Airplane Stability & Control

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n42

44 Airplane Stability & Control

Introduction^[1]

The stability and control characteristics of an aircraft or other flight vehicle are rather complex, partly because of the relatively difficult mathematics needed to explain its flight characteristics. Nevertheless, the subject’s basic principles can still be introduced without straying too far into the detailed mathematics. While the professional practice of stability and control requires specialist knowledge and considerable experience, it is nevertheless essential for all aerospace engineers to develop a first-level understanding of the fundamentals of this field and its importance to the aircraft design process.

The term “stability” in reference to a flight vehicle is the tendency for that vehicle to continue to fly in a prescribed flight condition, e.g., an aircraft is considered stable when it remains in the flight condition that the pilot intends it to be in. If the aircraft does something else, especially when the pilot removes their hands and feet from the flight controls, the aircraft would not be considered stable and may even be unstable in some cases. Most aircraft are inherently stable by design, but to a greater or lesser degree, so they can be safely flown by an average pilot without excessive effort and workload. The term “control” means the ability to change the flight attitude or other conditions of the flight vehicle, i.e., to make the vehicle do what is commanded of it. Naturally, a flight vehicle’s inherent or natural “stability” and the “control” of its flight by a pilot (or auto-pilot) are inextricably linked.

Learning Objectives

Appreciate the fundamentals of flight vehicle stability and control and why stability is essential for flight.
Understand the differences between a flight vehicle’s static and dynamic stability.
Become familiar with flight dynamic terms like short-period and long-period responses, phugoid, Dutch roll, and spiral divergence.
Be aware of an airplane’s primary design features contributing to its static and dynamic stability characteristics.
Understand what is meant by handling quality assessments.

Forces & Moments of Flight

Following Arthur Babister, flight dynamics concerns an aircraft’s flight characteristics and motion under six fundamental types of forces and moments:

Inertia effects that arise from the aircraft’s mass distribution and linear and angular accelerations.
Aerodynamic damping effects that arise from the aircraft’s angular velocities.
Aerodynamic effects that depend on the aircraft’s linear velocities.
Aerodynamic effects that depend on the application of flight controls.
Gravitational effects, which manifest from the magnitude and distribution of mass over the aircraft as well as its inertia.
Propulsive effects are produced by the engine(s) to propel the aircraft forward.

Other factors influencing flight dynamics are the atmospheric conditions, the flight altitude, and the flight airspeed and Mach number. These factors can all affect the aerodynamic characteristics of the aircraft and its propulsion system and so, either directly or indirectly, will impact the aircraft’s performance, flight dynamics, and handling characteristics.

1. Inertia Effects.

These effects arise from the aircraft’s mass distribution in response to linear and angular accelerations. These forces and moments are of two types, namely linear inertia effects and angular inertial effects. Linear inertial effects produce forces that arise from the aircraft’s mass in response to linear accelerations. Angular effects occur from the aircraft’s mass distribution and angular accelerations, which are governed by the aircraft’s moments of inertia. Higher moments of inertia are undesirable because they generally make the aircraft more “sluggish” and less agile, even more challenging to fly.

2. Aerodynamic Damping Effects

Aerodynamic damping forces and moments arise from the angular velocities of the aircraft and are sometimes called rotary forces and moments. Damping reduces the aircraft’s transient or oscillatory motion and is usually a desirable flight dynamic characteristic. An aircraft’s wing and horizontal and vertical tail surfaces primarily contribute to damping effects produced on an aircraft.

3. Aerodynamic Effects that Depend on Velocities

Aerodynamic forces and moments will depend on the aircraft’s linear velocities. These are often called static forces and moments because they depend on the aircraft’s instantaneous position in three-dimensional space. In particular, the lift and drag forces acting on an aircraft depend on its linear velocity through the air. Aerodynamic effects are also influenced by the aircraft’s angle of attack with respect to the flight velocities, the air density, and the wing and tail geometry. For high rates of change, unsteady aerodynamic effects may be significant because the instantaneous velocities may be insufficient to describe the aerodynamics, and what happened in prior time may be necessary, i.e., a hereditary effect.

4. Effects of Flight Controls

The application of flight controls can have significant effects on the aircraft’s aerodynamics. Examples of flight controls are the ailerons, elevator, and rudder. These flight controls can affect the motion and stability of the aircraft in roll, pitch, and yaw. Furthermore, the flaps and slats (if any) and spoilers (if any) can affect the aircraft’s flight characteristics, particularly at low airspeeds such as takeoff and landing.

5. Gravitational Effects

Gravity manifests as weight (which is a force) and the distribution of weight, i.e., the position of the center of gravity (c.g.). The c.g. is the point at which the aircraft’s weight can be considered to act, and it has an essential effect on the stability and control of the aircraft. If the c.g. is too far backward, the aircraft may become unstable and difficult to fly, while if the c.g. is too far forward, the aircraft may become excessively stable and difficult to maneuver. The fuel load, as well as passengers and cargo, need to be appropriately distributed to ensure the stability of the aircraft maintained during the entire flight.

6. Propulsive Effects

These are the effects produced by the engine(s) that propel the aircraft forward. The propulsive thrust affects the aircraft’s speed, acceleration, and overall performance. The magnitude of the thrust depends on several factors, including the type and design of the engine, the power setting, and the aircraft’s speed. For example, changes in thrust can produce pitching moments on the aircraft. For a multi-engine aircraft, losing one engine may cause the aircraft to yaw, pitch, or roll, affecting its flight characteristics and stability. Aircraft with large propellers may also have gyroscopic and slipstream effects that may influence their stability and control characteristics.

Stability Definitions

An airplane is just one type of aircraft, but its analysis forms an essential basis for understanding the flight dynamics and control characteristics of all flight vehicles. The issue of concern is with stability and control characteristics of the airplane about all three flight axes, as shown in the figure below, namely:

Longitudinal stability and control concerns the airplane’s response in the pitch degree of freedom.
Lateral stability and control relate to the lateral axis or rolling degree of freedom.
Directional stability and control relate to the yawing axis or directional (weathercock) degree of freedom.

An airplane can pitch, roll, and yaw. It is generally expected that the airplane is, by design, statically and dynamically stable about all three flight axes.

While the responses and controls of any airplane tend to be coupled about the three axes to some degree, it is found that its pitch motion is decoupled chiefly from the roll and yaw responses. However, an airplane’s lateral (roll) and directional (yaw) stability characteristics tend to be significantly more coupled; usually, one cannot be considered separately from the other for a stability and control analysis.

Coordinate System

For stability and control analysis, the airplane’s body axes are usually defined as a right-handed Cartesian coordinate system centered at the airplane’s center of gravity. The $x$ direction is defined as positive along the airplane’s longitudinal axis, with positive being forward. The $y$ direction is positive along the starboard wing, and $z$ is positive downward. For each of the coordinate axes, the forces, moments, and velocities can be summarised by the two tables below, one for the forces and the other for the moments. A right-handed rule defines the direction of positive rotations and positive angular rates. It is important to note here the standard symbol conflict between flight dynamic moments and aerodynamic coefficients, which must be carefully distinguished and reconciled in any analysis.

Summary of the conventions used for the forces and velocities.
Axis	Force	Linear Velocity	Description
$x$	$X$	$U$	Fore/aft
$y$	$Y$	$V$	Sideward
$z$	$Z$	$W$	Heave or plunge

Summary of the conventions used for the moments and angular velocities.
Axis	Moment	Moment Coefficient	Angular Displacement	Angular Velocity	Non-dimensional angular rate	Description
$x$	$L$	$C_l$	$\phi$	$P$	$\overline{p}$	Roll
$y$	$M$	$C_m$	$\theta$	$Q$	$\overline{q}$	Pitch
$z$	$N$	$C_n$	$\psi$	$R$	$\overline{r}$	Yaw

Trimmed Flight

For an airplane to be in static equilibrium or in trim at a particular flight condition, the net sum of all the forces and moments acting on the airplane must be zero, i.e., the position and attitude of the airplane will be in perfect balance about all three flight axes, namely pitch, roll, and yaw. If there are no net forces, then there will be no resultant accelerations on the airplane, so

(1) $\begin{equation*} \overbigdot{U} = \overbigdot{V} = \overbigdot{W} = \overbigdot{P} = Q = \overbigdot{R} = 0 \end{equation*}$

where the airspeed is $V_{\infty} = (U, V, W)$ . In trim, there are also no angular velocities about the flight axes, so

(2) $\begin{equation*} P = Q = R = 0 \end{equation*}$

Furthermore, for level, symmetric, coordinated flight with no yaw or sideslip, then

(3) $\begin{equation*} V = W = 0 \end{equation*}$

and if the wings are level, then $\phi=0$ .

