Aircraft flight dynamics
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.
[1] Roll, pitch and yaw refer to rotations about the respective axes starting from a defined steady flight equilibrium state.
The most common aeronautical convention defines roll as acting about the longitudinal axis, positive with the starboard (right) wing down.
To keep the analysis (relatively) simple, the control surfaces are assumed fixed throughout the motion, this is stick-fixed stability.
The expression to calculate the aerodynamic force is: where: projected on wind axes we obtain: where: In absence of thermal effects, there are three remarkable dimensionless numbers: where: According to λ there are three possible rarefaction grades and their corresponding motions are called: The motion of a body through a flow is considered, in flight dynamics, as continuum current.
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.
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.
A short, sharp pull back on the control column may be used, and will generally lead to oscillations about the new trim condition.
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.
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.
This is analyzed by assuming that the SSPO performs its proper function and maintains the angle of attack near its nominal value.
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.
The latter terms gives rise to cross products of small quantities (pq, pr, qr), which are later discarded.
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.
However a better intuitive understanding is to be gained by simply playing with a model airplane, and considering how the forces on each component are affected by changes in sideslip and angular velocity:
Anhedral wing and or stabilizer configurations can cause the sign of the side force to invert if the fin effect is swamped.
Sideslip in the absence of rudder input causes incidence on the fuselage and empennage, thus creating a yawing moment counteracted only by the directional stiffness which would tend to point the aircraft's nose back into the wind in horizontal flight conditions.
will tend to point the nose into the sideslip direction even without rudder input, causing a downward spiraling flight.
Lateral force components resulting from dihedral or anhedral wing lift differences has little effect on
opposes the inherent directional stiffness which tends to point the aircraft's nose back into the wind and always matches the sign of the yaw rate input.
upward rolling moment induced by the ensuing sideslip should return the aircraft to the horizontal unless exceeded in turn by the downward
Since Dutch roll is a handling mode, analogous to the short period pitch oscillation, any effect it might have on the trajectory may be ignored.
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 center causes a net change in roll orientation.
This takes place with insignificant changes in sideslip or yaw rate, so the equation of motion reduces to:
Simply holding the stick still, when starting with the wings near level, an aircraft will usually have a tendency to gradually veer off to one side of the straight flightpath.
The force equation of motion includes a component of weight:[citation needed] where g is the gravitational acceleration, and U is the speed.
[citation needed] Since the spiral mode has a long time constant, the pilot can intervene to effectively stabilize it, but an aircraft with an unstable Dutch roll would be difficult to fly.
