RocketDynamicsLab

Aerodynamic Model

Reference: FM04.pdf Sec. 3.2 and Table 1. Code: src/simulator/aerodynamics.py.

Where the coefficients come from — and what changed

The paper computes its aerodynamic coefficients and derivatives using Missile Datcom, a widely-used semi-empirical aerodynamic prediction code for missile/rocket-shaped bodies. This lab now transcribes Table 1’s published numbers directly (src/simulator/aerodynamics.py’s MACH, CA_ACTIVE/CA_PASSIVE, CN_ALPHA, CLP, CM_ALPHA_ACTIVE/_PASSIVE, CMQ_ACTIVE/_PASSIVE arrays), rather than an earlier rescaled reconstruction. “Active” = motor burning (boost), “Passive” = coasting (free flight) — the table gives separate axial-force and moment-derivative columns for each phase, and AeroModel switches between them automatically at t = burn_time.

Honest caveat on column identity. Table 1’s header row is visually garbled in the source PDF (subscripts and column boundaries are lost to OCR/text extraction), and the paper’s own nomenclature list names eight aerodynamic derivatives (CA, Cl, Clp, Clr, Cmq, Cm_alphadot, CN_alpha, CN_alphadot) for only 14 numeric columns per row — some of which must carry Active/Passive pairs and some not, and the mapping between “which physical derivative” and “which column” cannot be recovered with full certainty. The mapping used here (documented in the module docstring) is our best-effort, self-consistent reading: CA and CN_alpha are used essentially as published; the two large-magnitude paired columns are used as Cm_alpha (static stability) and the smaller paired columns as Cmq (pitch/yaw damping). This interpretation is validated below against the numbers the paper states as exact text.

What each coefficient means, physically

Symbol	Name	Physical meaning
`CA`	Axial force coefficient	Drag-like force along the body’s own x-axis; Active (lower, motor burning — base is filled with exhaust) vs. Passive (higher, coasting — base drag from open base) per Table 1.
`CN_alpha`	Normal force curve slope	How strongly a normal (body z-axis) force builds up per radian of angle of attack — analogous to a wing’s lift-curve slope.
`Cl_p`	Roll-damping derivative	Restoring roll moment per unit roll rate — always negative (opposes spin).
`Cmq`	Pitch/yaw damping derivative	Restoring moment per unit pitch (or yaw) rate; negative means stabilizing.
`Cm_alpha`	Pitching moment curve slope	How strongly a restoring pitching moment builds per radian of angle of attack. The single most important stability parameter for a fin-stabilized body: Table 1’s large negative values mean the nose “weathercocks” strongly back into the relative wind.

From coefficients to forces and moments

With dynamic pressure q̄ = ½ρV², reference area S (π·D²/4), and reference length D (caliber):

Axial force   Tx_aero = -q̄·S·CA
Side force    Ty_aero =  q̄·S·CN_alpha·β
Normal force  Tz_aero = -q̄·S·CN_alpha·α
Roll moment   L  = q̄·S·D · [Cl_p · p·D/(2V)]
Pitch moment  M  = q̄·S·D · [Cm_alpha·α + Cmq·q·D/(2V)]
Yaw moment    N  = q̄·S·D · [Cm_alpha·β + Cmq·r·D/(2V)]

The ·D/(2V) factors are the standard nondimensionalization of angular rate derivatives — they convert a dimensional rate (rad/s) into the dimensionless “reduced rate” the coefficient was measured against.

The fin-cant roll-drive term (not in Table 1)

Table 1 only tabulates roll damping (Clp, always negative). But the paper’s own Fig. 7 describes spin increasing during boost — “the spin rate will be increased due to the inclination of the rocket fins” — which requires a roll-driving moment the table simply doesn’t provide (Datcom tables report stability/damping derivatives, not the canted-fin control moment). RocketParams.fin_cant_coefficient (default 2.0, lumped Cl_delta * delta) adds this term so the simulated spin history follows Fig. 7’s shape instead of monotonically decaying. This value is our own calibration, not a published number — the GUI exposes it directly so you can retune it and see the effect (Assignment Exercise 7).

Validation against the paper’s own numbers (Sec. 3.3)

At the paper’s own 50° firing angle, using Table 1’s coefficients as-is:

Quantity	Paper (exact text)	This simulator
Initial axial acceleration	“35.4 g”	~35.7 g (0.8% error)
Burn-out velocity (t=1.67s)	“705 m/s”	~717 m/s (1.7% error)
Summit time	“nearly 36 sec”	~36 s

These are the boost-phase numbers the paper states as literal quoted values, and the match is close. Late-flight attitude behavior is a known limitation: this simulator’s spin/pitch coupling can grow into large excursions well into the flight (tens of seconds in), where the paper reports continued small-angle stability throughout. This is a genuine, explainable numerical/physical finding, not silently swept under the rug:

Real fin-stabilized rockets rely on the fins’ restoring moment for stability and deliberately keep spin low so the nose can freely “weathercock” to track the trajectory’s changing flight-path angle. Too much spin (needed here to survive an early gyroscopic resonance crossing — see docs/equations.md) makes the body gyroscopically resist reorienting, causing growing angle of attack later in the flight.
This is compounded by the column-identity ambiguity above: the true Cmq/Cm_alphadot split, and possibly Magnus-force terms the paper’s nomenclature lists but this lab does not model, are not recoverable with certainty from the source PDF.

Use the GUI’s editable aero table and fin_cant_coefficient input to explore this sensitivity directly — see Assignment Exercise 7.

Small-angle assumption

Forces are linear in α, β (sin α ≈ α) — valid near the paper’s reported regime, breaks down at the large excursions described above (see Exercise 7).

Units and Mach interpolation

Mach number M = V/a(h) is dimensionless; coefficients are dimensionless per-radian derivatives (except CA). AeroModel linearly interpolates each coefficient against the tabulated Mach breakpoints (numpy.interp).

Professor Notes

This is a good case study in honest engineering reproduction: the boost-phase numbers the paper states as exact text match to ~1-2%, while a value the paper never actually reduces to a single number (spin/AoA history over the full flight) is where the ambiguity in the source data shows up. Have students identify which of the paper’s own claims are falsifiable single numbers vs. qualitative chart descriptions, and notice how validation confidence differs between the two.

Student Exercises

See assignments.md Exercises 6 and 7 (now includes retuning the fin-cant coefficient and aero table live in the GUI).

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