Forces in Trimmed Flight

Consider now the equilibrium of an airplane in straight and level unaccelerated flight at a constant airspeed and altitude, i.e., in static equilibrium or trim, as shown in the figure below. In trim, the lift on the airplane equals its weight, and for most purposes, the weight can be considered to act at a center of gravity location. The thrust (from the propulsion system) equals the aerodynamic drag at that weight, airspeed, and altitude. Therefore, as previously defined, there can be no net forces or moments acting on the airplane about the center of gravity in the trim condition.

An airplane in static equilibrium or trimmed flight showing how the balance of forces and moments about the center of gravity must be zero.

The aerodynamic forces on the airplane can be considered to act at an effective location on each lifting component, i.e., the main wings and the horizontal tail. Of more significance is the lifting contributions, in aggregate, which can be assumed to act at a single point. The center of pressure is a convenient point because there is no net aerodynamic moment at this location. The center of gravity is usually located in front of the center of lift (for stability), and the horizontal tail and flight controls are needed to create the necessary aerodynamic forces (and hence moments) to reach a balanced pitch or trimmed flight condition.

The main wing produces most of the lift on the airplane, but the tail may make some small increments. Hence, the center of pressure is usually very close to the center of pressure of the wing by itself, which for the lift coefficients typical of flight, is near the 1/4-chord point. The horizontal tail acts like a smaller version of the main wing, and it can be used to give either positive or negative changes in the lift using the elevator control. Because of the typically long distance (arm) from the horizontal tail to the center of gravity location (but not always), only relatively small changes in the lift on the tail is required to produce significant longitudinal pitching moments.

Consider now what would happen if the center of gravity moves forward. In this case, a more significant nose-down gravitational moment would be produced on the airplane, which would need to be compensated for by increasing the downforce (negative lift) on the tail. Therefore, the elevator would need to be deflected up by the pilot by moving the control column aft, reducing the aerodynamic upward force on the tail and continuing to balance the net moments on the airplane to reestablish trim. Small changes in aerodynamic forces and moments from the control surfaces can be performed using trim tabs, as shown in the figure below, which can be actuated separately to trim the airplane and remove any residual forces from the pilot’s controls.

The use of the flight controls, elevator, ailerons, and rudder, also provide the necessary forces and moments to create a trimmed flight condition. The flight control surfaces may also have smaller auxiliary surfaces called “trim tabs” to allow the pilot to remove or “trim out” any residual forces on the controls.

The propulsion system can also affect the stability and control characteristics of the airplane. Propulsion will create a thrust vector, which may have a line of action offset from the center of gravity. Thrust can also produce a pitching moment, i.e., a thrust/pitch coupling; this latter effect being illustrated in the figure below. Airplanes with underslung engines with thrust vectors well below the center of gravity are prone to this type of coupling, which can also be interpreted as an airspeed coupling effect. In this latter regard, changes in thrust setting will also cause changes in airspeed.

The effect of thrust/pitch coupling is particularly pronounced on airplanes with underslung engines below the wing, which will change with changes in engine thrust and so power setting.

Both the center of gravity and the airplane’s center of lift may (and generally will) change during flight. As fuel is burned off and the airplane’s weight changes, the center of gravity may move forward or aft, depending on the type of airplane and how it is loaded with the payload. Therefore, the airplane’s stability characteristics can (and often will) change during flight, albeit slowly, and further trimming may be required. To reduce the trim drag on a large commercial airplane, fuel is pumped from one tank to another to manage the position of the longitudinal and lateral center of gravity during flight rather than accepting the increased drag from the trim tabs. As shown in the photograph below, all-flying horizontal tails may also be used on airliners to trim out the pitching moments.

Photo of the root of the trimmable horizontal stabilizer on an Embraer ERJ-170. The markings UP and DOWN refer to the angles needed for “nose up” trim and “nose down” trim, respectively

The center of lift may also change with airspeed, especially in high-speed flight at higher Mach numbers. Approaching transonic and into supersonic flight, the center of pressure typically migrates aft on the wing from near the 1/4-chord to closer to the 1/2-chord. The resulting effect is a pronounced nose-down pitching moment. This effect is called Mach tuck, and it can be a stability and control issue for a supersonic airplane as they transition from subsonic to supersonic flight. Of course, these effects can often be trimmed out by using the elevator (or a trimmable tail surface). Still, there will be a limit to this control capability depending on the combination of the center of gravity and/or center of lift movements during flight. On some larger airplanes, it is necessary to pump fuel longitudinally from one tank to another to keep the center of gravity between the required limits during supersonic flight, such as was done on the Concorde using trim tanks.

Stability Derivatives

The stability characteristics of an aircraft in response to disturbances from trimmed flight can be explained with the use of stability derivatives, i.e., the change in a specific force or moment with respect to specific type of disturbances. To this end, it can be assumed that the forces and moments on the aircraft are functions of the instantaneous values of the disturbance velocities (translational and angular), as well as their time rates of change. Unsteady or hereditary effects (i.e., what happened in previous time) can be ignored, which is often referred to as a quasi-steady assumption.

General Representations

Therefore, the quasi-steady forces on the aircraft can be expressed in general terms as

(4) $\begin{eqnarray*} X &= & f_X\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r} \right)\\ Y &= & f_Y\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r}\right)\\ Z &= & f_Z\left(u, \, \overbigdot{u}, \, v, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \overbigdot{r}\right) \end{eqnarray*}$

and for the corresponding moments, then

(5) $\begin{eqnarray*} L &= & f_L\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r}\right)\\ M &= & f_M\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r}\right)\\ N &= & f_N\left(u, \, \overbigdot{u}, \, u, \, \overbigdot{w}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, u, \, \overbigdot{r}, \, r \right) \end{eqnarray*}$

The function, $f$ , in each case represents the relationship between the aircraft’s instantaneous motions (or disturbances) and the resulting forces ( $X$ , $Y$ , and $Z$ ) and the corresponding moments ( $L$ , $M$ , and $N$ ) on the aircraft. With 12 dependencies in each case as well as potentially non-linear and interdependent (coupling) effects, then it becomes clear that this is one reason why the mathematical description of the flight dynamics of an aircraft tends to become complicated.

Furthermore, if the effects of the flight control deflections, $\delta$ , are added, where the subscript $a$ refers to the ailerons, $e$ refers to the elevator, and $r$ refers to the rudder, then there will be 18 dependencies in each case, i.e.,

(6) $\begin{eqnarray*} X &= &f_X\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right)\\ Y &= &f_Y\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right)\\ Z &= &f_Z\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right)\\ L &= &f_L\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right)\\ M &= &f_M\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right)\\ N &= &f_N\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w},\, p, \overbigdot{p}, \, q, \, \overbigdot{q}, r, \, \overbigdot{r}, \, \delta_a, \, \overbigdot{\delta}_a, \, \delta_e, \, \overbigdot{\delta}_e,\, \delta_r, \, \overbigdot{\delta}_r \right) \end{eqnarray*}$

More contributions could also be added to the list, such as for the effects of flaps, engine thrust, etc, so the problem quickly becomes somewhat formidable, and perhaps even intractable from any reasonable practical perspective.

Linearization Process

The method used in flight dynamics is to linearize the foregoing relationships with a Taylor series expansion about the trim state and then retain only the first-order derivatives. Therefore, now the perturbations to the forces and moments, i.e., $\Delta X \ldots \Delta N$ are now desired. The equilibrium is defined as $X_0\ldots N_0$ , so these represent the balanced forces and moments on the airplane in the trimmed flight condition. For example, for the $X$ force, then

(7) $\begin{eqnarray*} \begin{split}\begin{split}{X_0} + \Delta X = {X_0} + & \left[\pd{X}{u}{u} + \ppd{X}{u}\frac{u^2}{2!} + \pppd{X}{u}\frac{u^3}{3!} + \ldots\right] +\ldots \\ & + \left[\pd{X}{\overbigdot{u}}{\overbigdot{u}} + \ppd{X}{\overbigdot{u}}\frac{\overbigdot{u}^2}{2!} + \pppd{X}{\overbigdot{u}}\frac{\overbigdot{u}^3}{3!} + \ldots\right] +\ldots\\ & + \left[\pd{X}{v}{v} + \ppd{X}{v}\frac{v^2}{2!} + \pppd{X}{v}\frac{v^3}{3!} + \ldots\right] + \ldots\\ & \hspace{3cm}\vdots \\ & + \left[\pd{X}{\overbigdot{\delta_r}}{\overbigdot{\delta_r}} + \ppd{X}{\overbigdot{\delta_r}}\frac{\overbigdot{\delta_r}^2}{2!} + \pppd{X}{\overbigdot{\delta_r}}\frac{\overbigdot{\delta_r}^3}{3!} + \ldots\right] \end{split}\end{split} \end{eqnarray*}$

If the higher-order derivatives are now neglected, which is a reasonable assumption because they can be expected to be small in value, then

(8) $\begin{equation*} \Delta X = \left. \pd{X}{u}\right|_0 u + \left. \pd{X}{\overbigdot{u}}\right|_0 \overbigdot{u} + \left. \pd{X}{v}\right|_0 v + \ldots + \left. \pd{X}{\overbigdot{\delta_a}}\right|_0 \delta_a \end{equation*}$

The first-order partial derivatives in the foregoing equation are referred to as the stability derivatives and can be used to determine the aircraft’s response to disturbances or perturbations about the trim state. It is important to remember, however, that all of the stability derivative terms will be a function of the specific trim state of the aircraft during flight, as denoted by the subscript $0$ , so they are not necessarily constants.

