49 Airplane Stability & Control

Introduction[1]

The stability and control characteristics of an aircraft or other flight vehicle are rather complex, not just because of the relatively advanced mathematics needed to explain the flight characteristics. Nevertheless, the subject’s basic principles can still be introduced on a physical basis without straying too far into mathematics. While the professional practice of stability and control requires specialist knowledge and considerable experience, it is nevertheless essential for all aerospace engineers, regardless of their primary discipline, to develop a first-level understanding of the fundamentals of this field and to appreciate its importance in the aircraft design process.

The term “stability” in reference to a flight vehicle is the tendency for that vehicle to continue to fly in its prescribed or trimmed flight condition after being subjected to a disturbance, 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 releases hold of 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 “workload” is a measure of how easy or how difficult the aircraft is to fly. The term “control” means the ability to change the flight attitude or other state 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 that contribute to its static and dynamic stability characteristics.
  • Understand what is meant by aircraft handling qualities assessments.

Forces & Moments of Flight

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

  1. Inertia effects, which will arise from the aircraft’s mass distribution as well as its linear and angular accelerations.
  2. Aerodynamic forces and moments, which will depend on the aircraft’s linear (translational) velocities.
  3. Aerodynamic damping effects, which will arise from the aircraft’s angular velocities, i.e., pitch, roll, and yaw rates.
  4. Aerodynamic effects produced by the pilot’s application of the flight controls.
  5. Gravitational effects, which manifest from the magnitude and distribution of the aircraft’s mass and inertial properties about all three flight axes.
  6. Propulsive effects on the forces and moments on the aircraft produced by the engine(s) used to propel the aircraft forward.

Other factors influencing flight dynamics are atmospheric conditions, flight altitude (i.e., density altitude), flight airspeed (true airspeed), and corresponding Mach number. These factors can all affect the aerodynamic characteristics of the aircraft and its propulsion system, and so, either directly or indirectly, they 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 inertial 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,” less agile, and more challenging to fly.

2. Aerodynamic Damping Effects

Aerodynamic damping forces and moments arise from the aircraft’s angular velocities and are sometimes called rotary forces and moments. Damping reduces the transient or oscillatory motion and is usually a desirable flight dynamic characteristic in that it contributes to the aircraft’s stability. An aircraft’s wing and horizontal and vertical tail surfaces primarily contribute to the damping. For higher rates of change, apparent mass aerodynamic effects (also called added mass) may also be significant.

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. Therefore, what happened to the aircraft’s motion at a 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 within the design limits to ensure the stability of the aircraft maintained during the entire flight.

6. Propulsive Effects

These are the effects of 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 and/or pitch and/or roll, thereby 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:

  1. Longitudinal stability and control concerns the airplane’s response in the pitch or angle of attack degree of freedom.
  2. Lateral stability and control relate to the lateral axis or rolling degree of freedom.
  3. Directional stability and control relate to the yawing axis or directional (weathercock) degree of freedom.
An airplane can pitch, roll, and yaw. By design, it is generally expected to be 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 or angle of attack 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 the purposes of 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 in the direction of flight. 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 summarized by the two tables below, one for the forces and the other for the moments. A right-handed rule typically defines the direction of positive rotations and angular rates for stability and control analysis. It is essential to note the potential for symbol conflict between flight dynamic moments and aerodynamic coefficients, which must be carefully distinguished and properly reconciled in any analysis less undesirable outcomes manifest.

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 the equilibrium condition can be expressed as

(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 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, no net forces or moments can act on the airplane about the center of gravity when in the trim condition.

 

An airplane in static equilibrium or trimmed, unaccelerated, level flight, flight where 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 this location has no net aerodynamic moment. The center of gravity is usually located in front of the center of pressure (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 can 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 are required to produce significant longitudinal pitching moments.

Consider now what would happen if the center of gravity moved 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 that is offset from the location of the center of gravity. Thrust can also produce a pitching moment, i.e., a thrust/pitch coupling; this latter effect is 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 pressure (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 slowly during flight, and further trimming by the pilot or flight control system may be required. To reduce the trim drag on a large commercial airplane, fuel is pumped from one tank to another to manage the longitudinal and lateral center of gravity position during flight rather than accepting the increased drag from the application of 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 pressure 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 it transitions 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 pressure 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 using stability derivatives, i.e., the change in a specific force or moment with respect to specific types 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)\\[8pt] Y &= & f_Y\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r}\right)\\[8pt] 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)\\[12pt] M &= & f_M\left(u, \, \overbigdot{u}, \, v, \, \overbigdot{v}, \, w, \, \overbigdot{w}, \, p, \, \overbigdot{p}, \, q, \, \overbigdot{q}, \, r, \, \overbigdot{r}\right)\\[12pt] 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 and potentially non-linear and interdependent (coupling) effects, 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)\\[8pt] 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)\\[8pt] 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)\\[8pt] 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)\\[8pt] 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)\\[8pt] 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 formidable, and even intractable from any reasonable practical perspective.

