Rotorcraft Main Rotor Sizing

Main Rotor Sizing (flight-mechanics/performance/rotorcraft-main-rotor-sizing)

Use when you must size the main rotor of a single-main-rotor rotorcraft from the takeoff weight and the design ceilings, rather than analyze a rotor whose geometry is already given. The weight-borne hover thrust T = m * g0 against the chosen main-rotor-disk-loading ceiling sets the disk area and radius, the rotor-thrust-coefficient follows from momentum theory at the rotor tip speed, the rotor solidity closes from the ct-over-sigma hover design point, the blade area and the constant chord follow from the blade count on rectangular blades, and the rotor-tip-mach number checks the tip speed against the speed of sound. Closed-form sizing only, pure Python, stdlib only: no power term, no inflow solve, no figure of merit, no blade-element sections, no compressibility corrections beyond the tip Mach check. The geometry it produces feeds the rotorcraft performance leaves that consume a given rotor: hover power, forward flight, climb, autorotation, ground effect and turn. Pairs with flight-mechanics/performance/rotorcraft-tail-rotor- sizing, which applies the identical disk sizing inversion to the anti-torque rotor with its own torque-balance inputs.

Domain quick reference

• Weight-borne hover thrust: T = m * g0, g0 = 9.80665 m/s2, at the takeoff mass m.
• Disk area at the ceiling: A = T / DL_max, with DL_max the chosen main-rotor-disk-loading ceiling (250-500 Pa band, 350 Pa worked).
• Disk radius: R = sqrt(A / PI).
• Achieved disk loading: T / A equals DL_max exactly when the disk is sized with that ceiling.
• Hover thrust coefficient (momentum theory): CT = T / (rho * A * Vtip^2) with A = PI * R^2, Vtip the rotor tip speed (200-230 m/s band, 210 m/s worked), rho = 1.225 kg/m3 at sea level.
• Ceiling identity: at the sized disk CT = DL_max / (rho * Vtip^2), so the thrust coefficient is independent of the rotor size.
• Solidity closure: sigma = CT / (CT/sigma)_design from the ct-over- sigma hover design point (0.10-0.14 band, 0.12 worked); the closure round trip sigma * (CT/sigma)_design recovers CT.
• Blade area: A_b = sigma * A. Constant blade chord on rectangular blades: c = A_b / (b * R) from the blade count b; the solidity identity sigma = b * c * R / A recovers the input.
• Rotor tip Mach: M_tip = Vtip / a with a the speed of sound (340.3 m/s sea level).
• SI units throughout: N, m, m2, Pa, m/s.
• FAR-29 frames transport rotorcraft certification context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the design point: takeoff mass m and the design ceilings, the main-rotor-disk-loading ceiling, the ct-over-sigma hover design point, the blade count and the rotor tip speed, with the weight-borne hover thrust T = m * G0.
2. Size the disk from the weight against the ceiling with disk_area_and_radius and confirm the disk sits exactly at the main-rotor-disk-loading ceiling (thrust over area round trip and PI * radius**2 recovering the area).
3. Compute the rotor-thrust-coefficient at the rotor tip speed with hover_thrust_coefficient and cross-check the ceiling identity CT = disk_loading_max / (rho * tip_speed**2).
4. Close the rotor solidity from the ct-over-sigma design point with solidity_closure and check the design-point round trip CT / sigma.
5. Lay out the rectangular blades with blade_area_chord, total blade area and constant blade chord from the blade count, verifying the solidity identity sigma = b * c * R / A.
6. Check the rotor-tip-mach number with tip_mach against the speed of sound; subcritical at sea level, so no compressibility correction enters the sizing.
7. Close out with the deterministic contract test scripts/test_rotorcraft_main_rotor_sizing.py.

Worked example

Primary state: helicopter takeoff mass m = 4500 kg, main-rotor-disk- loading ceiling DL_max = 350 Pa, ct-over-sigma design point 0.12, blade count b = 4, rotor tip speed Vtip = 210 m/s, rho = 1.225 kg/m3, speed of sound a = 340.3 m/s. Module outputs:

• Thrust: T = W = 4500 * 9.80665 = 44129.925 N (weight-borne hover reference).
• Disk area: A = 44129.925 / 350 = 126.0855 m2.
• Disk radius: R = sqrt(126.0855 / PI) = 6.3352 m.
• Achieved disk loading: T / A = 350.0 Pa, exactly the ceiling.
• Thrust coefficient: CT = 44129.925 / (1.225 * 126.0855 * 210^2) = 0.006479, and the ceiling identity 350.0 / (1.225 * 210^2) gives the same 0.006479.
• Solidity closure: sigma = 0.006479 / 0.12 = 0.053990; the design point check CT / sigma = 0.1200.
• Blade area: A_b = 0.053990 * 126.0855 = 6.8073 m2.
• Blade chord: c = 6.8073 / (4 * 6.3352) = 0.2686 m, blade aspect ratio R / c = 23.58 (rectangular constant-chord blades); the solidity check b * c * R / A = 0.053990.
• Tip Mach: M_tip = 210.0 / 340.3 = 0.61710, comfortably subcritical at sea level, so no compressibility correction enters the sizing.