The full set of force perturbations can now be expressed as

(9) $\begin{eqnarray*} \Delta X & = & \allderivs{X} \\[18pt] \Delta Y & = & \allderivs{Y} \\[18pt] \Delta Z & = & \allderivs{Z} \end{eqnarray*}$

and the corresponding full set of moment perturbations will be

(10) $\begin{eqnarray*} \Delta L & = & \allderivs{L} \\[18pt] \Delta M & = & \allderivs{M} \\[18pt] \Delta N & = & \allderivs{N} \end{eqnarray*}$

Each of the terms involving a control deflection is often referred to as the control derivative. In total, it is apparent that there are 18 derivatives for each equation and so a total of 108 derivatives, which is unmanageable in any practical context.

Simplifications

Fortunately, the foregoing equations may be simplified using some reasonable assumptions. For a symmetric flight condition in the $x$ – $z$ plane, the asymmetric forces and moments, i.e., $Y$ , $L$ , $N$ , will be zero. Therefore, the derivatives of the asymmetric quantities and their derivatives with respect to $u$ , $\overbigdot{u}$ , $w$ , $\overbigdot{w}$ , $q$ , $\overbigdot{q}$ , $\delta_e$ , $\overbigdot{\delta_e}$ , will be zero. The reciprocal effect applies, so the derivatives of the symmetric forces and moments, i.e., $X$ , $Z$ , and $M$ , with respect to the asymmetric variables and their derivatives, $v, \overbigdot{v}, p, \overbigdot{p}, r, \overbigdot{r}, \delta_a, \overbigdot{\delta_a},\delta_r, \overbigdot{\delta_r}$ , are also zero. The control rate derivatives are considered small enough to be negligible. It has also been found from aircraft flight testing that all derivatives with respect to accelerations are negligible, so further simplifications are possible. Cumulatively, these simplifications now result in a much shorter subset of the the original equations.

Based on the linearization about the trim conditions and using the foregoing simplifications, then the perturbation forces now become

(11) $\begin{equation*} \Delta X = \left.\pd{X}{u}\right|_0u + \left.\pd{X}{w}\right|_0w \end{equation*}$

(12) $\begin{equation*} \Delta Y = \left.\pd{Y}{v}\right|_0v + \left.\pd{Y}{\delta_r}\right|_0\delta_r \end{equation*}$

(13) $\begin{equation*} \Delta Z = \left.\pd{Z}{u}\right|_0u + \left.\pd{Z}{w}\right|_0w + \left.\pd{Z}{\delta_e}\right|_0\delta_e \end{equation*}$

The corresponding perturbation moments are

(14) $\begin{equation*} \Delta L = \left.\pd{L}{v}\right|_0v + \left.\pd{L}{p}\right|_0p + \left.\pd{L}{r}\right|_0r + \left.\pd{L}{\delta_r}\right|_0\delta_r + \left.\pd{L}{\delta_a}\right|_0\delta_a \end{equation*}$

(15) $\begin{equation*} \Delta M = \left.\pd{M}{u}\right|_0u + \left.\pd{M}{w}\right|_0w + \left.\pd{M}{\overbigdot{w}}\right|_0\overbigdot{w} + \left.\pd{M}{q}\right|_0q + \left.\pd{M}{\delta_a}\right|_0\delta_a \end{equation*}$

(16) $\begin{equation*} \Delta N = \left.\pd{N}{v}\right|_0v + \left.\pd{N}{p}\right|_0 p + \left.\pd{N}{r}\right|_0r + \left.\pd{N}{\delta_r}\right|_0 \delta_r + \left.\pd{N}{\delta_a}\right|_0 \delta_a \end{equation*}$

The stability derivatives, i.e., the $\partial (-) / \partial (-)$ terms, which can be seen to be gradients or slopes, in the foregoing equations, represent the aircraft’s linearized response to small perturbations around an equilibrium trim state. The numerical values of the various derivatives can be estimated using simulations and perhaps wind tunnel tests, and then verified by flight testing. As previously mentioned, a complication is that the values of the derivatives will change with the trim state, and so are likely to depend on altitude, airspeed, Mach number, angle of attack, and the configuration of the aircraft as well as other things.

Linearized Equations

Finally, the linearized force and moment perturbations may be substituted into the linearized equations of motion of the aircraft about the trim condition, i.e., the net force in each direction equals the mass of the aircraft times its acceleration. For simplicity, only the three symmetric terms, i.e., for flight in the $x$ – $z$ plane, are retained, as shown in the figure below.

The stability analysis of an airplane in a vertical plane, i.e., in pitch, forms a good basis to understand the concepts because its motion is relatively uncoupled to roll and yaw.

In reference to the figure, then the perturbation forces in the $x$ direction is

(17) $\begin{equation*} \left.\pd{X}{u}\right|_0 u + \left.\pd{X}{w}\right|_0 - M g\cos\theta_0 = M \, \overbigdot{u} \end{equation*}$

For the perturbation forces in the $z$ direction, then

(18) $\begin{equation*} \left.\pd{Z}{u}\right|_0 u + \left.\pd{Z}{w}\right|_0 w + M U_0 q - M g \sin\theta_0 + \left.\pd{Z}{\delta_e}\right|_0 \delta_e = M \overbigdot{w} \end{equation*}$

Finally, for the perturbation in the pitching moment, then

(19) $\begin{equation*} \left.\pd{M}{u}\right|_0 u +\left.\pd{M}{w}\right|_0 w +\left.\pd{M}{\overbigdot{w}}\right|_0 \overbigdot{w} + \left.\pd{M}{q}\right|_0 q + \left.\pd{M}{\delta_e}\right|_0 \delta_e = I_{yy} \overbigdot{q} \end{equation*}$

Static Stability

It is initially convenient to describe the principles concerning the airplane’s longitudinal or pitching response because the responses in pitch are mostly clean and uncoupled from the responses in roll and yaw. Consider the situation when the balance of forces and moment in the trim state is disturbed, such as by a vertical gust caused by atmospheric turbulence. Gusts can come from virtually any direction and so can affect the airplane’s response about any axis, but the vertical or $w$ velocity gusts typically have the largest effects on the aircraft’s responses.

A vertically upward gust, $w$ , will cause an increase in the angle of attack of the wing so increasing its lift, and the airplane’s natural reaction is that its nose will pitch up slightly. The consequence of this effect is that the airplane is no longer in stable equilibrium and will have deviated from its trimmed flight condition, as shown in the figure below. If the subsequent forces and moments generated on the airplane from the gust disturbance tend to return it to its trimmed condition, the airplane’s response would be referred to as being statically stable, as shown in scenario (b). Mathematically, when expressed in terms of a stability derivative, then

(20) $\begin{equation*} \left. \frac{ \partial M}{\partial w} \right|_{\rm trim} < 0 \end{equation*}$

which must be negative to produce a restoring moment.

Principle of static stability, in this case with respect to pitch attitude: (a) Trimmed equilibrium flight, (b) Statically stable response, (c) Statically unstable response, (d) Statically neutral response.

However, if the forces and moments introduced by the disturbance tend to cause the nose to pitch up further, then the airplane would be considered statically unstable, which is the scenario (c). In this case,

(21) $\begin{equation*} \left. \frac{ \partial M}{\partial w} \right|_{\rm trim} > 0 \end{equation*}$

If the airplane is genuinely statically unstable, its subsequent motion may well cause a divergence of the flight path and, most likely, a departure from controlled flight. When the airplane remains indefinitely disturbed, as shown in scenario (d), then it is considered to have neutral static stability, i.e.,

(22) $\begin{equation*} \left. \frac{ \partial M}{\partial w} \right|_{\rm trim} = 0 \end{equation*}$

but this is not a common characteristic of an airplane.

