Phugoid Mode Period Calculator
Solve the classic phugoid period approximation Tph ≈ 2π·√2·V / g for period, velocity or gravitational acceleration — with a clean flight-path schematic and a simple period-versus-speed chart.
A slow interchange of height and speed
The phugoid is a long-period oscillation in which the aircraft trades altitude for airspeed and back again, all at nearly constant angle of attack. The flight path looks like a gentle roller-coaster with the aircraft pitching nose-down while descending and nose-up while climbing.
Period grows with speed
The formula shows the period is directly proportional to V. Doubling airspeed doubles the phugoid period — the oscillation becomes slower and lazier the faster you fly. That is why the phugoid is easy to miss at cruise but hard to ignore at approach speed.
Weaker gravity lengthens the period
Because g appears in the denominator, a lower gravitational acceleration produces a longer phugoid period. On the Moon (g ≈ 1.62 m/s²), the same aircraft at the same speed would have a phugoid period six times longer than on Earth.
A nearly-neutral mode
Because the phugoid involves almost no change in angle of attack, it is only lightly damped — the aircraft's natural damping is very small. Real designs add pitch damping augmentation or autopilot speed-hold loops to suppress it. This approximation captures the period but not the damping.
How to Use This Calculator
The phugoid period approximation links three quantities: the period Tph, the flight velocity V, and the gravitational acceleration g. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick Tph (period), V (velocity), or g (gravity) — the tool rearranges Tph ≈ 2π·√2·V/g automatically for whichever you choose.
2. Use realistic presets
Presets cover typical values: periods from 20 s to 120 s, velocities from 50 to 250 m/s, and gravitational accelerations for the Moon, Mars, and Earth.
3. Read the result and verdict
The result card shows the period in seconds plus the equivalent frequency, with a colour-coded verdict classifying it as typical transport, typical GA, or atypical and worth checking.
4. Check the flight-path schematic
The clean schematic shows the sinusoidal phugoid flight path with the aircraft moving along it, the crest and trough labelled, and the period marked on a dimension line beneath.
5. Switch to Advanced for depth
Advanced mode adds a chart of Tph vs. g, a sensitivity check on a +1% input change, and companion metrics including the frequency in Hz, cycles per minute, and altitude excursion estimate.
6. Cross-check with reference tables
The reference section includes a period lookup grid, typical phugoid periods by aircraft class, and a variable reference table with notes on when the approximation is valid.
What This Tool Offers
Built around the classic phugoid period approximation Tph ≈ 2π·√2·V/g, one of the simplest and most useful results in longitudinal flight dynamics.
Three-way solving
Solve for period, flight velocity, or gravitational acceleration. The calculator rearranges the relation automatically for whichever you choose.
Clean flight-path schematic
A simple sinusoidal flight path with the aircraft tracing the cycle, crest and trough labelled, and the period drawn as a dimension line beneath.
Period classification
Each result is classified against typical phugoid periods for different aircraft classes, with a coloured badge for immediate interpretation.
Multi-body gravity presets
Presets for Moon, Mars, and Earth gravity — useful for academic studies and for considering the mode on other planets.
Dual comparison charts
Period vs. velocity for three gravity values, plus an advanced period vs. gravity chart for three velocities.
Reference tables
A period lookup grid, typical values by aircraft class, and a variable reference with notes on the approximation's accuracy.
Reference Tables & Notes
Period lookup, typical values by aircraft class, and a variable reference table.
Phugoid Period Lookup
Computed from Tph = 2π·√2·V/g with g = 9.80665 m/s² (Earth). Values in seconds.
| V (m/s) | Earth g = 9.807 | Mars g = 3.71 | Moon g = 1.62 |
|---|
Typical Phugoid Periods by Aircraft Class
Representative values at cruise condition on Earth. Actual periods vary with speed and loading.
