodynamic coefficients. A 10 inch diameter is set as a reference diameter, then these variables are swept:
  1. Total Rocket Length (3m to 15m, 8 increments)
  2. Nozzle Exit Diameter (1in to 8in, 8 increments)
  3. Fin Scaling Factor (ratio of 0.2 to 5.0, 5 increments)

Effect of altitude

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Due to a lower density in the upper parts of the atmosphere, reynolds number decreases drammatically at higher altitudes, increasing drag coefficnet from ~0.4 to ~3.0. So, Aerodynamic data is collected at these altiudes:
  • (0.0 to 300,000.0 ft, 11 increments)

Configuration Assumptions (Fin geometry, nosecone geometry, protuberances, skin roughness)

Here are the documented assumptions of the rocket that is kept constant throughout the data collection process. These were selected to be conservative but realistically achievable.

Fin Geometry

  1. 3 Hexagonal Fins
  2. Thickness: 0.5 in
  3. Leading edge radius: 0.05 in
  4. Leading edge taper: 2 in
  5. Trailing edge taper: 2 in
  6. Span: 0.8 * OD * Fin Scaling Factor
  7. Root Chord: 1.5 * OD * Fin Scaling Factor
  8. Tip Chord: 0.75 * OD * Fin Scaling Factor
  9. Sweep: OD * Fin Scaling Factor
  10. Starting from rocket base

Nosecone Geometry

The nosecone is selected to always be Von Karman Ogive with a length of 5 times the diameter (50 in for a 10 inch diameter rocket) with a perfectly pointy tip

Protuberances

A 3 in sq streamlined protuberance is assumed (can include streamlined bolts, rail guides, raceway)

Surface finish

The worst case surface finish (0.01 in equivalent sand roughness) was chosen for conservatism.

Finless data

Finless rockets were also simulated at each length isolate fin specific aerodynamic coefficients

Defining the rocket, materials and launch site

Rocket definitions and assumptions Inputs:
  1. Rocket diameter
  2. Thrust
  3. Chamber Pressure
  4. Tank Pressure
  5. Sea level exit pressure
  6. Sea level ISP
  7. OF ratio
  8. Nozzle exit area / diameter
  9. Burn time
  10. Fin span Airframe:
  11. Ox tank material
  12. Fuel tank material
  13. Skin material
  14. Airframe thickness
  15. Fin thickness (assumed 10mmm)
Masses:
  1. Engine mass (assumed 10kg)
  2. Recovery (Parachutes, deployment hardware, etc) (assumed 5kg)
  3. Avionics (PCBs, batteries, wires) (aasumed 3kg)
Optionals:
  1. Fin cant angle
  2. Thrust misalignment angle
Materials are defined:
  1. Density
  2. Yield Strength
  3. Young's Modulus
  4. Shear Modulus
Launch site is defined:
  1. Ground elevation above sea level
  2. Launch rail length
  3. Wind speed at sea level
  4. Gust speed (a near step change at max Q from 0 to gust speed)

Dry rocket mass and length estimations

Ox and fuel tanks:
  1. 8% Ullage
  2. End cap volumes are neglected (conservative, can change)
  3. Tank wall thicknesses from 1.5x hoop or a minimum thickness (whichever is greater)
Pressurant tank:
  1. Assumed 300 bar
  2. Volume calculated with polytropic n=1.2 (between isothermal and adiabatic)
  1. Assumed tank diameter: 0.9 * rocket diameter, just an estimate
  2. Assumed tank mass = (8kg for 12V for vega, scaled linearly)
Masss are separated into sections and point masses
  1. Nosecone 2. Length: 5x diameter 3. Mass: Assumed cylinder with airframe thickness
  2. Rec Av section (Just the structure)
    1. Length: 30cm
    2. Mass: Tube with airframe thickness
  3. Pressurant tank 6. Length: Calculated below 7. Mass: Cylinder with airframe thickness + Tank mass + Nitrogen mass
  4. Plumbing Sections (x3) 9. Length: 50cm 10. Mass: Cylinder with airframe thickness + 8kg
  5. Endcaps (x4)
  6. Length: Hemispherical (0.5 * diameter)
  7. Mass: Airframe tube + hemispherical endcap with tank thickness * 1.5
  8. Ox tank
    1. Length: Calculated above
    2. Mass: Cylinder with ox tank thickness
  9. Fuel tank
    1. Length: Calculated above
    2. Mass: Cylinder with fuel tank thickness
  10. Engine 1. Length: 40cm 2. Mass: 10kg estimated
Point masses:
  1. Recovery location: 80% from tip of nosecone
  2. Avionics: 75% from top of RecAv bay
  3. Fins: With bottom of root chord aligning with bottom of rocket

