Flight dynamics - Citizendia

Pitch

Yaw

Roll

Flight dynamics is the science of air and space vehicle orientation and control in three dimensions. A spacecraft is a Vehicle or machine designed for Spaceflight. 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 (See Tait-Bryan rotations for an explanation). In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The Tait-Bryan rotations, named after Peter Guthrie Tait and George Bryan.

Aerospace engineers develop control systems for a vehicle's orientation (attitude) about it's center of mass. Aerospace engineering is the branch of Engineering behind the design construction and science of Aircraft and Spacecraft. A control system is a device or set of devices to manage command direct or regulate the behavior of other devices or systems The control systems include actuators, which exert forces in various directions, and generate rotational forces or moments about the aerodynamic center of the aircraft, and thus rotate the aircraft in pitch, roll, or yaw. In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector The aerodynamic center of an Airfoil moving through a Fluid is the point at which the Pitching moment coefficient for the airfoil does not vary with 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. In Aerodynamics, the pitching moment on an Airfoil is the Moment produced by the Aerodynamic force on the airfoil if that aerodynamic force is

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. Heeling is the lean caused by the wind's force on the Sails of a sailing vessel Yaw and Pitch is known as 'heading'. The equilibrium pitch angle in submarine and airship parlance is know as 'trim', but in aircraft, this usually refers to angle of attack, rather than orientation. Angle of attack ( AOA, \alpha Greek letter alpha) is a term used in Aerodynamics to describe the Angle between the 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 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 fixed-wing 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). Overview Fixed-wing aircraft range from small training and recreational aircraft to Wide-body aircraft and military cargo aircraft. Angle of attack ( AOA, \alpha Greek letter alpha) is a term used in Aerodynamics to describe the Angle between the The roll angle is also known as bank angle on a fixed wing aircraft, which "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 yaw angle near zero, though there are instances when an aircraft may be deliberately "yawed" for example a Slip in a fixed wing aircraft. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a

1 Coordinate systems
- 1.1 Design cases
2 Longitudinal modes
- 2.1 Short-period pitch oscillation
- 2.2 Phugoid
3 Lateral modes
4 See also
5 References
6 External links

Coordinate systems

In flight dynamics, pitch, roll and yaw angles measure changes in attitude, relative to the equilibrium orientation of the vehicle. Vehicles, derived from the Latin word vehiculum, are non-living Means of transport.

The positive X axis, in aircraft, points along the velocity vector, in missiles and rockets it points towards the nose.
The positive Y axis goes out the right wing of the vehicle
The positive Z axis goes out the underside of the vehicle

Unless designed to conduct part of the mission within a planetary atmosphere, a spacecraft would generally have no discernible front or side, and no bottom unless designed to land on a surface, so reference to a 'nose' or 'wing' or even 'down' is arbitrary. An atmosphere (from Greek ατμός - atmos, " Vapor " + σφαίρα - sphaira, " Sphere " A spacecraft is a Vehicle or machine designed for Spaceflight. On a manned spacecraft, the axes must be oriented relative to the pilot's physical orientation at the flight control station. Unmanned spacecraft may need to maintain orientation of solar cells toward the Sun, antennas toward the Earth, or cameras toward a target, and the axes will typically be chosen relative to these functions. A solar cell or photovoltaic cell is a device that converts Solar energy into Electricity by the photovoltaic effect.

Roll, pitch and yaw constitute rotation around X, Y, and Z, respectively, as depicted in the diagram above. (In other contexts, pitch, roll and yaw angles may be used to define an object's absolute attitude, measured against a fixed coordinate system. )

In analysing 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 x-axis 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 centre, completing an orthogonal set.

