Flight dynamics is the science of air and space 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 mass, known as pitch, roll and yaw (See Tait-Bryan rotations for an explanation).
Aerospace engineers develop control systems for a vehicle's orientation (attitude) about its center of mass. 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. 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.
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. Yaw and Pitch is known as 'heading'. The equilibrium pitch angle in submarine and airship parlance is known as 'trim', but in aircraft, this usually refers to angle of attack, rather than orientation. 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). 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 flight dynamics, pitch, roll and yaw angles measure changes
in attitude, relative to the equilibrium orientation of the vehicle
- 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. 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.
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 rotations, a quaternion, or a direction cosine matrix (rotation matrix). A rotation matrix is particularly convenient for converting velocity, force, angular velocity, and torque vectors between body and Earth coordinate frames.
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.
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.
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.
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.
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. 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.
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 transition is characterised 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.
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 centrifugal 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.
, 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 characterised 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:
Increasing the wing incidence without increasing the tail incidence produces a nose up moment, so is expected to be positive.
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
The equation for roll rate is identical. But the roll angle, (phi)is given by:
If p is a damped simple harmonic motion, so is , 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' and 'damping' terms to be positive. These are:
The denominator is dominated by , the roll damping derivative, which is always negative, so the denominators of these two expressions will be positive.
Considering the 'stiffness' term: will be positive because is always negative and is positive by design. is usually negative, whilst 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 .
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 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.
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:
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.
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 (mu). The body orientation is called the heading, denoted (psi). The force equation of motion includes a component of weight:
where g is the gravitational acceleration, and U is the speed.
Including the stability derivatives:
Roll rates and yaw rates are expected to be small, so the contributions of and will be ignored.
The sideslip and roll rate vary gradually, so their time derivatives are ignored. The yaw and roll equations reduce to:
Solving for and p:
Substituting for sideslip and roll rate in the force equation results in a first order equation in roll angle:
This is an exponential growth or decay, depending on whether the coefficient of is positive or negative. The denominator is usually negative, which requires (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. 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.
- Babister A W: Aircraft Dynamic Stability and Response. Elsevier 1980, ISBN 0-08-024768-799
- Stengel R F: Flight Dynamics. Princeton University Press 2004, ISBN 0-691-11407-2