Linearization Process

The method used in flight dynamics is to linearize the preceding relationships with a Taylor series expansion about the trim state and then retain only the first-order derivatives. Therefore, 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, representing 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 preceding equation are called the stability derivatives. They can be used to determine the aircraft’s response to disturbances or perturbations about the trim state. However, it is essential to remember 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, therefore, a total of 108 derivatives, which is unmanageable in any practical context.

Simplifications

Fortunately, the preceding 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. These simplifications now result in a much shorter subset of the original equations.

Based on the linearization about the trim conditions and using the preceding 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 as gradients or slopes in the preceding 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, 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 (i.e., density 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 an excellent 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 are

(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 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. However, gusts can come from virtually any direction, affecting the airplane’s response about any axis. Still, the vertical or w velocity gusts typically have the most significant effects on the aircraft’s responses.

A vertically upward gust, w, will cause an increase in the wing’s angle of attack, increasing its lift, and the airplane’s inherent 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 gust or other disturbance tend to cause the nose to pitch up further, then the airplane would be considered statically unstable, which is 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 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 length units, although it 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_{\rm HT} = W \end{equation*}

where W is the weight of the aircraft, L_{W} is the wing lift, and L_{\rm HT} is the tail lift. The wing lift can be written conventionally 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 for cambered (non-symmetric) wing sections if needed. 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). Still, 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_{\rm 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_{\rm 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) is the same as that of the main wing, so

(28)   \begin{equation*} L_{\rm HT} = \frac{1}{2} \varrho V_0^2 S_{\rm 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_{\rm HT} \, l_{\rm 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_{\rm 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_{\rm HT} \left( \frac{\partial L_{\rm 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_{\rm 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_{\rm HT}}{\partial \alpha} \right) \end{equation*}

Therefore, for pitch stability, it will be apparent that the sign of \partial M / \partial \alpha must be negative, i.e., for a positive change in \alpha. The moment change on the aircraft from its trimmed state must be negative to return its pitch attitude to the trim condition. Because \partial L_{W} / \partial \alpha and \partial L_{\rm 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 pressure.

Furthermore, 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_{\rm HT}}{\partial \alpha} = C_{L_{\alpha}} \end{equation*}

and so equating Eqs. 30 and 32 gives

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

The value of h is called the static margin.

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

where \overline{c} is the mean aerodynamic chord (MAC) of the main wing, i.e.,

(36)   \begin{equation*} \mbox{MAC} = \overline{c} = \frac{2 \int_0^s c^2 dy}{S} \end{equation*}

where s is semi-span and S is wing area. The grouping

(37)   \begin{equation*} {\cal{V}}_{\rm HT} = {\frac {l_{\rm HT}\, S_{\rm 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}}_{\rm HT} \le 0.7 for conventional types of airplanes. In the case of the vertical tail (VT), the tail volume coefficient is usually smaller in the range 0.2 \le {\cal{V}}_{\rm VT} \le 0.4.

From the use of 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

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

The neutral point is the fulcrum or pivot point 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 definitions and assumptions) as

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

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

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

The distance between the neutral point (n.p.) and the center of gravity (c.g.) 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 degree, the tail has a lift curve slope of 0.06 per degree, and an estimated downwash of 0.1 deg per degree. The horizontal tail volume coefficient is 0.8. The center of gravity is at 0.36 \overline{c} aft of the 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 the mean chord is given by

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

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

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

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

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

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

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

Therefore,

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

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 to 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 be reduced to unacceptable values, usually leading to adverse stability characteristics.

Sources of Longitudinal Stability

Notice from Eq. 35 that the horizontal tail significantly contributes to the static margin, with a more significant 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 center of pressure on 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 sign) are caused by the various components of the airplane, e.g., the wing, the fuselage, and the tail, which all produce different aerodynamic effects. Therefore, these components produce other moments 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.

Therefore, 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 will become apparent after flight testing. Adding dorsal fins is often a solution, which can be done with minimal design changes, weight, and cost.

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

In the preliminary design of a specific 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 historical values of the tail volume coefficients for this class of aircraft, {\cal{V}}_{\rm HT} = 0.80 and {\cal{V}}_{\rm VT} = 0.2, estimate the areas of both the horizontal and vertical tails if l_{\rm HT} = l_{\rm 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

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

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

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

The vertical tail volume coefficient is given by

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

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

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

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 yaw and roll will cause a return to the 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. 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, 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 exhibits neutral dynamic stability. While the pilot can dampen long-period responses by applying 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 called dynamically unstable. As with 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., 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 aircraft’s weight, the center of gravity location, and airspeed.

