Short Period Mode Analysis

Short-Period Mode Analysis (flight-mechanics/stability-control/short-period-mode-analysis)

Use when the task is the short-period longitudinal mode: the fast, well-damped pitch oscillation that dominates the response to elevator input and turbulence on transport aircraft. This leaf takes the dimensionless stability derivatives, converts them to dimensional per-unit-mass derivatives, and derives the short-period natural frequency and damping ratio from the two-DOF (alpha, q) pitch model, then checks the mode against Level 1 flying qualities damping and frequency boundaries. It complements the static pitch stability treatment of flight-mechanics/stability-control/longitudinal-stability and the full-mode treatment of flight-mechanics/stability-control/dynamic-stability.

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. Dimensionless stability coefficients (per rad): C_Z_alpha, C_Z_q, C_m_alpha, C_m_q, C_m_alphadot. Conversion to dimensional per-unit-mass derivatives uses the dynamic pressure q_bar, the wing area S, the mean chord c_bar, the mass m, the pitch inertia I_yy, the speed V, and the aerodynamic timescale tau = c_bar / (2 * V):

• Z_alpha = q_bar * S * C_Z_alpha / m, in m/s^2 per rad.
• Z_q = q_bar * S * c_bar * C_Z_q / (2 * V * m), in m/s per rad; the state matrix keeps the 1 + Z_q / V term only when |Z_q| / V is not negligible (z_q_negligible flags the case).
• M_alpha = q_bar * S * c_bar * C_m_alpha / I_yy, in 1/s^2 per rad.
• M_q = q_bar * S * c_bar^2 * C_m_q / (2 * V * I_yy), in 1/s per rad.
• M_alphadot = q_bar * S * c_bar^2 * C_m_alphadot / (2 * V * I_yy), in 1/s per rad.

The two-DOF pitch model with Z_q neglected has the state matrix

A = [[Z_alpha / V, 1], [M_alpha + M_alphadot * Z_alpha / V, M_q + M_alphadot]]

with det(A) = M_q * Z_alpha / V - M_alpha (the M_alphadot terms cancel in the determinant) and tr(A) = Z_alpha / V + M_q + M_alphadot. Therefore:

• omega_nsp = sqrt(M_q * Z_alpha / V - M_alpha), in rad/s.
• zeta_sp = -(Z_alpha / V + M_q + M_alphadot) / (2 * omega_nsp), dimensionless.

For a pitch-stable, damped configuration M_alpha < 0, M_q < 0 and Z_alpha < 0, so the radicand is positive. A non-positive radicand is categorized as a non-oscillatory or divergent (unstable) mode, and short_period_analysis returns it with level 3. Zero total damping gives zeta_sp = 0 (undamped oscillation) and negative total damping gives zeta_sp < 0 (divergent oscillation); both fail Level 1.

Phugoid separation: the Lanchester phugoid frequency is omega_np = sqrt(2) * g / V. The short-period approximation assumes the phugoid is much slower; phugoid_separation flags the mode as separated when omega_nsp / omega_np >= 5 (default).

Level 1 flying qualities (MIL-F-8785C style summary, referenced not reproduced): damping ratio bands 0.35-1.30 (category A), 0.30-2.00 (category B), 0.25-2.00 (category C); minimum natural frequency 0.28 rad/s (A and C) and 0.10 rad/s (B). Level 1 requires both criteria, boundaries inclusive. A damped oscillation outside the Level 1 band is Level 2; undamped, divergent, or non-oscillatory modes are Level 3.

Relation to the longitudinal-stability leaf: that leaf establishes static pitch stability from C_m_alpha (the pitch stiffness slope and the neutral point); this leaf treats the same configuration dynamically, where M_alpha contributes the spring term of the short-period oscillation while M_q and M_alphadot provide the damping.

Workflow

1. Collect the dynamic pressure, wing area, mean chord, speed, mass, pitch inertia, and the coefficient slopes C_Z_alpha, C_Z_q, C_m_alpha, C_m_q, C_m_alphadot.
2. Convert the coefficients to dimensional derivatives with dimensionless_derivative_conversion; check the resulting Z_q with z_q_negligible to confirm the approximation holds.
3. Compute the natural frequency with short_period_frequency and the damping ratio with short_period_damping.
4. Verify the mode separation with phugoid_separation against the Lanchester phugoid frequency.
5. Get the complete verdict with short_period_analysis, which returns stable, oscillatory, omega_n, zeta, and the Level 1/2/3 verdict with reasons; pick the flight phase category (A, B, or C) for the band check.
6. Confirm the deterministic checks with the contract test scripts/test_short_period_mode_analysis.py.

