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What is an acceptable settling time for the phugoid mode in a large aircraft?

As a more specific question, what is the settling time when the aircraft (say, a B747 or an A320) switches from Climb to Cruise phase at the top-of-climb?

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Here is the response of a 180 passenger high-subsonic twin-jet simulator: the first period takes about 87 seconds.

Conditions:

From the trimmed position, excite the phugoid mode by applying longitudinal control in one direction in order to change airspeed by approximately 10 kts, then release.

Initial conditions:

  • COG 0.2205 MAC
  • Pressure altitude 35,000 ft
  • Engine N1 rotor speed 87.47% (both engines)
  • Calibrated Airspeed 264.54 kts
  • Gross weight 66,360 kg

enter image description here

EDIT

To answer the additional question of when the resulting motion amplitude would be below 0.1 deg, I made a kind of a digital simulation according to this block diagram:

enter image description here

and then tuned spring stiffness C, inverse inertia 1/M, and damping constant K until the resulting response overlaid the original aircraft one. The resulting motion gets to a max. amplitude of 0.1 deg in about 1,600 sec - 27 minutes.

enter image description here

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  • $\begingroup$ What do you think is the approx. settling time, maybe when the amplitude goes below 0.1 deg? $\endgroup$
    – Raj
    Commented Nov 20, 2017 at 17:14
  • $\begingroup$ The peaks setlle down over time, a lime through the peaks looks like a first order response. I'll see if I can make a ROM on < 0.1 deg later in the day. $\endgroup$
    – Koyovis
    Commented Nov 21, 2017 at 0:05
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Normally this is in the tens of seconds.

An approximation for the undamped phygoid frequency is: $$\omega_P = \frac{2\cdot g}{v}\cdot \left(1-\frac{c_{mv}\cdot c_{L\alpha}}{c_L\cdot c_{m\alpha}}\right)$$

Nomenclature:
$g\;\;\;\;\;$Gravitational acceleration
$v\;\;\;\;\;$Flight speed
$c_{mv}\;\;$Speed stability (change in pitch moment over speed)
$c_{L\alpha}\;\;$Lift curve slope
$c_{L}\;\;\;$Lift coefficient
$c_{m\alpha}\;$Static stability (change in pitch moment over angle of attack)

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  • $\begingroup$ Isn't straight and level flight actually made up of infinitesimally small Phugoid oscillations? Do air-planes really fly perfectly straight? Could you enlighten me? $\endgroup$
    – user18035
    Commented Nov 20, 2017 at 13:28
  • $\begingroup$ Can you comment on the oscillation just after the top-of-climb? Is it phugoidal in nature? And it follows the same frequency formulae as is in the answer? $\endgroup$
    – Raj
    Commented Nov 20, 2017 at 17:18
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    $\begingroup$ @AnandS: I fail to see a difference between infinitesimally small oscillations and straight flight. Damping ensures that those oscillations die down (except in some gliders), so yes, a well-designed airplane does fly perfectly straight if we neglect that it follows the curvature of the earth. $\endgroup$
    – Peter Kämpf
    Commented Nov 21, 2017 at 8:10
  • $\begingroup$ @Raj: Those oscillations (if they occur at all) are short-period oscillations. $\endgroup$
    – Peter Kämpf
    Commented Nov 21, 2017 at 8:11
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Let's first address OP's first question:

What is an acceptable settling time for the phugoid mode in a large aircraft?

Phugoid is a hands-off (i.e. pilot and autopilot off) rigid-body mode, with or without Stability Augmentation System. MIL-STD-1797B specifies a damping ratio of at least 0.04 for Level 1 handling quality. The typical phugoid period is around 90 seconds. This converts to a settling time (within 2% of trim speed) of around 24 minutes. This is so long which makes settling time itself not a particularly good measure for phugoid.

Now onto,

What is the settling time when the aircraft (say, a B747 or an A320) switches from Climb to Cruise phase at the top-of-climb?

Leveling off from climb is not hands-off. A well-tuned autopilot should be deadbeat with minimal overshoot and oscillation. The time scale is also vastly different; we are talking about settling time on the order of a seconds (not half an hour). And this not phugoid.

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  • $\begingroup$ Mil-F-8785C is rescinded, however exact same requirements carried over to new milspec. So your answer still holds true. Strangely enough, civilian counterpart Part 25 section 25.181 is not even bothering to give any requirements for phugoid. I think that’s because phugoid is a nuisance mode. $\endgroup$
    – Kolom
    Commented Oct 31, 2019 at 21:45
  • $\begingroup$ @Kolom Is MIL-STD-1797A the replacement? $\endgroup$
    – JZYL
    Commented Oct 31, 2019 at 22:04
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    $\begingroup$ 1797B is the new one $\endgroup$
    – Kolom
    Commented Oct 31, 2019 at 22:11
  • $\begingroup$ This should be the accepted answer. $\endgroup$
    – Koyovis
    Commented Nov 4, 2019 at 5:37

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