Flight dynamics (fixed-wing aircraft)
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
(cg), known as pitch
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
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
increases or decreases the lift generated by the wings when it pitches nose up or down by increasing or decreasing the angle of attack
(AOA). The roll angle is also known as bank angle on a fixed-wing aircraft, which usually "banks" to change the horizontal direction of flight. An aircraft is 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.
, Cartesian coordinate systems
see frequent use in flight dynamics. The first coordinate system has an origin fixed in the reference frame of the Earth:
Origin - arbitrary, fixed relative to the surface of the Earth
xE axis - positive in the direction of north
yE axis - positive in the direction of east
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 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:
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 (generally, positive out the right wing)
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.
The wind frame is a convenient frame to express the aerodynamic forces and moments acting on an aircraft. In particular, the net aerodynamic force can be divided into components along the wind frame axes, with the drag force 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:
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.
To transform from the Earth frame to the body frame using Euler angles, the following rotations are done in the order prescribed. 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, the yaw angle ψ is the angle between north and the projection of the aircraft longitudinal axis onto the horizontal plane, the pitch angle θ is the angle between the aircraft longitudinal axis and horizontal, and the roll angle φ represents a rotation around the aircraft longitudinal axis after rotating by yaw and pitch.
To transform from the Earth frame to the wind frame, the three Euler angles are the bank angle μ, the flight path angle γ, and the heading angle σ. When performing the rotations described above to obtain the wind frame from the Earth frame, (μ,γ,σ) are analogous to (φ,θ,ψ), respectively. The heading angle σ is the angle between north and the horizontal component of the velocity vector, which describes which direction the aircraft is moving relative to cardinal directions. The flight path angle γ is the angle between horizontal and the velocity vector, which describes whether the aircraft is climbing or descending. The bank angle μ represents a rotation of the lift force around the velocity vector, which may indicate whether the airplane is turning.
To transform from the wind frame to the body frame, the two Euler angles are the angle of attack α and the sideslip angle β. When performing the rotations described earlier to obtain the body frame from the wind frame, (α,β) are analogous to (θ,ψ), respectively; the angle analogous to φ in this transformation is always zero. The sideslip angle β is the 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, also known as sideslip. The angle of attack α is the 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.
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
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.
Forces of flight
Three forces act on an aircraft in flight: weight, thrust, and the aerodynamic force.
Components of the aerodynamic force
The expression to calculate the aerodynamic force is:
:: Difference between static pressure and free current pressure
:: outer normal vector of the element of area
:: tangential stress vector practised by the air on the body
:: adequate reference surface
projected on wind axes we obtain:
:: Lateral force
Dynamic pressure of the free current
Proper reference surface (wing surface, in case of planes)
Lateral force coefficient
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:
Viscosity of the flow:
Rarefaction of the flow:
:: speed of sound
::: specific heat ratio
::: gas constant by mass unity
::: absolute temperature
:: mean free path
According to λ there are three possible rarefaction grades and their corresponding motions are called:
Continuum current (negligible rarefaction):
Transition current (moderate rarefaction):
Free molecular current (high rarefaction):
The motion of a body through a flow is considered, in flight dynamics, as continuum current. In the outer layer of the space that surrounds the body viscosity will be negligible. However viscosity effects will have to be considered when analysing the flow in the nearness of the boundary layer.
Depending on the compressibility of the flow, different kinds of currents can be considered:
Incompressible subsonic current:
Compressible subsonic current:
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:
:: angle of attack
:: considered point of the surface
Under these conditions, drag and lift coefficient are functions depending exclusively on the angle of attack of the body and Mach and Reynolds numbers. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well.
It is also possible to get the dependency of the drag coefficient respect to the lift coefficient. This relation is known as the drag coefficient equation:
:: drag coefficient equation
The aerodynamic efficiency has a maximum value, 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:
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 draft coefficient at zero lift. CDi is known as the induced drag coefficient and it is produced by the body lift.
Parabolic and generic drag coefficient
A good attempt for the induced drag coefficient is to assume a parabolic dependency of the lift
Aerodynamic efficiency is now calculated as:
If the configuration of the plane is symmetrical respect to the XY plane, minimum drag coefficient equals to the parasitic drag of the plane.
In case the configuration is asymmetrical respect to the XY plane, however, minimum flag 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.
Variation of parameters with the Mach number
The Coefficient of pressure varies with Mach number by the relation given below:
Cp is the compressible pressure coefficient
Cp0 is the incompressible pressure coefficient
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
Static stability and control
Longitudinal static stability
see Longitudinal static 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
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 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 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 mode of missile or rocket configurations. The motion involves mainly the pitch attitude (theta) and incidence (alpha). The direction of the velocity vector, relative to inertial axes is . The velocity vector is:
where , are the inertial axes components of velocity. According to Newton's Second Law, the accelerations are proportional to the forces, so the forces in inertial axes are:
where m is the mass.
By the nature of the motion, the speed variation is negligible over the period of the oscillation, so:
But the forces are generated by the pressure distribution on the body, and are referred to the velocity vector. But the velocity (wind) axes set is not an inertial frame so we must resolve the fixed axes forces into wind axes. Also, we are only concerned with the force along the z-axis:
In words, the wind axes force is equal to the centripetal acceleration.
The moment equation is the time derivative of the angular momentum:
where M is the pitching moment, and B is the moment of inertia about the pitch axis.
Let: , the pitch rate.
The equations of motion, with all forces and moments referred to wind axes are, therefore:
We are only concerned with perturbations in forces and moments, due to perturbations in the states and q, and their time derivatives. These are characterized by stability derivatives determined from the flight condition. The possible stability derivatives are:
:: Lift due to incidence, this is negative because the z-axis is downwards whilst positive incidence causes an upwards force.
:: Lift due to pitch rate, arises from the increase in tail incidence, hence is also negative, but small compared with .
:: Pitching moment due to incidence - the static stability term. Static stability requires this to be negative.
:: 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:
The delayed downwash effect gives the tail more lift and produces a nose down moment, so is expected to be negative.
The equations of motion, with small perturbation forces and moments become:
These may be manipulated to yield as second order linear differential equation in :
This represents a damped simple harmonic motion.
We should expect to be small compared with unity, so the coefficient of (the 'stiffness' term) will be positive, provided