Boeing 737 (Nonlinear 6-DoF Model)

tensoraerospace.aerospacemodel.b737.nonlinear — full nonlinear 6-DoF model of the Boeing 737 family, with two configurations covering both the original 737-100/200 (twin JT8D-9) and the 737-NG / 737-800 (twin CFM56-7B27).

Provenance

The 737 — unlike the 747, F-16 and X-15 — does not have a single canonical NASA technical paper publishing its full set of non-dimensional stability derivatives. The numerical values used here are consolidated from the open sources widely accepted in academic flight-mechanics work:

Source	Role
JSBSim 737 model (repo)	Primary numerical source — geometry, masses, inertias, full coefficient set
Roskam J. Airplane Flight Dynamics and Automatic Flight Controls (1995), Vol VI Appendix B	Original 737-100 derivatives that JSBSim transcribes
Hanke C. R. The Simulation of a Large Jet Transport Aircraft, NASA CR-114494 (1971)	Wind-tunnel methodology behind Roskam's tables
Cook M. V. Flight Dynamics Principles (3rd ed., Elsevier, 2013), Ch. 11	Cross-reference 737-100 example aircraft
NASA TM-86821 "Mach/CAS control law for the NASA TCV B737" (1986)	Validation reference for nonlinear-simulation envelope
FAA TCDS A16WE	Boeing 737 family weights, geometry, certified engine ratings
CFM International CFM56-3/-7B fact sheets	Engine performance

JSBSim is BSD-licensed and built explicitly on publicly available information for educational use; we transcribe its coefficient functional forms verbatim.

Parameter	Value (737-100)
Aerodynamic source	JSBSim 737-100 / Roskam Vol VI
Configurations	B737_100, B737_800
Engines	2 × Pratt & Whitney JT8D-9 (737-100) or 2 × CFM56-7B27 (737-800)
Coordinates	NED, body axis, ZYX 321 Euler
Control surfaces	elevator, aileron, rudder + throttle
Damage subsystem	Hooks open (parity with B-747); not yet wired up

Geometry & mass

                      737-100         737-800
S  (wing area)        1171 ft²        1341 ft²
b  (span)             94.7 ft         117.5 ft
c̄  (MAC)              12.31 ft        12.97 ft
W  (mid-cruise)       100 000 lb      140 000 lb
T_SLS (cluster)       29 000 lbf      54 600 lbf

Configuration	W, lb	Iₓ, slug·ft²	I_y	I_z	I_xz
B737_100	100 000	562 × 10³	1.473 × 10⁶	1.894 × 10⁶	8.0 × 10³
B737_800	140 000	820 × 10³	2.300 × 10⁶	3.000 × 10⁶	12.0 × 10³

State and control

State (12-D, body axis, NED, ZYX 321 Euler):

[u, v, w,           # body velocity, ft/s
 p, q, r,           # body angular rates, rad/s
 φ, θ, ψ,           # Euler angles, rad
 x_e, y_e, z_e]     # NED position, ft

Control (4-D):

[δ_e,  δ_a,  δ_r,  δ_T]
   ↓     ↓     ↓     ↓
elevator ail   rud   throttle
 (rad)  (rad) (rad)   [0, 1]

Limits (JSBSim 737.xml): \(|\delta_e| \le 17.2°\), \(|\delta_a|, |\delta_r| \le 20.1°\), all rate-limited at \(40\,°/s\).

Equations of motion

Standard Newton-Euler in body axis — identical to the B-747 nonlinear. The two differences from the 747 are:

1. Lighter airframe with smaller inertias. Short-period and Dutch-roll periods are shorter (~ 1–2 s vs the 747's 3–4 s).
2. Twin-engine configuration. Asymmetric-thrust events use the B-737 engine spanwise positions \(y_1 = -16.5\) ft, \(y_2 = +16.5\) ft (737-100 inboard pylons), tighter than the 747's outer engines at \(\pm 71.7\) ft.

Aerodynamic build (JSBSim functional forms)

Coefficients are computed at every ODE evaluation using the dimensional-decomposition approach from JSBSim:

Coefficient	Form
\(C_L\)	α-table (peak \(C_L \approx 1.45\) at \(\alpha = 13°\)) + linear \(C_{L_{\delta_e}} = 0.20\)
\(C_D\)	α-table + induced \(0.043 C_L^2\) + Mach compressibility table + sideslip + elevator drag
\(C_m\)	\(C_{m_\alpha} = -0.6\)/rad, Mach-dependent \(C_{m_{\delta_e}}\) (-1.20 to -0.30), pitch + α̇ damping
\(C_Y, C_l, C_n\)	Linear in (β, p̂, r̂, δa, δr); Mach-dependent \(C_{l_{\delta_a}}\)

The α-table gracefully degrades past stall (clamped to \(C_L = 0.6\) at \(\alpha = 35°\)) so the integrator does not blow up if the controller temporarily over-rotates. Ground / configuration effects (flap, gear, speedbrake) are exposed in the source but not applied in this MVP — the model is valid for the clean cruise envelope.

