Flight dynamics
Overview
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions
Dimensions
Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POVRay to render some of the animations, and the films are release under a Creative Commons licence....
about the vehicle's center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, known as pitch, roll and yaw (quite different from their use as TaitBryan angles).
Aerospace engineers develop control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
s for a vehicle's orientation (attitude) about its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
.
Unanswered Questions
Encyclopedia
Flight dynamics is the science of air
vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions
about the vehicle's center of mass
, known as pitch, roll and yaw (quite different from their use as TaitBryan angles).
Aerospace engineers develop control system
s for a vehicle's orientation (attitude) about its center of mass
. The control systems include actuators, which exert forces in various directions, and generate rotational forces or moment
s about the aerodynamic center
of the aircraft, and thus rotate the aircraft in pitch, roll, or yaw. For example, a pitching moment
is a vertical force applied at a distance forward or aft from the aerodynamic center of the aircraft
, causing the aircraft to pitch up or down.
Roll, pitch and yaw refer to rotations about the respective axes starting from a defined equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle, equivalent to a level heeling angle on a ship. Yaw is known as "heading". The equilibrium pitch angle
in submarine and airship parlance is known as "trim", but in aircraft, this usually refers to angle of attack
, rather than orientation. However, common usage ignores this distinction between equilibrium and dynamic cases.
The most common aeronautical convention defines the roll as acting about the longitudinal axis, positive with the starboard (right) wing down. The yaw is about the vertical body axis, positive with the nose to starboard. Pitch is about an axis perpendicular to the longitudinal plane of symmetry, positive nose up.
A fixedwing aircraft
increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack
(AOA). The roll angle is also known as bank angle on a fixed wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is usually streamlined from nose to tail to reduce drag
making it typically advantageous to keep the sideslip angle near zero, though there are instances when an aircraft may be deliberately "sideslipped" for example a slip
in a fixed wing aircraft.
For flight dynamics applications the earth axes are generally of minimal use, and hence will be ignored. The motions relevant to dynamic stability are usually too short in duration for the motion of the Earth itself to be considered relevant for aircraft
.
In flight dynamics
, pitch, roll and yaw angles measure both the absolute attitude angles (relative to the horizon/North) and changes in attitude angles, relative to the equilibrium orientation of the vehicle
. These are defined as:
In analyzing the dynamics, we are concerned both with rotation
and translation
of this axis set with respect to a fixed inertial frame. For all practical purposes a local Earth axis set is used, this has X and Y axis in the local horizontal plane, usually with the xaxis coinciding with the projection of the velocity vector at the start of the motion
, on to this plane. The z axis is vertical, pointing generally towards the Earth's center, completing an orthogonal set.
In general, the body axes are not aligned with the Earth axes. The body orientation may be defined by three Euler angles
, the TaitBryan rotations, a quaternion
, or a direction cosine matrix (rotation matrix). A rotation matrix is particularly convenient for converting velocity, force, angular velocity
, and torque
vectors between body and Earth coordinate frames.
Body axes tend to be used with missile and rocket configurations. Aircraft stability uses wind axes in which the xaxis points along the velocity vector. For straight and level flight this is found from body axes by rotating nose down through the angle of attack
.
Stability deals with small perturbations in angular displacements about the orientation at the start of the motion. This consists of two components; rotation about each axis, and angular displacements due change in orientation of each axis. The latter term is of second order for the purpose of stability analysis, and is ignored.
The speed, height and trim angle of attack are different for each flight condition, in addition, the aircraft will be configured differently, e.g. at low speed flaps
may be deployed and the undercarriage
may be down.
Except for asymmetric designs (or symmetric designs at significant sideslip), the longitudinal equations of motion (involving pitch and lift forces) may be treated independently of the lateral motion (involving roll and yaw).
The following considers perturbations about a nominal straight and level flight path.
To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stickfixed stability. Stickfree analysis requires the further complication of taking the motion of the control surfaces into account.
Furthermore, the flight is assumed to take place in still air, and the aircraft is treated as a rigid body
.
where:
projected on wind axes we obtain:
where:
Proper reference surface
(wing
surface, in case of planes
)
Pressure coefficient
Friction coefficient
Drag coefficient
Lateral force coefficient
Lift coefficient
It is necessary to know C_{p} and C_{f} in every point on the considered surface.
where:
According to λ there are three possible rarefaction grades and their corresponding motions are called:
The motion of a body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body viscosity
will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the boundary layer
.
Depending on the compressibility of the flow, different kinds of currents can be considered:
where: angle of attack
considered point of the surface
Under these conditions, drag
and lift coefficient
are functions depending exclusively on the angle of attack
of the body and Mach
and Reynolds numbers. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well.
It is also possible to get the dependency of the drag coefficient
respect to the lift coefficient
. This relation is known as the drag coefficient equation: drag coefficient equation
The aerodynamic efficiency has a maximum value, E_{max}, respect to C_{L} where the tangent line from the coordinate origin touches the drag coefficient equation plot.
The drag coefficient, C_{D}, can be decomposed in two ways. First typical decomposition separates pressure and friction effects:
There's a second typical decomposition taking into account the definition of the drag coefficient equation. This decomposition separates the effect of the lift coefficient
in the equation, obtaining two terms C_{D0} and C_{Di}. C_{D0} is known as the parasitic drag coefficient and it is the base draft coefficient at zero lift. C_{Di} is known as the induced drag coefficient and it is produced by the body lift.
Aerodynamic efficiency is now calculated as:
If the configuration of the pane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane.
