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in the disturbed state depend only on the instantaneous values of motion vari-
ables and evaluate them uvsing the method of Taylor series expansion. With these
approximations, the longitudinal and lateral-d:r9ctional equations of motion be-
come linearin all the motion variables. The aerodynamic coefficients appearing in
the Taylor series expansion are called stabiiity and control derivatives. Finally, we
will present engineering methods to evaluate these derivatives for typical airplane
configurations.
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
4.2 AxesSystems
Iri the formulation of flight dynamic problems, we need to introduce several co-
ordinate systems for specifying the position, velocity, forces, and moments acting
on the 'vehicle. However, the choice of a particular coordinate system in which
the equations of motion are written and solved is a matter of convenience to the
analyst. In the following, we will discuss some of the most commonly used axes
systems in fiight dynamics and present relations for transforming vectors from one
axes system into another.
4-2- I Inertialf.Axes System (OxrYizi)
For e'very flight dynamic problem,it is necessary to specify an inertial frame of
reference because Newton's laws of motion are valid only when the acceleration
is measured with respect to an inertial frame.ln other words, the acceleration of a
body in Newton's second law of motion, F -. ma,is the acceleration with respect
to an inertial frame of reference, which is actually at rest in the universe. While it
is a difficult task to find such an inertial reference system, for most of the flight
dynamic problems, a nonrotating reference system placed at the center of the Earth
(Fig. 4.1) is a reasonably good approximation for an inertial system of reference.
In this approximation, the orbital motion of the Earth around the sun is ignored.
However, for interplanetary motions Iike the mission to Mars, the orbital motion
of the Earth has to be considered and some other inertial frame of reference such
as one centered at the sun may have to be used.
4-2-2 Earth-Fixed Axes System(OxEyEzE)
Another axes system thatis usefulin flight dynanucs is an axes system fixed at
the center of the Earth and rotating with the Earth. The OxE yEzE system shown
in Fig. 4.1 is such a system. The angular velocity lr2e of the OxEyEzE system
with respect to the Ox,yizi system is directed along the Oz, or OzE axis. An
r;
yE
yi
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 321
Earth-fixed reference system is usefulin specifying the position and velocity of
the vehicle with respect to the rotating Earth.
4.2-3 Navigational System (OxeYeze)
The origin ofthis frame ofreference OxeYeze (Fig. 4.1) is located on the surface
of the Earth such that the Oze axis is directed towards the center of the spherical
Earth. The Oxe axis usually points towards the local north and Oye points towards
the local east to form a right-hand system. Usually, the location of the origin of
this coordinate system is chosen so thatit lies directly beneath the vehicle at t - 0.
With such a choice of the origin, this system of reference is useful to define the
position and velocity of the vehicle with respect to the launch point.
4.2-4 Body Axes System (Oxbybzb)
Stabdity axes system (OxsYszs). A special case of the body axes system is
the so-called stability axes system OxsYyzs (Fig. 4.2). Tlus axes system is widely
used in the study of airplane motion involving small disturbances from a steady
reference flight condition. The Oxs axis lies in the plane of symmetry and, if the
reference flight condition is symmetric (p = 0), points in the opposite direction
to the relatrve wind. If p +: 0, then the Oxs axis is chosen to coincide with the
projection of the relative velocity vector in the plane of symmetry. The Oy.,. axis
ZS,ZW
Fig 4 2 Body-f/xed axes systems.
':":~
. 55
322 PERFORMANCE, STABIUTY, DYNAMICS, AND CONTROL
is normal to the plane of symmetry and points to the right side or the starboard
wing and Ozs points downward to form a right-hand system. The angle between
Oxb and Oxs or that between the Ozz, and OZr axes is usually equal to the angle
of attack of the vehicle.
The angle that locates this axes system with respect to the body axes system
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