Static Margin

The conditions to ensure sufficient longitudinal static stability can now be more formally established, which leads to a parameter that quantifies the pitch stability known as the static margin. The static margin is a distance so it has units of length, although is usually quoted as a fraction or a percentage of the mean wing chord. Assume that in the trim condition, then $\theta_0 = 0$ . For vertical equilibrium, then

(23) $\begin{equation*} L_{W} + L_{HT} = W \end{equation*}$

where $W$ is the weight of the aircraft, $L_{W}$ is the wing lift, and $L_{HT}$ is the tail lift. The wing lift can be written in the conventional manner as

(24) $\begin{equation*} L_{W} = \frac{1}{2} \varrho V_0^2 S_{W} \frac {\partial C_{L}}{\partial \alpha } (\alpha - \alpha _{0}) \end{equation*}$

where $S_{W}$ is the wing area and $\alpha$ is its angle of attack. The zero-lift angle $\alpha _{0}$ can be included, if needed, for cambered (non-symmetric) wing sections. Notice that $\partial C_L / \partial \alpha$ is the aerodynamic lift-curve slope of the wing, i.e.,

(25) $\begin{equation*} \frac {\partial C_{L}}{\partial \alpha } \equiv \frac {d C_{L}}{d \alpha } = C_{L_{\alpha}} \end{equation*}$

so that

(26) $\begin{equation*} L_{W} = \frac{1}{2} \varrho V_0^2 S_{W} C_{L_{\alpha}} (\alpha - \alpha _{0}) \end{equation*}$

The lift force from the tailplane also depends on its angle of attack (which will be different), but it will additionally be affected by the upstream wing from a downwash that will lower its effective angle of attack, say by a value $\epsilon$ . The resulting lift force on the tail may act upward or downward, depending on the flight conditions.

Therefore, in trim then

(27) $\begin{equation*} L_{HT} = \frac{1}{2} \varrho V_0^2 S_{W} \Bigg( {\frac {\partial C_{L}}{\partial \alpha }}\left(\alpha -{\frac {\partial \epsilon }{\partial \alpha }}\alpha \right)+{\frac {\partial C_{L}}{\partial \delta_e }}\eta \Bigg) \end{equation*}$

where $S_{HT}$ is the horizontal tail area and $\delta_e$ is the elevator deflection angle. It is reasonable to assume that the lift-curve slope of the tail to changes in $\alpha$ and $\epsilon$ (but not $\delta_e$ ) are the same as that of the main wing, so

(28) $\begin{equation*} L_{HT} = \frac{1}{2} \varrho V_0^2 S_{HT} \left( C_{L_{\alpha}} \left( \alpha - \epsilon \right)+{\frac {\partial C_{L}}{\partial \delta_e }} \delta_e \right) \end{equation*}$

Taking moments (assumed to be positive nose-up) about the center of gravity gives

(29) $\begin{equation*} M = -L_{W} \, x_{cg} - L_{HT} \, l_{HT} = 0 \end{equation*}$

where $x_{cg}$ is the location of the center of gravity relative to the center of pressure on the wing, and $l_{HT}$ is the moment arm for the horizontal tail. For trim, then remember that $M = 0$ . Proceeding by differentiating the previous equation with respect to $\alpha$ gives

(30) $\begin{equation*} \frac {\partial M}{\partial \alpha} = -x_{cg} \left( \frac {\partial L_{W}}{\partial \alpha} \right) - l_{HT} \left( \frac{\partial L_{HT}}{\partial \alpha} \right) \end{equation*}$

It is convenient to treat total lift as acting at a distance $h$ behind the center of gravity, so that the moment equation may also be written as

(31) $\begin{equation*} M = h \left( L_{W} + L_{HT} \right) \end{equation*}$

Differentiating this latter equation with respect to $\alpha$ gives

(32) $\begin{equation*} \frac{\partial M}{\partial \alpha} = h \left( \frac{\partial L_{W}}{\partial \alpha} + \frac{\partial L_{HT}}{\partial \alpha} \right) \end{equation*}$

For stability in pitch, then the sign of $\partial M / \partial \alpha$ must be negative, i.e., for a positive change in $\alpha$ then the moment change on the aircraft from its trimmed state must be negative so as to return its pitch attitude to the trim condition. Because $\partial L_{W} / \partial \alpha$ and $\partial L_{HT} / \partial \alpha$ will both be positive, then $h$ must be negative, i.e., the center of gravity must be in front of the center of lift. Furthermore, because if the values of the lift-curve slopes of the wing and the tail are assumed to be the same, then

(33) $\begin{equation*} \frac{\partial L_{W}}{\partial \alpha} = \frac{\partial L_{HT}}{\partial \alpha} = C_{L_{\alpha}} \end{equation*}$

and so equating Eqs. 30 and 32 gives

(34) $\begin{equation*} h = -\left( x_{cg} + l_{HT} \right) \end{equation*}$

The value of $h$ is called the static margin. Also, assume that the flight controls (elevator in this case) are fixed and do not contribute to the aerodynamics, i.e., $\delta_e = \overbigdot{\delta}_e = 0$ , which is called the “stick-fixed” response. In non-dimensional terms, then

(35) $\begin{equation*} \frac {h}{\overline{c}} = - \Bigg( \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) \frac {l_{HT} \, S_{HT}}{\overline{c} \, S_{W}} \Bigg) = -\Bigg( \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \Bigg) {\cal{V}}_{HT} \right) \end{equation*}$

where $\overline{c}$ is the mean aerodynamic chord of the main wing. The grouping

(36) $\begin{equation*} {\cal{V}}_{HT} = {\frac {l_{HT}\, S_{HT}}{\overline{c} \, S_{W}}} \end{equation*}$

is a non-dimensional parameter known as the tail volume coefficient, in this case for the horizontal tail (HT). It typically has values in the range of $0.50 \le {\cal{V}}_{HT} \le 0.7$ for conventional types of airplanes. In the case of the vertical tail (VT), the tail volume coefficients is usually smaller in the range $0.2 \le {\cal{V}}_{VT} \le 0.4$ .

From Eq. 35, the location of the center of gravity (c.g.) that is just on the edge of static stability in pitch can be calculated, which is called the neutral point, $l_{\rm np}$ , as given by

(37) $\begin{equation*} \frac {l_{\rm np}}{\overline{c}} = -\left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) {\cal{V}}_{HT} \end{equation*}$

Basically, the neutral point can be considered as the fulcrum of aerodynamic forces. Therefore, the static margin is defined as the distance between the neutral point and the center of gravity, which can be written non-dimensionally (based on the preceding assumptions) as

(38) $\begin{equation*} \frac{h}{\overline{c}} = -\Bigg( \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \Bigg) {\cal{V}}_{HT} \right) \end{equation*}$

If the c.g. is ahead of the neutral point (n.p.), then the aircraft will be statically stable, and if the c.g. is behind the n.p., then the aircraft will be unstable, as illustrated below. For static stability then the value of the static margin, $h$ , must be negative. However, it value is often quoted such that positive static stability has a positive static margin, i.e.,

(39) $\begin{equation*} \frac{h}{\overline{c}} = \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) {\cal{V}}_{HT} \end{equation*}$

The distance between the neutral point (n.p.) and the center of gravity is called the static margin. The static margin must be positive such that the c.g. is in front of the n.p. to obtain static stability.

Worked Example #1 – Estimating the static margin

Consider an airplane with a conventional tail. The airplane is trimmed for straight, level and unaccelerated flight. The main wing has a lift curve slope of 0.08 per deg., and the tail has a lift curve slope of 0.06 per deg., and an estimated downwash of 0.1 deg per degree . The horizontal tail volume coefficient is 0.8. The center of gravity is located at 0.36 $\overline{c}$ aft of datum. Estimate the static margin relative to the position of the center of gravity. Will the aircraft be statically stable?

The static margin relative to the position of the center of gravity (c.g.) as a fraction of mean chord is given by

$\frac{h}{\overline{c}} = \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) {\cal{V}}_{HT}$

However, in this case there are different lift curve slopes for the wing and the tail, so the static margin must be written more generally as

$\frac{h}{\overline{c}} = \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) \left( \frac{ \displaystyle{ \frac{\partial C_{L_{HT}} }{\partial \alpha}}}{\displaystyle{\frac{C_{L_{W}}}{\partial \alpha}}} \right) {\cal{V}}_{HT}$

The values of all the terms in the second term on the right-hand-side of the equation is known. However, they must be converted into units of radians so that for the downwash

$\frac {\partial \epsilon}{\partial \alpha } = 0.1~\mbox{degrees per degree} = 0.1~\mbox{radians per radian}$

For the wing, then

$\frac{\partial C_{L_{W}} }{\partial \alpha} = 0.08 \left( \frac{180}{\pi} \right) = 4.58~\mbox{per radian}$

For the horizontal tail, then

$\frac{\partial C_{L_{HT}} }{\partial \alpha} = 0.06 \left( \frac{180}{\pi} \right) = 3.44~\mbox{per radian}$

Therefore,

$\frac{h}{\overline{c}} = \frac{x_{cg}}{\overline{c}} + \left( 1- \frac {\partial \epsilon}{\partial \alpha } \right) \left( \frac{ \displaystyle{ \frac{\partial C_{L_{HT}} }{\partial \alpha}}}{\displaystyle{\frac{C_{L_{W}}}{\partial \alpha}}} \right) {\cal{V}}_{HT}$

and entering the numerical values gives

$\frac{h}{\overline{c}} = \frac{x_{cg}}{\overline{c}} + \left( 1- 0.1 \right) \left( \frac{3.34}{4.58} \right) 0.8 = -0.36 + 0.54 = 0.18$

This result confirms that the aircraft has a positive static margin and so it will be statically stable. Notice that the static margin is affected by the tail volume coefficient, so if that value becomes too small the static margin can becomes too small or even negative, which will lead to adverse stability characteristics.