In case the configuration is asymmetrical respect to the XY plane, however, minimum flag difers from the parasitic drag. On these cases, a new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value.
to describe the longitudinal motion, and then factorize it approximately into a high frequency mode and a low frequency mode. This requires a level of algebraic manipulation which most readers will doubtless find tedious, and adds little to the understanding of aircraft dynamics. The approach adopted here is to use our qualitative knowledge of aircraft behavior to simplify the equations from the outset, reaching the same result by a more accessible route.
The two longitudinal motions (modes) are called the short period pitch oscillation (SPPO), and the phugoid
.
about the new trim. There is very little change in the trajectory over the time it takes for the oscillation to damp out.
Generally this oscillation is high frequency (hence short period) and is damped over a period of a few seconds. A realworld example would involve a pilot selecting a new climb attitude, for example 5º nose up from the original attitude. A short, sharp pull back on the control column may be used, and will generally lead to oscillations about the new trim condition. If the oscillations are poorly damped the aircraft will take a long period of time to settle at the new condition, potentially leading to Pilotinduced oscillation
. If the short period mode is unstable it will generally be impossible for the pilot to safely control the aircraft for any period of time.
This damped
harmonic motion
is called the short period pitch oscillation, it arises from the tendency of a stable aircraft to point in the general direction of flight. It is very similar in nature to the weathercock mode of missile or rocket configurations. The motion involves mainly the pitch attitude (theta) and incidence (alpha). The direction of the velocity vector, relative to inertial axes is . The velocity vector is:
where , are the inertial axes components of velocity. According to Newton's Second Law, the acceleration
s are proportional to the force
s, so the forces in inertial axes are:
where m is the mass
.
By the nature of the motion, the speed variation is negligible over the period of the oscillation, so:
But the forces are generated by the pressure
distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the zaxis:
Or:
In words, the wind axes force is equal to the centripetal acceleration.
The moment equation is the time derivative of the angular momentum
:
where M is the pitching moment, and B is the moment of inertia
about the pitch axis.
Let: , the pitch rate.
The equations of motion, with all forces and moments referred to wind axes are, therefore:
We are only concerned with perturbations in forces and moments, due to perturbations in the states and q, and their time derivatives. These are characterized by stability derivatives
determined from the flight condition. The possible stability derivatives are:
Since the tail is operating in the flowfield of the wing, changes in the wing incidence cause changes in the downwash, but there is a delay for the change in wing flowfield to affect the tail lift, this is represented as a moment proportional to the rate of change of incidence:
Increasing the wing incidence without increasing the tail incidence produces a nose up moment, so is expected to be positive.
The equations of motion, with small perturbation forces and moments become:
These may be manipulated to yield as second order linear differential equation
in :
This represents a damped
simple harmonic motion
.
We should expect to be small compared with unity, so the coefficient of (the 'stiffness' term) will be positive, provided . This expression is dominated by , which defines the longitudinal static stability
of the aircraft, it must be negative for stability. The damping term is reduced by the downwash effect, and it is difficult to design an aircraft with both rapid natural response and heavy damping. Usually, the response is underdamped but stable.
mode. This is analyzed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the climb angle (gamma) and speed. The small perturbation equations of motion are:
which means the centripetal force is equal to the perturbation in lift force.
For the speed, resolving along the trajectory:
where g is the acceleration due to gravity at the earths surface
. The acceleration along the trajectory is equal to the net xwise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the climb angle, so only and need be considered. is the drag increment with increased speed, it is negative, likewise is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the zaxis.
The equations of motion become:
These may be expressed as a second order equation in climb angle or speed perturbation:
Now lift is very nearly equal to weight:
where is the air density, is the wing area, W the weight and is the lift coefficient (assumed constant because the incidence is constant), we have, approximately:
The period of the phugoid, T, is obtained from the coefficient of u:
Or:
Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A propeller
with fixed speed would help. Heavy damping of the pitch rotation or a large rotational inertia increase the coupling between short period and phugoid modes, so that these will modify the phugoid.
in yaw is the same as the pitch stability; it resembles the short period pitch oscillation, with yaw plane equivalents to the pitch plane stability derivatives. For this reason pitch and yaw directional stability are collectively known as the "weathercock" stability of the missile.
Aircraft lack the symmetry between pitch and yaw, so that directional stability in yaw is derived from a different set of stability derivatives. The yaw plane equivalent to the short period pitch oscillation, which describes yaw plane directional stability is called Dutch roll. Unlike pitch plane motions, the lateral modes involve both roll and yaw motion.
Applying an impulse via the rudder pedals should induce Dutch roll
, which is the oscillation in roll and yaw, with the roll motion lagging yaw by a quarter cycle, so that the wing tips follow elliptical paths with respect to the aircraft.
The yaw plane translational equation, as in the pitch plane, equates the centripetal acceleration to the side force.
where (beta) is the sideslip angle
, Y the side force and r the yaw rate.
The moment equations are a bit trickier. The trim condition is with the aircraft at an angle of attack with respect to the airflow. The body xaxis does not align with the velocity vector, which is the reference direction for wind axes. In other words, wind axes are not principal axes
(the mass is not distributed symmetrically about the yaw and roll axes). Consider the motion of an element of mass in position z, x in the direction of the yaxis, i.e. into the plane of the paper.
If the roll rate is p, the velocity of the particle is:
Made up of two terms, the force on this particle is first the proportional to rate of v change, the second is due to the change in direction of this component of velocity as the body moves. The latter terms gives rise to cross products of small quantities (pq, pr,qr), which are later discarded. In this analysis, they are discarded from the outset for the sake of clarity. In effect, we assume that the direction of the velocity of the particle due to the simultaneous roll and yaw rates does not change significantly throughout the motion. With this simplifying assumption, the acceleration of the particle becomes:
