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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 gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
(cg), known as ''pitch'', ''roll'' and ''yaw''. These are collectively known as aircraft attitude, often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to
rotary aircraft A rotorcraft or rotary-wing aircraft is a heavier-than-air aircraft with rotary wings or rotor blades, which generate lift by rotating around a vertical mast. Several rotor blades mounted on a single mast are referred to as a rotor. The Internat ...
such as helicopters, and dirigibles, where the flight dynamics involved in establishing and controlling attitude are entirely different.
Control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...
s adjust the orientation of a vehicle about its cg. A control system includes control surfaces which, when deflected, generate a moment (or couple from ailerons) about the cg which rotates the aircraft in pitch, roll, and yaw. For example, a
pitching moment In aerodynamics, the pitching moment on an airfoil is the moment (or torque) 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 o ...
comes from a force applied at a distance forward or aft of the cg, causing the aircraft to pitch up or down. Roll, pitch and yaw refer to rotations about the respective axes starting from a defined steady flight equilibrium state. The equilibrium roll angle is known as wings level or zero bank angle. The most common aeronautical convention defines roll as acting about the longitudinal axis, positive with the starboard (right) wing down. 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
fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine, such as an airplane, which is capable of flight using wings that generate lift caused by the aircraft's forward airspeed and the shape of the wings. Fixed-wing aircraft are dist ...
increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the
angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
(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 streamlined from nose to tail to reduce drag making it advantageous to keep the sideslip angle near zero, though an aircraft may be deliberately "sideslipped" to increase drag and descent rate during landing, to keep aircraft heading same as runway heading during cross-wind landings and during flight with asymmetric power.


Introduction


Reference frames

Three
right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
, Cartesian coordinate systems see frequent use in flight dynamics. The first coordinate system has an origin fixed in the reference frame of the Earth: *Earth frame ** Origin - arbitrary, fixed relative to the surface of the Earth ** ''xE'' axis - positive in the direction of
north North is one of the four compass points or cardinal directions. It is the opposite of south and is perpendicular to east and west. ''North'' is a noun, adjective, or adverb indicating direction or geography. Etymology The word ''north ...
** ''yE'' axis - positive in the direction of
east East or Orient is one of the four cardinal directions or points of the compass. It is the opposite direction from west and is the direction from which the Sun rises on the Earth. Etymology As in other languages, the word is formed from the fac ...
** ''zE'' axis - positive towards the center of the Earth In many flight dynamics applications, the Earth frame is assumed to be inertial with a flat ''xE'',''yE''-plane, though the Earth frame can also be considered a
spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
with origin at the center of the Earth. The other two reference frames are body-fixed, with origins moving along with the aircraft, typically at the center of gravity. For an aircraft that is symmetric from right-to-left, the frames can be defined as: *Body frame ** Origin - airplane center of gravity ** ''xb'' axis - positive out the nose of the aircraft in the plane of symmetry of the aircraft ** ''zb'' axis - perpendicular to the ''xb'' axis, in the plane of symmetry of the aircraft, positive below the aircraft ** ''yb'' axis - perpendicular to the ''xb'',''zb''-plane, positive determined by the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of ...
(generally, positive out the right wing) *Wind frame ** Origin - airplane center of gravity ** ''xw'' axis - positive in the direction of the velocity vector of the aircraft relative to the air ** ''zw'' axis - perpendicular to the ''xw'' axis, in the plane of symmetry of the aircraft, positive below the aircraft ** ''yw'' axis - perpendicular to the ''xw'',''zw''-plane, positive determined by the right hand rule (generally, positive to the right) Asymmetric aircraft have analogous body-fixed frames, but different conventions must be used to choose the precise directions of the ''x'' and ''z'' axes. The Earth frame is a convenient frame to express aircraft translational and rotational kinematics. The Earth frame is also useful in that, under certain assumptions, it can be approximated as inertial. Additionally, one force acting on the aircraft, weight, is fixed in the +''zE'' direction. The body frame is often of interest because the origin and the axes remain fixed relative to the aircraft. This means that the relative orientation of the Earth and body frames describes the aircraft attitude. Also, the direction of the force of thrust is generally fixed in the body frame, though some aircraft can vary this direction, for example by
thrust vectoring Thrust vectoring, also known as thrust vector control (TVC), is the ability of an aircraft, rocket, or other vehicle to manipulate the direction of the thrust from its engine(s) or motor(s) to control the attitude or angular velocity of the ve ...
. The wind frame is a convenient frame to express the aerodynamic forces and moments acting on an aircraft. In particular, the net
aerodynamic force In fluid mechanics, an aerodynamic force is a force exerted on a body by the air (or other gas) in which the body is immersed, and is due to the relative motion between the body and the gas. Force There are two causes of aerodynamic force: ...
can be divided into components along the wind frame axes, with the
drag force In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
in the −''xw'' direction and the lift force in the −''zw'' direction. In addition to defining the reference frames, the relative orientation of the reference frames can be determined. The relative orientation can be expressed in a variety of forms, including: * Rotation matrices *
Direction cosine In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to ...
s *
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
*
Quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s The various Euler angles relating the three reference frames are important to flight dynamics. Many Euler angle conventions exist, but all of the rotation sequences presented below use the ''z-y'-x"'' convention. This convention corresponds to a type of Tait-Bryan angles, which are commonly referred to as Euler angles. This convention is described in detail below for the roll, pitch, and yaw Euler angles that describe the body frame orientation relative to the Earth frame. The other sets of Euler angles are described below by analogy.