Sources of Longitudinal Stability

Notice from Eq. 35 that the horizontal tail provides a significant contribution to the static margin, with a larger tail volume coefficient increasing the static margin. Indeed, it will be apparent that a larger tail area will contribute more to the static stability, as well as a longer distance between the tail and the wing.

The typical steady (static) pitching moment contributions about the center of gravity for a conventional airplane as a function of the angle of attack are shown in the figure below. In this regard, a conventional airplane is one with a single wing and tail combination. There is a net-zero pitching moment at the trim angle of attack, $\alpha_{\rm trim}$ . The sign convention is that positive moments are nose-up moments, i.e., $d M_{cg} /d\alpha$ , is positive nose-up, tending to increase the wing’s angle of attack and so having a destabilizing effect on the airplane. Notice that different pitching moment contributions (both in magnitude and in sign) are caused by the various components of the airplane, e.g., the wing, the fuselage, and the tail all produce different aerodynamic effects. Therefore, other moments are produced by these components about the center of gravity.

The primary components of an airplane that will affect the longitudinal stability.

The wing lift component by itself is destabilizing in that it produces a nose-up moment about the center of gravity, i.e., the slope of the moment curve, $d M_{cg} /d\alpha$ , is positive for the wing by itself. Likewise, the fuselage has a destabilizing effect. However, it can be seen that the horizontal tail produces a powerful nose-down moment about the center of gravity with a negative slope of the moment curve, providing a significant stabilizing effect, hence the name “horizontal stabilizer.”

The combined effect of all the components on the entire airplane is a negative $d M_{cg}/d\alpha$ slope, making the airplane statically stable. Generally, a larger horizontal tail will produce a more statically stable airplane, but the physical position of the tail on the fuselage relative to the center of gravity (and other things) plays an important role too. In practice, the area of the tail surfaces must be enough to give a sufficient pitch and directional stability to the airplane. Still, too much stability will also make the airplane less maneuverable as well as “tail-heavy” because of the larger sizes of the surfaces. A goal in airplane design is to give the tail surfaces sufficient area to obtain the needed stability characteristics but not to make them too big such that they adversely affect weight and center of gravity. It is not unusual for tail surfaces to be undersized during design, which only becomes apparent after flight testing.

Worked Example #2 – Horizontal and vertical tail sizing for static stability

In the preliminary design of a certain turboprop aircraft it is required to estimate the size (areas) of the horizontal and vertical tails to give the aircraft sufficient longitudinal and yaw (directional) static stability. Using historic values of the tail volume coefficients for this class of aircraft, ${\cal{V}}_{HT}$ = 0.80 and ${\cal{V}}_{VT}$ = 0.2, estimate the areas of both the horizontal and vertical tails if $l_{HT} = l_{VT} = 0.53 \, l_F$ where the length of the fuselage, $l_F$ , is 46 ft. Assume that $\epsilon = 0$ at the tail surfaces The reference wing area, $S_W$ , is 300 ft² with a mean chord, $\overline{c}$ , of 5.2 ft.

The horizontal tail volume coefficient is given by

${\cal{V}}_{HT} = \frac {l_{HT} \, S_{HT}}{\overline{c} \, S_{W}} = \frac {0.53 \, l_F \, S_{HT}}{\overline{c} \, S_{W}}$

and so the horizontal tail area needed for sufficient pitch stability will be

$S_{HT} = \frac{ {\cal{V}}_{HT} \, \overline{c} \, S_{W} }{0.53 \, l_F } = \frac{ 0.8 \times 5.2 \times 300.0} {0.53 \times 46.0} = 51.2~\mbox{ft$^2$}$

The vertical tail volume coefficient is given by

${\cal{V}}_{VT} = \frac {l_{VT}\, S_{VT}}{\overline{c} \, S_{W}} = \frac {0.53 \, l_F \, S_{VT}}{\overline{c} \, S_{W}}$

and so the vertical tail area needed for sufficient directional (yaw) stability will be

$S_{VT} = \frac{ {\cal{V}}_{VT} \, \overline{c} \, S_{W} }{0.53 \, l_F } = \frac{ 0.8 \times 5.2 \times 300.0} {0.53 \times 46.0} = 12.8~\mbox{ft$^2$}$

Dynamic Stability

If the airplane is statically stable in pitch, then the restoring forces and moments acting on it will cause the nose of the airplane to pitch down again after the initial disturbance. The same is true for yaw and roll disturbances in that disturbances and yaw and roll will cause a return to the undistrurbed trim state if the aircraft has positive static stability. However, this desirable static response does not necessarily mean the airplane will immediately settle and reestablish its original trimmed state. So, the question becomes: What happens to the airplane response(s) in subsequent time, i.e., the dynamic response?

To this end, several possibilities could happen:

1. The airplane may continue to pitch nose-down and overshoot the initial trimmed state. Then the nose comes back up and returns toward trim but overshoots again. This process may continue in a series of nose-up and nose-down pitching motions. Suppose these oscillatory motions eventually damp out over time and cause the airplane to return to the initial trim. In that case, this decaying oscillatory motion means the airplane is dynamically stable.

2. The airplane does not overshoot the trimmed state and settles out quickly to reestablish its trim, which is called subsidence. In this case, the airplane is dynamically stable, and the damping is said to be critically damped or to have a “deadbeat” response. Some airplanes exhibit this characteristic, but many do not because they would have to have larger than desirable tail surfaces, which becomes a weight issue.

3. The airplane may continue with a continuous nose-up and down pitching motion, with the subsequent oscillations in pitch remaining at an almost constant amplitude. In this case, the airplane’s resulting “roller-coaster” dynamic response is said to exhibit neutral dynamic stability. While long-period responses can be damped out by the pilot by the application of compensatory flight controls, it is still an undesirable response.

4. In a worst-case scenario, the airplane may respond with nose-up and nose-down pitching oscillations with increasing amplitude. This type of response would be referred to as being dynamically unstable. As in the case of weak or neutral damping, an unstable aircraft does not necessarily mean it is unsafe if the unstable tendency has a long period and can be controlled by the pilot.

An airplane must be statically stable to be dynamically stable, i.e., it is usually stated that a prerequisite for dynamic stability is static stability. Therefore, a statically unstable airplane will also be dynamically unstable. A statically and dynamically stable airplane is generally easier to fly and control. However, an airplane may be statically stable and dynamically unstable but still flyable, especially if the dynamic response is slow enough for the pilot to control employing appropriate flight control inputs. However, such an aircraft generally has inferior flying qualities and can impose a high workload on the pilot. The dynamic response may also depend on the weight of the aircraft and the center of gravity location, as well as airspeed.

Longitudinal (Pitch) Stability

There are two forms of longitudinal dynamic and oscillatory responses that are found on airplanes: The long period dynamic response and the short-period dynamic response, as shown in the figure below for the pitch motion. On the one hand, the short-period response is typically highly damped and lasts less than a second. On the other hand, the long-period or phugoid mode of oscillation is a slower, weakly damped oscillation of the aircraft’s flight path over many seconds or even minutes.

Short-Period Response

The short-period oscillatory response mode, as hown in the figure below, is higher frequency and highly damped oscillatory response, often appearing in the airplane’s dynamic response after encountering gusty air or from the application of quick elevator inputs, such as during landing. But the short-period response is usually unnoticed by the pilot and does not have to be controlled. All three flight axis will typically show a short period dynamic responsea, which in all cases are quickly damped out.

A representative dynamic response showing the short period (high frequency, heavily damped) and the long period (low frequency, lightly damped).

However, in some circumstances, the pilot may inadvertently excite the short-period responses, such as during landing or in severe turbulence when quick, deliberate movements of the control stick are being made to adjust the aircraft’s flight attitude. In particular, a landing requires significantly increased attention on the controls from the pilot. In some cases, the pilot’s control inputs may become entirely out of phase with the aircraft’s short-period response, resulting in a pilot-induced oscillation (PIO). A PIO is just one type of Aircraft-Pilot Coupling (APC) effect. A PIO is a hazardous flight condition because the pilot’s inputs may cause the short-period response to become quickly divergent, resulting in a mishap or a crash. Most pilots learning to fly sailplanes or jet fighters, which both have relatively sensitive flight controls, after flying other types of airplanes airplanes, initially tend to induce APC effects, the solution being simply to relax their grip on the flight controls.