The yawing moment is given by:
There is an additional yawing moment due to the offset of the particle in the y direction:
The yawing moment is found by summing over all particles of the body:
where N is the yawing moment, E is a product of inertia, and C is the moment of inertia about the yaw axis.
A similar reasoning yields the roll equation:
where L is the rolling moment and A the roll moment of inertia.
Sideslip generates a sideforce from the fin and the fuselage. In addition, if the wing has dihedral, side slip at a positive roll angle increases incidence on the starboard wing and reduces it on the port side, resulting in a net force component directly opposite to the sideslip direction. Sweep back of the wings has the same effect on incidence, but since the wings are not inclined in the vertical plane, backsweep alone does not affect . However, anhedral may be used with high backsweep angles in high performance aircraft to offset the wing incidence effects of sideslip. Oddly enough this does not reverse the sign of the wing configuration's contribution to (compared to the dihedral case).
Roll rate causes incidence at the fin, which generates a corresponding side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing has dihedral, this will result in a side force momentarily opposing the resultant sideslip tendency. Anhedral wing and or stabilizer configurations can cause the sign of the side force to invert if the fin effect is swamped.
Yawing generates side forces due to incidence at the rudder, fin and fuselage.
Sideslip in the absence of rudder input causes incidence on the fuselage and empennage
, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions. Under sideslip conditions at a given roll angle will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight.
Roll rate generates fin lift causing a yawing moment and also differentially alters the lift on the wings, thus affecting the induced drag contribution of each wing, causing a (small) yawing moment contribution. Positive roll generally causes positive values unless the empennage
is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on because the wing axis is normally closely aligned with the center of gravity.
Yaw rate input at any roll angle generates rudder, fin and fuselage force vectors which dominate the resultant yawing moment. Yawing also increases the speed of the outboard wing whilst slowing down the inboard wing, with corresponding changes in drag causing a (small) opposing yaw moment. opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input.
A positive sideslip angle generates empennage incidence which can cause positive or negative roll moment depending on its configuration. For any nonzero sideslip angle dihedral wings causes a rolling moment which tends to return the aircraft to the horizontal, as does back swept wings. With highly swept wings the resultant rolling moment may be excessive for all stability requirements and anhedral could be used to offset the effect of wing sweep induced rolling moment.
Yaw increases the speed of the outboard wing whilst reducing speed of the inboard one, causing a rolling moment to the inboard side. The contribution of the fin normally supports this inward rolling effect unless offset by anhedral stabilizer above the roll axis (or dihedral below the roll axis).
Roll creates counter rotational forces on both starboard and port wings whilst also generating such forces at the empennage. These opposing rolling moment effects have to be overcome by the aileron input in order to sustain the roll rate. If the roll is stopped at a nonzero roll angle the upward rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the downward rolling moment resulting from sideslip induced yaw rate. Longitudinal stability could be ensured or improved by minimizing the latter effect.
is a handling mode, analogous to the short period pitch oscillation, any effect it might have on the trajectory may be ignored. The body rate r is made up of the rate of change of sideslip angle and the rate of turn. Taking the latter as zero, assuming no effect on the trajectory, for the limited purpose of studying the Dutch roll:
The yaw and roll equations, with the stability derivatives become:
The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become:
This becomes a second order equation governing either roll rate or sideslip:
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions
Dimensions
Dimensions is a French project that makes educational movies about mathematics, focusing on spatial geometry. It uses POVRay to render some of the animations, and the films are release under a Creative Commons licence....
about the vehicle's center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, known as pitch, roll and yaw (quite different from their use as TaitBryan angles).
Aerospace engineers develop control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
s for a vehicle's orientation (attitude) about its center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
. The control systems include actuators, which exert forces in various directions, and generate rotational forces or moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
s about the aerodynamic center
Aerodynamic center
The torques or moments acting on an airfoil moving through a fluid can be accounted for by the net lift applied at some point on the foil, and a separate net pitching moment about that point whose magnitude varies with the choice of where the lift is chosen to be applied...
of the aircraft, and thus rotate the aircraft in pitch, roll, or yaw. For example, a pitching moment
Pitching moment
In aerodynamics, the pitching moment on an airfoil is the moment produced by the aerodynamic force on the airfoil if that aerodynamic force is considered to be applied, not at the center of pressure, but at the aerodynamic center of the airfoil...
is a vertical force applied at a distance forward or aft from the aerodynamic center of the aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
, causing the aircraft to pitch up or down.
Roll, pitch and yaw refer to rotations about the respective axes starting from a defined equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle, equivalent to a level heeling angle on a ship. Yaw is known as "heading". The equilibrium pitch angle
Pitch angle
The pitch angle of a charged particle is the angle between the particle's velocity vector and the local magnetic field. This is a common measurement and topic when studying the magnetosphere. See Aurora and Ring currentUsage: Particle motion:...
in submarine and airship parlance is known as "trim", but in aircraft, this usually refers to angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
, rather than orientation. However, common usage ignores this distinction between equilibrium and dynamic cases.
The most common aeronautical convention defines the roll as acting about the longitudinal axis, positive with the starboard (right) wing down. The yaw is about the vertical body axis, positive with the nose to starboard. Pitch is about an axis perpendicular to the longitudinal plane of symmetry, positive nose up.