Transformations (

Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
)


From Earth frame to body frame

* First, rotate the Earth frame axes ''xE'' and ''yE'' around the ''zE'' axis by the yaw angle ''ψ''. This results in an intermediate reference frame with axes denoted ''x'',y'',z'', where ''z'=zE''. * Second, rotate the ''x'' and ''z'' axes around the ''y'' axis by the pitch angle ''θ''. This results in another intermediate reference frame with axes denoted ''x",y",z"'', where ''y"=y''. * Finally, rotate the ''y"'' and ''z"'' axes around the ''x"'' axis by the roll angle ''φ''. The reference frame that results after the three rotations is the body frame. Based on the rotations and axes conventions above: * Yaw angle ''ψ:'' angle between north and the projection of the aircraft longitudinal axis onto the horizontal plane; * Pitch angle ''θ:'' angle between the aircraft longitudinal axis and horizontal; * Roll angle ''φ:'' rotation around the aircraft longitudinal axis after rotating by yaw and pitch.


From Earth frame to wind frame

* Heading angle ''σ:'' angle between north and the horizontal component of the velocity vector, which describes which direction the aircraft is moving relative to cardinal directions. *
Flight path In the United States, airways or air routes are defined by the Federal Aviation Administration (FAA) in two ways: "VOR Federal airways and Low/Medium Frequency (L/MF) (Colored) Federal airways" These are designated routes which aeroplanes f ...
angle ''γ: is the angle between horizontal and the velocity vector, which describes whether the aircraft is climbing or descending.'' * Bank angle ''μ: represents a rotation of the lift force around the velocity vector, which may indicate whether the airplane is
turning Turning is a machining process in which a cutting tool, typically a non-rotary tool bit, describes a helix toolpath by moving more or less linearly while the workpiece rotates. Usually the term "turning" is reserved for the generation ...
.'' When performing the rotations described above to obtain the body frame from the Earth frame, there is this analogy between angles: * ''σ, ψ'' (heading vs yaw) * ''γ, θ'' (Flight path vs pitch) * ''μ, φ'' (Bank vs Roll)


From wind frame to body frame

* sideslip angle ''β:'' angle between the velocity vector and the projection of the aircraft longitudinal axis onto the xw,yw-plane, which describes whether there is a lateral component to the aircraft velocity * angle of attack ''α'': angle between the ''xw'',''yw''-plane and the aircraft longitudinal axis and, among other things, is an important variable in determining the magnitude of the force of lift When performing the rotations described earlier to obtain the body frame from the Earth frame, there is this analogy between angles: * ''β, ψ'' (sideslip vs yaw) * ''α'', ''θ'' (attack vs pitch) * ''(φ = 0)'' (nothing vs roll)


Analogies

Between the three reference frames there are hence these analogies: * Yaw / Heading / Sideslip (Z axis, vertical) * Pitch / Flight path / Attack angle (Y axis, wing) * Roll / Bank / nothing (X axis, nose)


Design cases

In analyzing the stability of an aircraft, it is usual to consider perturbations about a nominal steady flight state. So the analysis would be applied, for example, assuming: ::Straight and level flight ::Turn at constant speed ::Approach and landing ::
Takeoff Takeoff is the phase of flight in which an aerospace vehicle leaves the ground and becomes airborne. For aircraft traveling vertically, this is known as liftoff. For aircraft that take off horizontally, this usually involves starting with a ...
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 stick-fixed stability. Stick-free 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 In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
.


