## This article was created by Felipe OrtegaEdit

### Aircraft StabilityEdit

Aircraft stability is the tendency of an aircraft to return to a state of equilibrium after a perturbation. Typically, a coordinate system is attached to the center of gravity of the aircraft in order to describe the dynamics or response to perturbations. This is done because forces acting on an airplane create moments and rotations naturally about the center of gravity. Also, measurements in flight data are on-body sensors, similar to what a pilot "feels".

## Static StabilityEdit

Static stability is concerned with the forces and moments produced by a small disturbance from a trim condition. It determines whether or not a body will return, of its own accord, to equilibrium once the perturbation is removed; also known as positive pitch stiffness [6: page 269 ]. A body is statically unstable when it tends to diverge from its equilibrium position and it possesses neutral static stability if it remains in the perturbed state [2]. As previously mentioned, the coordinate system constructed about the aircraft center of gravity is known as the body-fixed frame, illustrated below. If a gust causes a change in a nose up or down sense, this is known as a force which causes rotation about the x-axis, and requires analysis of longitudinal static stability.

**Longitudinal static stability** requires an aircraft to return to straight and level flight after a change in alpha, or angle of attack (α), is induced by a perturbation. In other words, a pitching moment is induced by a wind gust and if one were to analyze the contributing forces creating moments about the y-axis, they should revert the airplane back into trim. These forces are the airplanes weight (which may be idealized as a point mass located at the center of gravity), as well as the aerodynamic forces induced by the wings and empennage.

Figure 3 illustrates the **pitch moment coefficient** of two different airplanes as they experience a change in angle of attack. By convention, a positive pitch moment is one which results in a nose-up configuration of the airplane. The black point on the graph depicts the plane flying at a point of equilibrium and the other points show a perturbed state of both airplanes. Notice airplane 1 will continue on a path to a higher alpha given a nose-up perturbation, resulting in an even further divergence from equilibrium. Airplane 2 will pitch back up given its pitching moment coefficient, and return to equilibrium after a nose-down perturbation. Therefore airplane 2 is longitudinally stable.

Put simply, longitudinal static stability requires the rate of change of the pitching moment with respect to angle of attack be negative. There are several features which effect the pitch moment, both fixed aircraft structures and control surfaces. Wings, horizontal stabilizers and elevators (for the simplest airplane configuration) all contribute to the above equation.

## Static MarginEdit

Another crucial requirement for achieving longitudinal stability is the location of the aircrafts center of gravity (cg) with respect to its **neutral point**. The neutral point of an aircraflt is essentially the aerodynamic center of the whole aircraft [7]. The center of gravity must remain ahead of the neutral point throughout the entire flight. The distance between the center of gravity and the neutral point is the static margin. During long range flights or military missions, large changes in fuel weight may occur, often shifting the center of gravity. This must be taken into consideration. Fuel may often be shifted between tanks to maintain a positive static margin.

About the x-axis, the same principle is referred to as **lateral static stability**. This involves the rolling motion which results from a side wind gust. The features which effect aircraft lateral stability are again based mostly on the geometric structure of the airplane. Specifically, the location and orientation of the wing mounting and the design of the vertical tail. There are several configurations in which wing mountings may be considered and each will have uniquely effect the stability of the aircraft.

A term known as dihedral angle describes the angle the base mount of the wing makes with the horizontal orientation. Also, sweep back angle effects the roll stability. Typically, the effect of the wings and tail must be analyzed simultaneously because the two produce a coupled response. All of these variations lead to trade-offs in aircraft design. Structural features which may go unnoticed to the common spectator may be interpreted as design choices which effect the different aspects of flight stability.

Similar to the pitch moment coefficient analysis, a **roll moment coefficient **may be plotted versus beta ( ) or side-slip angle (positive if nose-left from x-axis). Through t he same analysis, one can conclude a negative rate of change of the roll moment with respect to beta is laterally stable. Conceptually, an airplane should return to straight and level flight when a side gust occurs.