A fixedwing aircraft
Fixedwing aircraft
A fixedwing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixedwing aircraft are distinct from rotarywing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...
increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
(AOA). The roll angle is also known as bank angle on a fixed wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is usually streamlined from nose to tail to reduce drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
making it typically advantageous to keep the sideslip angle near zero, though there are instances when an aircraft may be deliberately "sideslipped" for example a slip
Slip (aerodynamic)
A slip is an aerodynamic state where an aircraft is moving somewhat sideways as well as forward relative to the oncoming airflow. In other words, for a conventional aircraft, the nose will not be pointing directly into the relative wind .A slip is also a piloting maneuver where the pilot...
in a fixed wing aircraft.
Basic coordinate systems
The position (and hence motion) of an aircraft is generally defined relative to one of 3 sets of coordinate systems: Wind axes
 X axis  positive in the direction of the oncoming air (relative wind)
 Y axis  positive to right of X axis, perpendicularPerpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
to X axis  Z axis  positive downwards, perpendicular to XY plane
 Inertial axes (or body axes)  based about aircraft CG
 X axis  positive forward, through nose of aircraft
 Y axis  positive to right of X axis, perpendicular to X axis
 Z axis  positive downwards, perpendicular to XY plane
 Earth Axes
 X axis  positive in the direction of northNorthNorth is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....
 Y axis  positive in the direction of eastEastEast is a noun, adjective, or adverb indicating direction or geography.East is one of the four cardinal directions or compass points. It is the opposite of west and is perpendicular to north and south.By convention, the right side of a map is east....
(perpendicular to X axis)  Z axis  positive towards the center of Earth (perpendicular to XY plane)
 X axis  positive in the direction of north
For flight dynamics applications the earth axes are generally of minimal use, and hence will be ignored. The motions relevant to dynamic stability are usually too short in duration for the motion of the Earth itself to be considered relevant for aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
.
In flight dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
, pitch, roll and yaw angles measure both the absolute attitude angles (relative to the horizon/North) and changes in attitude angles, relative to the equilibrium orientation of the vehicle
Vehicle
A vehicle is a device that is designed or used to transport people or cargo. Most often vehicles are manufactured, such as bicycles, cars, motorcycles, trains, ships, boats, and aircraft....
. These are defined as:
 Pitch  angle of X body axis (nose) relative to horizon. Also a positive (nose up) rotation about Y body axis
 Roll  angle of Y body axis (wing) relative to horizon. Also a positive (right wing down) rotation about X body axis
 Yaw  angle of X body axis (nose) relative to North. Also a positive (nose right) rotation about Z body axis
In analyzing the dynamics, we are concerned both with rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and translation
Translation
Translation is the communication of the meaning of a sourcelanguage text by means of an equivalent targetlanguage text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of...
of this axis set with respect to a fixed inertial frame. For all practical purposes a local Earth axis set is used, this has X and Y axis in the local horizontal plane, usually with the xaxis coinciding with the projection of the velocity vector at the start of the motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...
, on to this plane. The z axis is vertical, pointing generally towards the Earth's center, completing an orthogonal set.
In general, the body axes are not aligned with the Earth axes. The body orientation may be defined by three Euler angles
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3dimensional Euclidean space three parameters are required...
, the TaitBryan rotations, a quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in threedimensional space...
, or a direction cosine matrix (rotation matrix). A rotation matrix is particularly convenient for converting velocity, force, angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
, and torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
vectors between body and Earth coordinate frames.
Body axes tend to be used with missile and rocket configurations. Aircraft stability uses wind axes in which the xaxis points along the velocity vector. For straight and level flight this is found from body axes by rotating nose down through the angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
.
Stability deals with small perturbations in angular displacements about the orientation at the start of the motion. This consists of two components; rotation about each axis, and angular displacements due change in orientation of each axis. The latter term is of second order for the purpose of stability analysis, and is ignored.
Design cases
In analyzing the stability of an aircraft, it is usual to consider perturbations about a nominal equilibrium position. So the analysis would be applied, for example, assuming:
 Straight and level flight
 Turn at constant speed
 Approach and landing
 TakeoffTakeoffTakeoff is the phase of flight in which an aerospace vehicle goes from the ground to flying in the air.For horizontal takeoff aircraft this usually involves starting with a transition from moving along the ground on a runway. For balloons, helicopters and some specialized fixedwing aircraft , no...
The speed, height and trim angle of attack are different for each flight condition, in addition, the aircraft will be configured differently, e.g. at low speed flaps
Flap (aircraft)
Flaps are normally hinged surfaces mounted on the trailing edges of the wings of a fixedwing aircraft to reduce the speed an aircraft can be safely flown at and to increase the angle of descent for landing without increasing air speed. They shorten takeoff and landing distances as well as...
may be deployed and the undercarriage
Undercarriage
The undercarriage or landing gear in aviation, is the structure that supports an aircraft on the ground and allows it to taxi, takeoff and land...
may be down.
Except for asymmetric designs (or symmetric designs at significant sideslip), the longitudinal equations of motion (involving pitch and lift forces) may be treated independently of the lateral motion (involving roll and yaw).
The following considers perturbations about a nominal straight and level flight path.
To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stickfixed stability. Stickfree analysis requires the further complication of taking the motion of the control surfaces into account.
Furthermore, the flight is assumed to take place in still air, and the aircraft is treated as a rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
.
Components of the aerodynamic force
The expression to calculate the aerodynamic force is:where:

