Stability

Static and dynamic stability

There are two ways of being stable: statically and dynamically.
Static stability means that your autogyro wants to go back to the original state after it encountered a disturbance. If, for instance, a gust makes the rotor tilt backward, a statically stable behaviour would be that the rotor tilted forward again
Dynamic stability means that the whole reaction to a disturbance is damped out. Examples of dynamic instability are "Pilot Induced Oscilation" (or PIO) and "ground resonance".

Things that can be (un-)stable

This is not as stupid as it sounds. Most people think of an autogyro as "stable" or "unstable", but there is more to it than that. The autogyro consists of a fuselage and a rotor that are largely independent on each other. In gusty situations, the fuselage and rotor could react in different ways. There are therefore three things to consider in stability questions: the fuselage, the rotor, and the whole aircraft. These three items have totally different mechanisms of stability, as we will discuss in the next sections.

Fuselage Stability

The fuselage hangs like a pendulum under the rotor. As long as the airspeed remains low and the aerodynamic forces therefore remain small, gravity and rotor thrust will be the main forces acting upon the fuselage. Note that this mechanism only leads to stability in pitch and roll, not in directional stability. Directional stability is completely governed by aerodynamic forces.
As the airspeed of the autogyro increases, the aerodynamic forces grow and the situation changes. It is comparable with a piece of rope that you hold by one end. If you hold it still in still air, it just hangs down. If you hold it in a firm wind, it will not only hang at an angle, it will also flutter! There are two ways to overcome this flutter:
  1. Add a weight at the free end, so that aerodynamic forces will be small with respect to the force of gravity. This only moves the instability to higher airspeeds.
  2. Give the piece of rope an aerodynamically stable shape.
So how do we make the fuselage stable? Simple: you can make anything stable with a proper tail. Have you ever seen an autogyro without a vertical tailplane? Well, that's my point. As I wrote earlier, the directional stability of the fuselage is completely dependent on aerodynamic forces. So every autogyro has a large vertical tailplane with a rudder that is located in the slipstream of the propeller. It is located in the propeller's slipstream, because the propeller will still blow the tail when the aircraft has no airspeed, keeping it controllable. The propeller's slipstream really helps to increase the tail's effectiveness. As a fixed-wing pilot (fixed wing aircraft do have "real" horizontal tailplanes), I have experienced that an aircraft is much more sensitive to gusts if you shut down the engine.
It is therefore a bit odd that many autogyros do not have large horizontal tailplanes. I think many autogyro manufacturers have relied on the gravitational forces to keep the fuselage stable in pitch. The never-exceed-speed (also denoted vne and visible as a red section on the airspeed indicator) in the autogyro's flight manual should keep the pilot away from airspeeds that cause instabilities. Especially Power-Pushover (PPO) problems have caused more and more people to ask for grown-up horizontal tailplanes. However, large horizontal tailplanes are not the ultimate answer to that problem. They do help stability, but PPO is more complicated and there are more forces involved.

Rotor Stability and Rotor Trim

You may have noticed that autogyro rotors are not located on top of the mast, but somewhat after the top on the torque tube (the torque tube is the tube that is attached to the mast and to the control rods). This helps the rotor to be stable: If, for some reasons, the load on the rotor increases, the torque tube is tilted forward and the rotor angle-of attack is decreased. Therefore the rotor load is decreased: the rotor "gives in". If the rotor was mounted in front of the mast on the torque tube, an increase in rotor load would cause a further increase in rotor load, wich would again increase the rotor load, until something would break.
With this mechanism, the rotor has a "preferred" angle-of-attack. The pilot can change this stable angle-of-attack by means of the rotor trim. The rotor trim usually consists of one or more springs that are attached to either the control rods or the rear end of the torque tube. Like the trim in a fixed-wing aircraft, this is used to "trim away" the forces that the pilot has to apply on the stick.