Mass, Pitching and Polar inertia

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Engine Performance Estimation

ISP and nozzle parameters are calculated from RocketCEA package for ISP, mdot, altitude varying thrust calculations. Film cooling is accounted for in the ISP estimation.
Here are the performance hits for conservatism:
  • 85% C* efficiency
  • Film cooling fraction: 15% of fuel flow
There are two O/F ratios defined:
  • OF is split between OF_core and OF.
  • OF_core is used to estimate ISP without film cooling
  • OF is used to estimate real ISP with film cooling
Where is the film cooling fraction from the fuel flow.

Flight simulation

States and initial conditions:
  1. World Position (x=0, y=0, z=0)
  2. World Velocity (Vx=0, Vy=0, Vz=0.001)
  3. Rocket Orientation (Quaternion) (qx=0, qy=-0.707, qz=0, qw=0.707) Body to world: (body x+ > world z+, body y+ > world y+, body z+ > world x-)
  4. Body Angular Velocity (p=0, q=0, r=0)
Main equations: Absolute to relative wind conversion: (Wind pointing negative x direction)
(Gust pointing negative y direction)
Gust speed profile:
Air velocity in body frame:
Air speed, angle of attack and crosswind vector:
With AoA, Mach and altitude, aerodynamic coefficients are interpolated from the database: , , , Center of Pressure location
Lateral wind:
Normalized y and z components of the crosswind vector:
slopes are interpolated/extrapolated from the three given AoA values (0, 2, 4). Finless data is used to isolate fin contributions to normal force slope. Slopes above 4 degrees are extrapolated linearly from the 2 to 4 degree slope.

Jet Damping Derivation:

Velocity vector of the exiting jet relative to the body:
Momentum flux from the exiting jet:
Angular momentum flux from the exiting jet:
As is aligned with , the term is zero, so:
From and and :
Assuming inertia is lost through a simplified radius of gyration:

Aero damping derivation:

Induced angle of attack from pitch rate:

Fin equation derivations

RasAero II provides CNalpha values for the entire rocket including fins. To isolate fin contributions, finless rockets are also simulated. The fin contribution to normal force slope is then:
The normal force can be expressed from the rocket perspective and the fin perspective:
Where and is the reference area of the rocket (pi * (D/2)^2). S_{fin}$ is the total planform area of all fins.
N is the number of fins and makes sense. The halving comes from the fact that only half the fins contribute to normal force in a given direction ad CNalpha given by Rasaero II is for the entire rocket in the normal direction. For example, with 3 fins, at a positive AoA, 2 fins produce positive normal force and 1 fin produces negative normal force, so the net effect is 1.5 fins worth of normal force (3 fins / 2).
Equating the two fin force equations:

Roll forcing

For roll forcing due to fin cant angle, as force per unit area is roughly constant, the mean force location is just the centroid of the fin planform area. (In literature they calculate this centroid, but let us just continue our derivation)
The force generated by a strip of a single fin at spanwise location r is:
For N fins, the total roll moment is:
And the integral is the first moment of area about the body axis.
That is already enough for the simulation, but for the sake of following literature to compare coefficients, we define forcing coefficient as:
Where is a reference length (rocket diameter is the conventional), and is the roll forcing coefficient per radian of fin cant angle.
And for a trapezoidal fin:
Where s is the fin span.