 Difference between static pressure and free current pressure
 outer normal vector of the element of area
 tangential stress vector practised by the air on the body
 adequate reference surface
projected on wind axes we obtain:
where:

 Drag
 Lateral force
 Lift
Aerodynamic coefficients
Dynamic pressure of the free currentProper reference surface
Surface
In mathematics, specifically in topology, a surface is a twodimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary threedimensional Euclidean space R3 — for example, the surface of a ball...
(wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...
surface, in case of planes
Fixedwing aircraft
A fixedwing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixedwing aircraft are distinct from rotarywing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...
)
Pressure coefficient
Pressure coefficient
The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics...
Friction coefficient
Drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
Lateral force coefficient
Lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...
It is necessary to know C_{p} and C_{f} in every point on the considered surface.
Dimensionless parameters and aerodynamic regimes
In absence of thermal effects, there are three remarkable dimensionless numbers: Compressibility of the flow:
 Mach numberMach numberMach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...
 Viscosity of the flow:
 Reynolds number
 Rarefaction of the flow:
 Knudsen numberKnudsen numberThe Knudsen number is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid...
where:

 speed of soundSoundSound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.Propagation of...
 gas constantGas constantThe gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
by mass unity  absolute temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
 gas constant
 mean free pathMean free pathIn physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.Derivation:...
 speed of sound
According to λ there are three possible rarefaction grades and their corresponding motions are called:
 Continuum current (negligible rarefaction):
 Transition current (moderate rarefaction):
 Free molecular current (high rarefaction):
The motion of a body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...
.
Depending on the compressibility of the flow, different kinds of currents can be considered:
 Incompressible subsonic currentIncompressible flowIn fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
:  Compressible subsonic currentCompressible flowCompressible flow is the area of fluid mechanics that deals with fluids in which the fluid density varies significantly in response to a change in pressure. Compressibility effects are typically considered significant if the Mach number of the flow exceeds 0.3, or if the fluid undergoes very large...
:  Transonic currentTransonicTransonic speed is an aeronautics term referring to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.2, i.e. 600–900 mph...
:  Supersonic currentSupersonicSupersonic speed is a rate of travel of an object that exceeds the speed of sound . For objects traveling in dry air of a temperature of 20 °C this speed is approximately 343 m/s, 1,125 ft/s, 768 mph or 1,235 km/h. Speeds greater than five times the speed of sound are often...
:  Hypersonic currentHypersonicIn aerodynamics, a hypersonic speed is one that is highly supersonic. Since the 1970s, the term has generally been assumed to refer to speeds of Mach 5 and above...
:
Drag coefficient equation and aerodynamic efficiency
If the geometry of the body is fixed and in case of symmetric flight (β=0 and Q=0), pressure and friction coefficients are functions depending on:where: angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
considered point of the surface
Under these conditions, drag
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
and lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...
are functions depending exclusively on the angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
of the body and Mach
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...
and Reynolds numbers. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well.
It is also possible to get the dependency of the drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
respect to the lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...
. This relation is known as the drag coefficient equation: drag coefficient equation
The aerodynamic efficiency has a maximum value, E_{max}, respect to C_{L} where the tangent line from the coordinate origin touches the drag coefficient equation plot.
The drag coefficient, C_{D}, can be decomposed in two ways. First typical decomposition separates pressure and friction effects:
There's a second typical decomposition taking into account the definition of the drag coefficient equation. This decomposition separates the effect of the lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...
in the equation, obtaining two terms C_{D0} and C_{Di}. C_{D0} is known as the parasitic drag coefficient and it is the base draft coefficient at zero lift. C_{Di} is known as the induced drag coefficient and it is produced by the body lift.
Parabolic and generic drag coefficient
A good attempt for the induced drag coefficient is to assume a parabolic dependency of the liftAerodynamic efficiency is now calculated as:
If the configuration of the pane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane.
In case the configuration is asymmetrical respect to the XY plane, however, minimum flag difers from the parasitic drag. On these cases, a new approximate parabolic drag equation can be traced leaving the minimum drag value at zero lift value.
Features and selection of the propeller
Longitudinal modes
It is common practice to derive a fourth order characteristic equationCharacteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
to describe the longitudinal motion, and then factorize it approximately into a high frequency mode and a low frequency mode. This requires a level of algebraic manipulation which most readers will doubtless find tedious, and adds little to the understanding of aircraft dynamics. The approach adopted here is to use our qualitative knowledge of aircraft behavior to simplify the equations from the outset, reaching the same result by a more accessible route.
The two longitudinal motions (modes) are called the short period pitch oscillation (SPPO), and the phugoid
Phugoid
A phugoid or fugoid is an aircraft motion where the vehicle pitches up and climbs, and then pitches down and descends, accompanied by speeding up and slowing down as it goes "uphill" and "downhill." This is one of the basic flight dynamics modes of an aircraft , and a classic example of a positive...
.
Shortperiod pitch oscillation
A short input (in control systems terminology an impulse) in pitch (generally via the elevator in a standard configuration fixed wing aircraft) will generally lead to overshoots about the trimmed condition. The transition is characterized by a damped simple harmonic motionSimple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
about the new trim. There is very little change in the trajectory over the time it takes for the oscillation to damp out.
Generally this oscillation is high frequency (hence short period) and is damped over a period of a few seconds. A realworld example would involve a pilot selecting a new climb attitude, for example 5º nose up from the original attitude. A short, sharp pull back on the control column may be used, and will generally lead to oscillations about the new trim condition. If the oscillations are poorly damped the aircraft will take a long period of time to settle at the new condition, potentially leading to Pilotinduced oscillation
Pilotinduced oscillation
Pilotinduced oscillations, as defined by MILHDBK1797A, are sustained or uncontrollable oscillations resulting from efforts of the pilot to control the aircraft and occurs when the pilot of an aircraft inadvertently commands an often increasing series of corrections in opposite directions, each...
. If the short period mode is unstable it will generally be impossible for the pilot to safely control the aircraft for any period of time.
This damped
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
harmonic motion
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
is called the short period pitch oscillation, it arises from the tendency of a stable aircraft to point in the general direction of flight. It is very similar in nature to the weathercock mode of missile or rocket configurations. The motion involves mainly the pitch attitude (theta) and incidence (alpha). The direction of the velocity vector, relative to inertial axes is . The velocity vector is:
where , are the inertial axes components of velocity. According to Newton's Second Law, the acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
s are proportional to the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s, so the forces in inertial axes are:
where m is the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
.
By the nature of the motion, the speed variation is negligible over the period of the oscillation, so:
But the forces are generated by the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the zaxis:
Or:
In words, the wind axes force is equal to the centripetal acceleration.
The moment equation is the time derivative of the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
:
where M is the pitching moment, and B is the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
about the pitch axis.
Let: , the pitch rate.
The equations of motion, with all forces and moments referred to wind axes are, therefore:
We are only concerned with perturbations in forces and moments, due to perturbations in the states and q, and their time derivatives. These are characterized by stability derivatives
Stability derivatives
Stability Derivatives, and also Control Derivatives, are measures of how particular forces and moments on an aircraft change as other parameters related to stability change . For a defined "trim" flight condition, changes and oscillations occur in these parameters...
determined from the flight condition. The possible stability derivatives are:


