Rotorcraft Cyclic Pitch Trim

Rotorcraft Cyclic-Pitch Trim (flight-mechanics/performance/rotorcraft-cyclic-pitch-trim)

Use when the task is the cyclic-pitch trim of a helicopter main rotor blade in forward flight: the longitudinal and lateral cyclic pitch, and the equivalent swashplate tilt, that hold the tip-path plane at a target longitudinal and lateral attitude under uniform inflow. This leaf extends the wave-46 collective-only first-harmonic flap equilibrium (the forward-flight flapping sibling) with the control channel theta(psi) = theta0 + theta1ccos(psi) + theta1ssin(psi), where theta1c is the lateral cyclic pitch (the cos(psi) harmonic) and theta1s the longitudinal cyclic pitch (the sin(psi) harmonic) in the standard rotor convention (Johnson, Helicopter Theory ch. 4 and Leishman, Principles of Helicopter Aerodynamics ch. 4, paraphrased, never reproduced), and solves the closed-form trim inversion for the cyclic command that holds a target tip-path-plane attitude, level disk or prescribed tilt. Pure Python, stdlib only, deterministic. It is the control-channel completion of flight-mechanics/performance/rotorcraft-forward-flight-flapping, which owns the collective-only equilibrium: at zero cyclic every closed form here reduces exactly to that sibling's.

Domain quick reference

• Blade element velocities in the hub plane frame: u_T = x + musin(psi) (tangential, x = r/R) and u_P = lambda + xbeta' + mubetacos(psi) (perpendicular), psi measured from the downwind blade position, advancing side at psi = pi/2, matching the wave-46 sibling convention.
• Blade pitch with the control channel: theta(psi) = theta0 + theta1ccos(psi) + theta1ssin(psi), theta1c the LATERAL cyclic pitch (peaks on the fore-aft line of the disk), theta1s the LONGITUDINAL cyclic pitch (peaks on the port-starboard line).
• First-harmonic flap ansatz: beta(psi) = a0 + a1scos(psi) + b1ssin(psi); flap equation beta'' + beta = (gamma/2)M(psi) with the dimensionless moment M(psi) = integral_0^1 x[u_T^2theta - u_Tu_P] dx. At 1/rev resonance the 1/rev content of beta'' + beta vanishes, so the equilibrium nulls the cos and sin projections of M and the steady projection fixes the coning.
• Closed forms (exact solution of the projection balances, all terms through mu^2): a0 = (gamma/2)(theta0(1 + mu^2)/4 - lambda/3 + mutheta1s/3), a1s = -[4mu*(2theta0/3 - lambda/2) + theta1s(1 + 3mu^2/2)]/(1 - mu^2/2), b1s = theta1c - (4mu/3)*a0/(1 + mu^2/2).
• a1s is gamma-free (the Lock number cancels in the 1/rev cos balance) and independent of theta1c; b1s carries theta1c with unity gain and gamma, mu and theta1s through the coning coupling. The coning gains the mean-lift term mu*theta1s/3 and stays free of theta1c, a1s and b1s.
• Zero-cyclic identity: at theta1c = theta1s = 0 the forms reduce EXACTLY to the wave-46 sibling closed forms, and a1s_free and b1s_free report that collective-only equilibrium.
• Hover limit mu = 0: a1s = -theta1s exactly and b1s = theta1c exactly (the 90-degree flap lag of the centrally hinged blade), so lateral cyclic controls the lateral disk orientation and longitudinal cyclic the longitudinal orientation.
• Sign convention: a1s < 0 is the tip-path-plane aft tilt; b1s < 0 means the advancing blade tip sits below the coning plane; theta1c and theta1s may be negative (trim routinely needs them so).
• Scope: 0 <= mu < 1 (singular at mu = 1, reverse flow out of scope), lambda > 0 (downward-positive convention), theta0 > 0, gamma > 0, centrally hinged, first harmonic only, small angles.
• Units are radians for the angle inputs and the _rad outputs; the _deg outputs are the _rad values times 180/pi.
• FAR-29 frames the transport-category rotorcraft certification context for rotor control loads; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the operating point: advance ratio mu, uniform inflow ratio lambda, collective theta0, blade Lock number gamma and the cyclic command (theta1c, theta1s) where one is applied.
2. Compute the cyclic-forced coning angle with coning_angle and confirm the cross-leaf identities: at zero cyclic it equals the wave-46 collective-only coning closed form, and at mu = 0 it equals the hover coning closed form 0.5gamma(theta0/4 - lambda/3), cyclic-free.
3. Compute the longitudinal flapping angle with longitudinal_flapping_angle (no gamma, no theta1c: neither enters the 1/rev cos balance) and read the tip-path-plane response to the longitudinal cyclic pitch, exactly -theta1s at mu = 0.
4. Compute the lateral flapping angle with lateral_flapping_angle and confirm the coning-coupling identity against coning_angle, exactly theta1c at mu = 0.
5. Run flap_response_summary for the one-call cyclic-forced equilibrium dict with all six documented keys in radians and degrees.
6. Run cyclic_response_gains for the affine control-to-flap gains: a1s_free and b1s_free (the wave-46 zero-cyclic flapping angles), d_a1s_d_theta1s, the exact zero and unity entries d_a1s_d_theta1c = 0.0 and d_b1s_d_theta1c = 1.0, and the coning-mediated coupling d_b1s_d_theta1s, quadratic in mu.
7. Solve the trim inversion with trim_cyclic for the cyclic command that holds the tip-path plane at the target attitude, level disk (a1s_target = b1s_target = 0.0) or prescribed tilt, and round-trip the command through flap_response_summary to confirm the disk lands on the target.
8. Report the equivalent ideal zero-phase swashplate plane tilt with trim_swashplate_tilt: the tilt components equal the trim cyclic harmonics and the magnitude is their hypot.
9. Confirm the deterministic checks with the contract test scripts/test_rotorcraft_cyclic_pitch_trim.py.