Aircraft Stability

I haven't written this section yet, but a friend of mine mailed me this article by Jean Fourcade of the PRA. This article describes the forces that play a role in the static stability of the autogyro as a whole. Please note that the "circle-with-blocks" symbol in the pictures denote the position of the center of gravity of the whole aircraft.

There are two ways of studying stability in aircraft science which are the "static stability" and the "dynamic stability".
Static stability, as the name implies, is not governed by mass or inertia characteristics of the aircraft. It's only a geometric criterion.
Dynamic stability is the most complete study of stability but is also, by far, the most complicated. It is called dynamic because you have to compute the full equations of motion of the aircraft in trimmed condition and see how the aircraft responds to an arbitrary perturbation.
The dynamic stability is the most complete because an aircraft could be statically stable while unstable from dynamic point of view.
Now what are technics used to compute static stability ?
It is a common use in engineer science that when you want to study stability relative to a given parameter, you plot a curve where the x coordinate is the parameter and the y coordinated is the acceleration of this parameter (second time derivative) or something proportional to this acceleration. In longitudinal stability there are mainly to parameters which are involved which are the air speed and the angle of attack of the aircraft. Let us consider only the angle of attack which is the most important.
To study the stability of the gyro relative to the angle of attack we have to plot the gyro pitching moment of forces which act on the aircraft and see how these moments varies. Sign convention is generally : positive values on y axis are nose up pitching moment, negative values are nose down pitching moment.
It is obvious that to flight in trimmed condition, the total pitching moment computed with all forces acting on the gyro is equal to zero. Then, the point on the previous curve on trimmed condition is on the x axis.
Now suppose that you have a perturbation which changes the angle of attack of your gyro (a vertical gust for example). As your angle of attack has changed, you are no longer in trimmed condition (you have moved on the curve). Then it will appear a non-zero pitching moment which will act on the gyro and on the angle of attack.
It can be easily understood that to be stable, the pithing moment which appears must act to bring back the gyro in its previous trimmed condition, i.e. if we consider an increase of the angle of attack the pitching moment must reduce the angle of attack and therefore must be negative (nose down) ; on the opposite, if we consider a decrease of the angle of attack, the pitching moment must be positive (nose up). In others words the slope of the curve (or derivative of the pitching moment relative to the angle of attack) must be negative.
Now that we have define the condition of static stability we can apply it to the forces acting on the gyro. There are mainly 4 forces acting on a gyro which are :
  1. the engine thrust,
  2. the horizontal stabilizer forces (lift and drag),
  3. the body drag,
  4. the rotor thrust (lift and drag).
To compute the stability given by each of theses forces we have to evaluate derivative of their pitching moment and see what is the best placement of the CG so that these derivatives are negatives.
  1. Engine thrust.
    The thrust of the engine depends of the speed of the gyro but is not very sensitive to the angle of attack. We can consider, at first order, that the moment of the engine thrust is independent of the angle of attack and therefore the derivative of this moment is equal to zero.
    Therefore, the engine, by itself has no impact on longitudinal stability. The CG may be up or down the thrust line or before or behind the propeller. We will go back on that assertion later on.
  2. Body drag.
    It can be demonstrated that to have a negative derivative moment, the center of pressure (the point where the drag is applied) must be behind the CG. This condition is also very important in lateral stability. To do that, we must have a sufficient vertical tail surface. It is so important for lateral stability that all gyros respect this condition.
  3. Horizontal stabilizer.
    Everybody will agree that an horizontal stabilizer adds stability. The efficiency of the stabilizer is greater when you have a long cross arm and when the speed increase because the lift is proportional to the square of the air speed. As speeds of gyro are generally not very high it is better to place the stabilizer in the slipstream of the propeller. We will go back on the use of the horizontal stabilizer later on.
  4. Rotor thrust
    This is the point because the main problem (and the main difference between low profile gyros and high profile gyros) is coming from the horizontal placement of the CG relative to the rotor thrust line. The phenomena which is involved is called "instability of the rotor relative to the angle of attack" and is well known in helicopter world.
    We have to study two cases. First case : the CG is in front of the rotor thrust line, second case the CG is behind the thrust line. The first configuration is stable, the second is unstable.
    download chstab2.gif (0.9K)
    unstable C.G. position
    Let's consider that the CG of the gyro is behind the rotor thrust line and that you are in forward flight. Suppose that you have a gust which increases the angle of attack. An increase of the angle of attack of the rotor will increase the thrust of the rotor. It will also increase the difference of thrust between the advancing blade and the retreating blade which will then increase the flapping angle.
    Now what will append on the rotor pitching moment ?
    If the CG is behind the thrust line of the rotor, the pitching moment induce by the rotor is positive (nose up). An increase of the thrust line will increase the moment. The rotor thrust is, at first order, perpendicular to the tip path plane, and then an increase of the flapping angle will rock the thrust behind. This will increase the cross arm of the moment and then will also increase the moment. Therefore, the derivative of the moment relative to the angle of attack is positive. You are in unstable condition.
    To summarize when CG is behind the thrust line :
    Increase AOA => increase thrust and flapping => both increase moment => increase AOA : unstable