 Lift due to incidence, this is negative because the zaxis is downwards whilst positive incidence causes an upwards force.



 Lift due to pitch rate, arises from the increase in tail incidence, hence is also negative, but small compared with .



 Pitching momentPitching momentIn aerodynamics, the pitching moment on an airfoil is the moment produced by the aerodynamic force on the airfoil if that aerodynamic force is considered to be applied, not at the center of pressure, but at the aerodynamic center of the airfoil...
due to incidence  the static stability term. Static stabilityLongitudinal static stabilityLongitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane during static conditions. This characteristic is important in determining whether an aircraft will be able to fly as intended...
requires this to be negative.
 Pitching moment



 Pitching moment due to pitch rate  the pitch damping term, this is always negative.

Since the tail is operating in the flowfield of the wing, changes in the wing incidence cause changes in the downwash, but there is a delay for the change in wing flowfield to affect the tail lift, this is represented as a moment proportional to the rate of change of incidence:
Increasing the wing incidence without increasing the tail incidence produces a nose up moment, so is expected to be positive.
The equations of motion, with small perturbation forces and moments become:
These may be manipulated to yield as second order linear differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
in :
This represents a damped
Damping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
simple harmonic motion
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
.
We should expect to be small compared with unity, so the coefficient of (the 'stiffness' term) will be positive, provided . This expression is dominated by , which defines the longitudinal static stability
Longitudinal static stability
Longitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane during static conditions. This characteristic is important in determining whether an aircraft will be able to fly as intended...
of the aircraft, it must be negative for stability. The damping term is reduced by the downwash effect, and it is difficult to design an aircraft with both rapid natural response and heavy damping. Usually, the response is underdamped but stable.
Phugoid
If the stick is held fixed, the aircraft will not maintain straight and level flight, but will start to dive, level out and climb again. It will repeat this cycle until the pilot intervenes. This long period oscillation in speed and height is called the phugoidPhugoid
A phugoid or fugoid is an aircraft motion where the vehicle pitches up and climbs, and then pitches down and descends, accompanied by speeding up and slowing down as it goes "uphill" and "downhill." This is one of the basic flight dynamics modes of an aircraft , and a classic example of a positive...
mode. This is analyzed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the climb angle (gamma) and speed. The small perturbation equations of motion are:
which means the centripetal force is equal to the perturbation in lift force.
For the speed, resolving along the trajectory:
where g is the acceleration due to gravity at the earths surface
Standard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...
. The acceleration along the trajectory is equal to the net xwise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the climb angle, so only and need be considered. is the drag increment with increased speed, it is negative, likewise is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the zaxis.
The equations of motion become:
These may be expressed as a second order equation in climb angle or speed perturbation:
Now lift is very nearly equal to weight:
where is the air density, is the wing area, W the weight and is the lift coefficient (assumed constant because the incidence is constant), we have, approximately:
The period of the phugoid, T, is obtained from the coefficient of u:
Or:
Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A propeller
Propeller (aircraft)
Aircraft propellers or airscrews convert rotary motion from piston engines or turboprops to provide propulsive force. They may be fixed or variable pitch. Early aircraft propellers were carved by hand from solid or laminated wood with later propellers being constructed from metal...
with fixed speed would help. Heavy damping of the pitch rotation or a large rotational inertia increase the coupling between short period and phugoid modes, so that these will modify the phugoid.
Lateral modes
With a symmetrical rocket or missile, the directional stabilityDirectional stability
Directional stability is stability of a moving body or vehicle about an axis which is perpendicular to its direction of motion. Stability of a vehicle concerns itself with the tendency of a vehicle to return to its original direction in relation to the oncoming medium when disturbed away from...
in yaw is the same as the pitch stability; it resembles the short period pitch oscillation, with yaw plane equivalents to the pitch plane stability derivatives. For this reason pitch and yaw directional stability are collectively known as the "weathercock" stability of the missile.
Aircraft lack the symmetry between pitch and yaw, so that directional stability in yaw is derived from a different set of stability derivatives. The yaw plane equivalent to the short period pitch oscillation, which describes yaw plane directional stability is called Dutch roll. Unlike pitch plane motions, the lateral modes involve both roll and yaw motion.
Dutch roll
It is customary to derive the equations of motion by formal manipulation in what, to the engineer, amounts to a piece of mathematical sleight of hand. The current approach follows the pitch plane analysis in formulating the equations in terms of concepts which are reasonably familiar.Applying an impulse via the rudder pedals should induce Dutch roll
Dutch roll
Dutch roll is a type of aircraft motion, consisting of an outofphase combination of "tailwagging" and rocking from side to side. This yawroll coupling is one of the basic flight dynamic modes...
, which is the oscillation in roll and yaw, with the roll motion lagging yaw by a quarter cycle, so that the wing tips follow elliptical paths with respect to the aircraft.
The yaw plane translational equation, as in the pitch plane, equates the centripetal acceleration to the side force.
where (beta) is the sideslip angle
Sideslip angle
Sideslip angle, also called angle of sideslip , is a term used in fluid dynamics and aerodynamics and aviation. It relates to the rotation of the aircraft centerline from the relative wind...
, Y the side force and r the yaw rate.
The moment equations are a bit trickier. The trim condition is with the aircraft at an angle of attack with respect to the airflow. The body xaxis does not align with the velocity vector, which is the reference direction for wind axes. In other words, wind axes are not principal axes
Aircraft principal axes
An aircraft in flight is free to rotate in three dimensions: pitch, nose up or down about an axis running from wing to wing), yaw, nose left or right about an axis running up and down; and roll, rotation about an axis running from nose to tail. The axes are alternatively designated as lateral,...
(the mass is not distributed symmetrically about the yaw and roll axes). Consider the motion of an element of mass in position z, x in the direction of the yaxis, i.e. into the plane of the paper.
If the roll rate is p, the velocity of the particle is:
Made up of two terms, the force on this particle is first the proportional to rate of v change, the second is due to the change in direction of this component of velocity as the body moves. The latter terms gives rise to cross products of small quantities (pq, pr,qr), which are later discarded. In this analysis, they are discarded from the outset for the sake of clarity. In effect, we assume that the direction of the velocity of the particle due to the simultaneous roll and yaw rates does not change significantly throughout the motion. With this simplifying assumption, the acceleration of the particle becomes:
The yawing moment is given by:
There is an additional yawing moment due to the offset of the particle in the y direction:
The yawing moment is found by summing over all particles of the body:
where N is the yawing moment, E is a product of inertia, and C is the moment of inertia about the yaw axis.
A similar reasoning yields the roll equation:
where L is the rolling moment and A the roll moment of inertia.
Lateral and longitudinal stability derivatives
The states are (sideslip), r (yaw rate) and p (roll rate), with moments N (yaw) and L (roll), and force Y (sideways). There are nine stability derivatives relevant to this motion, the following explains how they originate. However a better intuitive understanding is to be gained by simply playing with a model airplane, and considering how the forces on each component are affected by changes in sideslip and angular velocity:

