Rotorcraft Forward Flight Climb Test

Rotorcraft Forward-Flight Climb Test (flight-test-operations/performance/rotorcraft-forward-flight-climb-test)

Use when the task is reducing a FAR 29 rotorcraft forward-flight climb flight test from MEASURED data: steady-climb or level-accelerated runs across an airspeed sweep, each run a window of recorded pressure altitude against time, reduced to the measured rate of climb by least squares, corrected to the reference weight and the standard day with the rho/weight ratio law, and read for the best-rate-of-climb speed Vy and the maximum corrected rate of climb. Vy-schedule runs flown across density altitudes reduce further to the climb ceilings. Every input is a flight test measurement: recorder altitude and time samples, calibrated airspeed, test and reference gross weights, pressure altitude and outside air temperature; no RNG anywhere. It is the rotorcraft forward-flight axis of the measured climb reduction trio in this pack: the level-flight polar reduction from torque and rotor speed (rotorcraft-forward-flight-performance-test) and the hover-axis reduction with the vertical rate correction (rotorcraft-performance-flight-test) are separate leaves, as are the fixed-wing climb reduction in feet per minute (climb-performance-flight-test), the analytic flight-mechanics predictions of the vertical climb and of the climb ceilings, and the airspeed-trace total-energy reduction (level-acceleration-test), whose per-run measured rates enter this leaf as inputs when a sweep is flown as level accelerations.

Domain quick reference

• ISA troposphere at pressure altitude h_p: theta = 1 - LAPSE*h_p/T0, delta = thetaK_DELTA, sigma_ISA = thetaK_SIGMA. Module constants T0 = 288.15 K, LAPSE = 0.0065 K/m, G0 = 9.80665 m/s^2, R_AIR = 287.053 J/(kg K), tropopause 11000 m, K_DELTA = 5.25588..., K_SIGMA = 4.25588..., ALT_INV = 44330.77 m.
• Test-day density ratio from the recorded OAT: sigma_test = delta(h_p) * T0/(OAT + 273.15), the altitude AND temperature content of the day; sigma_ref = sigma_ISA(h_p) is the standard day at the same pressure altitude.
• Density altitude of the test day: invert sigma_ISA, theta_da = sigma_test**(1/K_SIGMA), h_da = ALT_INV*(1 - theta_da).
• True airspeed: V = CAS/sqrt(sigma_test). The airspeed axis stays in kt because the sweep is flown and quoted in knots; altitudes in m, OAT in C, weights in N, rates of climb in m/s.
• Measured rate of climb: least-squares slope of the pressure altitude window against time, slope = (nsum(th) - sum(t)sum(h)) / (nsum(t^2) - (sum(t))^2) in m/s (positive while climbing), with r_squared = 1 - ss_res/ss_tot (1.0 when ss_tot is 0).
• The rho/weight ratio law: ROC_corr = ROC_meas * (W_test/W_ref) * (sigma_ref/sigma_test)**0.5. The weight enters linearly through the excess specific power (ROC = P_ex/W); the day's density deviation enters to the half power because near the best-rate speed the rotor induced power, the dominant term of the climb power at fixed weight and airspeed, scales with the inverse square root of the density ratio while the flat-rated turboshaft available power does not lapse within its rating band. Identity on the standard day at the reference weight; a hot day (sigma_test below sigma_ref) corrects the measured rate UP.
• Vy: the discrete maximum of the corrected ROC over the TAS sweep with a neighbour check; an interior peak above both neighbours refines with the vertex of the parabola through the peak triplet (Newton form, vertex at xv = (x0 + x1)/2 - d1/(2*a) when the curvature a < 0 and xv lies inside the triplet). An endpoint peak reports Vy at the band edge with peak_bracketed False.
• Climb ceilings: the corrected best-rate points across density altitudes interpolate linearly to the altitude where the corrected ROC crosses 0.5 m/s (service ceiling, the FAR 29 climb-demonstration framing, about 100 ft/min) and where it crosses zero (absolute ceiling); no crossing inside the tested band returns None and the report flags it.
• FAR 29 frames the rotorcraft climb demonstration; the relations above are standard engineering methodology, name and convention only.

