Speed Stability

Static Speed Stability (flight-mechanics/performance/speed-stability)

Use when the task is the static speed stability assessment of a fixed-wing aircraft from its trim drag balance. Below the minimum drag speed the aircraft flies on the back side of the thrust required curve, the region of reversed command: drag rises as the airplane slows, so a small speed disturbance from the trimmed level flight point produces no restoring thrust imbalance and the trim point is speed unstable. This leaf computes the thrust required curve versus true airspeed from the drag polar, derives its analytic slope dT/dV, classifies each candidate trim speed, locates the minimum drag speed boundary, and reports the slow flight stability margin. It pairs with performance/thrust-required, which computes the same drag polar family, and it feeds the excess thrust users climb-performance and turn-performance. This is the performance trim concept of speed stability, not a control surface or pitch handling check.

Domain quick reference

• Dynamic pressure: q = 0.5 * rho * v^2, with rho the air density and v the true airspeed.
• Induced drag factor: k = 1 / (pi * oswald_e * aspect_ratio), from the Oswald span efficiency and the aspect ratio.
• Thrust required for level unaccelerated flight: T = cd0 * q * S + k * W^2 / (q * S), the sum of the parasite drag and the induced drag, with S the reference wing area and W the aircraft weight.
• Thrust curve slope, exact derivative of T with respect to v: dT/dv = cd0 * rho * v * S - 4 * k * W^2 / (rho * v^3 * S). The induced term carries coefficient 4 so that the slope zero lands exactly on the closed-form minimum drag speed (wave-27 ops approved this exact derivative over the draft's factor-2 term; the test file records the adaptation).
• Minimum drag speed: v_md = (2 * W / (rho * S))^0.5 * (k / cd0)^0.25. At v_md the parasite drag equals the induced drag and dT/dv is zero.
• Speed stability verdict: stable when dT/dv > 0 (front side, above the back side), unstable when dT/dv < 0 (back side, region of reversed command), neutral when |dT/dv| < 1e-9 N/(m/s).
• Slow flight stability margin: margin_ms = v - v_md. A negative margin with unstable_below True means the candidate speed sits on the region of reversed command.
• Units are SI throughout: N, m/s, kg/m^3, m^2.

Workflow

1. Gather the aircraft state: weight_n, wing_area_m2, the air density rho_kg_m3 (default 1.225 at sea level), the drag polar terms cd0, oswald_e and aspect_ratio, and the candidate trim speeds trim_speeds_ms.
2. Compute the induced drag factor with the module constant relation k = 1 / (pi * oswald_e * aspect_ratio).
3. Get the minimum drag speed boundary with min_drag_speed; it splits the speed range into the stable front side and the unstable back side.
4. Build the thrust required curve with thrust_required over the swept range from 0.5 * v_md to 1.5 * v_md (25 points via analyze).
5. Derive the slope dT/dV at each candidate speed with d_thrust_dv and classify the trim point with speed_stability_verdict: stable, unstable, or neutral.
6. Compute the slow flight stability margin with margin_to_back_side for each candidate speed (v_md, unstable_below, margin_ms).
7. Run analyze for the full assessment dict: v_md_ms, trim_classifications with {speed, dT_dv, verdict}, margins with {speed, margin_ms, unstable_below}, and the curve points.
8. Confirm the deterministic checks with the contract test scripts/test_speed_stability.py.

Worked example

Transport at sea level: W = 600000 N, S = 120 m2, rho = 1.225 kg/m3, cd0 = 0.02, e = 0.8, AR = 9.

• Induced drag factor: k = 1 / (pi * 0.8 * 9) = 0.0442097.
• Minimum drag speed: v_md = (2 * 600000 / (1.225 * 120))^0.5 * (0.0442097 / 0.02)^0.25 = 110.17 m/s (module value 110.1676 m/s, within 0.01 of the worked anchor).
• Slope at 80 m/s (below v_md): dT/dv = -610.65 N/(m/s) < 0, verdict unstable, the trim point sits on the back side.
• Slope at 130 m/s (above v_md): dT/dv = +185.08 N/(m/s) > 0, verdict stable.
• Slope at v_md: zero to machine precision (5.7e-14), verdict neutral.
• Slow flight stability margin at 80 m/s: margin_ms = 80 - 110.17 = -30.17 m/s, unstable_below True. The 80 m/s approach trim speed sits below the minimum drag speed on the region of reversed command.
• Thrust at v_md: T = 2 * cd0 * q_md * S = 35682.5 N, with the parasite drag equal to the induced drag at the minimum drag point.
• Curve: 25 points from 55.08 m/s (0.5 * v_md) to 165.25 m/s (1.5 * v_md), minimum thrust at the v_md sample, monotonic thrust on the stable branch.