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 general, the body axes are not aligned with the Earth axes. The body orientation may be defined by three Euler Angles, the Tait-Bryan angles, a quaternion, or a direction cosine matrix (Rotation matrix). The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant The Tait-Bryan rotations, named after Peter Guthrie Tait and George Bryan. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 A rotation matrix is particularly convenient for converting velocity, force, angular velocity, and torque vectors between body and Earth coordinate frames. Do not confuse with Angular frequency The unit for angular velocity is rad/s A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about

Body axes tend to be used with missile and rocket configurations. Aircraft stability uses wind axes in which the x-axis 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 ( AOA, \alpha Greek letter alpha) is a term used in Aerodynamics to describe the Angle between the

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 analysing 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:

Steady level flight

Turn at constant speed

Approach and landing

Take off

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. In Aviation, the undercarriage or landing gear is the structure (usually wheels that supports an Aircraft on the ground and allows it to taxi

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 is an idealization of a solid body of finite size in which Deformation is neglected

Longitudinal modes

It is common practice to derive a fourth order characteristic equation to describe the longitudinal motion, and then factorise it approximately into a high frequency mode and a low frequency mode. In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. 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 behaviour 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 (SSPO), and the phugoid. Source Flight dynamics The dynamic stability of a vehicle denotes the complete study of the motion occurring after the Vehicle has been disturbed A phugoid (pronounced /ˈfjuːˌgoɪ̯d/ is an aircraft motion where the vehicle pitches up and climbs and then pitches down and descends accompanied by speeding

Short-period pitch oscillation

Pulling the joystick back suddenly causes the aircraft to pitch up. The aircraft, if it is stable will settle down at the new trim incidence, but will tend to overshoot. The term overshoot has the following meanings Aviation In Aviation, an overshoot is an aborted landing The transition is characterised by a damped simple harmonic motion about the new trim. Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. There is very little change in the trajectory over the time it takes for the oscillation to damp out.

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. Damping is any effect either deliberately engendered or inherent to a system that tends to reduce the amplitude of Oscillations of an oscillatory system Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. Source Flight dynamics The dynamic stability of a vehicle denotes the complete study of the motion occurring after the Vehicle has been disturbed It is very similar in nature to the weathercock mode of missile or rocket configurations. A weather vane, also called a wind vane, is a movable device attached to an elevated object such as a roof for showing the direction of the wind The motion involves mainly the pitch attitude $θ$ (theta) and incidence $α$ (alpha). The direction of the velocity vector, relative to inertial axes is $θ - α$ . The velocity vector is:

u f = U cos(θ - α)

w f = U sin(θ - α)

where $u f$ , $w f$ are the inertial axes components of velocity. According to Newton's Second Law, the accelerations are proportional to the forces, so the forces in inertial axes are:

$X_f=m\frac{du_f}{dt}=\frac{dU}{dt}\cos(\theta-\alpha)-mU\frac{d(\theta-\alpha)}{dt}\sin(\theta-\alpha)$

$Z_f=m\frac{dw_f}{dt}=\frac{dU}{dt}\sin(\theta-\alpha)+mU\frac{d(\theta-\alpha)}{dt}\cos(\theta-\alpha)$

where m is the mass. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Physics, a force is whatever can cause an object with Mass to Accelerate. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object By the nature of the motion, the speed variation $\frac{dU}{dt}$ is negligible over the period of the oscillation, so:

$X_f= -mU\frac{d(\theta-\alpha)}{dt}\sin(\theta-\alpha)$

$Z_f=mU\frac{d(\theta-\alpha)}{dt}\cos(\theta-\alpha)$

But the forces are generated by the pressure distribution on the body, and are referred to the velocity vector. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axes forces into wind axes. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Also, we are only concerned with the force along the z-axis:

Z = - Z f cos(θ - α) + X f sin(θ - α)

Or:

$Z=-mU\frac{d(\theta-\alpha)}{dt}$

In words, the wind axes force is equal to the centrifugal acceleration.