Forces of flight

Three forces act on an aircraft in flight:
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
,
thrust Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
, and the
aerodynamic force In fluid mechanics, an aerodynamic force is a force exerted on a body by the air (or other gas) in which the body is immersed, and is due to the relative motion between the body and the gas. Force There are two causes of aerodynamic force: ...
.


Aerodynamic force


Components of the aerodynamic force

The expression to calculate the aerodynamic force is: :: \mathbf_A = \int_\Sigma (-\Delta p \mathbf + \mathbf) \,d\sigma where: :: \Delta p \equiv Difference between static pressure and free current pressure :: \mathbf \equiv outer normal vector of the element of area :: \mathbf \equiv tangential stress vector practised by the air on the body :: \Sigma \equiv adequate reference surface projected on wind axes we obtain: :: \mathbf_A = -(\mathbf_w D + \mathbf_w Q + \mathbf_w L) where: :: D \equiv Drag :: Q \equiv Lateral force :: L \equiv Lift


Aerodynamic coefficients

Dynamic pressure In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ( ...
of the free current \equiv q = \tfrac12\, \rho\, V^ Proper reference
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
(
wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
surface, in case of planes) \equiv S
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. Every point in a fluid flow field has its own ...
\equiv C_p = \dfrac Friction coefficient \equiv C_f = \dfrac
Drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 e ...
\equiv C_d = \dfrac = - \dfrac \int_\Sigma (-C_p) \mathbf \bullet \mathbf + C_f \mathbf \bullet \mathbf\,d\sigma Lateral force coefficient \equiv C_Q = \dfrac = - \dfrac \int_\Sigma (-C_p) \mathbf \bullet \mathbf + C_f \mathbf \bullet \mathbf\,d\sigma Lift coefficient \equiv C_L = \dfrac = - \dfrac \int_\Sigma (-C_p) \mathbf \bullet \mathbf + C_f \mathbf \bullet \mathbf\,d\sigma It is necessary to know Cp and Cf 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 number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \f ...
\equiv M = \dfrac * Viscosity of the flow: :
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
\equiv Re = \dfrac * Rarefaction of the flow: :
Knudsen number The Knudsen number (Kn) 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. The number is name ...
\equiv Kn = \dfrac where: :: a = \sqrt \equiv speed of
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
::: k\equiv
specific heat ratio In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
::: R\equiv
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
by mass unity ::: \theta \equiv absolute
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
:: \lambda = \dfrac \sqrt = \dfrac \sqrt \equiv
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as ...
According to λ there are three possible rarefaction grades and their corresponding motions are called: * Continuum current (negligible rarefaction): \dfrac \ll 1 * Transition current (moderate rarefaction): \dfrac \approx 1 * Free molecular current (high rarefaction): \dfrac \gg 1 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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
. Depending on the compressibility of the flow, different kinds of currents can be considered: * Incompressible subsonic current: 0 < M < 0.3 * Compressible subsonic current: 0.3 < M < 0.8 * Transonic current: 0.8 < M < 1.2 * Supersonic current: 1.2 < M < 5 * Hypersonic current: 5 < M


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: :: C_p = C_p ( \alpha , M , Re , P) :: C_f = C_f ( \alpha , M , Re , P) where: :: \alpha \equiv
angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
:: P \equiv considered point of the surface Under these conditions, drag and lift coefficient are functions depending exclusively on the
angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is m ...
of the body and Mach and
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
s. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well. :: \begin C_D = C_D ( \alpha , M , Re) \\ C_L = C_L ( \alpha , M , Re) \\ E = E ( \alpha , M , Re) = \dfrac \\ \end It is also possible to get the dependency of the
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) 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 e ...
respect to the lift coefficient. This relation is known as the drag coefficient equation: :: C_D = C_D ( C_L , M , Re ) \equiv drag coefficient equation The aerodynamic efficiency has a maximum value, Emax, respect to CL where the tangent line from the coordinate origin touches the drag coefficient equation plot. The drag coefficient, CD, can be decomposed in two ways. First typical decomposition separates pressure and friction effects: :: C_D = C_ + C_ \begin C_ = \dfrac = - \dfrac \int_\Sigma C_f \mathbf \bullet \mathbf \,d\sigma \\ C_ = \dfrac = - \dfrac \int_\Sigma (-C_p) \mathbf \bullet \mathbf \,d\sigma \end 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 CD0 and CDi. CD0 is known as the parasitic drag coefficient and it is the base drag coefficient at zero lift. CDi is known as the induced drag coefficient and it is produced by the body lift. :: C_D = C_ + C_ \begin C_ = (C_D)_ \\ C_ \end