Long-Period Response

The term “phugoid” was initially coined by Frederick Lanchester for the dynamic pitch response of an aircraft; the word has a literal translation from the Greek meaning “fleeing,” so it is really a misnomer. The phugoid response is typically a weakly damped dynamic response, as shown in the figure above, but it is easily damped out by pilot control inputs and so is easily controlled even if the response is weakly divergent. The phugoid frequency depends on the airspeed of the airplane, and damping in the pitch mode depends on the aircraft’s lift-to-drag ratio. The phugoid response tends to be very pronounced and weakly damped on airplanes with high lift-to-drag ratios, e.g., sailplanes, but is also easily controlled.

Analysing the Long-Period Response

The long-period response is one of the most interesting aspects of the flight dynamic behavior of any aircraft, which is typically is a weakly damped oscillatory motion. Representative in-flight measurements are shown the figure below, although such results could also come from a flight dynamic simulation using solutions from the equations of motion. These types of responses are obtained by disturbing the aircraft about its trimmed condition by the sudden (abrupt) application of flight controls, and then measuring the displacements using suitable instrumentation, such as angle of attack sensors and accelerometers.

Representative in-flight measurements of the dynamic response of an airplane to an aerodynamic perturbation.

For disturbances about a mean or trim angle, the dynamic motion can be described using the equation

(40) $\begin{equation*} \theta(t) = \theta_0 + A e^{-\zeta \omega_n t} \sin( \omega_d t + \phi ) \end{equation*}$

where $\theta$ is the angular displacement, $\theta_0$ is the trim angle, $\omega_n$ is the undamped natural frequency of the dynamic motion, and $\omega_d$ is the damped frequency of the motion. The constants $A$ and $\phi$ are constants that can be determined from the conditions at the application of the initial disturbance, which are arbitrary.

The damped natural frequency is related to the undamped frequency using

(41) $\begin{equation*} \omega_d = \omega_n \sqrt{1 - \zeta^2} \end{equation*}$

where $\zeta$ is the damping. Notice that the damped frequency is always less than the undamped frequency.The parameter of primary interest from flight-test measurements is the damping of the dynamic response, which can be expressed as a logarithmic decrement.

The logarithmic decrement represents the rate at which the amplitude of the phugoid displacement decreases. It can be defined as the ratio of any two of the successive amplitudes in the time history of the motion. These amplitudes could be peaks or valleys in the motion, and so different methods of estimating the damping can be devised. In the ideal cases, measurements of the complete time history of the aircraft dynamic motion will be available. In other cases, only the times of the peaks and valleys in the dynamic response may be available for analysis. If any of these results are available, the damping of the dynamic motion can be estimated.

Peak-to-Peak Method

Let $t_1$ and $t_2$ be the times corresponding to two peaks in the phugoid displacement that are measured one cycle apart. Notice that the displacements should be measured about the trim state. The time period of the dynamic response is $T_P$ which is equal to $t_2 - t_1$ , as shown in the figure below.

The displacements at these two times can be written as

(42) $\begin{equation*} \theta_1 = A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi ) \end{equation*}$

and

(43) $\begin{equation*} \theta_2 = A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi ) \end{equation*}$

Therefore, the ratio of the displacements is

(44) $\begin{equation*} \frac{\theta_2}{\theta_1} = \frac{ A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi )}{A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi)} \end{equation*}$

Because the two peaks are defined to be exactly one cycle apart, then

(45) $\begin{equation*} t_2 = t_1 + T_p = t_1 + \left( \frac{2\pi}{\omega_d} \right) \end{equation*}$

where $T_p$ is the time period of the dynamic response. Therefore,

(46) $\begin{equation*} \sin (\omega_d t_2 + \phi) = \sin (2\pi + \omega_d t_1 + \phi) = \sin (\omega_d t_1 + \phi) \end{equation*}$

Using this result, the ratio of the peaks can now be written as

(47) $\begin{equation*} \frac{\theta_2}{\theta_1} = \frac{ A e^{-\zeta \omega_n t_2} }{A e^{-\zeta \omega_n t_1} } = \frac{ A e^{-\zeta \omega_n (t_1 + T_p) } }{A e^{-\zeta \omega_n t_1 } } = e^{-\zeta \omega_n T_p} \end{equation*}$

To improve the accuracy of the estimated damping, an average can be obtained of the ratios formed at successive peaks in the motion. This average is called the Transient Peak Ratio or $T\!P\!R$ . Therefore the $T\!P\!R$ can be written as

(48) $\begin{equation*} T\!P\!R = \left( \frac{\theta_2}{\theta_1} + \frac{\theta_3}{\theta_2} + \frac{\theta_4}{\theta_3} + ... \right) / N \end{equation*}$

where $N$ is the number of ratios formed, i.e.,

(49) $\begin{equation*} T\!P\!R = \frac{1}{N} \sum_{i=1}^{N} \frac{\theta_{i+1}}{\theta_{i}} \end{equation*}$

Therefore, using Eq. 47 the logarithmic decrement can be written as

(50) $\begin{equation*} \delta = \ln \left( T\!P\!R \right) = -\zeta \omega_n T_{p} = -\zeta \omega_n \left( \frac{2\pi}{\omega_d} \right) \end{equation*}$

Recall that the damped and undamped natural frequencies are related using $\omega_d = \omega_n \sqrt{1-\zeta^2}$ , so $\delta$ can be written as

(51) $\begin{equation*} \delta = -\zeta \omega_n \left( \frac{2 \pi}{\omega_n \sqrt{1-\zeta^2} } \right) = -\frac{ 2\pi \zeta}{\sqrt{1-\zeta^2}} \end{equation*}$

Rearranging this equation gives the average damping, $\zeta$ , as

(52) $\begin{equation*} \zeta = \bigg| \frac{\delta}{\sqrt{(2\pi)^2 + \delta^2} } \bigg| \end{equation*}$

where $\delta = \ln (T\!P\!R)$ . Therefore, with a knowledge of the $T\!P\!R$ , the damping, $\zeta$ , can be determined.

As a byproduct of the above, an average of the time period for one cycle of the long-period dynamic response can be found using

(53) $\begin{equation*} T_{p_{av}} = \frac{1}{N} \sum_{i=1}^{N} T_{p_{i}} \end{equation*}$

and so an average value of the damped frequency is found from

(54) $\begin{equation*} \omega_{d_{av}} = \frac{2\pi}{T_{p_{av}}} \end{equation*}$

Finally, the undamped frequency of the dynamic motion can be found using

(55) $\begin{equation*} \omega_n = \frac{ \omega_{d_{av}}}{\sqrt{1-\zeta^2}} \end{equation*}$

Peak-to-Valley Method

Let $t_1$ and $t_2$ be the times corresponding to a successive peak and valley or a successive valley and a peak in the dynamic response. Again, notice that the displacements should be measured about the trim angle. These peaks and valleys are measured half a cycle apart, i.e., $t_2 - t_1 = T_p/2$ , as shown in the figure below. The displacements at these two times are

(56) $\begin{equation*} \theta_1 = A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi ) \end{equation*}$

and

(57) $\begin{equation*} \theta_2 = A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi ) \end{equation*}$

Application of “Peak-to-Valley Method” to determine oscillatory damping.