| Class | Typical cruise V (m/s) | Approx. Tph (s) | Frequency (Hz) |
|---|---|---|---|
| Light GA (Cessna 172 class) | 55 – 65 | 25 – 30 | 0.033 – 0.040 |
| Regional turboprop | 120 – 140 | 55 – 65 | 0.015 – 0.018 |
| Business jet | 200 – 230 | 90 – 105 | 0.0095 – 0.011 |
| Narrow-body transport | 230 – 250 | 105 – 115 | 0.0087 – 0.0095 |
| Wide-body transport | 250 – 260 | 115 – 120 | 0.0083 – 0.0087 |
| Supersonic transport | 600 | ≈ 275 | ≈ 0.0036 |
| Sailplane (slow flight) | 25 – 35 | 11 – 16 | 0.062 – 0.090 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| Tph | Phugoid period | 10 – 300 | s |
| V | Flight velocity (true airspeed) | 25 – 600 | m/s |
| g | Gravitational acceleration | 1.62 – 9.81 | m/s² |
| 2π·√2 / g | Conversion factor (Earth) | ≈ 0.9064 | s per (m/s) |
| ωph | Phugoid natural frequency | 0.02 – 0.6 | rad/s |
Notes on the Approximation
The formula Tph ≈ 2π·√2·V/g is derived from Lanchester's classic analysis of the phugoid and assumes the oscillation occurs at constant angle of attack, that the aircraft is in level flight, and that thrust remains constant during the cycle. It ignores damping and the pitch-damping coupling, so it predicts the period accurately to within roughly 10–20% for most conventional aircraft but says nothing about how quickly the oscillation decays.
The full linearised expression (from the longitudinal characteristic equation) also involves lift-curve slope, drag, and thrust terms, but the simpler form is remarkably accurate because the phugoid is dominated by the aircraft's inertial exchange between kinetic energy (speed) and potential energy (altitude). The quantity 2π√2 is approximately 8.886, so the period is roughly 0.906 × V seconds when V is in m/s and g is Earth-standard.
What Does T_ph ≈ 2π·√2·V / g Measure?
The phugoid is a slow longitudinal oscillation in which an aircraft exchanges altitude for airspeed while holding its angle of attack nearly constant. The phugoid period Tph is the time for one complete cycle of this oscillation. It is one of the two natural longitudinal modes of an aircraft — the other being the short period — and it is by far the slower and more lightly damped of the two.
Phugoid Period Formula
The classic approximation is:
Rearranged to solve for the other two unknowns:
g = 2π · √2 · V / Tph
Since 2π·√2 ≈ 8.8858, this is often written as Tph ≈ 8.886 · V / g. For Earth gravity (g = 9.80665 m/s²), the constant becomes Tph ≈ 0.9064 · V, meaning the period in seconds is roughly 0.906 times the airspeed in m/s.
Phugoid Variables
| Symbol | Meaning | Notes |
|---|---|---|
| Tph | Phugoid period | Time for one complete altitude-airspeed cycle |
| V | Flight velocity (true airspeed) | Level-flight trim speed |
| g | Gravitational acceleration | 9.80665 m/s² on Earth by convention |
| ωph | Phugoid natural frequency | ωph = 2π / Tph |
How Velocity and Gravity Affect the Period
Because the relation is linear in V and inversely linear in g, the phugoid period responds predictably to both. Doubling the flight velocity doubles the period — the oscillation becomes slower and lazier. Doubling the gravitational acceleration halves the period — the oscillation becomes faster because the restoring force per unit mass is larger. On the Moon (g = 1.62 m/s²), the period is roughly six times longer than on Earth; on Mars (g = 3.71 m/s²) it is about 2.6 times longer.
In practice, the phugoid is heavily influenced by flight condition. It is longest at high-speed cruise and shortest at low-speed approach, and its damping changes dramatically with thrust setting, drag, and speed stability. Autopilots typically suppress it with a pitch-attitude-and-airspeed-hold loop, and many aircraft have a speed-stability augmentation system that artificially increases the damping to give the mode acceptable handling qualities.
