Rotorcraft Blade Flapping Dynamics

Rotorcraft Blade Flapping Dynamics (flight-mechanics/performance/rotorcraft-blade-flapping-dynamics)

Use when the task is the basic blade-flapping dynamics of a helicopter main rotor: the Lock number that fixes the ratio of aerodynamic flap moment to centrifugal restoring moment, the steady hover coning angle of an untwisted centrally hinged blade under uniform inflow, and the rotating flap natural frequency ratio for a flap hinge offset. This leaf implements the classical articulated-rotor flap model (Johnson, Helicopter Theory ch.4 and Leishman, Principles of Helicopter Aerodynamics ch.4, paraphrased, never reproduced) in pure Python, stdlib only, deterministic. It is the first blade-dynamics leaf in the rotorcraft subdomain: the momentum-theory POWER leaves (rotorcraft-hover-performance and siblings) own hover power, induced velocity and figure of merit, while this leaf owns flap motion only. Flap dynamics here covers coning and frequency ratio, not ground resonance or lag dynamics.

Domain quick reference

• Lock number: gamma = rho * a * c * R^4 / I_beta, where rho is the air density, a the section lift-curve slope (typical 5.73 /rad), c the blade chord, R the rotor radius and I_beta the blade flap moment of inertia. The Lock number is the ratio of the aerodynamic flap moment to the centrifugal restoring moment; published rotor values fall in the 5-12 band.
• Uniform-blade flap inertia: I_beta = m_b * R^2 / 3 for a blade of mass m_b flapping about the rotation axis.
• Hover coning angle: a0 = 0.5 * gamma * (theta0 / 4 - lambda / 3) rad, from the steady flap-moment balance. The aerodynamic flap moment 0.5 * rho * a * c * Omega^2 * R^4 * (theta0/4 - lambda/3) equals the centrifugal restoring moment I_beta * Omega^2 * a0 for an untwisted centrally hinged blade with uniform inflow, where theta0 is the collective pitch and lambda the uniform inflow ratio. Hover coning is typically 3-8 deg.
• Rotating flap frequency ratio: nu = sqrt(1 + 1.5 * e / (1 - e)), with e the flap hinge offset as a fraction of rotor radius. This is algebraically identical to nu^2 = (1 - 3e/2 + e^3/2) / (1 - e)^3 for a uniform blade. The central hinge limit e = 0 gives exactly 1.0 (1/rev); articulated flap frequencies run about 1.02-1.08 per rev.
• Flap frequency per revolution: because the flapping natural frequency scales with rotor speed, the ratio nu already expresses the frequency in units of rotor revolutions and no rotor speed input is required.
• Units are SI throughout: kg, m, rad, /rad, dimensionless ratios.
• FAR-29 frames the transport-category rotorcraft certification context for rotor loads; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the blade geometry and operating point: blade mass m_b, radius R, chord c, collective theta0, uniform inflow ratio lambda, hinge offset fraction e, and optionally rho and the lift-curve slope a.
2. Get the flap inertia with blade_flap_inertia_uniform (uniform blade about the rotation axis); the module order is mass, radius, chord, theta0, lambda, e, then the optional lift_slope and rho defaults.
3. Compute the Lock number with lock_number from rho, a, c, R and I_beta, and sanity check it against the published 5-12 band.
4. Compute the hover coning with hover_coning_angle(gamma, theta0, lambda); a0 near zero means collective and inflow nearly balance.
5. Compute the flap frequency with flap_frequency_ratio(e); e = 0 is the central hinge 1/rev limit, larger offsets stiffen the blade.
6. Run blade_flapping_summary for the one-call assessment dict with all six documented keys (lock_number, flap_inertia_kg_m2, coning_angle_rad, coning_angle_deg, flap_frequency_ratio, flap_frequency_per_rev).
7. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_blade_flapping_dynamics.py.