The final component of static stability is **directional stability**. This is a resulting balance of forces and moments about the z-axis, also known as the "yaw" of the aircraft. The contributing structures to this analysis are mainly the fuselage and vertical tail of the aircraft. Basically the fuselage is a destabilizing structure in the presence of a side gust, and the tail is a stabilizing structure. Th is time, with the definition of side-slip beta and the **coefficient of yaw moment** as positive as a nose-right moment, the following graph is generated to characterize two different responses. Here it is clear that airplane 1 will return to a point of equilibrium and airplane 2 will amplify the effect of the side gust and become more unstable.

For each of the cases presented here, a more complex derivation of the structural components and control surface contributions would be added to the master equations in order to understand how the system as a whole meets the requirements of stability.

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## Dynamic StabilityEdit

Understanding the dynamic stability of an aircraft will require more detail. Apart from the response of an aircraft to small perturbations during straight and level cruise flight, we would also like to understand the response to an aircraft when it undergoes a maneuver about its trim configuration. A series of equations of motion (EOM) are utilized to describe the dynamic response of the aircraft to these maneuvers. The motion usually consists of oscillations about the equilibrium position. A body is dynamically stable if the oscillations eventually damp out [2].

### Modeling Physical SystemsEdit

Looking at the source of these equations, one must be aware that the complete set of information includes a description of forces, moments, position and orientation (attitude). Each of these four sets must be derived in the three directions of the body-centered coordinate system; this alone requires the transformation of coordinate axis from other orientations not mentioned here, but are often more convenient to describe specific forces, etc. The forces of thrust, drag and weight are decomposed into the most convenient coordinate system and applied through Newton's second law. Moments are calculated by relating the inertia tensor to angular momentum, again in three dimensions. This yields terms which represent the angular acceleration, gyroscopic effects, and the coupling of different dynamic systems. The position equations are quite self-explanatory The orientation information requires the translation of angular information (as described by the body-centered coordinate system). Thus the complete dynamic response is modeled by twelve highly coupled, non-linear, differential equations [4].

As a side note, these twelve EOM are only capable of being solved by computational numerical methods. Aeronautic software such as flight simulator employ these equations as an extremely accurate model. As is common in many engineering practices, it is convenient to simplify these parameters in order to use non-numerical methods to interpret these dynamics. By making the assumptions of small angles and considering maneuvers are first order perturbations about the trim condition, we may reduce the system to twelve equations which naturally decouple into six equations with only longitudinal terms and six with both directional and lateral terms. At this point in the analysis, it is also convenient to recall the operations of linear algebra in regards to solving systems of ordinary differential equations. The purpose of this decomposition is to make more specific assumptions about the particular conditions to break the dynamic responses into flight modes. As a last assumption, we ignore the instantaneous position of the aircraft at the time of calculation, since its dynamic response does not depend on whether it is flying over point A or B, as well as its yaw angle (for convenience), and the system reduces into common flight modes well understood in aerospace industries. These modifications are quite involved and are the basics of any *Aircraft Stability and Control*

To re-iterate, the decomposition is as follows:

- 12 non-linear highly coupled ODE for accurate flight dynamics

- 12 linear decoupled ODE for trim maneuvering

- 2 sets of 6 ODE for longitudinal/lateral-directional dynamics

- 2 sets of 4 ODE by neglecting position/yaw angle

- decompose into modes

The dynamics of the airplane response can actually be modeled in the same manner as a mass, spring, damper system. In general, there are four types of solutions.

A stable response is indicated in the bottom cases. Thus the eigenvalues of the characteristic must either be real and negative, or complex with a negative real component. The oscillatory nature of the complex solution is either diverging or converging within an exponential envelope (red line). These variations in response lead to different flight mode interpretations [4].

## Flight ModesEdit

### Longitudinal ModesEdit

Again, the roots of the characteristic equation dictate the response of the system, and that response may take one of these four forms. In regards to longitudinal flight modes, there are two forms; the damped short period mode and the lightly damped phugoid mode. The short period mode involves mainly pitching motion with no change in forward speed [1: page 75]. Neglecting the gravitational component, the matrix of ODE decomposes into a direct relationship between pitch rate and change in aoa. This may be interpreted as a pilot command to quickly pitch up or down, then the aircraft response lagging the command, perhaps overshooting and correcting, however dampening this effect very quickly. In other words, this response may be thought of as a phase lag between pitch rate and rate of change of aoa.