    download chstab1.gif (0.9K)
    stable C.G. position
    Now imagine that the CG is in front of the thrust line of the rotor. This time the rotor induces a negative pitching moment (nose down). When a gust increases the angle of attack the rotor reacts the same way as before. We have an increase of the thrust and the flapping angle. But how comes it this time on the pitching moment ?
    An increase of thrust will increase the absolute value of the moment (more nose down moment). As the moment is negative, this will lower the moment.
    The flapping angle will reduce the cross arm and then decrease the absolute value of the moment. This time, the two phenomena don't act in the same way but it can be demonstrated that it is the variation of thrust which is most important. So we are in stable condition.
    Increase AOA => increase thrust and flapping => decrease moment => decrease AOA : stable.

Now what is the relation between what we said and the vertical placement of the CG relative to the engine thrust line ?
It comes from the trimmed conditions. Let's suppose, to be more simple, that there are only two forces acting on the gyro which are the engine thrust and the rotor thrust. If you have a CG below the engine thrust line (as traditional Bensen gyro), the engine gives your gyro a nose down pitching moment. To be in trimmed condition, the rotor must induce a nose up (positive) pitching moment and for that, the CG must be behind the thrust line of the rotor. Bensen gyros are therefore instable in AOA.
On the opposite if you have a CG which is above the engine thrust (just a little above ; maybe one inch ; so nearly a center line thrust) then the moment coming from the engine is a nose up moment and to trim the gyro the rotor must induce a nose down moment. For that the CG must be in front of the rotor thrust line. You are in stable condition.
And when you are in stable condition, you have reduced risk of PIO (pilot induced oscilation).
Now I would like to add few words about PPO (power-pushover).
From my point of view PPO is the most dangerous phenomena in gyroplanes. PIO can be handled quite well if you have sufficient training, even in less stable gyros. But PPO can occur suddenly when your are flying in a very windy condition without alert (that is the reason why I think it is the most dangerous phenomenon). There is an article of Chuck in an old rotorcraft magazine where he computes the time to a Bensen gyro to do a 180 roll over when you have a down vertical gust and no horizontal stabilizer. I did some similar computation and find consistent results : It is less than 1 second. If this occurs when your left hand is not on the throttle, it's gone ... The reason for that is that the gust unloads the rotor blades. You have no more rotor thrust and the great nose down moment coming from the engine will roll the gyro.
What can we do to avoid that ?
First, we MUST add an horizontal stabilizer. This stabilizer will create an opposite moment to reduce the roll. Second, we must avoid the engine to create a nose down moment when the rotor blades are unloaded. For that we must put the engine thrust line close to the CG and if possible a little bit below to create a nose up moment to load again the rotor blades.
There is a chance that the solution to reduce PPO is the same as the one to reduce PIO.
So, and to answer more directly to your question, centerline thrust and horizontal stabilizers are complementary and acts on the same ways : reduce PIO, avoid PPO.
I hope these remarks will help you to understand stability of gyroplane.