Worked example

Cruise case in the corpus band: advance ratio mu = 0.3, uniform inflow ratio lambda = 0.06, collective theta0 = 0.14 rad, Lock number gamma = 6.0 (the exact worked point of the wave-46 leaf, so the cross-leaf identities are direct). Real module outputs:

• Zero-cyclic limit: flap_response_summary(0.3, 0.06, 0.14, 6.0, 0.0, 0.0) returns coning_angle_rad 0.05445 (3.11976 deg), longitudinal_flapping_rad -0.07958 (-4.55966 deg) and lateral_flapping_rad -0.02084 (-1.19416 deg), bit-identical to the wave-46 sibling worked values.
• Forward map under a cyclic command: at theta1c = 0.02 rad and theta1s = -0.04 rad the coning relaxes to 0.04245 rad (2.43221 deg, the negative longitudinal cyclic unloading the advancing-side mean lift), the aft tilt eases to -0.03204 rad (-1.83586 deg) and the lateral flapping becomes +0.00375 rad (0.21493 deg, the lateral cyclic overcoming the coning coupling).
• Cyclic response gains: cyclic_response_gains(0.3, 0.06, 0.14, 6.0) returns a1s_free -0.07958, b1s_free -0.02084, d_a1s_d_theta1s -1.18848, d_a1s_d_theta1c 0.0, d_b1s_d_theta1c 1.0 and d_b1s_d_theta1s -0.11483: one degree of longitudinal cyclic moves the longitudinal tilt 1.19 deg, one degree of lateral cyclic moves the lateral tilt 1.00 deg, and the coning-mediated cross coupling moves the lateral tilt only 0.115 deg per degree of longitudinal cyclic.
• Level-disk trim: trim_cyclic(0.3, 0.06, 0.14, 6.0, 0.0, 0.0) returns longitudinal_cyclic_pitch_rad -0.06696 (-3.83655 deg) and lateral_cyclic_pitch_rad 0.01315 (0.75360 deg): holding the tip-path plane level at mu = 0.3 needs -3.84 deg of longitudinal cyclic and +0.75 deg of lateral cyclic. Round trip: the forward map at that command returns a1s = 0.0 and b1s = 0.0 exactly.
• Prescribed attitude trim: trim_cyclic(0.3, 0.06, 0.14, 6.0, -0.02, 0.0), holding the disk at a relaxed aft tilt of -0.02 rad (-1.14592 deg), returns longitudinal_cyclic_pitch_rad -0.05013 (-2.87236 deg) and lateral_cyclic_pitch_rad 0.01509 (0.86432 deg). Round trip returns a1s = -0.02 and b1s = 0.0 within 1e-9.
• Free-equilibrium identity: trim_cyclic at the a1s_free and b1s_free attitude returns theta1s = -0.0 and theta1c = 0.0 exactly: the wave-46 collective-only equilibrium is held with zero cyclic, as it must be.
• Level-disk swashplate report: trim_swashplate_tilt under the ideal zero-phase idealization returns swashplate_longitudinal_tilt_rad 0.01315 (0.75360 deg), swashplate_lateral_tilt_rad -0.06696 (-3.83655 deg) and swashplate_tilt_magnitude_rad 0.06824 (3.90986 deg): the swashplate plane stands tilted 3.91 deg from level at the level-disk cruise trim.