Parabolic and generic drag coefficient

A good attempt for the induced drag coefficient is to assume a parabolic dependency of the lift : C_ = k C_L^2 \Rightarrow C_D = C_ + k C_L^2 Aerodynamic efficiency is now calculated as: : E = \dfrac \Rightarrow \begin E_ = \dfrac \\ (C_L)_ = \sqrt \\ (C_)_ = C_ \end If the configuration of the plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane. : C_ = (C_D)_ = C_ In case the configuration is asymmetrical respect to the XY plane, however, minimum drag differs 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. : C_ = C_ \neq (C_D)_ : C_D = C_ + k (C_L - C_)^2


Variation of parameters with the Mach number

The
Coefficient of pressure 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. Every point in a fluid flow field has its own ...
varies with
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \f ...
by the relation given below: ::C_ = \frac where * Cp is the compressible
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. Every point in a fluid flow field has its own ...
* Cp0 is the incompressible
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. Every point in a fluid flow field has its own ...
* ''M'' is the freestream Mach number. This relation is reasonably accurate for 0.3 < M < 0.7 and when ''M = 1'' it becomes ∞ which is impossible physical situation and is called Prandtl–Glauert singularity.


Aerodynamic force in a specified atmosphere

see
Aerodynamic force In fluid mechanics, an aerodynamic force is a force exerted on a body by the air (or other gas) in which the body is immersed, and is due to the relative motion between the body and the gas. Force There are two causes of aerodynamic force: ...


Stability

Stability refers to the ability of the aircraft to counteract disturbances to its flight path. According to David P. Davies, there are six types of aircraft stability: speed stability, stick free static longitudinal stability, static lateral stability, directional stability, oscillatory stability, and spiral stability.


Speed stability

An aircraft in
cruise flight Cruise is the phase of aircraft flight that starts when the aircraft levels off after a climb, until it begins to descend for landing. Cruising usually consumes the majority of a flight, and it may include changes in heading (direction of fligh ...
is typically speed stable. If speed increases, drag increases, which will reduce the speed back to equilibrium for its configuration and thrust setting. If speed decreases, drag decreases, and the aircraft will accelerate back to its equilibrium speed where thrust equals drag. However, in slow flight, due to
lift-induced drag In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings o ...
, as speed decreases, drag increases (and vice versa). This is known as the "back of the
drag curve The drag curve or drag polar is the relationship between the drag on an aircraft and other variables, such as lift, the coefficient of lift, angle-of-attack or speed. It may be described by an equation or displayed as a graph (sometimes called a " ...
". The aircraft will be speed unstable, because a decrease in speed will cause a further decrease in speed.


Static stability and control


Longitudinal static stability

Longitudinal stability refers to the stability of an aircraft in pitch. For a stable aircraft, if the aircraft pitches up, the wings and tail create a pitch-down moment which tends to restore the aircraft to its original attitude. For an unstable aircraft, a disturbance in pitch will lead to an increasing pitching moment. Longitudinal static stability is the ability of an aircraft to recover from an initial disturbance. Longitudinal dynamic stability refers to the damping of these stabilizing moments, which prevents persistent or increasing oscillations in pitch.


Directional stability

Directional or weathercock stability is concerned with the static stability of the airplane about the z axis. Just as in the case of longitudinal stability it is desirable that the aircraft should tend to return to an equilibrium condition when subjected to some form of yawing disturbance. For this the slope of the yawing moment curve must be positive. An airplane possessing this mode of stability will always point towards the relative wind, hence the name weathercock stability.


Dynamic stability and control


Longitudinal modes

It is common practice to derive a fourth order characteristic equation to describe the longitudinal motion, and then factorize it approximately into a high frequency mode and a low frequency mode. The approach adopted here is using qualitative knowledge of aircraft behavior to simplify the equations from the outset, reaching the result by a more accessible route. The two longitudinal motions (modes) are called the short period pitch oscillation (SPPO), and the phugoid.