Longitudinal (Pitch) Stability

Two forms of longitudinal dynamic and oscillatory responses 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.

 

Types of dynamic longitudinal responses: (a) Short-period response; (b) Phugoid (long period) response.

Short-Period Response

As shown in the figure below, the short-period oscillatory response mode is a higher frequency and highly damped oscillatory response, often appearing in the airplane’s dynamic response after encountering gusty air or applying quick elevator inputs, such as during landing. However, the short-period response is usually unnoticed by the pilot and does not have to be controlled. All three flight axes will typically show short-period dynamic responses, 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 from the pilot on the controls. 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 often 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 have relatively sensitive flight controls, initially tend to induce APC effects after flying other airplanes. The solution to a PIO is for the pilot 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 Greek meaning “fleeing,” so it is a misnomer. As shown in the figure above, the phugoid response is typically a weakly damped dynamic response. Still, it is easily damped out by pilot control inputs and so is easily controlled by the pilot even if the response is weakly divergent. The phugoid frequency, \omega_{\rm ph}, depends on the airspeed of the airplane, V_{\infty} = U_0, where U_0 is the initial trimmed airspeed, according to

(41)   \begin{equation*} \omega_{\rm ph} = \frac{\sqrt{2} \, g}{U_0} \end{equation*}

a result for small perturbations first given by Lanchester in his 1908 book titled Aerodonetics. The damping in the pitch mode depends on the aircraft’s lift-to-drag ratio, as given by

(42)   \begin{equation*} \zeta_{\rm ph} = \frac{1}{\sqrt{2}} \left( \frac{C_L}{C_D} \right) \end{equation*}

Therefore, the phugoid response tends to be very pronounced and weakly damped on airplanes with high lift-to-drag ratios, e.g., sailplanes, but the response is also easily controlled by the pilot with corrective elevator inputs because it occurs over long periods.

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 a weakly damped oscillatory motion. Representative in-flight measurements are shown in 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. If sufficiently large, the resulting angle of attack excursions can often resemble a roller-coaster ride!

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

Characteristic Response

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

(43)   \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

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

where \zeta is the damping coefficient. 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, so different methods of estimating the damping can be devised. In the ideal cases, measurements of the complete time history of the aircraft’s 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.

Method of Analysis

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. These displacements should be measured about the trim angle and are measured half a cycle apart, i.e., t_2 - t_1 = T_p/2, as shown in the figure below.

“Peak-to-Valley Method” to determine oscillatory frequency and damping.

The displacements at these two times are

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

and

(46)   \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

(47)   \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

(48)   \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,

(49)   \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

(50)   \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 TPR method. The value of the TPR can be written as

(51)   \begin{equation*} TPR = \frac{\left( \displaystyle{\frac{\theta_2}{\theta_1} + \frac{\theta_3}{\theta_2} + \frac{\theta_4}{\theta_3} + ... + \frac{\theta_{N}}{\theta_{N-1}}} \right)}{N} \end{equation*}

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

(52)   \begin{equation*} TPR = \frac{1}{N} \sum_{i=2}^{N} \frac{\theta_{i}}{\theta_{i-1}} \end{equation*}

Therefore, using Eq. 50 the logarithmic decrement can be written as

(53)   \begin{equation*} \delta = \ln \left( TPR \right) = -\zeta \omega_n T_{p} = -\zeta \omega_n \left( \frac{2\pi}{\omega_d} \right) \end{equation*}

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

(54)   \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

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

where \delta = \ln (TPR). Therefore, with a knowledge of the value of the TPR, the damping, \zeta, can be determined.

To find the value of the TPR, let the net amplitudes of successive peaks and valleys or valleys and peaks be D\!A_1 = \big| \theta_1 \big| + \big|\theta_2 \big|,  D\!A_2 = \big|\theta_2 \big| + \big|\theta_3\big|, D\!A_3 = \big|\theta_3\big|+ \big|\theta_4\big|, etc. For example, taking the D\!A_1 term gives

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

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

(57)   \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 or period of the dynamic response. Therefore,

(58)   \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

(59)   \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 ) \\[12pt] & = & 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

(60)   \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

(61)   \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*}

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

(62)   \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, the TPR can be written as

(63)   \begin{equation*} TPR = \frac{ \left(\displaystyle{ \frac{D\!A_2}{D\!A_1} + \frac{D\!A_3}{D\!A_2} + \frac{D\!A_4}{D\!A_3} + ... + \frac{D\!A_{N}}{D\!A_{N-1}}}\right) }{N} \end{equation*}

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

(64)   \begin{equation*} TPR = \frac{2}{N} \sum_{i=1}^{N} \frac{D\!A_{i}}{D\!A_{i-1}} \end{equation*}

The value of \delta is given by

(65)   \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 so the damping, \zeta, by

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

where \delta = \ln (TPR).