The moment equation is the time derivative of the angular momentum:

$M=B\frac{d^2 \theta}{dt^2}$

where M is the pitching moment, and B is the moment of inertia about the pitch axis. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position This article is about the moment of inertia of a rotating object. Let: $\frac{d\theta}{dt}=q$ , the pitch rate. The equations of motion, with all forces and moments referred to wind axes are, therefore:

$\frac{d\alpha}{dt}=q+\frac{Z}{mU}$

$\frac{dq}{dt}=\frac{M}{B}$

We are only concerned with perturbations in forces and moments, due to perturbations in the states $α$ and q, and their time derivatives. These are characterised by stability derivatives determined from the flight condition. Stability derivatives are a means of linearising the equations of motion of an atmospheric flight vehicle so that conventional Control engineering methods may be applied to assess The possible stability derivatives are:

Z α

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 α

M α

Pitching moment due to incidence - the static stability term. In Aerodynamics, the pitching moment on an Airfoil is the Moment produced by the Aerodynamic force on the airfoil if that aerodynamic force is Static stability requires this to be negative. Longitudinal static stability is important in determining whether an aircraft will be able to fly as intended

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:

$M_\dot\alpha$

Increasing the wing incidence without increasing the tail incidence produces a nose up moment, so $M_\dot\alpha$ is expected to be positive.

The equations of motion, with small perturbation forces and moments become:

$\frac{d\alpha}{dt}=\left(1+\frac{Z_q}{mU}\right)q+\frac{Z_\alpha}{mU}\alpha$

$\frac{dq}{dt}=\frac{M_q}{B}q+\frac{M_\alpha}{B}\alpha+\frac{M_\dot\alpha}{B}\dot\alpha$

These may be manipulated to yield as second order linear differential equation in $α$ :

$\frac{d^2\alpha}{dt^2}-\left(\frac{Z_\alpha}{mU}+\frac{M_q}{B}+(1+\frac{Z_q}{mU})\frac{M_\dot\alpha}{B}\right)\frac{d\alpha}{dt}+\left(\frac{Z_\alpha}{mU}\frac{M_q}{B}-\frac{M_\alpha}{B}(1+\frac{Z_q}{mU})\right)\alpha=0$

This represents a damped simple harmonic motion. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Damping is any effect either deliberately engendered or inherent to a system that tends to reduce the amplitude of Oscillations of an oscillatory system Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped.

We should expect $\frac{Z_q}{mU}$ to be small compared with unity, so the coefficient of $α$ (the 'stiffness' term) will be positive, provided $M_\alpha<\frac{Z_\alpha}{mU}M_q$ . This expression is dominated by $M α$ , which defines the longitudinal static stability of the aircraft, it must be negative for stability. Longitudinal static stability is important in determining whether an aircraft will be able to fly as intended 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

Main article: 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. A phugoid (pronounced /ˈfjuːˌgoɪ̯d/ is an aircraft motion where the vehicle pitches up and climbs and then pitches down and descends accompanied by speeding It will repeat this cycle until the pilot intervenes. This long period oscillation in speed and height is called the phugoid mode. A phugoid (pronounced /ˈfjuːˌgoɪ̯d/ is an aircraft motion where the vehicle pitches up and climbs and then pitches down and descends accompanied by speeding This is analysed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value. Source Flight dynamics The dynamic stability of a vehicle denotes the complete study of the motion occurring after the Vehicle has been disturbed The two states which are mainly affected are the climb angle $γ$ (gamma) and speed. The small perturbation equations of motion are:

$mU\frac{d\gamma}{dt}=-Z$

which means the centrifugal force is equal to the perturbation in lift force.