=Short-period pitch oscillation

= A short input (in
control systems A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
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 motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
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 real-world example would involve a pilot selecting a new climb attitude, for example 5° 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 Pilot-induced 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 A wind vane, weather vane, or weathercock is an instrument used for showing the direction of the wind. It is typically used as an architectural ornament to the highest point of a building. The word ''vane'' comes from the Old English word , m ...
mode of missile or rocket configurations. The motion involves mainly the pitch attitude \theta (theta) and incidence \alpha (alpha). The direction of the velocity vector, relative to inertial axes is \theta-\alpha. The velocity vector is: ::u_f=U\cos(\theta-\alpha) ::w_f=U\sin(\theta-\alpha) where u_f, w_f are the inertial axes components of velocity. According to
Newton's Second Law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in mo ...
, the
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
s are proportional to the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s, so the forces in inertial axes are: ::X_f=m\frac=m\frac\cos(\theta-\alpha)-mU\frac\sin(\theta-\alpha) ::Z_f=m\frac=m\frac\sin(\theta-\alpha)+mU\frac\cos(\theta-\alpha) where ''m'' is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
. By the nature of the motion, the speed variation m\frac is negligible over the period of the oscillation, so: ::X_f= -mU\frac\sin(\theta-\alpha) ::Z_f=mU\frac\cos(\theta-\alpha) But the forces are generated by the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an
inertial In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the z-axis: ::Z=-Z_f\cos(\theta-\alpha)+X_f\sin(\theta-\alpha) Or: ::Z=-mU\frac In words, the wind axes force is equal to the centripetal acceleration. The moment equation is the time derivative of the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
: ::M=B\frac where M is the pitching moment, and B is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
about the pitch axis. Let: \frac=q, the pitch rate. The equations of motion, with all forces and moments referred to wind axes are, therefore: ::\frac=q+\frac ::\frac=\frac We are only concerned with perturbations in forces and moments, due to perturbations in the states \alpha and q, and their time derivatives. These are characterized by
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 (parameters such as airspeed, altitude, angle of attack, etc.). For a def ...
determined from the flight condition. The possible stability derivatives are: ::Z_\alpha Lift due to incidence, this is negative because the z-axis is downwards whilst positive incidence causes an upwards force. ::Z_q Lift due to pitch rate, arises from the increase in tail incidence, hence is also negative, but small compared with Z_\alpha. ::M_\alpha
Pitching moment In aerodynamics, the pitching moment on an airfoil is the moment (or torque) 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 o ...
due to incidence - the static stability term. Static stability requires this to be negative. ::M_q 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: ::M_\dot\alpha The delayed downwash effect gives the tail more lift and produces a nose down moment, so M_\dot\alpha is expected to be negative. The equations of motion, with small perturbation forces and moments become: ::\frac=\left(1+\frac\right)q+\frac\alpha ::\frac=\fracq+\frac\alpha+\frac\dot\alpha These may be manipulated to yield as second order linear
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
in \alpha: ::\frac-\left(\frac+\frac+(1+\frac)\frac\right)\frac+\left(\frac\frac-\frac(1+\frac)\right)\alpha=0 This represents a damped simple harmonic motion. We should expect \frac to be small compared with unity, so the coefficient of \alpha (the 'stiffness' term) will be positive, provided M_\alpha<\fracM_q. This expression is dominated by M_\alpha, which defines the
longitudinal static stability In flight dynamics, longitudinal stability is the stability of an aircraft in the longitudinal, or pitching, plane. This characteristic is important in determining whether an aircraft pilot will be able to control the aircraft in the pitching p ...
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 (except in the unlikely case that it happens to be perfectly trimmed for level flight at its current altitude and thrust setting), 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 phugoid mode. This is analyzed by assuming that the
SSPO SCO-spondin is a protein that in humans is encoded by the ''SSPO'' gene. SCO-spondin is secreted by the subcommissural organ, and contributes to commissural axon growth and the formation of Reissner's fiber Reissner's fiber ''(named after E ...
performs its proper function and maintains the angle of attack near its nominal value. The two states which are mainly affected are the flight path angle \gamma (gamma) and speed. The small perturbation equations of motion are: ::mU\frac=-Z which means the centripetal force is equal to the perturbation in lift force. For the speed, resolving along the trajectory: ::m\frac=X-mg\gamma where g is the acceleration due to gravity at the Earth's surface. The acceleration along the trajectory is equal to the net x-wise force minus the component of weight. We should not expect significant aerodynamic derivatives to depend on the flight path angle, so only X_u and Z_u need be considered. X_u is the drag increment with increased speed, it is negative, likewise Z_u is the lift increment due to speed increment, it is also negative because lift acts in the opposite sense to the z-axis. The equations of motion become: :: mU\frac=-Z_u u :: m\frac=X_u u -mg\gamma These may be expressed as a second order equation in flight path angle or speed perturbation: ::\frac-\frac\frac-\fracu=0 Now lift is very nearly equal to weight: ::Z=\frac\rho U^2 c_L S_w=W where \rho is the air density, S_w is the wing area, W the weight and c_L is the lift coefficient (assumed constant because the incidence is constant), we have, approximately: ::Z_u=\frac=\frac The period of the phugoid, T, is obtained from the coefficient of u: ::\frac=\sqrt Or: ::T=\frac Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A
propeller A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...
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 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 ...
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 is a type of aircraft motion consisting of an out-of- phase combination of "tail-wagging" (yaw) and rocking from side to side (roll). This yaw-roll coupling is one of the basic flight dynamic modes (others include phugoid, short ...
, 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. ::\frac=\frac-r where \beta (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 x-axis 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 y-axis, i.e. into the plane of the paper. If the roll rate is p, the velocity of the particle is: :::v=-pz+xr 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: :::\frac=-\fracz+\fracx The yawing moment is given by: :::\delta m x \frac=-\fracxz\delta m + \fracx^2\delta m There is an additional yawing moment due to the offset of the particle in the y direction:\fracy^2\delta m The yawing moment is found by summing over all particles of the body: :::N=-\frac\int xz dm +\frac\int x^2 + y^2 dm =-E\frac+C\frac 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: :::L=A\frac-E\frac where L is the rolling moment and A the roll moment of inertia.