Worked example

A transport configuration at cruise: q_bar = 14000 Pa, S = 30 m^2, c_bar = 2.5 m, V = 150 m/s, m = 12000 kg, I_yy = 85000 kg m^2, C_Z_alpha = -5.0, C_Z_q = -3.0, C_m_alpha = -0.6, C_m_q = -12.0, C_m_alphadot = -3.0.

• tau = 2.5 / 300 = 0.00833 s.
• Z_alpha = 14000 * 30 * (-5) / 12000 = -175 m/s^2 per rad, so Z_alpha / V = -1.167 /s.
• M_alpha = 14000 * 30 * 2.5 * (-0.6) / 85000 = -7.41 /s^2.
• M_q = 14000 * 30 * 6.25 * (-12) / (2 * 150 * 85000) = -1.235 /s.
• M_alphadot = 14000 * 30 * 6.25 * (-3) / (2 * 150 * 85000) = -0.309 /s.
• omega_nsp = sqrt((-1.235)(-175)/150 - (-7.41)) = sqrt(8.853) = 2.98 rad/s (period about 2.1 s).
• zeta_sp = -(-1.167 - 1.235 - 0.309) / (2 * 2.98) = 0.456.
• Level 1 check (category A): zeta 0.456 is within [0.35, 1.30] and omega_n 2.98 rad/s is above 0.28 rad/s, so the mode is Level 1.
• Phugoid separation: omega_np = sqrt(2) * 9.80665 / 150 = 0.0925 rad/s; the ratio omega_nsp / omega_np = 32.2, well above the minimum of 5, so the approximation is valid.
• Z_q negligibility: |Z_q| / V = 0.875 / 150 = 0.0058, below 0.05, so neglecting Z_q in the state matrix is justified.

Pitfalls

• Taking the square root of a non-positive radicand: omega_nsp = sqrt(M_q*Z_alpha/V - M_alpha) is real only for a pitch-stable damped configuration; a non-positive radicand means a non-oscillatory or divergent mode and short_period_analysis reports level 3, so do not force a frequency from it.
• Neglecting Z_q without the check: the state matrix keeps the 1 + Z_q/V term only when |Z_q|/V is not negligible (0.05 rule); z_q_negligible flags the case before you drop the term.
• Reading the Level 1 band as open: the boundaries are inclusive (0.35-1.30 category A damping, 0.28 rad/s minimum frequency) and Level 1 requires both damping and frequency; zero total damping (zeta = 0) and negative damping are Level 3, not Level 2.
• Checking the wrong category band: the damping bands differ by flight phase category (A 0.35-1.30, B 0.30-2.00, C 0.25-2.00) with frequency floors 0.28 (A, C) and 0.10 (B) rad/s.
• Forgetting phugoid separation: the approximation assumes omega_nsp/omega_np >= 5 against the Lanchester frequency sqrt(2)*g/V; phugoid_separation flags when the mode separation assumption does not hold.

Related leaves

• flight-mechanics/stability-control/longitudinal-stability: static pitch stability, C_m_alpha sign, and neutral point; the static counterpart of this leaf.
• flight-mechanics/stability-control/dynamic-stability: the full longitudinal and lateral-directional mode set including the short-period band criterion at the aircraft level.
• flight-mechanics/stability-control/trim-analysis: the trimmed flight condition (q_bar, alpha) that fixes the operating point for the derivative evaluation.
• flight-mechanics/stability-control/stability-derivatives-avl: estimating the dimensionless coefficient slopes this leaf consumes.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_short_period_mode_analysis.py

The test covers the derivative conversion, the natural frequency and damping anchors, the M_alphadot effect on damping, zero and negative damping, unstable non-oscillatory modes, the Level 1 boundaries at the band edges and frequency floors, the category-dependent bands, phugoid separation, the Z_q negligibility check, and invalid-input edge cases.

Compliance

• Standards referenced, not reproduced: FAR-25.181 and CS-25.181 require heavily damped short period oscillations for transport aeroplanes; the damping and frequency bands are a MIL-F-8785C style summary only, per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.