# What does it mean when an aircraft is statically stable but dynamically unstable?

I am reading this book: "Flight Stability and Automatic Control", second edition, Dr. Robert C. Nelson. In the chapter 2, page 42, the author wrote:

Note that the vehicle can be statically stable but dynamically unstable. Static stability, therefore, does not guarantee dynamic stability. However, for the vehicle to dynamically stable it must be statically stable.

I understand static stability, but I am not sure I understand dynamic stability. Can someone help me understand the above texts and give me an example of this case?

## 4 Answers

If an aircraft is statically stable, it will always return to equilibrium after a disturbance. But what happens after can either show instability or stability. This is where the dynamic stability comes in.

You can think of an aircraft at equilibrium at a particular speed, altitude and angle of attack and it is suddenly faced with a disturbance which changes its speed, altitude and angle of attack. If the aircraft has static stability, it will immediately seek its equilibrium state. If the said aircraft also is dynamically stable, the amplitude of its motion will reduce in time. This is called subsidence. One of the major factors that affect dynamic stability is the amount of damping in the system. From now on we will consider the aircraft already is statically stable. When there is enough damping, an aircraft will slowly in time, reduce its amplitude until the amplitude goes to zero. Here the aircraft is said to be dynamically stable. If there is less damping the oscillations increases with time and the amplitude of motion also increases. This is called divergent oscillation. In this situation, the aircraft is said to be statically stable but dynamically unstable.

The aircraft has less dynamic stability at high altitudes where aerodynamic damping is lower. A pilot can also reduce the dynamic stability of the aircraft. If his/ her inputs are close to the natural frequency of the aircraft it can add energy to the system and the divergence increases. This is called Pilot induced oscillation (PIO).

Statically and dynamically stable.

Statically stable but dynamically unstable.

It is important to keep in mind that an aircraft, while it can be statically stable and dynamically unstable, it cannot be the other way around. That is, an aircraft can never be statically unstable and dynamically stable.

• Possible improvement-- first sentence-- change "it will always return to equilibrium" to "it will always seek to return to equilibrium"... if it has truly returned to equilibrium then it will stay there till the next disturbance-- – quiet flyer Jan 7 at 15:44
• Would it be possible for an aircraft to settle into an oscillation which reaches a certain amplitude and then remains there? Would that be classified as dynamically stable or unstable? – supercat Jan 7 at 20:47
• @supercat not for control theory, however all material components have maximum deflections etc, and once those are reached the control system cannot react further. This might mean that a wing bends in such a shape that the dynamic stability is removed. Another reason might be that you have a secondary control system acting "over" the basic system: which adds stability but works at much lower frequency. But those effects are ignored when discussing stability of an aircraft: you just don't want these to happen or depend on them. – paul23 Jan 7 at 21:02
• @paul23: In a system of linear differential equations, a system will either be overdamped, underdamped, or critically damped; even when flying within normal parameters, however, an airplane's behavior won't be perfectly modeled by a system of linear differential equation. Many real-world non-linear systems (including most purpose-designed oscillators) can oscillate with closed-loop gains that are greater than 1 for part of each cycle and less than one for other parts. – supercat Jan 7 at 21:09
• @supercat If the aircraft reaches a disturbance and stays at the point where the disturbance threw it to, the aircraft is said to be both statically and dynamically neutral. Airbus aircraft control laws are actually designed this way. It behaves as if it has neutral stability. Within the protection envelope if you make an input the aircraft remains there until the pilot brings in a change. – Anas Maaz Jan 8 at 4:06

Static stability means that a deviation from a trimmed state produces forces which return the system to this trimmed state.

If these forces produce an overshoot which increases over time, such that the system oscillates around this trim point with increasing amplitude, the system is dynamically unstable. The long period oscillation (phygoid) of gliders is often unstable because their L/D is high enough to push them into unstable territory.

Dynamic stability means that the oscillations die down over time. Without static stability the system would simply leave the trimmed state without any tendency to return. This tendency to return is the prerequisite for the oscillation, whether stable or unstable.

