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.

Mast, torque tube and rotor head

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 :

a. the engine thrust,

b. the horizontal stabilizer forces (lift and drag),

c. the body drag,

d. 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.

a. 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.

b. 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.

c. 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.

d. 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.

e. 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

f.

g. 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.

h.

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.

unstable C.G. position

stable C.G. position

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.

When in forward flight, there is an important difference to the circumstances of a helicopter in a hover, namely its airspeed due to its forward movement. We will now look at the effects of this additional airspeed on the rotors in the rotordisc. To assist us in discussing this subject, we will make the following assumptions: the rotors are

rotating anti-clockwise, seen from above; and the extra airspeed is generated solely because of the forward speed of the helicopter. We will discriminate between the advancing and the retreating blade, the former being the blade that moves in the same direction as the helicopter, whereas the latter moves in the opposite direction. Note that in one complete revolution, each blade will play both the advancing and the retreating roles (both roles for half a revolution).

The effect of forward speed on the rotors is easily explained. Firstly, think about the rotordisc when the helicopter is in a hover (no forward speed). In this scenario, each rotor-blade encounters the same airspeed, which is defined by the speed of rotation of the rotorsystem. As a result, all of the blades in the rotordisc contribute the same amount of lift. Note that we can't speak of an advancing or a retreating blade in these circumstances.

So, let’s take forward speed into account. The advancing blade now encounters a higher relative airspeed because, as a result of rotation, the forward speed must be added to the airspeed. Using the same line of reasoning, the retreating blade will encounter a lower relative airspeed. Because the lift of a blade depends on the squared relative airspeed, the advancing blade will produce substantially more lift than the retreating one. The result is a rotordisc which suffers from asymmetrical lift distribution over the disc.

Asymmetrical lift distribution along the rotordisc is one reason why the blades must be able to flap up and down. This can be realized either by hinges in the rotorhead, or by designing rotorblades which are flexible at the root.

Flap Back

We can now look again at the helicopter in forward flight, but this time taking into account that the lift distribution is asymmetrical, while the blades can also flap. So, what will happen? The advancing blade will flap up and the retreating one down. This has the effect that the rotordisc orientates itself backwards, slowing the helicopter down. This is known as 'flap back', and must be compensated for by applying (forward) cyclic. This example highlights that the effects of asymmetrical lift production are not merely compensated for by blade flapping, but also by applying corrective cyclic input (pilot action). You should be aware that there are a lot of textbooks which don't mention this (correctly).

Flapping and Angle of Attack

When the advancing blade flaps up, its angle of attack decreases, which reduces its lift considerably. When the blade flaps down, however, the angle of attack increases, which produces more lift. When the blades have enough freedom to flap up and down, this mechanism will then balance itself out automatically: blade flapping is dampened down by aerodynamic forces.

To understand why the angle of attack changes when a blade flaps up or down, one has to realize that the up and down flapping involves a vertical movement of the blade, which influences the relative airflow (RAF). Imagine the hypothetical situation in which there is no horizontal airflow (which, of course, is not possible with the blades rotating). In this situation, there would only be vertical airflow due to the blade flapping. When the blade flaps up, the relative airflow would come from above, and the angle of attack would equal -90 degrees. Now, we must make the situation realistic by adding the vertical airspeed to the horizontal airflow component that we first left out. Now, the relative airflow has a smaller angle of attack when compared to the situation where there is no upward blade movement (no flapping). This explains the smaller angle of attack. When the blade flaps down, the same mechanism increases the angle of attack (by introducing a vertical airspeed component that goes up instead of down).

Coriolis Forces due to Blade Flapping

When the blades flap up and down, a Coriolis force will work on them. When a blade flaps up, this Coriolis force will act in a forwards direction, accelerating the blade forwards. The blade will lead ('move ahead'). When the blade flaps back again, the opposite occurs and the blade will lag ('stay behind'). This explanation, which is based on Coriolis, is only valid in a rotating reference frame; that is, the reference frame rotates with the same angular velocity as the rotor blade.

When we assume a non-rotating reference frame (inertial reference frame), the lead and lag movements are explained by the law of conservation of angular momentum. When a blade flaps up, its centre of mass moves towards the shaft. In order to conserve angular momentum, the speed of the blade has to increase. When the blade flaps down, the opposite happens and the blade slows down. Both explanations are equally valid.

This lead and lagging is an important reason why rotorheads and/or rotorblades are designed to allow for the rotational freedom of the blades in the plane of rotation around the shaft. Another reason is the Hookes’ joint effect.