Aircraft 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 (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 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 (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.
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.
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 (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.Flightwise Volume 2 - Aircraft Stability And Control, Chris Carpenter 1997, Airlife Publishing Ltd., , Figure 2.6
Reference frames
Three right-handed, 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 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: 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 (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.
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:
Rotation matrices Direction cosines Euler angles Quaternions
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)
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 (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
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.
Aerodynamic force
Components of the aerodynamic force
The expression to calculate the aerodynamic force is:
where: Difference between static pressure and free current pressure outer normal vector of the element of area tangential stress vector practised by the air on the body adequate reference surface
projected on wind axes we obtain:
where: Drag Lateral force Lift
Aerodynamic coefficients
Dynamic pressure of the free current
Proper reference surface (wing surface, in case of planes)
Pressure coefficient
Friction coefficient
Drag coefficient
Lateral force coefficient
Lift coefficient
It is necessary to know 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 Viscosity of the flow: Reynolds number Rarefaction of the flow: Knudsen number
where:
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: 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 (β=0 and Q=0), pressure and friction coefficients are functions depending on:
where: angle of attack considered point of the surface
Under these conditions, drag and lift coefficient are functions depending exclusively on the angle of attack of the body and Mach and Reynolds numbers. Aerodynamic efficiency, defined as the relation between lift and drag coefficients, will depend on those parameters as well.