Workflow

1. Fix the test configuration and the test-day atmosphere: the test gross weight W_test, the reference weight W_ref, the test pressure altitude h_p and the recorded OAT. pressure_altitude_sigma gives the test-day sigma_test, isa_sigma the standard-day sigma_ref, and density_altitude_m the density altitude of the day.
2. Reduce every run window to the measured rate of climb: roc_from_altitude_samples on the pressure-altitude-versus-time window of each steady climb run returns the measured rate of climb in m/s with the fit quality r_squared.
3. Convert the recorded airspeeds to true airspeed: tas_from_cas at sigma_test turns each recorded calibrated airspeed into the TAS of the run on the sweep axis.
4. Correct the measured rates of climb to the reference condition: corrected_roc applies the rho/weight ratio law to every measured rate, weight ratio linearly and the day's density-ratio deviation to the half power.
5. Identify the best-rate-of-climb speed Vy: vy_and_max_roc takes the corrected ROC-versus-TAS curve, locates the discrete maximum with the neighbour check, refines an interior peak with the parabola vertex, and returns vy_kt, vy_mps (through KTAS_TO_MPS), the maximum corrected rate of climb, peak_bracketed and vertex_used.
6. Run the one-call sweep reduction: reduce_forward_flight_climb_sweep chains steps 1, 3, 4 and 5 into the report dict with the keys sigma_test, sigma_ref, density_altitude_m, tas_kt, roc_corr_mps, vy_kt, vy_mps, max_roc_corr_mps, peak_bracketed, vertex_used. The per-run measured rates come from step 2 on the run windows, or directly from level-accelerated runs pre-reduced by the total-energy method of the level-acceleration-test leaf.
7. Reduce the Vy-schedule runs to the climb ceilings: repeat steps 2 and 4 for the Vy-schedule climbs flown across density altitudes, then climb_ceilings interpolates the corrected best-rate curve to the service ceiling at 0.5 m/s and the absolute ceiling at zero (None outside the tested band, flagged in the report).
8. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_forward_flight_climb_test.py.

Worked example

Measured forward-flight climb sweep of a light single-rotor helicopter at W_test 21,800 N, W_ref 21,000 N: nine steady-climb runs across the airspeed band at a fixed maximum power setting near h_p = 1000 m with recorded OAT 15.00 C (ISA there is 8.5 C, so a warm day). Every run is a window of five pressure-altitude samples at t = [0, 2, 4, 6, 8] s. Values below are the module's real outputs, identical under the system python3 and the pyenv 3.13.12 hook interpreter.