Roll damping

The force generated is similar to the fin forcing case, except now the angle of attack is induced from the roll rate:
And now instead of the first moment of area, we have the second moment of area about the body axis.
The nondimensionalizing roll damping coefficient is defined as:
The 2 is included to match literature conventions.

Adding all the forces and moments together

Aerodynamic forces:
Aerodynamic moments:
Jet Damping Moments:
Aero Damping Moments:
Fin rolling moments:

Body forces and moments to inertial accelerations and angular accelerations

Body forces
Body moments
Inertial forces
Angular accelerations from Euler's equations:
Quarternion kinematics: I'm not deriving this here, see references, but the final equation is:
Where and is the quaternion multiplication operator.
All derivatives of the states have now been found:

Structural simulation

  • Rocket is discretized to 100 elements from 0 to L_rocket
  • Mass of sections are known, point masses are assumed to be 1m long at their location.

Force distribution

RasAero II only gives total axial and normal forces. For this reason, three RasAero II aerodynamic tables are generated:
  1. Full rocket with fins
  2. Finless rocket
  3. Nosecone only
CN is separated:
Force of each section is assumed to be constant along its length.
  • Per unit length force distributions:
  • is then constructed as a piecewise function of the above 3 distributions. Where is the root chord length of the fin.

D'Alembert’s Principle

Because the aerodynamic coefficients are obtained seperately, the error in force distribution is minimized to match the first two equations.
  • Force distribution is assumed correct
  • Moment distribution is scaled to match total moment from flight simulation
Now, a(x) is updated with the new angular acceleration, and the inertial load distribution is recalculated.
Shear and bending moment diagrams from only normal distributed load

Axial loading

Axial force is assumed to come from thrust and drag. Drag is a point load at the tip of the rocket, and thrust is a point load at the base of the rocket. D'Alembert's principle is applied similarly to the normal loading case to get axial inertial loads. This is conservative as even fin drag and skin friction drag is assumed to be applied at the tip of the rocket.
is negative and has to be made positive, so it has to be subtracted.

Beam Column Analysis

The beam column equation is:

Imperfections

  • The rocket is assumed to have an initial imperfection of a half sine wave with an amplitude of L/500.
  • The thrust is also assumed to have an eccentricity of D/100 (radial offset from the centerline) which adds to the bending moment.
This changes the total moment to:

Rearranging for second order equation

With the moment known, the beam column equation can be rearranged to:

Finite Difference Method

Integrating using finite difference method with boundary conditions v(0)=0, M(0)=0, v(L)=0, M(L)=0 gives deflection and slope along the rocket.
Using the difference equation:
Substituting into the beam column equation and rearranging gives:
Which can be arranged in a tridiagonal matrix system to be solved for v.
This is then just a matter of solving the linear system:
The boundary conditions are applied by setting A[0,0]=1, A[n,n]=1 and b[0]=0, b[n]=0. This works as the matrix equation is and which are the boundary conditions.

AI Beam Column Buckling

Strain energy U:
Work done by axial load W:
Total potential energy:
The critical buckling load occurs the system is on the verge of instability, which is when the strain energy is equal to the work done by the axial load.
The safety factor against buckling is calculated as:

Amplified Stresses

With a known deflection profile, the bending stresses with true deflection lever arm can be calculated as:
Airframe stress can finally be calculated from

Shell Buckling

Using NASA SP-8007 (eq 8-10):
This is conservative as it doesn't take into account internal pressure which helps resist buckling.

Fin flutter

NASA TN-4197 gives a back of the envelope method for estimating fin flutter velocity.
Where:

Fin stress

Fins feel two main aerodynamic loads:
  1. Rolling force from fin cant angle and damping
  2. Normal force from rocket angle of attack
I'm going to place the conservative assumption that the point forces are applied at half the fin span.
Where is the planform area of a single fin.
The N/2 is explained in the fin forces derivation section.
Fin is conservatively modelled as a hexagon at its tip chord, with
Where
  • H is the fin thickness
  • b is the fin tip chord
  • a = b / 3