• Hover anchors at mu = 0.0 with theta1c = 0.02, theta1s = -0.04: a1s = 0.04000 (exactly -theta1s), b1s = 0.02 (exactly theta1c) and a0 = 0.04500 (the hover coning closed form, cyclic-free); a level disk at hover needs zero cyclic.
• Ordering anchor: the level longitudinal cyclic magnitude 0.06696 rad sits below the free aft tilt 0.07958 rad, because the leveling denominator (1 + 3*mu^2/2) exceeds the free-flap denominator (1 - mu^2/2).

Verification

• Confirm flap_response_summary(0.3, 0.06, 0.14, 6.0, 0.0, 0.0) matches the worked values and that each component equals the wave-46 sibling closed form computed from the printed formulas within 1e-12.
• Confirm the hover cyclic identities: longitudinal_flapping_angle(0.0, 0.06, 0.14, -0.04) = 0.04000 (equals -theta1s), lateral_flapping_angle (0.0, 0.06, 0.14, 6.0, 0.02, -0.04) = 0.02 (equals theta1c) and coning_angle(0.0, 0.06, 0.14, 6.0, -0.04) = 0.04500 (the hover coning closed form), all within 1e-9.
• Confirm both trim round trips: the level-disk and prescribed-attitude trims return the target attitude through flap_response_summary within 1e-9, and the free-equilibrium trim returns both cyclic pitches exactly 0.0 (math.isclose at 1e-15).
• Confirm the gain structure: cyclic_response_gains(0.3, 0.06, 0.14, 6.0) gives d_a1s_d_theta1s -1.18848, d_a1s_d_theta1c exactly 0.0, d_b1s_d_theta1c exactly 1.0 and d_b1s_d_theta1s -0.11483, each gain matching its closed form within 1e-12.
• Confirm the ordering identity |theta1s_level| = 0.06696 rad is below |a1s_free| = 0.07958 rad (strict comparison).
• Confirm the two-point inflow linearity of the level trim: over lam = 0.06 versus 0.05 the theta1s difference 0.0052863 rad and the theta1c difference -0.0032207 rad equal the exact closed-form slopes within 1e-12.
• Confirm the degree outputs each equal the _rad value times 180.0/pi within 1e-12 and that each summary, trim and swashplate dict returns exactly its documented keys.
• Confirm the coning decoupling: coning_angle depends on theta1s only through the mu*theta1s/3 mean-lift term and is unchanged by any finite theta1c.
• Confirm the independent Fourier self-consistency: the closed-form equilibrium nulls the steady, cos and sin projections of (gamma/2)*M - (beta'' + beta) by direct quadrature (at least 1024 by 128 points) with absolute residuals below 1e-5.
• Confirm ValueError rejection of mu < 0, mu >= 1, lambda <= 0, theta0 <= 0, gamma <= 0 and any non-finite argument (mu = 0.0, negative cyclic pitches and negative a1s_target are valid inputs).
• Run the contract test offline: python3 scripts/test_rotorcraft_cyclic_pitch_trim.py (41 tests, deterministic, no network).

