Phugoid Damping⚠ unverified

Understanding the Parameters

• What the phugoid is — a long-period oscillation (typically 20–60 s) at roughly constant angle of attack, in which the aircraft alternately dives (gaining speed, then extra lift pulls it back up), climbs (losing speed, then falling lift lets it sink again), and repeats. The nose barely changes attitude relative to the flight path; it is the speed and height that oscillate. Contrast the fast short-period mode, which is an angle-of-attack oscillation at nearly constant speed.

• Why damping $\sim 1/(L/D)$ — the oscillation is damped by drag: as the aircraft speeds up in the dive, drag rises and removes energy; as it slows in the climb, drag falls. This drag modulation bleeds energy from the swap, damping the oscillation. A high-$L/D$ aircraft (a sailplane, a clean jet) has little drag to do the damping, so its phugoid is very lightly damped and persists for many cycles; a draggy aircraft damps it quickly. Hence $\zeta_{ph} \propto C_D/C_L = 1/(L/D)$.

• Trim lift coefficient $C_{L0}$ — the lift coefficient at the flight condition, which for level flight encodes the speed: high $C_{L0}$ means slow flight (near stall), low $C_{L0}$ means fast. Since it sits in the denominator, the phugoid is less damped at low speed / high $C_{L0}$.

• Why light damping is usually acceptable — the phugoid is so slow that a pilot corrects it without even noticing, and even undamped it is not dangerous (the aircraft does not diverge, it oscillates). Handling specs therefore tolerate low or even slightly negative phugoid damping, unlike the short period, which must be well damped. Autopilots suppress it trivially.

• Zero-lift vs total drag — the registry's use of $C_{D0}$ (parasite drag only) instead of the full $C_D = C_{D0} + C_{Di}$ slightly under-estimates the damping, since induced drag also contributes. Combined with the $\tfrac12$-vs-$\tfrac{1}{\sqrt2}$ constant, treat the output as a rough indicator, not a precise damping ratio.

Derivation (Approaching a Proof)

Lanchester's energy picture (1908). Frederick Lanchester modelled the phugoid as an exchange between kinetic and potential energy at constant angle of attack and constant total energy (to first order). If lift always equals the value needed for the instantaneous speed, an aircraft displaced in speed/height oscillates like a bead on a frictionless track — a conservative oscillator. Working out the restoring dynamics gives the phugoid natural frequency

$$\omega_{ph} \approx \frac{g\sqrt2}{V},$$

where $V$ is the trim speed — remarkably, independent of the aircraft, depending only on speed and gravity. The period is $T_{ph} = 2\pi/\omega_{ph} \approx \dfrac{\pi\sqrt2\,V}{g}$ (about 0.14 s per m/s of speed — e.g. ~36 s at 80 m/s).

Adding drag to get damping. The idealised phugoid is undamped; damping comes from drag doing net work over a cycle. Including the drag force in the energy balance and linearising, the damping ratio works out to

$$\zeta_{ph} = \frac{1}{\sqrt2}\,\frac{C_D}{C_L} = \frac{1}{\sqrt2}\,\frac{1}{L/D}.$$

The result is beautifully simple: phugoid damping is inversely proportional to the lift-to-drag ratio. The registry adopts the same $\propto C_D/C_L$ form but writes the constant as $\tfrac12$ and uses $C_{D0}$:

$$\zeta_{ph} \approx \frac{C_{D0}}{2\,C_{L0}}. \qquad\blacksquare$$

The two agree to within the $\tfrac12$-vs-$\tfrac{1}{\sqrt2}$ factor ($\approx 30\%$) and the drag definition — close enough to show the physics (low $L/D$ damps the phugoid), not close enough for precise dynamics.

Dimensional check. $C_{D0}$ and $C_{L0}$ are both dimensionless, so their ratio — and $\zeta_{ph}$ — is dimensionless ✓, as a damping ratio must be.

History and Development

• Lanchester coins the phugoid (1908). Frederick W. Lanchester, in Aerodonetics, first analysed this mode and named it. He intended "phugoid" from a Greek root he believed meant "flight"; it actually derives from a word for fleeing, making the name a well-known etymological mistake — but it stuck.

• A triumph of simple modelling. The result $\omega_{ph} \approx g\sqrt2/V$ and $\zeta_{ph} \approx 1/(\sqrt2\,L/D)$ is a celebrated example of extracting the essential physics from a complex system with an energy argument — taught in every flight-dynamics course as the archetype of the "constant-$\alpha$, energy-exchange" approximation.

• Handling qualities. Because the phugoid is slow and benign, handling-qualities standards (Cooper–Harper, MIL-F-8785/MIL-STD-1797) set only loose requirements on it — typically that it be at worst mildly divergent — in sharp contrast to the tight damping demanded of the short-period mode. Modern flight-control systems damp it as an afterthought, but its low natural damping remains why a hand-flown aircraft gently hunts in pitch.

Related Concepts: Static Margin, Dutch Roll Wn, Damping Ratio, Lift-to-Drag Ratio, Drag Force, Lift Force, Natural Frequency mass-spring

Notes: The phugoid = slow (20–60 s) longitudinal oscillation exchanging altitude ↔ airspeed at ~constant $\alpha$; damped by drag, so $\zeta_{ph}\sim1/(L/D)$. Lanchester: $\omega_{ph}\approx g\sqrt2/V$, $\zeta_{ph}=\tfrac{1}{\sqrt2}C_D/C_L$. Registry uses $C_{D0}/(2C_{L0})$ — same form, constant $\tfrac12$ vs $\tfrac1{\sqrt2}$, and zero-lift drag vs total (both make it a rough estimate). Lightly damped but benign; handling specs are lenient on it.