Limit Cycle Oscillation

Limit Cycle Oscillation (flight-test-operations/flutter/limit-cycle-oscillation)

Use when the task is limit cycle oscillation (LCO) assessment for a flight test flutter clearance campaign: damping ratio from the log decrement, amplitude growth rate from the amplitude time history, sustained versus diverging oscillation at a fixed airspeed, amplitude margin against the limit amplitude, and the effect of control surface freeplay and nonlinear damping on LCO onset.

Domain quick reference

• Log decrement from a decaying amplitude history: delta = (1/n) * ln(A_0 / A_n), with A_0 the amplitude at the start and A_n the amplitude n oscillation cycles later (m or deg; the ratio makes the decrement dimensionless). Example: amplitude falling from 5.0 mm to 2.5 mm in one cycle gives delta = ln(2) = 0.693.
• Damping ratio from the log decrement: zeta = delta / sqrt(4 * pi^2
  • delta^2), dimensionless. Example: delta = 1.0 gives zeta = 1 / sqrt(4 * pi^2 + 1) = 0.157; for small delta the ratio approaches delta / (2 * pi); delta = 0 gives zeta = 0 (undamped). The formula is valid only for a decaying response, so delta must be non- negative.
• Amplitude growth rate: the linear least squares slope of amplitude versus time, slope = (N * sum(t*A) - sum(t) * sum(A)) / (N * sum(t^2) - (sum(t))^2), with amplitude in m and time in s, so the slope is in m/s (or deg/s for angular amplitude). A positive slope means the amplitude is growing (flutter-like divergence), a slope within tolerance at a fixed airspeed means the amplitude is stabilizing (the LCO signature), and a negative slope means the response is decaying.
• LCO amplitude margin: margin = (A_limit - A) / A_limit, with A the measured sustained amplitude and A_limit the limit amplitude of the clearance basis, both in the same unit. Positive margin means below the limit, zero means at the limit, negative means above the limit.
• Sustained LCO verdict: the oscillation is sustained when the amplitude stabilizes (|growth rate| within tolerance) at a fixed airspeed with the amplitude at or below the limit amplitude. Clearance requires a positive margin to the limit amplitude, no growing trend, and airspeed at or below the limit speed.
• Freeplay effect: freeplay in a control surface hinge creates a deadband of near-zero hinge stiffness below the freeplay angle; once the oscillation amplitude exceeds the deadband the effective stiffness drops, and LCO can onset at airspeeds below the linear flutter speed, so the linear extrapolation is not conservative.
• FAR-25 (14 CFR Part 25) and CS-25 set the flutter clearance context (25.629 family: the airframe must be free of flutter and sustained oscillations within the envelope with margin); the LCO relations above are common flight test practice feeding that clearance.

Workflow

1. Extract the amplitude time history at the test airspeed from accelerometer, gyro, or strain data band-passed at the mode frequency, and record the test airspeed against the limit speed.
2. Estimate the damping from decaying segments of the response with log_decrement(amplitudes, cycles), then convert with damping_ratio_from_log_decrement(delta).
3. Fit the amplitude trend with amplitude_growth_rate(amplitudes, times) and classify it as growing, stable (sustained), or decaying.
4. Compute the margin with lco_amplitude_margin(amplitude, amplitude_limit) and compare the sustained amplitude with the limit amplitude of the clearance basis.
5. Combine airspeed band, amplitude band, and damping trend with lco_verdict(airspeed, limit_speed, amplitude, amplitude_limit, damping_slope, ...); the sustained_lco flag marks the LCO signature and the clearance field applies the margin requirement.
6. Flag control surface freeplay with freeplay_lco_onset_risk(freeplay_angle_deg) and re-check the LCO onset speed when freeplay is present, since the linear flutter speed is not conservative.

Pitfalls

• Using the flutter speed extrapolation where LCO amplitude assessment belongs: LCO is bounded and flutter is divergent, so the damping trend and amplitude stabilization separate them, not the peak amplitude alone.
• Treating a single large spike as sustained LCO: sustained means the amplitude stabilizes within tolerance at a fixed airspeed; one transient peak is not an LCO and needs a longer record.
• Feeding a negative log decrement to the damping ratio: a negative decrement means the amplitude is growing, and the damping ratio formula is undefined for it (ValueError); re-check the data segment instead of forcing a damping value.
• Mixing units: amplitudes in mm and m, or time in minutes with amplitude in m; keep everything SI (m, s) so the growth rate is m/s and the margin is a clean ratio.
• Clearing an oscillation sitting exactly at the limit amplitude: a zero margin at the limit is not clearance; the verdict applies the required margin above the limit amplitude, not equality.
• Ignoring freeplay: a control surface with freeplay can show LCO onset below the linear flutter speed, so the linear flutter speed extrapolation alone is not conservative for the clearance.
• Passing zero or negative inputs: amplitude limits and speeds must be positive, and amplitudes non-negative; the module raises ValueError instead of returning a nonsense margin or verdict.

Behavior contract (gate 3)

The log decrement, damping ratio, amplitude growth rate, amplitude margin, LCO verdict, and freeplay onset risk logic is exercised by the gate 3 contract test: scripts/test_limit_cycle_oscillation.py against scripts/limit_cycle_oscillation.py (stdlib unittest, offline). Run: python3 scripts/test_limit_cycle_oscillation.py

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain) and CS-25 is a free EASA download; the LCO assessment methodology (log decrement damping, amplitude growth rate, amplitude margin, freeplay effects) is common flight test practice in the flutter clearance context of the 25.629 family, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.