Worked example

A typical light-to-medium helicopter blade: rho = 1.225 kg/m3, a = 5.73 /rad, chord c = 0.50 m, radius R = 6.0 m, blade mass m_b = 50 kg (I_beta = 600 kg m2), collective theta0 = 0.170 rad, uniform inflow ratio lambda = 0.050, hinge offset e = 0.05. Real module outputs:

• blade_flap_inertia_uniform(50, 6.0) = 600.0 kg m2 exactly (50 * 36 / 3).
• lock_number(1.225, 5.73, 0.5, 6.0, 600.0) = 7.58079, inside the published 6-10 rotor band (hand check 1.225 * 5.73 * 0.5 * 1296 / 600 = 7.58079).
• hover_coning_angle(7.58079, 0.170, 0.050) = 0.09792 rad = 5.61032 deg, inside the published 3-8 deg hover coning band.
• flap_frequency_ratio(0.05) = 1.03872, inside the published 1.02-1.08 per rev articulated flap band.
• blade_flapping_summary(50, 6.0, 0.50, 0.170, 0.050, 0.05) returns lock_number 7.58079, flap_inertia_kg_m2 600.0, coning_angle_rad 0.09792, coning_angle_deg 5.61032, flap_frequency_ratio 1.03872 and flap_frequency_per_rev 1.03872.

Verification

• Confirm blade_flap_inertia_uniform(50, 6.0) returns 600.0 exactly.
• Confirm lock_number(1.225, 5.73, 0.5, 6.0, 600.0) returns 7.58079 and falls in the 6-10 band.
• Confirm hover_coning_angle(7.58079, 0.170, 0.050) returns about 0.09792 rad and 5.61 deg, inside 3-8 deg.
• Confirm the coning limiting behaviour: theta0/4 equal to lambda/3 gives a0 = 0.0, higher collective raises a0, higher inflow lowers it.
• Confirm flap_frequency_ratio(0.0) equals exactly 1.0, e = 0.05 gives about 1.0387, and e = 0.5 gives sqrt(2.5) = 1.5811; larger offsets always raise nu.
• Confirm the summary dict contains exactly the six documented keys and that flap_frequency_per_rev equals flap_frequency_ratio.
• Confirm ValueError rejection of non-positive mass, radius, chord, lift slope, density, inertia and gamma, of negative collective or inflow ratio, and of hinge offset below 0 or at or above 1.
• Run the contract test offline: python3 scripts/test_rotorcraft_blade_flapping_dynamics.py (34 tests, deterministic, no network).

Pitfalls

• Using a non-uniform flap inertia without saying so: blade_flap_inertia_uniform assumes I_beta = m_b*R^2/3 about the rotation axis; the Lock number (and coning) is only as good as the inertia input.
• Sanity-checking the Lock number against the wrong band: published rotor values run 5-12 (the worked example lands at 7.58); treat the band as a plausibility check, not a pass/fail gate.
• Expecting coning sign intuition to hold: a0 = 0.5gamma(theta0/4 - lambda/3), so higher inflow lowers coning and theta0/4 = lambda/3 gives exactly zero; a 'negative coning' reading means collective and inflow are out of balance, not a sign error in the formula.
• Passing hinge offset e at or above 1 (or negative): flap_frequency_ratio raises ValueError there; e = 0 is the exact central-hinge 1/rev limit and larger offsets stiffen the blade.
• Forgetting the units convention: SI throughout (kg, m, rad) with gamma dimensionless; a negative collective or inflow ratio raises ValueError, and coning_angle_deg is the only degree output.

Related leaves

• flight-mechanics/performance/rotorcraft-hover-performance: the momentum-theory hover power leaf; its blade geometry inputs (radius, chord, solidity) are shared with this leaf.
• flight-mechanics/performance/rotorcraft-tail-rotor-sizing: tail rotor anti-torque sizing, the power-side complement to main rotor dynamics.
• flight-mechanics/stability-control/spin-recovery: fixed-wing autorotation of a stalled wing is a different topic from rotor blade flapping and lives in the fixed-wing stability leaves.

Behavior contract (gate 3)

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

python3 scripts/test_rotorcraft_blade_flapping_dynamics.py

The test covers the worked example magnitude bounds (Lock number 6-10 about 7.58, coning about 0.0979 rad = 5.61 deg inside 3-8 deg, flap frequency ratio about 1.0387 inside 1.02-1.08 per rev), the exact uniform-blade inertia value, the flap-moment balance zero, monotonic coning responses, the closed-form flap frequency identities including the exact central hinge limit, scaling relations, ValueError rejection of every non-physical input, and run-to-run determinism.

Compliance

• Standards referenced, not reproduced: FAR-29 is the FAA transport-category rotorcraft airworthiness standard (ecfr.gov); the flap relations above are standard engineering methodology (Johnson, Leishman), summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.