 Side force due to side slip (in absence of yaw).

Sideslip generates a sideforce from the fin and the fuselage. In addition, if the wing has dihedral, side slip at a positive roll angle increases incidence on the starboard wing and reduces it on the port side, resulting in a net force component directly opposite to the sideslip direction. Sweep back of the wings has the same effect on incidence, but since the wings are not inclined in the vertical plane, backsweep alone does not affect . However, anhedral may be used with high backsweep angles in high performance aircraft to offset the wing incidence effects of sideslip. Oddly enough this does not reverse the sign of the wing configuration's contribution to (compared to the dihedral case).


 Side force due to roll rate.

Roll rate causes incidence at the fin, which generates a corresponding side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing has dihedral, this will result in a side force momentarily opposing the resultant sideslip tendency. Anhedral wing and or stabilizer configurations can cause the sign of the side force to invert if the fin effect is swamped.


 Side force due to yaw rate.

Yawing generates side forces due to incidence at the rudder, fin and fuselage.


 Yawing moment due to sideslip forces.

Sideslip in the absence of rudder input causes incidence on the fuselage and empennage
Empennage
The empennage , also known as the tail or tail assembly, of most aircraft gives stability to the aircraft, in a similar way to the feathers on an arrow...
, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions. Under sideslip conditions at a given roll angle will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight.


 Yawing moment due to roll rate.

Roll rate generates fin lift causing a yawing moment and also differentially alters the lift on the wings, thus affecting the induced drag contribution of each wing, causing a (small) yawing moment contribution. Positive roll generally causes positive values unless the empennage
Empennage
The empennage , also known as the tail or tail assembly, of most aircraft gives stability to the aircraft, in a similar way to the feathers on an arrow...
is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on because the wing axis is normally closely aligned with the center of gravity.


 Yawing moment due to yaw rate.

Yaw rate input at any roll angle generates rudder, fin and fuselage force vectors which dominate the resultant yawing moment. Yawing also increases the speed of the outboard wing whilst slowing down the inboard wing, with corresponding changes in drag causing a (small) opposing yaw moment. opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input.


 Rolling moment due to sideslip.

A positive sideslip angle generates empennage incidence which can cause positive or negative roll moment depending on its configuration. For any nonzero sideslip angle dihedral wings causes a rolling moment which tends to return the aircraft to the horizontal, as does back swept wings. With highly swept wings the resultant rolling moment may be excessive for all stability requirements and anhedral could be used to offset the effect of wing sweep induced rolling moment.


 Rolling moment due to yaw rate.

Yaw increases the speed of the outboard wing whilst reducing speed of the inboard one, causing a rolling moment to the inboard side. The contribution of the fin normally supports this inward rolling effect unless offset by anhedral stabilizer above the roll axis (or dihedral below the roll axis).


 Rolling moment due to roll rate.

Roll creates counter rotational forces on both starboard and port wings whilst also generating such forces at the empennage. These opposing rolling moment effects have to be overcome by the aileron input in order to sustain the roll rate. If the roll is stopped at a nonzero roll angle the upward rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the downward rolling moment resulting from sideslip induced yaw rate. Longitudinal stability could be ensured or improved by minimizing the latter effect.
Equations of motion
Since Dutch rollDutch roll
Dutch roll is a type of aircraft motion, consisting of an outofphase combination of "tailwagging" and rocking from side to side. This yawroll coupling is one of the basic flight dynamic modes...
is a handling mode, analogous to the short period pitch oscillation, any effect it might have on the trajectory may be ignored. The body rate r is made up of the rate of change of sideslip angle and the rate of turn. Taking the latter as zero, assuming no effect on the trajectory, for the limited purpose of studying the Dutch roll:
The yaw and roll equations, with the stability derivatives become:

 (yaw)

 (roll)
The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become:
This becomes a second order equation governing either roll rate or sideslip:

The equation for roll rate is identical. But the roll angle, (phi) is given by:
If p is a damped simple harmonic motion, so is , but the roll must be in quadrature with the roll rate, and hence also with the sideslip. The motion consists of oscillations in roll and yaw, with the roll motion lagging 90 degrees behind the yaw. The wing tips trace out elliptical paths.
Stability requires the "stiffnessStiffnessStiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.Calculations:...
" and "damping" terms to be positive. These are:


 (damping)


