Aircraft flight dynamics


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, 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 such as helicopters, and dirigibles, where the flight dynamics involved in establishing and controlling attitude are entirely different.
Control systems adjust the orientation of a vehicle about its cg. A control system includes control surfaces which, when deflected, generate a moment 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.
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. 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.

Background

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 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.

Transformations ([Euler angles])

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 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.
When performing the rotations described above to obtain the body frame from the Earth frame, there is this analogy between angles:
  • σ, ψ
  • γ, θ
  • ''μ, φ''

    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
  • α'', θ
  • ''''

    Analogies

Between the three reference frames there are hence these analogies:
  • Yaw / Heading / Sideslip
  • Pitch / Flight path / Attack angle
  • Roll / Bank / nothing

    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:
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, the longitudinal equations of motion may be treated independently of the lateral motion.
The following considers perturbations about a nominal straight and level flight path.
To keep the analysis 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.

Aerodynamic force

Components of the aerodynamic force

The expression to calculate the aerodynamic force is:
where:
projected on wind axes we obtain:
where:

Aerodynamic coefficients

In absence of thermal effects, there are three remarkable dimensionless numbers:
  • Compressibility of the flow:
  • Viscosity of the flow:
  • Rarefaction of the flow:
where:
According to λ there are three possible rarefaction grades and their corresponding motions are called:
  • Continuum current :
  • Transition current :
  • Free molecular current :
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:
  • Transonic current:
  • Supersonic current:
  • Hypersonic current:

    Drag coefficient equation and aerodynamic efficiency

If the geometry of the body is fixed and in case of symmetric flight, pressure and friction coefficients are functions depending on:
where:
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:
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 is 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.

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 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.

Variation of parameters with the Mach number

The Coefficient of pressure varies with Mach number by the relation given below:
where
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.