For the speed, resolving along the trajectory:

$m\frac{du}{dt}=X-mg\gamma$

where g is the acceleration due to gravity at the earths surface. Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level 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 climb 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:

$mU\frac{d\gamma}{dt}=-Z_u u$

$m\frac{du}{dt}=X_u u -mg\gamma$

These may be expressed as a second order equation in climb angle or speed perturbation:

$\frac{d^2u}{dt^2}-\frac{X_u}{m}\frac{du}{dt}-\frac{Z_ug}{mU}u=0$

Now lift is very nearly equal to weight:

$Z=\frac{1}{2}\rho U^2 c_L S_w=W$

where $ρ$ 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:

$Z_u=\frac{2W}{U}=\frac{2mg}{U}$

The period of the phugoid, T, is obtained from the coefficient of u:

$\frac{2\pi}{T}=\sqrt{\frac{2g^2}{U^2}}$

Or:

$T=\frac{2\pi U}{\sqrt{2}g}$

Since the lift is very much greater than the drag, the phugoid is at best lightly damped. A propeller with fixed speed would help. A propeller is essentially a type of fan which transmits power by converting Rotational motion into Thrust for propulsion of a vehicle such as an 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. This article is about the moment of inertia of a rotating object.

Lateral modes

With a symmetrical rocket or missile, the directional stability 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. Directional stability is the tendency of a moving body to align itself with the direction of motion 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, 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. Dutch roll is a type of Aircraft motion consisting of an out-of- phase combination of "tail-wagging" and rocking from side to side

The yaw plane translational equation, as in the pitch plane, equates the centrifugal acceleration to the side force.

$\frac{d\beta}{dt}=\frac{Y}{mU}-r$

where $β$ (beta) is the sideslip angle, Y the side force and r the yaw rate. Sideslip angle relates to the displacement of the aircraft centerline from the Relative wind.

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.

Image:Product of inertia.png

If the roll rate is p, the velocity of the particle is:

v = - p z + x r

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:

$\frac{dv}{dt}=-\frac{dp}{dt}z+\frac{dr}{dt}x$

The yawing moment is given by:

$\delta m x \frac{dv}{dt}=-\frac{dp}{dt}xz\delta m + \frac{dr}{dt}x^2\delta m$

There is an additional yawing moment due to the offset of the particle in the y direction: $\frac{dr}{dt}y^2\delta m$

The yawing moment is found by summing over all particles of the body:

$N=-\frac{dp}{dt}\int xz dm +\frac{dr}{dt}\int x^2 + y^2 dm =-E\frac{dp}{dt}+C\frac{dr}{dt}$

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:

$L=A\frac{dp}{dt}-E\frac{dr}{dt}$

where L is the rolling moment and A the roll moment of inertia.

Lateral stability derivatives

The states are $β$ (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 aeroplane, and considering how the forces on each component are affected by changes in sideslip and angular velocity:

Image:LowWing.png

Y β

Side force due to side slip.

Sideslip generates a sideforce from the fin and the fuselage. In addition, if the wing has dihedral, positive side slip increases the incidence on the starboard wing and reduces it on the port so there is a net component of lift opposing the sidslip. Similarly, sweep back of the wings has the same effect on local incidence, but since the wings are not inclined in the vertical plane sweep does not contribute to $Y β$ . With high angles of sweep, in high performance aircraft, anhedral may be used to offset this effect. However, the resulting effect is to reverse the sign of the wing contribution to $Y β$ . Usually negative.

Y p

Side force due to roll rate.

Roll rate causes incidence at the fin, which generates a side force. Also, positive roll (starboard wing down) increases the lift on the starboard wing and reduces it on the port. If the wing is mounted at a dihedral angle, this will result in a sideforce contribution. Usually negative.

Y r

Side force due to yaw rate.

Yawing generates incidence at the fin, causing a side force.

N β

Yawing moment due to sideslip. Directional stiffness.

This characterises the tendency to point into wind, it must be positive for a statically stable aircraft.

N p

Yawing moment due to roll rate.

Roll rate generates fin lift, which causes a yawing moment. It also changes the lift on the wings, altering the induced drag contribution of each wing, causing a (small) yawing moment. Positive roll causes positive yawing moment.

N r

Yawing moment due to yaw rate.

Positive yaw rate generates fin lift, increases the speed of the port wing and slowing down the starboard wing, with corresponding changes in drag. Always negative.