Engine model

Two-engine cluster with Mach-altitude-derated installed thrust following Mattingly §8.6.4 (same form as the JT9D model in the B-747 module):

\[ T_{inst}(M, h, \delta_T) = T_{SLS} \cdot \sigma(h)^{n_h} \cdot \eta_{ram}(M) \cdot \mathrm{PLA}_{eff} \]

with \(\eta_{ram}(M) = 1 - 0.49\sqrt{M}\) (clamped to 0.05) and \(n_h = 0.7\) below tropopause / \(1.0\) above. The same correlation serves both JT8D-9 (737-100) and CFM56-7B (737-800) — cross-checked against FAA TCDS A16WE certified ratings.

Configuration	\(T_{SLS}\), lbf	per-engine	Engine type
737-100	29 000	14 500	P&W JT8D-9
737-800	54 600	27 300	CFM56-7B27

Trim finder

tensoraerospace.aerospacemodel.b737.nonlinear.trim(h, V) solves \(\dot u = \dot w = \dot q = 0\) via Newton-Raphson, returning trimmed \((\alpha, \delta_e, \delta_T)\). Unlike the X-15, the 737 is a proper transport with air-breathing engines that scale with Mach and altitude — cruise trim converges across the entire normal flight envelope:

Configuration	h, ft	M	V, ft/s	α	δ_e	δ_T
B737-100	25 000	0.74	738	1.0°	-0.6°	0.92
B737-800	35 000	0.83	820	2.4°	-1.6°	0.91

Residual norms reach \(10^{-13}\) — i.e. machine-precision trim.

Gymnasium env

Registered as "NonlinearB737-v0". Two initialisation modes:

import gymnasium as gym
import tensoraerospace  # registers the env

# 1. Trim-finder at any (h, V)
env = gym.make("NonlinearB737-v0",
    trim_at=(25_000.0, 738.0), number_time_steps=2000)

# 2. Arbitrary initial state
import numpy as np
env = gym.make("NonlinearB737-v0",
    initial_state=np.array([738, 0, 13, 0,0,0, 0, 0.017, 0,
                            0, 0, -25_000]),
    number_time_steps=2000)

For the 737-NG / 737-800, pass config=B737Configuration.B737_800 through the constructor (or directly when you instantiate NonlinearB737Env(...)).

Action-space: either "virtual" (physical units) or "normalized" (for RL: [-1, 1]^4).

Scope and limitations

• High-lift devices not modelled — flaps, slats, ground effect, gear, speedbrake aerodynamic increments are exposed in the source but not applied. The model is valid for clean cruise; for approach / landing scenarios you'd need to enable these.
• Damage subsystem hooks open but no events — parity with the B-747 architecture (engines_mu, flap_jam_config, etc.) is prepared, but no concrete events are wired up. Adding them is straightforward since the engine model already accepts an engines_mu dict.
• 737-NG aerodynamics use 737-100 derivatives scaled by geometry — Roskam Vol VI does not separately publish 737-800 derivatives, so the dimensional CL/CD/Cm functions are evaluated at the new reference area / span / chord. Acceptable for control-design work; for performance studies a re-derivation is recommended.

Related modules

• Boeing 747-100 (Nonlinear 6-DoF) — heavier 4-engine air-breather with the canonical NASA CR-2144 derivative bank. Same code patterns.
• F-16 (Nonlinear longitudinal) — fighter, mid-Mach envelope.
• X-15 (Nonlinear hypersonic) — research rocketplane, hypersonic envelope. Same code patterns, different engine model.

References

• JSBSim — Berndt J. S. "JSBSim: An Open Source Flight Dynamics Model in C++", AIAA Modeling and Simulation Technologies Conference, 2004 (repo).
• Roskam J. Airplane Flight Dynamics and Automatic Flight Controls, Roskam Aviation, 1995. Vol VI Appendix B.
• Hanke C. R. The Simulation of a Large Jet Transport Aircraft, NASA CR-114494, 1971.
• Cook M. V. Flight Dynamics Principles, Elsevier 3rd ed., 2013, Chapter 11.
• NASA TM-86821 — Bahm C. M., Sivolell P. "Design and verification by nonlinear simulation of a Mach/CAS control law for the NASA TCV B737 aircraft", 1986 (NTRS 19870010857).
• FAA TCDS A16WE — Boeing 737 type certificate data sheet.
• Mattingly J. D. Aircraft Engine Design, AIAA Education Series, 2nd ed., 2002, §8.6.4 (installed-thrust lapse model).