Vmc Determination

Minimum Control Speed Vmc (flight-test-operations/envelope/vmc-determination)

Use when the task is the Vmc prediction and demonstration check for a multi-engine airplane in the spirit of the FAR/CS 25.149 minimum-control-speed method, summary-only. This leaf identifies the critical engine from the engine-out yawing-moment geometry, builds the asymmetric yawing moment (operating-engine thrust at the failed-engine lateral arm plus the failed-engine windmilling drag contribution), computes the rudder deflection required to balance it at each airspeed, solves the authority-limited speed where the required deflection reaches the rudder limit, evaluates the rudder pedal-force criterion at that speed with the boost factor, applies the stall-protection guard on the reference stall speed, and returns the predicted Vmc with the governing-limit label and the flight-test go verdict. Model simplifications: first-order yaw balance without sideslip coupling, sea-level density for the dynamic pressure, operating engines at maximum takeoff thrust. It pairs with flight-test-operations v-speeds (Vmc sits beside the V-speed set), flight-mechanics oei-climb-gradient (the engine-out performance sibling), and vehicle-design control-surface-sizing (the rudder sizing sibling).

Domain quick reference

• Critical engine: the engine whose failure gives the largest adverse yawing moment, ranked by the |thrust * y_m| product of its lateral arm about the CG; ties resolve to the lower index.
• Asymmetric yawing moment: N_asym = T_op * y_fail + 0.5 * RHO_SL * V^2 * S_f * Cd * y_fail, where T_op is the sum of the operating (non-failed) engine thrusts and y_fail the signed lateral arm of the failed engine.
• Rudder deflection required: delta = |N_asym| / (q * S_v * l_v * C_Lv_delta_r), with q = 0.5 * RHO_SL * V^2, S_v the fin area, l_v the CG-to-fin arm and C_Lv_delta_r the effective fin lift slope per rudder deflection (includes the rudder-to-fin area ratio).
• Rudder-authority limit: the speed where delta reaches delta_r_max. Closed form q* = T_op * |y_fail| / (S_v * l_v * C_Lv_delta_r * delta_r_max_rad - S_f * Cd * |y_fail|); V_auth = sqrt(2 * q* / RHO_SL). If the denominator is not positive the configuration is never authority limited (V_auth = None).
• Rudder pedal force: F = q * S_r * c_r * |C_h_delta_r| * min(delta_req, delta_r_max_rad) * boost_factor / pedal_arm_m, with C_h_delta_r the hinge moment coefficient per radian (magnitude used, negative for a restoring hinge), S_r and c_r the rudder area and chord, and the boost factor scaling felt force (1.0 manual, smaller for power-boosted systems).
• Pedal-force limit: F_lim = 667 N (150 lbf criterion of the test condition, paraphrased). The force-limited speed V_force follows from the quadratic F(V) at the deflection limit, V_force = sqrt(2 * q_F / RHO_SL).
• Vmc: vmc = max(V_auth, V_force); the governing limit label is "rudder-authority" when V_auth >= V_force else "pedal-force".
• Stall guard: stall_guard_speed = stall_guard * v_s1g (default guard factor 1.05). When vmc falls below it the flight test must demonstrate above the guard; vmc is reported unchanged with the guard_verdict "stall-guard-governs".
• Units are SI: N, m, m/s, rad, deg, N m; knots = m/s * 1.94384.
• FAR/CS 25.149 frames the minimum-control-speed test condition; the relations above are standard engineering methodology, summary-only.

Workflow

1. Collect the engine layout: engines as a list of {thrust_N, y_m} dicts (y_m positive on the right), the failed engine index, and the failed-engine windmilling drag area S_f * Cd at speed.
2. Confirm the critical engine with critical_engine_index; the failed engine is usually the critical one (the leaf reports whichever index is given).
3. Build the geometry set: S_v, l_v, C_Lv_delta_r, delta_r_max, S_r, c_r, C_h_delta_r, pedal arm, boost factor, v_s1g, and the pedal force limit and stall guard defaults.
4. Compute the authority-limited speed with authority_limited_speed and the force-limited speed from the pedal-force quadratic (inside vmc_predict as v_force_m_s).
5. Run the full prediction with vmc_predict and read vmc_m_s, vmc_kt, governing, force_at_vmc_N, force_ok, guard_verdict and flight_test_go.
6. Inspect the asymmetric moment model at Vmc with asymmetric_yaw_moment_Nm for the moment value carried by the rudder.
7. If force_ok is False or guard_verdict is "stall-guard-governs", the engine-inoperative flight test cannot clear the Vmc demonstration as configured; revisit the boost factor or the demonstration speed.
8. Confirm the deterministic checks with the contract test scripts/test_vmc_determination.py.

Worked example

Twin jet, 65000 N engines at y = +-8.0 m, S_v 25 m^2, l_v 16 m, rudder effectiveness 0.8/rad, delta_r_max 30 deg (0.52360 rad), S_r 8.5 m^2, c_r 1.3 m, C_h_delta_r -0.045, pedal arm 0.35 m, boost factor 0.15 (power-boosted), v_s1g 66 m/s, windmilling drag neglected.

