Phugoid Mode Analysis

Phugoid Mode Analysis (flight-mechanics/stability-control/phugoid-mode-analysis)

Use when the task is the phugoid, the long-period longitudinal mode that shows up after a speed or thrust disturbance at cruise: the aircraft slowly oscillates in airspeed and height while the angle of attack stays nearly constant. This leaf implements the Lanchester approximation, which needs only the cruise speed V and the lift-to-drag ratio L/D, and derives the phugoid natural frequency, period, drag-damping ratio, damped frequency, and the time to half amplitude in seconds and in cycles. It complements the fast-mode treatment of flight-mechanics/stability-control/short-period-mode-analysis, the derivative-based full evaluation of flight-mechanics/stability-control/dynamic-stability, and the static pitch stability background of flight-mechanics/stability-control/longitudinal-stability.

Domain quick reference

Physics: the phugoid trades kinetic and potential energy along the flight path at roughly constant angle of attack. After a speed disturbance the airplane climbs as it slows, then dives and accelerates back, so the disturbance shows up as a slow airspeed and height-velocity exchange (height up when speed down) rather than as a pitch oscillation. Lanchester modeled this exchange for small perturbations at constant cruise speed V and constant lift coefficient; drag does work each cycle and provides the damping. With g0 = 9.80665 m/s^2:

• Natural frequency: omega_p = g0 * sqrt(2) / V, in rad/s.
• Period: T_p = 2 * pi / omega_p = 2 * pi * V / (g0 * sqrt(2)), in s.
• Damping ratio (drag-damping approximation): zeta_p = 1 / (sqrt(2) * (L/D)), dimensionless. Valid for large L/D; the model assumes zeta_p is small (zeta_p at L/D 8 is about 0.088).
• Damped frequency: omega_d = omega_p * sqrt(1 - zeta_p^2), in rad/s. For cruise L/D the damping is light, so omega_d is practically equal to omega_p.
• Time to half amplitude: t_half = ln(2) / (zeta_p * omega_p), in s. The decay-rate identity zeta_p * omega_p = g0 / (V * (L/D)) gives the closed form t_half = ln(2) * V * (L/D) / g0.
• Cycles to half amplitude: N_half = t_half / T_p = ln(2) * sqrt(2) * (L/D) / (2 * pi), about 0.156 * (L/D). Independent of speed: only the lift-to-drag ratio sets how many cycles the oscillation survives.
• Validity: L/D must be at least 8 for the small-damping approximation to be credible; omega_p is assumed to sit well below the short period frequency (the standard two-timescale longitudinal split, stated as an assumption because this leaf does not compute the short period mode).

Workflow

1. Collect the cruise speed V in m/s and the cruise lift-to-drag ratio L/D at the trim condition (from flight-mechanics/stability-control/trim-analysis or cruise performance data).
2. Compute the natural frequency with phugoid_frequency and the period with phugoid_period.
3. Compute the drag-damping ratio with phugoid_damping_ratio and the damped frequency with damped_frequency.
4. Get the decay metrics: time_to_half_amplitude and cycles_to_half_amplitude.
5. Check validity with ld_valid_for_small_damping (L/D 8 floor) and record the separation assumption that omega_p is well below the short period frequency.
6. Get the complete summary with phugoid_characteristics, which returns omega_p, period, zeta_p, omega_d, t_half, cycles_half, small_damping_valid, and the separation_assumption string.
7. Confirm the deterministic checks with the contract test scripts/test_phugoid_mode_analysis.py.

Worked example

A transport cruise configuration: V = 250 m/s, L/D = 18.

• omega_p = 9.80665 * 1.41421356237 / 250 = 0.05547 rad/s.
• T_p = 2 * pi / 0.05547 = 113.3 s.
• zeta_p = 1 / (1.41421356237 * 18) = 0.03928.
• Decay rate: zeta_p * omega_p = 0.03928 * 0.05547 = 0.002179 /s, equal to g0 / (250 * 18) = 0.002179 /s.
• t_half = ln(2) / 0.002179 = 318.1 s.
• Cycles to half amplitude: N_half = 318.1 / 113.3 = 2.81.
• Validity: L/D = 18 is well above 8, and omega_p = 0.0555 rad/s is about two orders below a typical short period frequency of several rad/s, so the Lanchester model and the mode separation hold.