• +1 for the reference to L/D, I've often wondered whether when cloud-flying in a glider especially without artificial horizon, it would be helpful to lower the gear and possibly also crack the spoilers open a bit? Maybe make some sort of block or other mechanical aid to hold the spoilers slightly open without constantly occupying the pilot's left hand? (Just a theoretical question for me at present but I know in some other parts of the world cloud-flying in gliders is still somewhat routine...) – quiet flyer Jan 7 at 14:43
• That'll just make more noise as you become disoriented and go into a spiral dive past Vne, which is guaranteed if you don't have gyro instruments, and likely eventually even if u had an electric turn coordinator.If the glider has speed limiting dive brakes/spoilers, meaning the plane is incapable of exceeding Vne even straight down, and most modern glider do, your best bet is to just dump full spoilers and let it go and wait to come out of the bottom. If you try to maintain controlled flight in cloud for any length of time with spoilers in or partly out, you will come out the bottom in pieces. – John K Jan 7 at 16:37
• Well, I don't know about all that, flying w/ turn rate indicator as sole gyro instrument has long been common practice for many decades (like since the 1940's) in gliders in some countries in Europe and the UK. – quiet flyer Jan 7 at 17:42
• @JohnK: I met an old pilot who claimed he could maintain controlled flight in clouds without a gyro, but required extreme concentration. I wouldn't want to test it, but I can see how it might be possible. Mathematically, it's integrating over the bank indicator (otherwise known as the turn coordination indicator) and the compass. – Joshua Jan 7 at 20:29
• @Joshua You have to have some kind of reference to tell your eyes you're turning, because the inner ear is useless for that purpose. If you sustain a right turn say for a full circle, and the cochleal fluid stabilizes with the rotation, when you roll out you get an intense sensation of turning the other way and almost all pilots will unconsciously start to roll back into the turn. It's supposedly possible using the compass to tell you you are turning, but you would have to be very good at anticipating and accounting for turning errors, and if there was any turbulence you'd be screwed. – John K Jan 7 at 22:26

Lots of good information in answers posted so far but I think it is also useful to point out that with no static stability (in the pitch axis), the aircraft wouldn't be trimmable. With positive static stability (in the pitch axis), you can trim for a given airspeed, and if you then pull the stick aft, you'll feel an increasing forward pressure on the stick, and if you then release the stick, the nose will drop and the airspeed will increase. Likewise, if you trim for a given airspeed and then push the stick forward, you'll feel an increasing aft pressure on the stick, and if you then release the stick, the nose will rise and the airspeed will decrease. Static stability makes that possible.

But that's not sufficient to ensure that the aircraft will actually settle down at the trim airspeed after a disturbance, if you continue to exert no forward or aft pressure on the stick or yoke. As other answers have noted, if the aircraft is statically stable but dynamically unstable (in the pitch axis), after an initial disturbance, it can enter a pitch "phugoid" oscillation that keeps getting bigger, rather than damping out, unless the pilot intervenes. Here's a video showing such a thing.

It's worth noting that these effect can be subtle enough that a pilot can accumulate many hours of hands-on flight time in some particular aircraft, without actually having any idea whether it is dynamically stable or unstable in the hands-off case, in any given configuration.

• why does that real airplane not have dynamic stability? I thought engineers would design an airplane that has ocsillation damped out? – Dat Jan 8 at 4:19
• @Dat -- it's a glider, and see the reference to gliders in other answer here aviation.stackexchange.com/a/83369/34686 – quiet flyer Jan 8 at 13:48

Static stability is the initial response of a plane to an instant impulse (like a turbulence), while dynamic stability is how a plane responds over time to a disturbance.

Dynamic stability can be verified by pulling/pushing one flight control surface and instantly letting it go: oscillations on the related axis can increase in amplitude, decrease in amplitude or maintain the same amplitude.

If, over time, the plane is stable then, the initial response is stable, but you can't say the opposite.

I don't know to explain it mathematically (if it's what you are looking for).