• Critical engine: index 0 (tie, lower index).
• T_op * y_fail = 65000 * 8.0 = 520000 N m.
• Authority limit: q* = 520000 / (25 * 16 * 0.8 * 0.52360) = 3103.5 Pa, V_auth = sqrt(2 * 3103.5 / 1.225) = 71.18 m/s.
• Force limit: q_F = 667.0 * 0.35 / (8.5 * 1.3 * 0.045 * 0.52360 * 0.15) = 5977.6 Pa, V_force = sqrt(2 * 5977.6 / 1.225) = 98.79 m/s.
• Vmc = max(71.18, 98.79) = 98.79 m/s = 192.0 kt, governing "pedal-force".
• Pedal force at Vmc: delta_req = 520000 / (5977.6 * 320) = 0.2718 rad (below the limit), F = 346.2 N, force_ok True (below 667 N).
• Stall guard: 1.05 * 66 = 69.3 m/s below Vmc, guard_verdict "stall-guard-ok"; flight_test_go True.
• Manual-rudder case (boost 1.0): force at V_auth 71.18 m/s reaches 2308.6 N above the 667 N limit, force_ok False and flight_test_go False; V_force drops to 38.26 m/s so Vmc = 71.18 m/s governed by "rudder-authority".
• Windmilling variant (S_f * Cd = 1.5 m^2): the denominator shrinks and V_auth rises to 73.87 m/s; the pedal-force limit still binds at 98.79 m/s.

Pitfalls

• Reading the governed limit off the wrong constraint: Vmc is the max of the authority and force limits (max(71.18, 98.79) = 98.79 m/s governed by "pedal-force"), and with manual rudder the force limit drops to 38.26 m/s so the authority limit governs at 71.18 m/s.
• Quoting the pedal force without the limit: the force at Vmc is 346.2 N against the 667 N limit (force_ok True), while the manual-rudder case reaches 2308.6 N with force_ok False and flight_test_go False.
• Neglecting windmilling drag: with S_f * Cd = 1.5 m2 the denominator shrinks and V_auth rises from 71.18 to 73.87 m/s, so the no-windmilling assumption understates the critical speed.
• Forgetting the stall guard: Vmc must stay above 1.05 * v_s1g (69.3 m/s here); if the stall guard exceeded Vmc the guard verdict would not be "stall-guard-ok" and the flight test would not go.
• Assuming a symmetric critical engine: with two engines at y = +-8.0 m the critical engine is index 0 by the tie rule (lower index), and an out-of-range failed index raises ValueError.
• Feeding non-physical inputs: empty engine lists, thrust_N <= 0, negative windmilling area, delta_r_max outside (0, 60] deg, boost factor outside (0, 1], zero hinge moment coefficient, and v_s1g <= 0 all raise ValueError.

Verification

• Confirm vmc_predict on the worked twin returns vmc 98.79 m/s within 0.05, vmc_kt 192.0 within 0.2, v_auth_m_s 71.18 within 0.05, and governing "pedal-force".
• Confirm force_at_vmc_N is within 2 N of 346.2, force_ok True, and the manual-rudder case (boost 1.0) gives 2308.6 N within 5 with force_ok False and flight_test_go False.
• Confirm the windmilling case (1.5 m^2) gives v_auth_m_s 73.87 within 0.1 and above the no-windmilling 71.18.
• Confirm every non-physical input raises ValueError: empty engine list, thrust_N <= 0, negative windmilling area, S_v <= 0, l_v <= 0, effectiveness <= 0, delta_r_max outside (0, 60] deg, S_r <= 0, c_r <= 0, zero hinge moment coefficient, pedal arm <= 0, boost factor outside (0, 1], v_s1g <= 0, out-of-range failed index, and non-positive airspeed for the moment and deflection functions.
• Run the contract test offline: python3 scripts/test_vmc_determination.py (46 tests, deterministic).

Related leaves

• flight-test-operations/envelope/v-speeds: the V-speed set beside which the Vmc result sits.
• flight-mechanics/performance/oei-climb-gradient: the engine-out climb performance sibling; this leaf does not compute climb gradients.
• vehicle-design/sizing/control-surface-sizing: sizing the rudder area from the yaw moment requirement, upstream of this check.
• flight-mechanics/stability-control/control-surface-effectiveness: elevator hinge moments and stick force, out of scope here.
• flight-test-operations/envelope/stall-characteristics-testing: stall demonstration testing behind the stall-protection guard.

Behavior contract (gate 3)

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

python3 scripts/test_vmc_determination.py

The test covers the critical-engine ranking (tie, outboard, left-side asymmetric and magnitude cases), the asymmetric yawing moment values including the windmilling contribution and the negative-side failure, the required deflection at fixed speeds with the inverse-q scaling and the deflection-to-moment round trip, the authority-limited and force-limited closed forms against the worked anchors (71.18 and 98.79 m/s), the Vmc value and governing label, the pedal force at Vmc (346.2 N boosted, 2308.6 N manual), the force-ok and flight-test-go flags, the stall-guard verdicts, the knots conversion (192.0 kt), the windmilling shift of V_auth to 73.87 m/s, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: FAR/CS 25.149 frames the minimum-control-speed method and its 150 lbf pedal-force test condition; the relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.