Phugoid Period Calculation Examples
2π · √2 = 8.88577
Tph = 8.88577 × 85 / 9.80665
Tph = 77.01 s
V = 60 × 9.80665 / 8.88577
V = 66.23 m/s (= 238 km/h)
g = 8.88577 × 100 / 100
g = 8.886 m/s² (a Mars-like gravity)
Frequently Asked Questions
The phugoid is a slow longitudinal oscillation in which the aircraft climbs and descends in a series of gentle waves while its airspeed rises and falls. The angle of attack stays nearly constant, so the aircraft stays at essentially the same trim condition. Pilots feel it as a slow porpoising motion. Because the damping is usually very light, it can persist for many cycles if not actively corrected.
Because the phugoid is fundamentally an energy-exchange oscillation. During the cycle, kinetic energy (½mV²) is traded for potential energy (mgh) and back. The rate at which this exchange happens scales with the flight speed V, and the restoring force scales with g. A dimensional analysis of these two quantities gives the period as a constant times V/g, and the constant turns out to be 2π√2 when the full derivation is done.
For conventional subsonic aircraft in level flight, the formula is typically accurate to within 10–20%. It becomes less accurate when the aircraft is very heavily loaded, flying at speeds close to the stall, or operating at high angles of attack. It also says nothing about the damping, which is a separate and equally important handling-quality question.
The short period is fast (period of a few seconds) and involves mostly angle of attack with little speed change. The phugoid is slow (period of 30–120 seconds for most aircraft) and involves mostly speed and altitude with little angle-of-attack change. The two modes are widely separated in frequency, which is why they can be analysed independently and why linearised flight-dynamics models treat them as decoupled from each other.
Because the period is proportional to V, the smallest realistic periods occur at low speed. A sailplane flying at 25 m/s has a phugoid period of around 23 s; a slow-flying training aircraft at 40 m/s has one around 36 s. Going below about 20 s requires either very low airspeed (below stall for most aircraft) or artificially high gravity.
Yes. Certification requirements such as MIL-F-8785C and FAR/JAR Part 25 set minimum phugoid damping for different aircraft classes. When damping is too low, the aircraft takes many cycles to settle after a disturbance, which is annoying and can lead to airsickness. Aircraft with naturally low damping (large transports, in particular) use yaw dampers or speed-stability augmentation systems to improve it.
The amplitude depends on the initial disturbance and the damping, not on the period itself. For a lightly damped aircraft with a 1 m/s perturbation in vertical speed, the resulting altitude excursion is on the order of V/(g·√2) metres — a few metres per metre-per-second of initial perturbation. A hard elevator pull followed by a hold can produce excursions of tens of metres, which is why the phugoid is visible in flight-test traces even when it is not felt by the pilot.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T_ph | Phugoid period | s |
| V | Trim airspeed | m/s |
| g | Gravitational acceleration | m/s2 |
What it means
The phugoid mode is a low‑frequency, usually poorly damped oscillation involving exchange between kinetic and potential energy. Its period is approximately proportional to speed, independent of altitude. The approximation assumes small perturbations and no drag. The phugoid is important for pilot control and for autopilot design. It is often lightly damped and may require active damping. Understanding phugoid behaviour is essential for flight test and for evaluating longitudinal stability.
Worked example
Phugoid Period – Two Examples
Real‑World| Parameter | Value |
|---|---|
| V | 60 m/s |
| Parameter | Value |
|---|---|
| V | 100 |
Common mistakes
- Phugoid mode period (approx.): T_ph ≈ 2π·√2 · V / g.
- V: Trim speed (m/s).
- g: Gravitational acceleration.
- Long‑period oscillation, typically poorly damped.
- Approximate formula; full analysis includes derivatives.
Applications
The phugoid mode period, T_ph ≈ 2π√2·V/g, is the long‑period oscillatory mode involving exchange of kinetic and potential energy. It is a slow, lightly damped mode that affects speed and altitude variations. Engineers use this to ensure adequate damping (through auto‑throttle or pilot input) and to assess ride quality. The period depends on airspeed; faster aircraft have longer periods. By understanding phugoid dynamics, aerospace engineers can design control laws that either damp the mode or make it acceptable to pilots, improving flight safety and comfort.
- Longitudinal dynamic stability analysis
- Auto‑throttle and speed control system design
- Pilot‑induced oscillation avoidance
- Flight test mode identification
- Ride quality and passenger comfort assessment
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