The interpretation: after a speed disturbance the airspeed oscillation decays to half amplitude in about 5.3 minutes (2.81 cycles of a roughly 1.9 minute period). Damping is light because the L/D of 18 means drag does little work per cycle.

Verification

• Confirm phugoid_frequency(250.0) returns 0.0554748 rad/s and phugoid_period(250.0) returns 113.2620 s, both within 1% of the worked example values.
• Confirm phugoid_damping_ratio(18.0) returns 0.0392837, t_half via time_to_half_amplitude(250.0, 18.0) returns 318.0660 s (within 1% of 318.1 s), and cycles_to_half_amplitude(18.0) returns 2.8082 (within 1% of 2.81).
• Confirm the identities: phugoid_frequency * phugoid_period equals 2 * pi; zeta_p * omega_p equals g0 / (V * (L/D)); the closed form ln(2) * V * (L/D) / g0 equals the log form ln(2) / (zeta_p * omega_p); cycles_half equals t_half / T_p; and omega_d is at or below omega_p.
• Confirm ld_valid_for_small_damping returns True at and above L/D 8 and False below.
• Confirm ValueError on V <= 0, L/D < 1 (below the oscillatory floor), non-numeric inputs, and non-positive gravity, across the frequency, damping, time, and summary functions.
• Run the contract test offline: python3 scripts/test_phugoid_mode_analysis.py (34 tests, deterministic).

Pitfalls

• Applying the model below L/D 8: the drag-damping approximation assumes small zeta, and ld_valid_for_small_damping returns False below L/D 8; the numbers are not credible there (L/D below 1 raises ValueError outright).
• Quoting t_half without the closed form check: time to half amplitude must satisfy ln(2)/(zetaomega) = ln(2)V(L/D)/g0; the decay-rate identity zeta_pomega_p = g0/(V*(L/D)) is the cross-check.
• Reading cycles to half amplitude as speed dependent: N_half = t_half/T_p depends only on L/D (about 0.156*(L/D)), so it is constant across cruise speeds for a given configuration.
• Forgetting the two-timescale assumption: the leaf assumes omega_p sits well below the short period frequency and records it as the separation_assumption string, but does not compute the short period mode itself.
• Non-positive V, L/D below 1, non-numeric inputs and non-positive gravity raise ValueError across the frequency, damping, time and summary functions.

Related leaves

• flight-mechanics/stability-control/longitudinal-stability: the static pitch stability context (C_m_alpha, neutral point) in which the phugoid operates.
• flight-mechanics/stability-control/short-period-mode-analysis: the fast pitch mode of the same longitudinal split; its separation check uses the Lanchester frequency from this leaf's model.
• flight-mechanics/stability-control/dynamic-stability: the full longitudinal mode evaluation from stability derivatives, where the phugoid frequency and damping are assessed against the non-divergent criterion.
• flight-mechanics/stability-control/trim-analysis: the trimmed cruise condition that fixes the speed and lift-to-drag inputs.

Behavior contract (gate 3)

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

python3 scripts/test_phugoid_mode_analysis.py

The test covers the worked-example anchors (natural frequency 0.05547 rad/s, period 113.3 s, damping 0.03928, time to half 318.1 s, cycles to half 2.81, all within 1%), the frequency and period round trip, the speed scaling of frequency and period, the L/D 1 oscillatory boundary, the decay-rate identity zeta_p * omega_p = g0 / (V * (L/D)), the closed-form time to half, the cycles-to-period ratio, the L/D 8 validity floor, the summary dict consistency, and ValueError rejection of non-positive and non-numeric inputs.

Compliance

• Standards referenced, not reproduced: FAR-25.181 and CS-25.181 require longitudinal dynamic stability characteristics for transport aeroplanes, with the phugoid to be non-divergent and not to increase pilot workload; the Lanchester model above is classical engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.