(58) $\begin{equation*} D\!A_1 = \bigg|\theta_1\bigg| + \bigg|\theta_2\bigg| = \bigg| A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi ) \bigg| + \bigg| A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi ) \bigg| \end{equation*}$

Because $t_1$ and $t_2$ are exactly one-half cycle apart, then

(59) $\begin{equation*} t_2 = t_1 + \frac{T_p}{2} = t_1 + \left( \frac{\pi}{\omega_d} \right) \end{equation*}$

where $T_p$ is the cycle time of the dynamic response. Therefore,

(60) $\begin{equation*} \sin (\omega_d t_2 + \phi) = \sin (\pi + \omega_d t_1 + \phi) = -\sin (\omega_d t_1 + \phi) \end{equation*}$

and $D\!A_1$ is given by

(61) $\begin{eqnarray*} D\!A_1 & = & A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi ) + A e^{-\zeta \omega_n (t_1 + T_p/2)} \sin( \omega_d t_1 + \phi ) \\ & = & A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi ) \left[ 1 + e^{-\zeta \omega_n T_p/2} \right] \end{eqnarray*}$

In a similar way, $D\!A_2$ is given by

(62) $\begin{equation*} D\!A_2 = A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi ) \left[ 1 + e^{-\zeta \omega_n T_p/2} \right] \end{equation*}$

The ratio of successive $D\!A$ displacements will be

(63) $\begin{equation*} \frac{D\!A_2}{D\!A_1} = \frac{ A e^{-\zeta \omega_n t_2} \sin( \omega_d t_2 + \phi )}{A e^{-\zeta \omega_n t_1} \sin( \omega_d t_1 + \phi)} \end{equation*}$

Again, using the result that $\sin (\omega_d t_2 + \phi) = \sin (\pi + \omega_d t_1 + \phi) = -\sin (\omega_d t_1 + \phi)$ , the ratio of the displacements can be written as

(64) $\begin{equation*} \frac{D\!A_2}{D\!A_1} = -\frac{ A e^{-\zeta \omega_n t_2} }{A e^{-\zeta \omega_n t_1} } = -\frac{ A e^{-\zeta \omega_n (t_1 + T_p/2) } }{A e^{-\zeta \omega_n t_2 } } = -e^{-\zeta \omega_n T_p/2} \end{equation*}$

Therefore, in this case the $T\!P\!R$ can be written as

(65) $\begin{equation*} T\!P\!R = \left( \frac{D\!A_2}{D\!A_1} + \frac{D\!A_3}{D\!A_2} + \frac{D\!A_4}{D\!A_3} + ... \right) / N \end{equation*}$

where $N$ is the number of ratios formed, i.e.,

(66) $\begin{equation*} T\!P\!R = \frac{1}{N} \sum_{i=1}^{N} \frac{D\!A_{i+1}}{D\!A_{i}} \end{equation*}$

Therefore, in this case $\delta$ is

(67) $\begin{equation*} \delta = \zeta \omega_n \left( \frac{\pi}{\omega_n \sqrt{1-\zeta^2} } \right) = \frac{\pi \zeta}{\sqrt{1-\zeta^2}} \end{equation*}$

and the damping, $\zeta$ , as

(68) $\begin{equation*} \zeta = \bigg| \frac{\delta}{\sqrt{\pi^2 + \delta^2} } \bigg| \end{equation*}$

where $\delta = \ln (T\!P\!R)$ . Generally, this approach is found to give a slightly better approximation to the damping, although at the expense of slightly more computational effort.

Again, an average of the time period for one cycle of the dynamic response can be found using

(69) $\begin{equation*} T_{p_{av}} = \frac{1}{N} \sum_{i=1}^{N} T_{p_{i}} \end{equation*}$

and so an average value of the damped frequency is

(70) $\begin{equation*} \omega_{d_{av}} = \frac{2\pi}{T_{p_{av}}} \end{equation*}$

with the undamped frequency (found using the estimated damping) being given by

(71) $\begin{equation*} \omega_n = \frac{ \omega_{d_{av}}}{\sqrt{1-\zeta^2}} \end{equation*}$

Center of Gravity Effects & Limits

As previously discussed, the center of gravity location is critical on an airplane because it has a powerful effect on its stability and control characteristics, e.g., if the center of gravity moves with respect to the neutral point or if the neutral point moves (because of compressibility effects) with respect to the center of gravity. If the center of gravity moves progressively aft (toward the tail), such as when fuel is burned off, then the moment curve slope becomes less negative and will eventually become zero at the neutral point; in this case, the airplane will have neutral static stability.

If the center of gravity is moved even further back, then the airplane will become unstable. On some airplanes, this behavior can become a severe problem if the center of gravity moves too far rearward such as when a load is discharged in flight, e.g., weapons, cargo, parachutists, etc. Likewise, if the center of gravity moves toward the nose, the moments will need to be trimmed out by using up elevator control inputs or a horizontal tail with trim capability. Eventually, suppose the center of gravity moves too far forward. so that the upward elevator displacements on the tail surfaces will not be enough to compensate. In this case, the airplane cannot be trimmed and will become unflyable, nosing down toward the ground and building up airspeed, often with a catastrophic outcome.

The movement of the center of gravity of the airplane will also affect its static stability, with an excessively aft location making the aircraft unstable.

It is clear, therefore, that engineers must carefully establish the center of gravity limits (fore and aft) on an airplane to ensure that it falls within an acceptable range for safe flight and that any center of gravity movements during the flight will not compromise its stability and control characteristics. An example of a center of gravity chart for an airplane is shown in the chart below. Based on the estimated takeoff weight and the calculated center of gravity location, the values must lie within the aircraft’s defined and certified envelope to be safe to fly.

Example of an allowable center of gravity location for an airplane, which will be verified by flight test before certification.

The pilot must be sure that the airplane is loaded correctly before the flight with all passengers and cargo, etc., takes off and that the center of gravity location is within limits. Indeed, many airplanes have crashed because they were not loaded correctly, and the airplane either became unstable during flight or the control forces became too high.

Lateral (Roll) Stability

Lateral stability and control refer to displacements about the longitudinal or roll axes. For example, an airplane has lateral static stability if, after a disturbance is applied, the aircraft rolls and acquires a bank angle but simultaneously generates new aerodynamic forces and moments that tend to reduce the bank angle and bring the airplane back to the initially trimmed flight condition. All airplanes have at least some inherent lateral stability, although because roll and yaw responses are coupled, the resulting dynamic response characteristics tend to be more complicated.

It is well known that using dihedral on the wings is a powerful means of providing an airplane with increased static lateral stability, as shown in the figure below. Just a few degrees of dihedral can make marked improvements to lateral stability. The horizontal tail may also have some dihedral and may contribute to the lateral stability and limit any coupling to the yaw response.

Dihedral is a powerful source of lateral stability for an airplane.

However, using dihedral tends to enhance the coupling between yaw and roll control inputs (i.e., rudder and aileron, respectively), so when the airplane yaws in one direction and develops a sideslip angle, it also tends to roll in that same direction. The application of ailerons to produce a roll response and initiate the turn is also accompanied by different changes in lift and drag on each wing (port and starboard). The consequence is to cause a yaw response in the opposite direction to the turn. The resulting behavior is called adverse yaw and is particularly pronounced on airplanes with long wingspans.

The piloting solution is to first lead the turn using the rudder to compensate for the adverse yaw response when the ailerons are applied. A sailplane, with its long wings, is particularly susceptible to adverse yaw effects and must be flown with constant use of rudder inputs to lead the roll (aileron) inputs so as to prevent adverse yaw.The position of the wing relative to the center of gravity also affects lateral stability, i.e., airplanes with a high-mounted wing versus a low wing. A high-wing airplane design naturally tends to have better lateral stability than a low-mounted wing because the center of gravity is below the wing’s center of pressure, thereby giving a form of pendulum stability. This behavior is why dihedral is not a common design feature on high-wing airplanes. Some airplanes with high-mounted wings may use wings with only partial dihedral, such as on the outer wing panels, but this feature is rare in airplane designs.

Wing sweep is used on high-speed airplanes to reduce compressibility drag, but wing sweep gives significant increases in lateral stability. In fact, a combination of wing dihedral and wing sweep often gives an airplane too much lateral stability. Therefore, airplanes with highly swept wings may use anhedral to negate the inherent lateral stability caused by wing sweep. Furthermore, on a large/heavy airplane with a high-mounted wing configuration (e.g., the C-5 Galaxy), there is usually excess pendular roll stability, so the use of anhedral wings is common.

The use of anhedral on a fighter aircraft is needed to maintain its agility and maneuverability, as shown in the photo below. Fuselage and vertical tail effects may contribute to or detract from the airplane’s lateral stability, but this depends on the shape of the fuselage and the side of the vertical tail.

The use of anhedral wings is often needed on higher-speed aircraft to limit the amount of lateral stability caused by wing sweep and so allowing the aircraft to have better agility.

Directional Stability

Dynamic responses are also associated with an airplane’s directional (yaw) or weathercock motion. However, as already mentioned, the static lateral (roll) and directional (yaw) stability characteristics of an airplane are coupled, i.e., a roll response causes a yaw response and vice-versa. This coupling also affects the dynamic response, i.e., what happens to the airplane at longer times after a disturbance in roll or yaw. This so-called cross-coupling between the static directional and lateral static stability can give rise to three important dynamic responses: a directional divergence mode, spiral divergence mode, and the “Dutch Roll” mode.

Directional Divergence

Directional divergence results from a directionally unstable airplane, as shown in the figure below. When the airplane yaws or rolls into a sideslip, side forces on the airplane are generated, and the yawing moments that arise can continue to increase the sideslip and result in significant yaw angles. Recovery is accomplished by the normal use of flight controls. However, the concern here also is that the vertical tail can stall for steep angles, which reduces its aerodynamic effectiveness to rudder inputs, thereby making recovery difficult. For this reason, not only is the sizing of the vertical tail important to give sufficient directional stability, i.e., in providing adequate surface area, as well as controlling its high yaw angle of attack behavior.