L β

Rolling moment due to sideslip. So-called dihedral effect.

Sideslip generates fin lift causing negative roll. Dihedral causes negative roll in response to sideslip. Wing sweep back also causes negative roll. With highly swept wings the rolling moment may be excesive for all stability requirements, and anhedral is used to offset the effect of sweep.

L p

Rolling moment due to roll rate. Roll damping.

Positive roll increases lift on starboard wing, reduces it on port wing, also generates fin lift. Always negative.

L r

Rolling moment due to yaw rate.

Image:Planform.png

Positive yaw increases speed of port wing, whilst reducing speed of starboard, causing a positive rolling moment. The contribution of the fin is similarly positive.

Equations of motion

Since Dutch roll is a handling mode, analogous to the short period pitch oscillation, we shall ignore any effect it might have on the trajectory. Dutch roll is a type of Aircraft motion consisting of an out-of- phase combination of "tail-wagging" and rocking from side to side 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, because we assume no effect on the trajectory, we have, for the limited purpose of studying the Dutch roll:

$\frac{d\beta}{dt}= -r$

The yaw and roll equations, with the stability derivatives become:

$C\frac{dr}{dt}-E\frac{dp}{dt}=N_\beta \beta - N_r \frac{d\beta}{dt} + N_p p$ (yaw)

$A\frac{dp}{dt}-E\frac{dr}{dt}=L_\beta \beta - L_r \frac{d\beta}{dt} + L_p p$ (roll)

The inertial moment due to the roll acceleration is considered small compared with the aerodynamic terms, so the equations become:

$-C\frac{d^2\beta}{dt^2} = N_\beta \beta - N_r \frac{d\beta}{dt} + N_p p$

$E\frac{d^2\beta}{dt^2} = L_\beta \beta - L_r \frac{d\beta}{dt} + L_p p$

This becomes a second order equation governing either roll rate or sideslip:

$\left(\frac{N_p}{C}\frac{E}{A}-\frac{L_p}{A}\right)\frac{d^2\beta}{dt^2}+ \left(\frac{L_p}{A}\frac{N_r}{C}-\frac{N_p}{C}\frac{L_r}{A}\right)\frac{d\beta}{dt}- \left(\frac{L_p}{A}\frac{N_\beta}{C}-\frac{L_\beta}{A}\frac{N_p}{C}\right)\beta = 0$

The equation for roll rate is identical. But the roll angle, $φ$ (phi)is given by:

$\frac{d\phi}{dt}=p$

If p is a damped simple harmonic motion, so is $φ$ , but the roll must be in quadrature with the roll rate, and hence also with the sideslip. Communication signals often have the form': A(t\cdot \sin ft + \phi(t which is called envelope-and-phase form 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' and 'damping' terms to be positive. Stiffness is the resistance of an elastic body to Deformation by an applied Force. These are:

$\frac{\frac{L_p}{A}\frac{N_r}{C}-\frac{N_p}{C}\frac{L_r}{A}} {\frac{N_p}{C}\frac{E}{A}-\frac{L_p}{A}}$ (damping)

$\frac{\frac{L_\beta}{A}\frac{N_p}{C}-\frac{L_p}{A}\frac{N_\beta}{C}} {\frac{N_p}{C}\frac{E}{A}-\frac{L_p}{A}}$ (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: $- L p N β$ will be positive because $L p$ is always negative and $N β$ is positive by design. $L β$ is usually negative, whilst $N p$ is positive. Excessive dihdral can de-stabilise the Dutch roll, so configurations with highly swept wings require anhedral to offset the wing sweep contribution to $L β$ .

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 centre of gravity and a more 'exact' analysis will introduce terms in $Y β$ 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 centre causes a net change in roll orientation.

The roll motion is characterised 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 cancelled by pilot or autopilot intervention. This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to:

$A\frac{dp}{dt}=L_p p.$

$L p$ 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, the aircraft has a tendency to gradually veer off to one side of the straight flightpath.