 (stiffness)
The denominator is dominated by , the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive.
Considering the "stiffness" term: will be positive because is always negative and is positive by design. is usually negative, whilst is positive. Excessive dihedral can destabilize the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to .
The damping term is dominated by the product of the roll damping and the yaw damping derivatives, these are both negative, so their product is positive. The Dutch roll should therefore be damped.
The motion is accompanied by slight lateral motion of the center of gravity and a more "exact" analysis will introduce terms in etc. In view of the accuracy with which stability derivatives can be calculated, this is an unnecessary pedantry, which serves to obscure the relationship between aircraft geometry and handling, which is the fundamental objective of this article.
Roll subsidence
Jerking the stick sideways and returning it to center causes a net change in roll orientation.
The roll motion is characterized by an absence of natural stability, there are no stability derivatives which generate moments in response to the inertial roll angle. A roll disturbance induces a roll rate which is only canceled by pilot or autopilotAutopilotAn autopilot is a mechanical, electrical, or hydraulic system used to guide a vehicle without assistance from a human being. An autopilot can refer specifically to aircraft, selfsteering gear for boats, or auto guidance of space craft and missiles...
intervention. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to:
is negative, so the roll rate will decay with time. The roll rate reduces to zero, but there is no direct control over the roll angle.
Spiral mode
Simply holding the stick still, when starting with the wings near level, an aircraft will usually have a tendency to gradually veer off to one side of the straight flightpath. This is the (slightly unstable) spiral mode.
Spiral mode trajectory
In studying the trajectory, it is the direction of the velocity vector, rather than that of the body, which is of interest. The direction of the velocity vector when projected on to the horizontal will be called the track, denoted (muMu (letter)Carlos Alberto Vives Restrepo is a Grammy Award and threetime Latin Grammy Award winningColombian singer, composer and actor.Biography:...
). The body orientation is called the heading, denoted (psi). The force equation of motion includes a component of weight:
where g is the gravitational acceleration, and U is the speed.
Including the stability derivatives:
Roll rates and yaw rates are expected to be small, so the contributions of and will be ignored.
The sideslip and roll rate vary gradually, so their time derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s are ignored. The yaw and roll equations reduce to:

 (yaw)

 (roll)
Solving for and p:
Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle:
This is an exponentialExponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
growth or decay, depending on whether the coefficient of is positive or negative. The denominator is usually negative, which requires (both products are positive). This is in direct conflict with the Dutch roll stability requirement, and it is difficult to design an aircraft for which both the Dutch roll and spiral mode are inherently stable.
Since the spiral mode has a long time constant, the pilot can intervene to effectively stabilize it, but an aircraft with an unstable Dutch roll would be difficult to fly. It is usual to design the aircraft with a stable Dutch roll mode, but slightly unstable spiral mode.
See also
 Acronyms and abbreviations in avionicsAcronyms and abbreviations in avionicsA:*ACARS: Aircraft Communications Addressing and Reporting System.*ACAS: Airborne Collision Avoidance System.*ACP: Audio Control Panel.*ACS: Audio Control System.*ADAHRS: Air Data and Attitude Heading Reference System.*ADC: Air Data Computer....
 1902 Wright Glider
 AeronauticsAeronauticsAeronautics is the science involved with the study, design, and manufacturing of airflightcapable machines, or the techniques of operating aircraft and rocketry within the atmosphere...
 Aircraft heading
 Aircraft attitude
 Aircraft bank
 Helicopter dynamicsHelicopter dynamicsHelicopter dynamics is a field within aerospace engineering concerned with theoretical and practical aspects of helicopter flight. Its purpose is the knowledge of forces and torques which appear in helicopter flight.History:...
 Aircraft flight mechanicsAircraft flight mechanicsIn aeronautics, aircraft flight mechanics is the study of the forces that act on an aircraft in flight, and the way the aircraft responds to those forces.Aircraft flight mechanics are relevant to gliders, helicopters and aeroplanes....
 Attitude control
 Crosswind landingCrosswind landingA crosswind landing is a landing maneuver in which a significant component of the prevailing wind is perpendicular to the runway center line.Significance:Aircraft in flight are subject to the direction of the winds in which the aircraft is operating...
 Dynamic positioningDynamic positioningDynamic positioning is a computer controlled system to automatically maintain a vessel's position and heading by using its own propellers and thrusters...
 JSBSimJSBSimJSBSim is an open source Flight Dynamics Model software library that models the flight dynamics of an aerospace vehicle. The library has been incorporated into the flight simulation packages FlightGear and OpenEaagles and a commercial . It can also be called from a small standalone program to...
(An open source flight dynamics software model)
 Longitudinal static stabilityLongitudinal static stabilityLongitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane during static conditions. This characteristic is important in determining whether an aircraft will be able to fly as intended...
 Rigid body dynamicsRigid body dynamicsIn physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...
 Rotation matrix
 Ship motionsShip motionsShip motions are defined by the six degrees of freedom that a ship, boat or any other craft can experience. Translation :HeaveSwaySurgeVertical axis:Vertical axis, or yaw axis — an axis drawn from top to bottom, and perpendicular to the other two axes...
 Stability derivativesStability derivativesStability Derivatives, and also Control Derivatives, are measures of how particular forces and moments on an aircraft change as other parameters related to stability change . For a defined "trim" flight condition, changes and oscillations occur in these parameters...
 Static marginStatic marginStatic margin is a concept used to characterize the static stability and controllability of aircraft and missiles.*In aircraft analysis, static margin is defined as the distance between the center of gravity and the neutral point of the aircraft....
 VariableResponse Research Aircraft
 Weathervane effectWeathervane effectWeathervaning or weathercocking is a phenomenon experienced by aircraft on the ground.Aircraft on the ground have a natural pivoting point on an axis through the main landing gear contact points [disregarding the effects of toe in/toe out of the main gear]...
External links

 (stiffness)


 (damping)