An average of the period for one cycle of the long-period dynamic response can be found using

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

and also average value of the damped frequency is found from

(68)   \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

(69)   \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 further back, the airplane will become unstable. This behavior can become a severe problem on some airplanes 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, suppose the center of gravity moves toward the nose. In that case, the moments must be trimmed out using 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, contributing somewhat to the lateral stability.

 

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 first to lead the turn using the rudder to compensate for the adverse yaw response when the ailerons are applied. With its long wings, a sailplane has powerful adverse yaw effects and must be flown with constant use of rudder inputs to lead the roll (aileron) inputs susceptible 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, giving a form of pendular 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. 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 will negate some of that.

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, an airplane’s static lateral (roll) and directional (yaw) stability characteristics 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 is that the vertical tail can stall for steep angles, reducing its aerodynamic effectiveness to rudder inputs and making recovery difficult. For this reason, not only is the sizing of the vertical tail essential to give sufficient directional stability, i.e., in providing adequate surface area, but also in 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, as shown in the figure below.

Directional or “spiral” divergence can result from an undersized vertical fin, i.e., the vertical fin has an insufficient area to give adequate “damped” directional stability.

Most airplanes exhibit a spiral instability mode, but it is usually slow to develop, and the airplane can be quickly recovered to a level flight attitude through the regular use of appropriate flight control inputs. Of course, this assumes that the pilot has visual references to the horizon and the ground, or the equivalent in terms of instrument readings. Unfortunately, if a spiral divergence occurs in the clouds and the pilot is unable to interpret (or believe) the instruments correctly, the divergence can continue with increasing airspeed and bank angle into what pilots know as a “Graveyard spiral.”

A contributing human factor, in this case, is the possibility of spatial disorientation resulting from the vestibular response in the pilot’s ears (which tell the brain about balance) that the forces on the aircraft are in equilibrium even though airspeed and bank angle are increasing. Corrective control inputs, in this case by pulling back on the controls to decrease airspeed, serve only to tighten the radius of the spiral and increase the rate of descent, and the aircraft will eventually crash into the ground.

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 relatively common approach to save airframe weight; airplane designs generally always become tail-heavy. Notice that the term “Dutch Roll,” describing this particular dynamic response of an airplane in analogy with ice skating, was coined by Jerome C. Hunsaker in 1916.

If a sideslip disturbance occurs, 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. Most jet aircraft, which will have swept wings, require a yaw damper to prevent an excessive Dutch Roll response.

 

The Dutch-roll dynamic mode is a coupled roll-yaw behavior usually well damped on most airplanes but can also be damped out 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 the aircraft yaws after a disturbance is applied but simultaneously generates new aerodynamic forces and moments that tend to dampen 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 this case, the fuselage produces a side force that tends to increase that angle, similar to what the fuselage does for the pitch response. 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 the 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 so reduce the tendency of the aircraft to develop a Dutch roll.

Indeed, dorsal or ventral fins are simple, lightweight structures and can be resized quickly and inexpensively during flight testing. 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 in 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, adding so many separate vertical tail surfaces seems almost a design 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 greatly emphasize an aircraft’s handling qualities 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, which test pilots use to assess how easy (or difficult) aircraft are to fly.

The scale is subjective, so several test pilots and engineers are usually used to evaluate 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 also by 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 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

Sufficient stability and control are crucial factors in the design of any flight vehicle. Aerospace engineers must understand the basics of stability and control to ensure safe and efficient flight. Initially, the term stability is viewed as a static stability, which is the aircraft’s initial response to a disturbance, such as a gust of wind. If the aircraft returns to its original state after removing the disturbance, it is statically stable. For an airplane, static stability about all axes is necessary for safe flight and good handling, which is usually mandated during the design stage. Dynamic stability refers to the aircraft’s response over time to a disturbance. If the aircraft oscillates around its original trim 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 its dynamic stability. Aerodynamic surfaces like ventral or dorsal fins, increasing the surface area and the resulting aerodynamic forces, may also be added to an aircraft to increase damping and so improve the stability characteristics. 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 specific type of flight path, e.g., an instrument approach. Good handling qualities are critical for safe and efficient flight operations, as they allow the pilot to maintain positive control of the aircraft in all 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 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, visit the following websites:

  1. The author acknowledges his teacher, Arthur W. Babister. Dr. Babister was a flight dynamics engineer at the Blackburn Aircraft Company and contributed to the design of the Buccaneer fighter-bomber and the Beverly transport airplane. He became a professor of aeronautical 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 to master. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 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