The lightly damped phugoid mode requires neglecting the components of the equations with respect to changes in aoa and quick changes in pitch (pitch rate). This leads to a correlation between forward velocity and altitude. This response may be conceptually understood as a form of energy conservation. A slight change in forward velocity will lead to increased lift, transferring kinetic energy into potential energy. At the peak of the oscillation, the airplane will descend and consequently speed up again. This oscillation is very slow, hence lightly damped, and if not corrected may have a period of several minutes for commercial aircraft.

### Lateral-directional ModesEdit

Lateral-directional modes for subsonic aircraft include roll, spiral and dutch-roll. The derivation for all flight modes may be rather involved and so only a conceptual overview is provided here. For in depth formulations, see the references listed below.

Roll and spiral modes are non-oscillatory responses. An input to the ailerons of an airplane will excite the roll mode. Its motion is almost purely a rolling return to trim with slight yaw and side-slip. A stable response is known as "roll convergence" which is analogous to roll mode. An unstable response is "roll divergence". The spiral mode is more complex and interesting. Consider a wind gust which induces a positive roll moment. This roll angle increase will cause an increase in yaw rate due to the reorientation of aerodynamic forces relative to a constant weight vector, and at the same time the airplane is speeding up as it loses altitude. The result is a descending spiral in which the spirals get tighter and faster.

The final mode is the dutch-roll and it is an oscillatory response which couples a rolling, yaw, side-slip motion. The rolling dominates the motion. Basically, the nose of the aircraft will yaw side to side without much change in the direction of the flight, then a roll "corrects" the shift with a slight lag. The nose of the aircraft essentially traces figure eights in space. This mode is actually quite complex and very dangerous to pilots if they are on a landing approach for example.

In order to understand the scale of these responses, the following figures illustrate the phugoid, short period and dutch roll responses for a Boeing 747. Note the time scales for the responses. The phugoid response may last several minute, however the highly damped short period response lasts for a matter of second. These responses are often corrected by computer control systems [4].

## Final NoteEdit

Careful analysis of the derivation of stability requirements and modal decomposition leads to an interesting aircraft design dilemna. The structural components which ensure static stability often worsen the response to unstable flight modes. For example, a large vertical tail lends a directionally stable airplane, however it will worsen the spiral mode of the same airplane. Therefore, careful consideration in the design constraints must increase in scope to include both static and dynamic stability. Also, aircraft with a great deal of stability are often not very maneuverable. There is a definite trade off between the two qualities. This principle is analogous to the driving characteristics of a school bus for example; very stable but not very exciting to drive.

These flight modes lead to important physical characteristics in aviation design. Decades ago, the ability to provide control to these modes was entirely done by the pilot. This fact could lead to pilot error and may become quite tiring. Today, although there is no substitute for the instincts and ability of a well trained pilot, computer systems offer much support in controlling aircraft. More unconventional designs which may not have been piloted in the past, are today, more feasible due to these computer controlled systems [4].

## ReferencesEdit

[1] Babister, A. *Aircraft Dynamic Stability and Response*. Pergamon Press Inc. 1980

[2] Dickinson, B. *Aircraft Stability and Control for Pilots and Engineers*. Page 24. Sir Isaac Pitman and Sons LTD 1968.

[3] Irving, F.G. *An Introduction to the Longitudinal Static Stability of Low-speed Aircraft*. Chapter 1. Pergamon Press 1966.

[4] Lind, Richard. Associate Professor. University of Florida department of Aerospace Engineering. Fall 2008.

[5] Allerton, David. *The ACSE Flight Simulator*. Department of Automatic Control and Systems Engineering.04-2006.

[6] Stevens, B. Lewis, F.* Aircraft Control and Simulation*. John Wiley & Sons. New Jersey. 2003.

[7] *Aerodynamics*. Stability Concepts. http://adamone.rchomepage.com/index5.htm. 10-09.