The initiation directional or spiral is from a perturbation in side force, such as from a gust for rudder input.

Spiral Divergence

The spiral divergence mode tends to be pronounced on an airplane that is very stable directionally (about the axis yaw) but is not as stable laterally. This behavior relates to the size of the vertical fin versus the amount of dihedral. The tendency to exhibit spiral divergence is reduced by increasing the dihedral on the wing. When the airplane is in a bank, the aerodynamic forces tend to turn the plane more deeply into the bank, and the nose drops, resulting in an ever-tightening downward spiral with increasing airspeed. However, the use of wing dihedral tends to slow the development of a spiral instability mode. Most airplanes exhibit a spiral instability mode, but it is usually slow to develop, and the airplane is easily recovered to a level flight attitude by the normal use of appropriate flight control inputs.

Dutch Roll

The so-called “Dutch Roll” is a weaving dynamic response mode coupled with both the directional and spiral divergence modes. For most airplanes, the lateral stability is always fairly good, whereas the directional stability can be much weaker. This behavior is especially so if the tail is even slightly undersized during design, which tends to be a rather common approach to save airframe weight; airplane designs generally always become tail-heavy.

If a sideslip disturbance occurs, then the airplane yaws in one direction and the airplane rolls the other away in the form of a weaving motion, as illustrated in the figure below. While this motion is fairly low-frequency and usually well-damped, it is incredibly annoying, if not uncomfortable, for the crew and passengers. In addition, on some airplanes, the Dutch roll mode can become weakly divergent. In this case, a yaw damper, which is part of the flight control system (autopilot), is used to automatically damp out the Dutch roll mode and improve the flying qualities; the damper is switched on soon after takeoff and switched off again just before landing.

The Dutch-roll dynamic mode is a coupled roll-yaw behavior but is usually well damped on most airplanes but can also be damped out almost completely by the action of flight control system called a yaw damper.

The vertical tail provides most of the static directional stability of an airplane. An airplane is said to possess lateral directional or yaw stability if, after a disturbance is applied, the aircraft yaws but simultaneously generates new aerodynamic forces and moments that tend to damp out the yaw displacement. However, the combined size and depth of the fuselage and the vertical tail’s height, area, and shape ultimately affect an airplane’s directional stability characteristics. For example, suppose an airplane experiences a sideslip angle. In that case, the fuselage produces a side force that tends to increase that angle, similar to what the fuselage does for the pitch response.

In this regard, deep fuselages with boxy or elliptical cross-sections tend to be much worse in terms of directional stability than fuselages with circular cross-sections. During the design process, such issues are often explored by wind tunnel testing. A design goal is to shape the fuselage to minimize the static instability and so minimize the size of the tail, hence saving on airframe weight. Fuselages with more area forward of the airplane’s center of gravity tend to have less directional stability. Good examples are the Boeing 747 and the C-5 Galaxy, which have large vertical tails to compensate for the large “hump” on the forward fuselages. The Airbus A380 had similar design issues. The somewhat “disproportionately” large vertical fin and horizontal tail sizes of the A380 or the shorter fuselage on Boeing 747SP are also distinctive, as shown in the photograph below.

The disproportionally large tail on the shorter fuselage Boeing 747SP is needed to give the aircraft sufficient static and dynamic stability.

Ventral and/or dorsal fins are used on some airplanes to augment the directional stability and/or reduce the tendency to develop a Dutch roll, especially at high airspeeds. Adding more vertical tail area using a dorsal fin extension or ventral tail area, as shown in the figure below, provides increased directional stability but at the price of some minor structural weight.

Diagram of how the axis of an airplane can pitch, roll, and yaw. — Ventral and/or dorsal fins are used to augment the directional stability and/or reduce the tendency to develop a Dutch roll, especially at high airspeeds.

Nevertheless, dorsal or ventral fins are simple, lightweight structures and can be resized quickly and inexpensively during flight testing. As a result, they are often helpful in achieving the needed levels of directional stability for the airplane without having to embark on a complete redesign of the vertical tail to resolve deficiencies with the flight dynamics. The photograph below, for example, suggests that additions were made to the aircraft after the first flights to improve its flight stability characteristics. In this case, the addition of so many tail surfaces seems almost an afterthought.

The addition of ventral and dorsal fins can help to achieve better levels of directional stability without increasing the size and weight of the primary stability surfaces.

Handling Qualities Assessments

Handling or flying qualities is used in the study and evaluation of an aircraft’s stability and control characteristics. Assessments of handling qualities are critical to the flight of the aircraft and are related to the ease of controlling an airplane in steady flight and various types of maneuvers. The “ease” of controlling the aircraft will include the forces needed to be applied to the various controls that the pilot must move or otherwise actuate during flight.

Poor handling qualities can lead to pilot error, loss of control, and accidents. Therefore, aircraft manufacturers and regulatory bodies place great emphasis on the handling qualities of an aircraft during its certification and operational phases. Test pilots and flight test engineers use the Cooper-Harper handling qualities rating scale to assess aircraft handling and flying qualities in setting. The numerical scale ranges from 1 to 10, with a value of 1 indicating the best handling characteristics and a value of 10 being the worst, i.e., unflyable.

The Cooper-Harper aircraft handling qualities rating scale.

The scale is subjective, so several test pilots and engineers are usually used in the evaluation of aircraft handling qualities. Specific mission task elements (MTEs) are usually defined for the aircraft in question (based on its intended purpose), which are a set of “role-relatable” or representative tasks. The test pilots and flight test engineers then evaluate the aircraft’s handling qualities for these MTEs, amongst other criteria.

The handling qualities are judged not just by reference to the aircraft’s role but by reference to the MTE and the skill level expected of an average pilot. Usually, handling qualities assessments rated less than Level 3 are considered unacceptable for a modern aircraft, and changes to the aircraft and/or the flight control system will likely be required. Particular attention is usually focused on exploring deficient handling qualities in the form of pilot-induced oscillations (PIO) or, in general, the possibility of any type of Aircraft-Pilot Coupling (APC) effects. In addition, there are derivative MTEs used for specialist military flight operations such as air-to-air refueling, operations from ships, etc.

Summary & Closure

Stability and control are crucial factors in the design of any flight vehicle. Aerospace engineers need to understand the basics of stability and control to ensure safe and efficient flight. Static stability refers to the aircraft’s initial response to a disturbance, such as a gust of wind. If the aircraft returns to its original state after the disturbance is removed, it is said to be statically stable. For an airplane, static stability about all axes is necessary for safe flight and good handling, which is usually achieved during the design stage.

Dynamic stability refers to the aircraft’s response over time to a disturbance. If the aircraft oscillates around its original state before returning to a stable state, it is said to be dynamically stable. Dynamic stability is explored and documented during flight testing, and changes to the design may be made if needed to improve stability. Aerodynamic surfaces like ventral or dorsal fins may also be added to an aircraft to improve stability characteristics. These surfaces can provide additional stability and control to the aircraft by increasing the surface area and the resulting aerodynamic forces. Aircraft handling qualities refer to the ease and precision with which a pilot can control an aircraft to perform a specific task, such as maneuvering, landing, or following a flight path. Good handling qualities are critical for safe and efficient flight operations, as they allow the pilot to maintain control of the aircraft in various flight conditions.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Review and comment on the tail design for the Beech 1900 turboprop airplane.
Can you fly a statically unstable aircraft? Explain carefully.
During flight testing, it was found that a new military fighter airplane has a very pronounced spiral instability mode. Consider the options as to whether this is a problem and whether it should be mitigated.
It is found during flight testing of an airplane that, under some conditions, the phugoid mode is mildly divergent. Discuss whether this is a problem or not.
Besides PIO, do some research to find out more about Aircraft-Pilot Coupling (APC) events.
If the yaw damper on a jet aircraft fails during flight, are there likely to be any issues of concern?
Consider some types of handling qualities assessments that might be needed for an uncrewed aerial vehicle.

Other Useful Online Resources

To dive further into the stability and control of aircraft, then visit the following web sites:

A great website on aircraft stability & control.
A fantastic educational old film on the Secret of Flight 5: Stability and Control
A nice set of course notes on stability and control from Cornell University.
A good website on stability and control more from the piloting perspective.

The author acknowledges his teacher, Arthur W. Babister. Dr. Babister was a flight dynamics engineer at the Blackburn Aircraft Company, contributing to the design of the Buccaneer fighter-bomber and the Beverly transport. He became a professor of engineering at the University of Glasgow in 1965. Using his well-known book, "Aircraft Dynamic Stability and Response," he taught countless student engineers the intricacies of flight dynamics with his characteristic patience, always appreciating that it was not the most straightforward subject for a student to master. ↵

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/ https://doi.org/10.15394/eaglepub.2022.1066.n42

Introduction[1]