Pitfalls

• Passing gamma or theta1c to longitudinal_flapping_angle: the function takes no Lock number and no lateral cyclic argument by design, because gamma cancels in the 1/rev cos balance and the cos-harmonic pitch never projects onto the longitudinal balance.
• Reading a1s as the cyclic command: a1s is the flapping RESPONSE, not the input. The longitudinal cyclic theta1s that holds a level disk comes out NEGATIVE at cruise (it cancels the free aft tilt), and negative cyclic values are valid, expected inputs.
• Confusing the cyclic harmonics: theta1c (the cos(psi) harmonic) is the LATERAL cyclic pitch and theta1s (the sin(psi) harmonic) the LONGITUDINAL cyclic pitch in the pinned standard rotor convention; at hover the response is a1s = -theta1s and b1s = theta1c, the 90-degree flap lag.
• Using this leaf for the collective-only equilibrium: the zero-cyclic flapping angles themselves, the collective-only forward-flight equilibrium and the tip-path-plane-tilt surface belong to rotorcraft-forward-flight-flapping; this leaf reports them only as the zero-cyclic limit and as a1s_free and b1s_free.
• Treating the swashplate tilt as rigging geometry: the report uses the ideal zero-control-phase swashplate idealization with unit pitch gearing; real rotors phase their control horns, and that rigging geometry is out of scope.
• Extrapolating past mu = 1: the uniform-inflow model is singular at mu = 1 and reverse flow is out of scope; ValueError rejects mu >= 1.0.
• Expecting the trim to be gamma-free: theta1s is gamma-free by cancellation, but theta1c carries gamma through the coning coupling (the d_b1s_d_theta1s gain is quadratic in mu and linear in gamma).

Related leaves

• flight-mechanics/performance/rotorcraft-forward-flight-flapping: the collective-only sibling whose equilibrium this leaf reproduces exactly at zero cyclic; its first-harmonic flap response is the free-flapping state the trim cyclic here commands against.
• flight-mechanics/performance/rotorcraft-blade-flapping-dynamics: the hover-state sibling that owns the Lock number, the hover coning angle and the flap frequency ratio.
• flight-mechanics/performance/rotorcraft-forward-flight-performance: the power and best-speeds leaf of forward flight; it never touches blade motion or rotor control.
• flight-mechanics/performance/rotorcraft-lead-lag-dynamics: the in-plane blade motion sibling, distinct from the out-of-plane flap control channel here.

Behavior contract (gate 3)

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

python3 scripts/test_rotorcraft_cyclic_pitch_trim.py

The test covers the zero-cyclic cross-leaf identity against the wave-46 closed forms, the hover cyclic response identities (a1s = -theta1s, b1s = theta1c, cyclic-free hover coning), the hover no-cyclic zero, the level-disk and prescribed-attitude trim inversions with their round trips through flap_response_summary, the free-equilibrium inversion identity, the exact gain structure, the ordering identity, the two-point inflow linearity of the trim, the degree-output and dict-key discipline of every function, the coning decoupling, the independent Fourier self-consistency of the closed forms, ValueError rejection of every non-physical or non-finite input, and run-to-run determinism.

Compliance

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