Aeronautical Engineering cheat sheet

Newton–Raphson root finding: x_{n+1} = x_n − f(x_n)/f′(x_n); convergence is quadratic near a simple root and fails when f′ ≈ 0 or the initial guess is remote from the solution.

Composite Simpson 1/3 rule: ∫ ≈ (h/3)[f₀ + 4(f₁ + f₃ + …) + 2(f₂ + f₄ + …) + f_n] with n even; truncation error is O(h⁴) and the rule is exact for polynomials up to cubic.

Laplace pairs to memorise: L{e^{at}} = 1/(s − a); L{f′(t)} = sF(s) − f(0); the final-value theorem lim sF(s) as s→0 holds only when every pole of sF(s) lies in the open left half-plane.

Second-order response: percent overshoot = 100·exp(−ζπ/√(1 − ζ²)); 2 percent settling time ≈ 4/(ζω_n); ζ ≥ 1 gives no overshoot, and rise time shortens as ω_n grows.

Plane-stress transformation: σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]; maximum in-plane shear equals the Mohr circle radius, (σ₁ − σ₂)/2.

Beam bending σ = My/I and shear τ = VQ/(It); bending stress peaks at the extreme fibre, transverse shear peaks at the neutral axis and vanishes at free surfaces.

Column buckling: P_cr = π²EI/(KL)², with effective-length factors K = 1.0 pinned-pinned, 0.7 fixed-pinned, 0.5 fixed-fixed and 2.0 cantilever; I is the least second moment of area.

Dimensional homogeneity is mandatory in any valid physical equation; the Buckingham π theorem reduces n variables described by m primary dimensions to n − m independent dimensionless groups.

Conversions used constantly in aircraft work: 1 ft = 0.3048 m, 1 kt = 0.514444 m/s, 1 lbf = 4.44822 N, 1 hp = 745.7 W, 1 slug = 14.5939 kg.

Lift from dynamic pressure: L = q S C_L with q = ½ρV²; for a thin airfoil in incompressible two-dimensional flow C_L = 2π(α − α_L0) per radian.

Parabolic drag polar: C_D = C_D0 + C_L²/(πAe) with aspect ratio A = b²/S; maximum L/D occurs at C_L = √(C_D0·πAe) and equals 1/(2√(C_D0/(πAe))).

Compressibility rules: M = V/a and a = √(γRT); subsonic lift scales as C_L = C_L0/√(1 − M²), and wing sweep raises the drag-rise Mach number roughly in proportion to 1/cos Λ.

Isentropic stagnation relations: T₀/T = 1 + (γ − 1)M²/2 and p₀/p = (T₀/T)^{γ/(γ−1)}; a convergent nozzle chokes at p_e/p₀ = (2/(γ + 1))^{γ/(γ−1)}, about 0.528 in air.

Jet thrust F = ṁ(V_j − V₀) + (p_e − p_a)A_e; thrust specific fuel consumption TSFC = ṁ_f/F, and propulsive efficiency improves as exhaust velocity approaches flight velocity.

Ideal Brayton efficiency = 1 − 1/π_c^{(γ−1)/γ}; raising compressor pressure ratio improves thermal efficiency, while raising bypass ratio improves propulsive efficiency and lowers jet noise.

Thin-walled closed-section torsion: shear flow q = T/(2A_m) and skin shear stress τ = q/t; open sections have far lower torsional stiffness and warp markedly under torque.

Fatigue assessment: S-N curves give allowable stress amplitude for a life; mean stress is handled by the Goodman line σ_a/S_e + σ_m/S_u = 1 and cumulative damage by Miner's rule Σ n_i/N_i = 1.

Longitudinal static stability: the aircraft is pitch-stable when the centre of gravity lies forward of the neutral point (positive static margin); changes in C_m with C_L are negative in the stable regime.

The 1944 Chicago Convention created ICAO; core Annexes include 1 Personnel Licensing, 2 Rules of the Air, 8 Airworthiness of Aircraft, 14 Aerodromes and 16 Environmental Protection.

US certification framework: 14 CFR Part 21 for procedures, Part 23 and Part 25 for aircraft airworthiness, Part 33 and Part 35 for engines and propellers, Part 91 for operating rules; EASA parallels these with Part-21 and CS-23/CS-25/CS-E.

Certificate chain: type certificate approves the design, production certificate approves manufacture for conformity, and each aircraft receives its own airworthiness certificate; design changes require approval before further flight.

Traffic rights: first freedom overflight, second technical stop, third and fourth set down or pick up traffic in the home state, fifth carry revenue traffic between two foreign states - all conferred only by bilateral air service agreements.

Export control: ITAR governs USML defense articles and associated technical data, the EAR governs dual-use items by ECCN classification; releasing controlled data to a foreign national inside the country is a deemed export.

Professional codes such as NSPE and AIAA place public safety, health and welfare first; engineers sign only work within their competence that they have personally reviewed, and disclose conflicts of interest and known hazards.

Negligence requires four elements: duty, breach, causation and damage, judged against what a reasonable competent engineer would do; some jurisdictions also impose strict product liability without proof of fault.

Just-culture reporting: confidential schemes such as ASRS grant immunity for inadvertent violations, but reckless, intentional or substance-related conduct remains subject to enforcement action.

Continuing airworthiness obligations: comply with airworthiness directives, maintain under approved organisations and records, and report unapproved parts or suspected falsified documentation to the competent authority.