Verification

• Confirm v_md = 110.1676 m/s: it matches the closed form to 1e-9 and sits within 0.01 of the 110.17 m/s worked value. The exact radicals are sqrt(2W/(rho S)) = 90.351 and (k/cd0)^0.25 = 1.21933.
• Confirm the derivative zero lands on v_md: |dT/dv(v_md)| < 1e-6, and the analytic slope matches a central finite difference at 80 and 130 m/s within 0.1 percent.
• Confirm the verdicts: unstable at 80 m/s, stable at 130 m/s, neutral at v_md, unstable below v_md and stable above it by 0.1 percent.
• Confirm the margin at 80 m/s: -30.17 m/s with unstable_below True.
• Confirm thrust at v_md equals 2 * cd0 * q_md * S (parasite equals induced), and that thrust rises as the airplane slows on the back side.
• Confirm every non-positive weight, wing area, density, cd0, induced factor or trim speed, every Oswald efficiency outside (0, 1], and an empty trim list raises ValueError.
• Run the contract test offline: python3 scripts/test_speed_stability.py (38 tests, deterministic).

Pitfalls

• Judging stability from the thrust curve shape instead of the slope: the verdict is dT/dv at the trim speed (positive = stable front side, negative = back side, |dT/dv| < 1e-9 = neutral), and the analytic derivative carries coefficient 4 on the induced term so that its zero lands exactly on v_md.
• Calling the minimum-drag speed 'neutral' in the wrong sense: at v_md the slope is zero to machine precision and the verdict is neutral; below v_md the trim is speed unstable (region of reversed command), not merely 'low performance'.
• Quoting a slow-flight margin without its sign: margin_ms = v - v_md is negative below the boundary with unstable_below True (about -30.2 m/s at 80 m/s for the worked transport); a negative margin on the back side is the warning, not a number to minimize.
• Evaluating far outside the swept band: the curve is swept over 0.5v_md to 1.5v_md (25 points) and the drag polar is parabolic about that trim family.
• Non-positive weight, wing area, density, cd0, induced factor or trim speed, Oswald efficiency outside (0, 1], and an empty trim list raise ValueError.

Related leaves

• flight-mechanics/performance/thrust-required: the same drag polar family, thrust analysis for level flight, the sibling this leaf follows on.
• flight-mechanics/performance/climb-performance and flight-mechanics/performance/turn-performance: the excess thrust users of this trim analysis.
• aerodynamics/drag-polars/drag-polar and aerodynamics/drag-polars/parasite-drag: the drag polar coefficient source for the trim drag balance.
• flight-mechanics/stability-control/longitudinal-stability: the distinct pitch stability concept about the aerodynamic center, not the performance speed stability treated here.

Behavior contract (gate 3)

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

python3 scripts/test_speed_stability.py

The test covers the induced factor, the closed-form v_md and its 110.17 m/s worked anchor, the derivative zero on v_md, the analytic slope against central finite differences at 80 and 130 m/s, verdicts (stable, unstable, neutral) with boundary checks, margins at 80 m/s (-30.17 m/s) and 130 m/s, thrust curve checks at v_md (parasite equals induced, thrust minimum and monotonic stable branch, 25 swept points), the analyze output structure for the default trim list, and ValueError rejection of non-physical inputs.

Compliance

• Standards referenced, not reproduced: the FAR 25 and CS 25 speed-stability context frames the assessment (reference-only per standards-map.yaml); the drag polar relations above are standard engineering methodology, summary-only.
• compliance: STANDARDS-REF, gated: false.