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Longitudinal Stability of Gyroplanes
Jean Fourcade
Jean Fourcade works for the French Space Agency where he is involved in the computation of satellite trajectories. He has been flying gyros in the South West of France for five years and one of his hobbies is to study flight mechanics and aerodynamics of the autogyro.
The longitudinal stability of the gyroplane has always been a subject of passionate debate between pilots. Phenomena involved, which are Pilot Induced Oscillation (PIO) and Power Push Over (PPO), were discussed in some articles of Rotorcraft magazine and in some classical gyro books. However, the University of Glasgow has made a complete mathematical study of this problem only recently. This work was funded under the UK Civil Aviation Authority and consisted of a parametric study of the longitudinal stability.
This study concluded, the longitudinal stability of the gyroplane is largely insensitive to a wide range in design characteristics.
However, an exception was found to be the vertical location of the propeller thrust line in relation to the center-of-mass. Stable or unstable configurations could be found depending on the height of the propeller thrust line.
A complete study of longitudinal stability of the gyroplane is beyond the scope of this article. However, some simple considerations can be shown, so that everyone can understand the principles of stability and how it affects gyro design.
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 concerned with mass or inertia characteristics of the aircraft. It is only a geometric criterion.
Dynamic stability is the most complete study of stability but is also, by far, the most complex. It requires writing equations of motion of the trimmed aircraft, and seeing how the aircraft responds to an arbitrary perturbation (disturbance).
The dynamic stability of gyroplanes is not very different from the dynamic stability of other aircraft. As for airplanes and helicopters, there are two oscillating modes; the short period mode and the long period mode (also called phugoid mode). The short period mode is an oscillation of the pitch of the aircraft at mainly constant speed while the phugoid mode is an oscillation of the pitch at mainly constant angle of attack.
The point which differs from other aircraft (even from helicopters) is that there is one more degree of freedom which is the rotorspeed and it can be shown that there is some coupling between rotorspeed and phugoid mode which indicates a potential handling problem as the phugoid mode can be unstable.
The purpose of this article is to deal only with static stability which can be understood without the need of an extensive mathematical background but is of great importance in understanding the correct placement of the CG relative to forces involved in the longitudinal motion.
Let us examine the definition of the static stability.
It is common usage in engineering science that when you want to study stability relative to a given parameter, you plot a curve where the x coordinate represents the parameter involved and the y coordinate the acceleration (or something proportional) of this parameter.
The longitudinal motion of a gyroplane is described by five parameters which are the air speed, the angle of attack of the fuselage, the pitch attitude, the pitch angular velocity and the rotorspeed. Therefore static stability can be studied for each of these parameters.
We will consider in this study only the angle of attack which is one of the most important.
The longitudinal pitch motion of the gyro is due only to the pitch moment of forces acting in the longitudinal plane. Therefore, to study the static stability relative to the angle of attack we have to plot pitching moments versus the angle of attack and see how these moments vary.
The sign convention is generally: positive angle of attack when nose up (and then positive pitching moments are nose up moments).
It is obvious that to fly in trimmed condition, the pitch must be constant and for that, the total moment computed with all forces involved must be equal to zero. Therefore, trimmed points belong to the x-axis.
The definition of the static stability can be defined this way: we will say that the gyro is statically stable with angle of attack when a variation of the trimmed angle of attack will induce a moment which tends to return the gyro to its previous trimmed condition.
Let us examine case (a) of Figure 1 where the trim is point A. Let us consider a disturbance which increases the angle of attack up to point B (a vertical gust for example). As the angle of attack has changed, the pitching moment is no longer equal to zero and we can see in Figure 1 that point B is not trimmed because it is not on the x axis. As the slope of the curve in this example is positive, it appears as a nose up pitching moment that will increase the pitch of the gyro and then the angle of attack. Therefore, case (a) is unstable because when a disturbance increases the angle of attack, the gyro reaction is to magnify this phenomenon.
On the other hand, case (b) is stable because when a disturbance increases the angle of attack up to point B, the pitching moment which appears is negative and acts in a way to reduce the angle of attack until the gyro returns to its previous trimmed condition (point A).
Therefore, the static stability depends on the slope of the curve on points of the x-axis or in others words, in the derivative of the pitching moment relative to the angle of attack. The condition of static stability is that the derivative must be negative.
Let us apply this rule to the forces acting on the gyro.
There are mainly 4 longitudinal forces acting on a gyro which are:
- The propeller thrust,
- The horizontal stabilizer thrust (lift and drag),
- The body drag,
- The rotor thrust (lift and drag).
To compute the stability given by each of these forces we have to evaluate the derivative of their pitching moment and see what is the best placement of the CG so that these derivatives are negative.
Propeller thrust:
For a given engine RPM, the propeller thrust depends on 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 propeller thrust is independent of the angle of attack and then, that the derivative of this moment is equal to zero.
Therefore, the propeller thrust, by itself has no impact on longitudinal stability (we say, in this case, that the trim is indifferent).
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) Horizontal stabilizer:
It is well known that the horizontal stabilizer must be placed at the tail of an aircraft. As a matter of fact, it can be demonstrated that the derivative of the pitching moment is negative when the center of pressure of the thrust (the point where the drag and the lift are applied) is behind the CG.
Therefore, a horizontal stabilizer adds stability.
The efficiency of the stabilizer is greater when the moment arm is longer and when the aerodynamic lift is greater.
To increase the moment arm we have to place the stabilizer far behind the CG and to increase the lift we have to increase the surface area of the stabilizer.
It is important to note that the lift is proportional to the square of the air speed while it is only proportional to the angle of attack. It is therefore preferable to place the stabilizer in the slipstream of the propeller to use the benefit of more air speed. It is especially important in a gyro as these machines are not flying fast.
c) Body drag:
The body drag is an aerodynamic force. As a horizontal stabilizer, the center of pressure must be behind the CG to have a negative derivative moment.
For a given curve and at a given point of this curve, the derivative is defined as follows: we consider DX a small (I should say an infinitesimal) variation on the x coordinates around the given point and we compute DY the corresponding variation given by the curve of the y coordinates. The derivative is the value: DY/DX.
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Figure 1: Pitching moment as a function of angle of attack
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