• Test day: sigma_test 0.886993020 (warm day, below sigma_ref 0.907463301), density altitude 1231.677263 m.
• Run windows (m) with the measured and corrected rates: run 1 CAS 56.51 kt window 920.30 930.66 941.86 952.32 963.53 gives ROC_meas 5.406000 m/s (r2 0.999827847) and ROC_corr 5.676331; run 2 CAS 65.93 kt window 931.75 944.40 956.04 968.49 980.28 gives 6.057500 and 6.360409; run 3 CAS 75.34 kt window 944.30 956.37 969.29 981.46 994.39 gives 6.263500 and 6.576710; run 4 CAS 84.76 kt window 955.75 968.20 979.66 991.91 1003.52 gives 5.962500 and 6.260659; run 5 CAS 94.18 kt window 968.30 978.28 989.10 999.18 1010.00 gives 5.215000 and 5.475779; run 6 CAS 103.60 kt window 979.75 988.68 996.61 1005.34 1013.42 gives 4.200000 and 4.410024; run 7 CAS 113.02 kt window 992.30 998.09 1004.72 1010.61 1017.24 gives 3.120000 and 3.276018; run 8 CAS 122.43 kt window 1003.75 1008.68 1012.61 1017.34 1021.42 gives 2.200000 and 2.310012; run 9 CAS 131.85 kt window 1016.30 1018.66 1021.86 1024.32 1027.53 gives 1.406000 and 1.476308. The r_squared values sit in [0.997460981, 0.999868624].
• True airspeed per run (CAS/sqrt(sigma_test)): 60.001928 to 139.997420 kt, the 60 to 140 KTAS sweep band. The corrected curve rises from 5.676331 m/s at 60 kt to a peak of 6.576710 m/s at 80 kt and decays to 1.476308 m/s at 140 kt.
• Vy identification: the discrete peak sits at run 3 (TAS 79.995492 kt, interior with both neighbours lower); the vertex of the parabola through the 70-80-90 kt triplet refines it to vy_kt 79.064126 (vy_mps 40.674100) with max_roc_corr_mps 6.579021, peak_bracketed True, vertex_used True. The maximum corrected rate of climb is about 6.58 m/s (about 1295 ft/min) at about 79 kt, representative for the 21,000 N class.
• One-call reducer: reduce_forward_flight_climb_sweep on the recorded CAS list and the nine measured rates returns exactly the dict {sigma_test 0.886993020, sigma_ref 0.907463301, density_altitude_m 1231.677263, tas_kt as listed, roc_corr_mps as listed, vy_kt 79.064126, vy_mps 40.674100, max_roc_corr_mps 6.579021, peak_bracketed True, vertex_used True}.
• Ceiling series: six Vy-schedule climbs (the IAS of the best-rate run, about 75 kt CAS) flown at h_p 0 to 6000 m on days about 5 C warmer than ISA reduce to corrected best-rate points 6.676074041 (DA 178.8336 m), 6.511011229 (DA 1678.7675 m), 4.876046983 (DA 3178.6967 m), 3.110151870 (DA 4678.6206 m), 1.676274964 (DA 5678.5667 m) and 0.429868544 m/s (DA 6178.5387 m), bracketing the 0.5 m/s threshold. climb_ceilings returns service_ceiling_m 6150.406816 m (about 20,180 ft, in the plausible band for the 21,000 N class) and absolute_ceiling_m None: the zero crossing lies above the highest tested density altitude, so the report flags the absolute ceiling as outside the tested band.

Verification

• Confirm pressure_altitude_sigma(1000.0, 15.0) returns 0.886993020, isa_sigma(1000.0) 0.907463301 and density_altitude_m(1000.0, 15.0) 1231.677263 m.
• Confirm the nine window reductions return the measured rates above with r_squared in [0.997, 1.0], and the regression identity: a window on a perfect 1000 + 5.0*t line recovers 5.0 m/s with r_squared 1.0.
• Confirm the rho/weight law identities: corrected_roc returns the input unchanged at the reference weight and equal sigmas (6.000000000), exactly roc*W_test/W_ref when only the weight ratio acts (6.228571429 at 21,800 to 21,000 N), and a hot day corrects UP (6.068938790 for 6.0 m/s at sigma 0.8870 against 0.9075).
• Confirm the airspeed round trip: tas_from_cas(75.34, 0.886994) gives 79.995448 kt and tas*sqrt(sigma) recovers the calibrated airspeed.
• Confirm Vy: vy_and_max_roc returns vy_kt 79.064126 within 0.05, vy_mps 40.674100 within 0.03 and max_roc_mps 6.579021 within 1e-3; uniform scaling of every rate leaves vy_kt unchanged (scale-free peak); an endpoint peak reports peak_bracketed False.
• Confirm the ceiling interpolation: the worked density-altitude series crosses 0.5 m/s at service_ceiling_m 6150.406816 m and never crosses zero (absolute_ceiling_m None); a series that stays above the threshold returns None for the service ceiling.
• Confirm ValueError rejection of every non-physical input: pressure altitude outside [0, 11000] m, OAT outside [-60, 60] C, a density altitude above the tropopause, non-positive CAS or sigma, unequal or single-sample windows, all-equal time samples, flat or descending windows, non-positive weights or sigmas, fewer than 3 speed points, non-increasing TAS, all-sink sweeps, sweeps outside 4-40 runs, non-increasing CAS, mismatched or single-point or non-monotone ceiling lists, and negative density altitudes.
• Confirm the reducer is deterministic across reruns and imports nothing beyond math.