= Lateral and longitudinal stability derivatives

= The states are \beta (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: :::Y_\beta 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 Y_\beta. 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 Y_\beta (compared to the dihedral case). :::Y_p 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. :::Y_r Side force due to yaw rate. Yawing generates side forces due to incidence at the rudder, fin and fuselage. :::N_\beta Yawing moment due to sideslip forces. Sideslip in the absence of rudder input causes incidence on the fuselage and
empennage The empennage ( or ), also known as the tail or tail assembly, is a structure at the rear of an aircraft that provides stability during flight, in a way similar to the feathers on an arrow.Crane, Dale: ''Dictionary of Aeronautical Terms, third e ...
, 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 N_\beta will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight. :::N_p 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 N_p values unless the
empennage The empennage ( or ), also known as the tail or tail assembly, is a structure at the rear of an aircraft that provides stability during flight, in a way similar to the feathers on an arrow.Crane, Dale: ''Dictionary of Aeronautical Terms, third e ...
is anhedral or fin is below the roll axis. Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on N_p because the wing axis is normally closely aligned with the center of gravity. :::N_r 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. N_r 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. :::L_\beta 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 non-zero 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. :::L_r 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). :::L_p 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 non-zero roll angle the L_\beta ''upward'' rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the ''downward'' L_r 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 roll Dutch roll is a type of aircraft motion consisting of an out-of- phase combination of "tail-wagging" (yaw) and rocking from side to side (roll). This yaw-roll coupling is one of the basic flight dynamic modes (others include phugoid, short ...
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: :::\frac= -r The yaw and roll equations, with the stability derivatives become: ::C\frac-E\frac=N_\beta \beta - N_r \frac + N_p p (yaw) ::A\frac-E\frac=L_\beta \beta - L_r \frac + L_p p (roll) The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become: ::-C\frac = N_\beta \beta - N_r \frac + N_p p ::E\frac = L_\beta \beta - L_r \frac + L_p p This becomes a second order equation governing either roll rate or sideslip: ::\left(\frac\frac-\frac\right)\frac+ \left(\frac\frac-\frac\frac\right)\frac- \left(\frac\frac-\frac\frac\right)\beta = 0 The equation for roll rate is identical. But the roll angle, ''\phi'' (phi) is given by: :::\frac=p If ''p'' is a damped simple harmonic motion, so is ''\phi'', 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 "
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a ...
" and "damping" terms to be positive. These are: :::\frac (damping) :::\frac (stiffness) The denominator is dominated by L_p, the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive. Considering the "stiffness" term: -L_p N_\beta will be positive because L_p is always negative and N_\beta is positive by design. L_\beta is usually negative, whilst N_p is positive. Excessive dihedral can destabilize the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to L_\beta. 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 Y_\beta 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
autopilot An autopilot is a system used to control the path of an aircraft, marine craft or spacecraft without requiring constant manual control by a human operator. Autopilots do not replace human operators. Instead, the autopilot assists the operator' ...