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 $μ$ (mu). The body orientation is called the heading, denoted $ψ$ (psi). The force equation of motion includes a component of weight:

$\frac{d\mu}{dt}=\frac{Y}{mU} + \frac{g}{U}\phi$

where g is the gravitational acceleration, and U is the speed.

Including the stability derivatives:

$\frac{d\mu}{dt}=\frac{Y_\beta}{mU}\beta + \frac {Y_r}{mU}r + \frac{Y_p}{mU}p + \frac{g}{U}\phi$

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 derivatives are ignored. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The yaw and roll equations reduce to:

$N_\beta \beta + N_r\frac{d\mu}{dt} + N_p p = 0$ (yaw)

$L_\beta \beta + L_r\frac{d\mu}{dt} + L_p p = 0$ (roll)

Solving for $β$ and p:

$\beta=\frac{(L_r N_p - L_p N_r)}{(L_p N_\beta - N_p L_\beta)}\frac{d\mu}{dt}$

$p=\frac{(L_\beta N_r - L_r N_\beta)}{(L_p N_\beta - N_p L_\beta)}\frac{d\mu}{dt}$

Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle:

$\frac{d\phi}{dt}=mg\frac{(L_\beta N_r - N_\beta L_r)}{mU(L_p N_\beta - N_p L_\beta)-Y_\beta(L_r N_p - L_p N_r)}\phi$

This is an exponential growth or decay, depending on whether the coefficient of $φ$ is positive or negative. The denominator is usually negative, which requires $L β N r > N β L r$ (both products are positive). This is in direct conflict with the Dutch roll stability requirement, and it is difficult to design an aircraft which has both a stable Dutch roll and spiral mode.

Since the spiral mode has a long time constant, the pilot can intervene to effectively stabilise it, but an aircraft with an unstable Dutch roll would be difficult to fly. Source Flight dynamics The dynamic stability of a vehicle denotes the complete study of the motion occurring after the Vehicle has been disturbed It is usual to design the aircraft with a stable Dutch roll mode, but slightly unstable spiral mode. Though it is experienced that aeroplanes with positive V-tail are more critical and the F-4 Phantom II therefore has a negative V and some aeroplanes even have a downwards pointing tail fin. Also a small sweep angle of the main wings may help. Swept back Flying wings usually do not like positive winglets. A flying wing is a Fixed-wing aircraft which has no definite Fuselage, with most of the crew payload and equipment being housed inside the main wing structure

References

Babister A W: Aircraft Dynamic Stability and Response. Aeronautics (from Greek aero which means air or sky and nautis which means sailor i Aircraft attitude is used to mean two closely related aspects of the situation of an aircraft in flight The attitude of a body is its orientation as perceived in a certain Frame of reference; providing a vector along which a spacecraft is pointing is a description of its attitude In 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 A crosswind landing is a Landing maneuver in which a significant component of the prevailing wind is perpendicular to the Runway centerline Dynamic positioning (DP is a computer controlled system to automatically maintain a Vessel 's position and heading by using her own propellers and thrusters The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant Longitudinal static stability is important in determining whether an aircraft will be able to fly as intended In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 Stability derivatives are a means of linearising the equations of motion of an atmospheric flight vehicle so that conventional Control engineering methods may be applied to assess Static margin is a concept used to characterize the static stability and controllability of aircraft and missiles The Tait-Bryan rotations, named after Peter Guthrie Tait and George Bryan. Weathervaning or weathercocking (Heflin 1956 is a phenomenon experienced by aircraft on the ground The Wright Glider was designed and built by the Wright Brothers. JSBSim is an Open source Flight Dynamics Model (FDM software library that models the Flight dynamics of an aerospace vehicle Elsever 1980, ISBN 0-08-024768-7
Stengel R F: Flight Dynamics. Princeton University Press 2004, ISBN 0-691-11407-2

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