The light 350 Pa ceiling gives a large disk (about 126 m2, R = 6.34 m) and hence a low hover thrust coefficient, so the 0.12 design point closes at sigma about 0.054 with a 0.269 m chord. At fixed tip speed and design point the solidity scales linearly with the ceiling, sigma = DL_max / (rho * Vtip^2 * (CT/sigma)_design), about 0.092 at a 600 Pa ceiling.

Secondary state: m = 3000 kg, DL_max = 300 Pa, ct_over_sigma = 0.10, b = 4, Vtip = 210 m/s: W = 29419.950 N, A = 98.0665 m2, R = 5.5871 m, achieved disk loading 300.0 Pa, CT = 0.005553, sigma = 0.055532 (CT / sigma = 0.1000), blade area 5.4459 m2, chord 0.2437 m, and M_tip = 0.61710 unchanged at the fixed tip speed.

Verification

• Confirm disk_area_and_radius(44129.925, 350.0) returns (126.0855, 6.3352) and that thrust / area equals 350.0 Pa exactly, the disk sized at the ceiling; PI * radius**2 recovers the area.
• Confirm hover_thrust_coefficient(44129.925, 1.225, 6.3352, 210.0) returns 0.006479, equal to the ceiling identity 350.0 / (1.225 * 210^2); the value is independent of the rotor size at a fixed ceiling and tip speed.
• Confirm solidity_closure(0.006479, 0.12) returns 0.053990 and that the closure round trip sigma * 0.12 recovers the thrust coefficient (design-point check CT / sigma = 0.1200).
• Confirm blade_area_chord(0.053990, 126.0855, 4, 6.3352) returns (6.8073, 0.2686) and the solidity identity 4 * 0.2686 * 6.3352 / 126.0855 recovers 0.053990.
• Confirm tip_mach(210.0, 340.3) returns 0.61710.
• Confirm the disk radius scales with the square root of the mass at a fixed ceiling (6.3352 m at 4500 kg against 5.5871 m at 3000 kg) and that the tip Mach is unchanged when only the mass or the ceiling changes at a fixed tip speed.
• Convention cross-check: disk_area_and_radius(1851.8519, 300.0) returns (6.1728, 1.4017), matching the rotorcraft-tail-rotor-sizing worked example for the shared A = T / DL_max inversion applied to its own anti-torque thrust; the torque-balance inputs never enter here.
• Confirm every non-positive thrust, disk loading, air density, radius, tip speed, thrust coefficient, ct-over-sigma design point, solidity, area and speed of sound, every blade count below 1 and every fractional blade count raises ValueError (15 rejection classes).
• No RNG anywhere: repeated runs give identical floats.
• Run the contract test offline: python3 scripts/test_rotorcraft_main_rotor_sizing.py (35 tests, deterministic).

Pitfalls

• Sizing on a radius that is already given: this leaf derives the disk radius from the takeoff weight and the disk-loading ceiling; the rotorcraft performance siblings (hover, forward flight, climb, turn) all consume a given radius and solidity and never size the rotor.
• Writing the ceiling with the hover sibling's token: the sizing ceiling is the main-rotor-disk-loading compound; rotor-disk-loading belongs to rotorcraft-hover-performance for a given rotor.
• Closing solidity from a rounded thrust coefficient: the closure runs on the module thrust coefficient of the sized rotor (0.006479 into 0.12 gives 0.053990); round CT to fewer decimals first and the division drifts out of the 1e-6 window.
• Forgetting the blade count in the chord: the chord is the total blade area spread over b * R, so doubling the blade count halves the chord at a fixed solidity and disk.
• Reading the achieved disk loading as below the ceiling: A = T/DL_max sizes the disk at the ceiling, so the achieved T / A equals DL_max exactly and the CT ceiling identity holds.
• Non-positive thrust, disk loading, rho, radius, tip speed, thrust coefficient, design point, solidity, area or speed of sound, and blade counts below 1 or fractional raise ValueError; runs are deterministic (no RNG).

Related leaves

• flight-mechanics/performance/rotorcraft-hover-performance: the OGE hover power and figure of merit for the main rotor sized here.
• flight-mechanics/performance/rotorcraft-tail-rotor-sizing: the identical disk sizing inversion applied to the anti-torque rotor from the torque balance.
• flight-mechanics/performance/rotorcraft-forward-flight-performance: forward flight power of the sized main rotor.
• flight-mechanics/performance/rotorcraft-vertical-climb-performance: climb power states of the sized main rotor.
• flight-mechanics/performance/rotorcraft-turn-performance: the turn closure that reuses a given hover rotor at n times the weight.

Behavior contract (gate 3)

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

python3 scripts/test_rotorcraft_main_rotor_sizing.py

The test covers the primary worked example chain (4500 kg at the 350 Pa ceiling, 0.12 design point, 4 blades, 210 m/s) and the secondary 3000 kg / 300 Pa / 0.10 state against the spec anchors, every function against its defining equation, the ceiling and radius round trips, the ceiling identity of the thrust coefficient, the solidity closure round trip, the blade solidity identity, the tip Mach invariance, the shared inversion cross-check with the sibling worked anchor, run-to-run determinism, and ValueError rejection of every non-physical input class.

Compliance

• Standards referenced, not reproduced: FAR-29 is a regulatory standard; the main rotor sizing relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.