intervention. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to: ::A\frac=L_p p. L_p 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 ''\mu'' ( mu). The body orientation is called the heading, denoted ''\psi'' (psi). The force equation of motion includes a component of weight: ::\frac=\frac + \frac\phi where ''g'' is the gravitational acceleration, and ''U'' is the speed. Including the stability derivatives: ::\frac=\frac\beta + \frac r + \fracp + \frac\phi Roll rates and yaw rates are expected to be small, so the contributions of Y_r and Y_p will be ignored. The sideslip and roll rate vary gradually, so their time
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s are ignored. The yaw and roll equations reduce to: ::N_\beta \beta + N_r\frac + N_p p = 0 (yaw) ::L_\beta \beta + L_r\frac + L_p p = 0 (roll) Solving for ''\beta'' and ''p'': :::\beta=\frac\frac :::p=\frac\frac Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle: :::\frac=mg\frac\phi This is an
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
or decay, depending on whether the coefficient of ''\phi'' is positive or negative. The denominator is usually negative, which requires L_\beta N_r > N_\beta L_r (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 avionics Below are abbreviations used in aviation, avionics, aerospace and aeronautics. A B C D E F G H I J K L M N N numbers (turbines) O P Q R S T U V V speeds W X Y Z See also * List of avia ...
*
Aeronautics Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identif ...
* Steady flight *
Aircraft flight control system A conventional fixed-wing aircraft flight control system consists of flight control surfaces, the respective cockpit controls, connecting linkages, and the necessary operating mechanisms to control an aircraft's direction in flight. Aircraft ...
*
Aircraft flight mechanics Aircraft flight mechanics are relevant to fixed wing ( gliders, aeroplanes) and rotary wing (helicopters) aircraft. An aeroplane (''airplane'' in US usage), is defined in ICAO Document 9110 as, "a power-driven heavier than air aircraft, deriving i ...
*
Aircraft heading In navigation, the heading of a vessel or aircraft is the compass direction in which the craft's bow or nose is pointed. Note that the heading may not necessarily be the direction that the vehicle actually travels, which is known as its ''cours ...
* Aircraft bank *
Crosswind landing In aviation, a 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 ...
* Dynamic positioning *
Flight control surfaces Aircraft flight control surfaces are aerodynamic devices allowing a pilot to adjust and control the aircraft's flight attitude. Development of an effective set of flight control surfaces was a critical advance in the development of aircraft. Ea ...
* Helicopter dynamics * JSBSim (An open source flight dynamics software model) *
Longitudinal static stability In flight dynamics, longitudinal stability is the stability of an aircraft in the longitudinal, or pitching, plane. This characteristic is important in determining whether an aircraft pilot will be able to control the aircraft in the pitching p ...
*
Rigid body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of ...
*
Rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \ ...
* Ship motions *
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 (parameters such as airspeed, altitude, angle of attack, etc.). For a def ...
*
Static margin In flight dynamics, longitudinal stability is the stability of an aircraft in the longitudinal, or pitching, plane. This characteristic is important in determining whether an aircraft pilot will be able to control the aircraft in the pitching pl ...
*
Weathervane effect Weathervaning or weathercocking is a phenomenon experienced by aircraft on the ground and rotorcraft on the ground and when hovering. Aircraft on the ground have a natural pivoting point on a plane through the main landing gear contact points i ...
* 1902 Wright Glider


References


Notes


Bibliography

* NK Sinha and N Ananthkrishnan (2013), ''Elementary Flight Dynamics with an Introduction to Bifurcation and Continuation Methods'', CRC Press, Taylor & Francis. *


External links


Open source simulation framework in C++


{{DEFAULTSORT:Flight Dynamics Aerodynamics Avionics Flight control systems