Related leaves

• skills/flight-test-operations/performance/rotorcraft-forward-flight- performance-test: the level-flight polar reduction from torque and rotor speed with the best-endurance, best-range and Vh read-offs, the sibling measured reduction of this pack.
• skills/flight-test-operations/performance/rotorcraft-performance- flight-test: the hover-axis measured reduction with figure of merit, hover power corrections and hover ceilings, and the vertical-rate weight correction of the hover climb.
• skills/flight-test-operations/performance/climb-performance-flight- test: the fixed-wing climb flight test in feet per minute with the 100 ft/min ceiling convention.
• skills/flight-test-operations/performance/level-acceleration-test: the total-energy airspeed-trace reduction whose per-run measured rates enter this leaf when the climb sweep is flown as level accelerations.
• skills/flight-test-operations/performance/rotorcraft-autorotation- flight-test: the rotorcraft descent-axis flight test of this pack.
• skills/flight-mechanics/performance/rotorcraft-vertical-climb- performance: the hover-axis axial momentum theory prediction of vertical climb, the analytic counterpart of the measured hover climb.
• skills/flight-mechanics/performance/climb-performance: the fixed-wing climb prediction leaf for the rate of climb and ceilings.

Pitfalls

• Quoting the corrected maximum rate as if it were measured: the corrected curve peak 6.579021 m/s is the standard-day value at the reference weight; the measured sweep peaked at 6.263500 m/s on the test day at the heavier weight, and reporting the measured peak without the correction understates the climb capability.
• Applying a linear density correction: the day's density-ratio deviation enters to the HALF power in the rho/weight ratio law ((sigma_ref/sigma_test)**0.5), not linearly; the weight ratio is the only linear factor.
• Treating an endpoint peak as the best-rate speed: an endpoint peak reports Vy at the band edge with peak_bracketed False, which means the sweep did not bracket Vy; extend the sweep before quoting the best-rate speed.
• Extrapolating a ceiling outside the tested band: climb_ceilings returns None when the corrected best-rate curve does not cross the threshold or zero inside the measured density-altitude band; the report must flag the ceiling as outside the tested band rather than guessing an extrapolation.
• Mixing this leaf with the fixed-wing climb reduction: this leaf is in m/s with the 0.5 m/s rotorcraft service-ceiling convention and the Vy-schedule basis; the fixed-wing leaf works in feet per minute with climb gradient and time to climb, and the level-acceleration leaf owns the airspeed-trace reduction itself.
• Feeding hover or vertical climb data: this leaf reduces the forward-flight climb sweep only; hover-axis data with the vertical rate correction belongs to the rotorcraft-performance-flight-test leaf.

Behavior contract (gate 3)

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

python3 skills/flight-test-operations/performance/\

rotorcraft-forward-flight-climb-test/scripts/
test_rotorcraft_forward_flight_climb_test.py

The test covers the worked-example contract with the module's real outputs as targets: the step-1 atmosphere reduction of the test day, the step-2 measured rate of climb of each least-squares pressure-altitude window with fit quality, the step-3 true airspeed conversion of the recorded calibrated airspeeds, the step-4 corrected rates of climb through the rho/weight ratio law, the step-5 Vy identification with the parabola vertex refinement, the step-6 one-call sweep reduction dict with all ten keys and its determinism, the step-7 Vy-schedule climb ceilings over the density-altitude band, the closed-form identities (regression recovery, correction identity, weight-ratio-only scaling, airspeed round trip, uniform-scale Vy invariance), the pinned module constants, and ValueError rejection of every non-physical input in the spec validation list. It also passes under the pyenv 3.13.12 hook interpreter.

Compliance

• Standards referenced, not reproduced: FAR 29 frames the rotorcraft climb demonstration; this leaf implements standard flight-test reduction methodology, summary-only per standards-map.yaml, no verbatim rule text.
• compliance: STANDARDS-REF, gated: false.