ashfordeOU Dynamic Stability (flight-mechanics/stability-control/dynamic-stability)
Use when the task is dynamic stability analysis: the longitudinal modes (short period, phugoid), the lateral-directional mode classification, and the damping and frequency criteria.
Domain quick reference
Documented convention (stability axes): x forward, y out the right wing, z down. Perturbation states are the angle of attack alpha and the pitch rate q. All stability derivatives below are per unit mass: M_alpha and M_q use the pitch inertia I_yy, Z_alpha uses the mass m. The pitch stiffness M_alpha, the pitch damping M_q, and the vertical force derivative Z_alpha are negative for a pitch-stable, damped configuration; all angles in radians.
- Short period: the reduced-order (alpha, q) model with the state matrix A = [[Z_alpha/V, 1], [M_alpha, M_q]] gives the natural frequency omega_ns = sqrt(det A) = sqrt(M_q * Z_alpha / V - M_alpha) and the damping ratio from tr A = Z_alpha / V + M_q = -2 * zeta_s * omega_ns. The short period is a fast, well damped pitch oscillation, roughly 1-4 s on transport aircraft.
- Phugoid: the slow pitch-speed oscillation. The Lanchester approximation gives omega_np = sqrt(2) * g / V, the period T_p = sqrt(2) * pi * V / g (roughly 30-60 s), and the damping ratio zeta_p = 1 / (sqrt(2) * (L/D)). The phugoid is lightly damped.
- Stability derivatives: Z_alpha = -(q_bar * S * C_Lalpha) / m, M_alpha = (q_bar * S * c_bar * C_malpha) / I_yy, and M_q = (q_bar * S * c_bar^2 * C_mq) / (2 * V * I_yy), with the dynamic pressure q_bar, the wing area S, the mean chord c_bar, and the lift and moment coefficient slopes C_Lalpha, C_malpha, C_mq.
- Lateral-directional modes: a complex eigenvalue pair (non-zero imaginary part) is an oscillatory mode: a damped Dutch roll when the real part is negative, a divergent oscillation when it is positive. Real negative eigenvalues are convergent non-oscillatory modes: the fast roll subsidence (roll mode, time constant tau = -1 / L_p) and the slow stable spiral. A real positive eigenvalue is a divergent spiral. For a complex pair lambda = re + im * j the natural frequency is omega_n = |lambda| and the damping ratio is zeta = -re / |lambda|.
- Metrics: a divergent real root doubles amplitude in T2 = ln(2) / lambda; a convergent real root halves amplitude in T_half = ln(2) / |lambda|.
- Criteria: FAR-25.181 requires short period oscillations to be heavily damped and phugoid oscillations not to grow in amplitude. Common level 1 handling-quality criteria (MIL-F-8785C style, summary only, not reproduced): short period damping ratio in [0.3, 2.0]; Dutch roll damping ratio at least 0.08 with zeta * omega_n at least 0.15; roll subsidence time constant at most 1.0 s; divergent spiral time to double at least 20 s; phugoid damping ratio positive.
Workflow
- Collect the dynamic pressure, wing area, mean chord, mass, pitch inertia, speed, and the coefficient slopes C_Lalpha, C_malpha, C_mq.
- Compute the derivatives with z_alpha, m_alpha, and m_q.
- Compute the short period frequency and damping ratio with short_period_frequency and short_period_damping; check the band with short_period_damping_adequate.
- Compute the phugoid frequency, period, and damping ratio with phugoid_frequency, phugoid_period, and phugoid_damping; check the verdict with phugoid_acceptable.
- Collect the lateral eigenvalues, classify each with classify_mode, and derive the damping ratio with damping_ratio, the time to double with time_to_double, or the time to half with time_to_half.
- Check the Dutch roll criterion with dutch_roll_adequate, the roll subsidence limit with roll_mode_acceptable, and the spiral with spiral_acceptable.
- Gate the dynamic stability assessment on the short period band, the phugoid verdict, and the three lateral criteria.
Pitfalls
- Confusing the sign of the pitch stiffness: M_alpha negative is pitch-stable; a positive M_alpha drives the short period radicand toward zero or negative, which the model rejects.
- Forgetting the per-unit-mass normalization: M_alpha and M_q need I_yy in the denominator, Z_alpha needs m.
- Mixing units: the short period and phugoid formulas assume V in m/s, g in m/s^2, and radian-based derivatives; feeding degrees misstates every result.
- Reading the spiral criterion backwards: a divergent spiral passes only when the time to double is at least 20 s; a stable spiral (negative root) always passes.
- Confusing time to double with time to half: time_to_double needs a positive divergent root, time_to_half needs a negative convergent root, and each rejects the wrong sign.
- Treating every complex pair as damped: the real part decides; a positive real part is a divergent oscillation regardless of the imaginary part.
Behavior contract (gate 3)
The dynamic stability logic is exercised by the gate 3 contract test: scripts/test_dynamic_stability.py against scripts/dynamic_stability_logic.py (stdlib unittest, offline). Run:
python3 scripts/test_dynamic_stability.py
Compliance
- Standards referenced, not reproduced: FAR-25.181 and CS-25.181 require heavily damped short period oscillations and non-growing phugoid oscillations for transport aeroplanes; the derivative and mode computations are common flight mechanics methodology, summary-only per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.