Longitudinal stability of airships: How is the critical speed defined?

Airships are unstable in pitch above a certain speed, called the critical speed. What causes the instability, how does it manifest itself for the pilot, and how is the critical speed defined and determined?

• I've got the feeling you are going to answer this one yourself – DeltaLima Dec 22 '14 at 21:58
• hmmmm, there's no Airship Flying Handbook to go along with Airplane, Glider, Helicopter, Rotorcraft (Gyro), and Instrument – rbp Dec 22 '14 at 23:28

On page 43 of the Technical Manual of Airship Aerodynamics I found the following image:

Here $F_e$ is the resultant airforce on the hull, or the dynamical lift of the hull. As this is before the CoG, it is destabilizing. This source mentions Munk's formula being: $$M_e = Volume \cdot 0.5 \rho V^2 (k_2-k_1) \sin{2\theta}$$ Where $k_2$ and $k_1$ indicate the effect of gas movement inside the airship. I tried to trace the origin of this formula in Munk's paper, available here, but I can't immediately find a clear explanation.

The critical speed is reached when this resultant airforce equals the moment induced by the weight: $$Wh\sin{(\alpha + \theta)}=Volume \cdot 0.5 \rho V^2 (k_2-k_1) \sin{2\theta}$$

In this sense, the critical speed is the maximum speed at which the gravity moment is able to compensate the moment induced by dynamical lift. As such, one should stay below the critical speed.

• Thanks for the link to the Technical Manual! If you read on, on page 51 the topic is mentioned. What I called "critical speed" is called "reversing speed" in the manual, however. – Peter Kämpf Dec 23 '14 at 13:38
• Do you fully understand the reason mentioned? As I read it, it says that the moment induced by the elevator is quickly compensated by the static equilirbium moment, but the downward force of the elevator will push the airship down, is this correct? – ROIMaison Dec 23 '14 at 13:50
• Is this just the gas sloshing, like a long jar or tube with some water in it over a centered fulcrum? – rbp Dec 23 '14 at 17:48
• @ROIMaison: Yes. Pretty simple, right? But not intuitive. – Peter Kämpf Dec 23 '14 at 18:28
• @rbp: Soft ballonets will indeed exacerbate a static imbalance, but the real reason for the instability is the low thrust at low speed. – Peter Kämpf Dec 23 '14 at 18:30

Airships have a center of lift well above their center of mass, since engines and the gondola are mounted at the bottom. This gives them a strong static pitch stability: They will try to assume an attitude where the center of mass is right below the center of lift. Normally, the airship will be carefully trimmed to ensure that the hull is level at this attitude.

During flight both the mass and the lift will not stay constant: Sun might shine on the hull, heating and expanding the gas inside and increasing lift, the hull might be wetted by rain and become heavier, or fuel burn will lighten the ship. To stay at the desired altitude, the airship can compensate for any imbalance by creating dynamic lift. By pitching the hull up or down, up to 10% of the needed lift can be added or subtracted this way. Note that this will not only create aerodynamic lift, but it will also point engine thrust slightly up- or downwards, which will add to the lift change. If the ship is fast enough, a negative deflection of the horizontal tail surface or a shifting of masses backward will increase lift.

At low speed, however, both thrust and aerodynamic lift are much lower, but both mass and static lift are unchanged. Now the ship needs considerably more elevator deflection for the same pitch attitude change, but aerodynamic lift and engine thrust will be a lot less. The downforce at the horizontal tail must still be equal to that at high speed, since the static moment has not changed, and the lower dynamic pressure at low speed is compensated by deflecting the elevator by a bigger angle.

Below the critical speed, the elevator downforce will become dominant, and pitching the ship up will decrease total lift. The critical speed is characterized as the speed at which attitude changes will not result in lift changes, and when crossing this speed, the elevator effect on lift reverses. For this reason some books use the term "reversal speed" for the critical speed. No airship captain wants to cruise at this speed, since he will be bereft of means to change lift quickly!

This effect can create a dangerous situation if a heavy ship comes in for landing. In cruise, the missing static lift could easily be compensated by raising the nose, but now the ship has to decelerate. It will start to pick up sink speed, and raising the elevator will make matters worse. The only way to avoid a crash in this situation is the application of reverse thrust combined with a positive elevator deflection which lifts up the tail. Proper timing is of the essence: The ship must touch down at the desired location just when the reverse thrust has stopped both the vertical and the horizontal speed. This requires a trained crew and a skilled commander. If the propellers cannot run in reverse, this situation must be avoided at all cost!

The critical speed is higher the blunter the airship is. The slender Zeppelins of the first World War had a fairly low critical speed (maybe 8 m/s), but the Cargolifter's critical speed was around 20 m/s. During the design phase the required power had to be increased by 40% to raise the cruise speed by a sufficient margin above the critical speed.

Both the vertical and the horizontal tail surfaces of airships are much smaller than what would be required for natural stability, and they typically were just big enough to keep the ship under control. Aerodynamic lift is mostly created at the forward part of the hull, and lifting the nose will increase the pitch-up moment. The same goes for the sideforce: In both axes the helmsman needs to constantly correct the control surface deflection to keep the ship at the desired attitude and heading. Below the critical speed, the aerodynamic forces and the pitch-up due to the low location of the propellers will not be sufficient to overcome the static stability, so pitch excursions will be limited. Above the critical speed the aerodynamic instability will become dominant and the helmsman needs to be alert and responsive.