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equations of motiort. We will consider the influence ofits mass and inertia on the
motion. We will also consider aerodynamic damping effects. However, we will not
be considering the aeroelastic effects and,instead, we will assume that the airplane
functions like a rigid body.
   The foundaOons of the airplane dynamic stability and response were laid by
the pioneering work of Bryan.l His'9ormulation was based on two principal as-
sumptions: 1) the instantaneous aerodynamic forces and moments depend only
on instantaneous values of the motionyvariables and 2) the aerodynamic forces
and moments vary linearly with motion variables. This approach ofjBryan,l intro-
duced more than 80 years ago,is used even today in the study of dynamic stability,
control, and response of the airplanc and forms the basis of the subject matter
discussed in this chapter.
       To begin with, we will discuss various axes systems used in the study of airplane
dynamics and present relations for transforming vectors from one coordinate sys-
tem into another system. We will then formulate the problem of airplane dynamics
and derrve equations of motion for six-degree-of-fre9dom analyses. Because these
equations are, in general, coupled and nonlinear, it is difficult to obtain analytical
solutions. In view of this, we will assume that the motion following a disturbance
is one of small amplitudes in all the disturbed variables. With this assumption
and the usual approximation that the airplane has a vertical plane of symmetry, it
is possible to linearize and decouple the equations of motion into two sets, one
for the longitudinal motion and a:other for lateral-directional motion. Then we
use the method of Bryanl and assume that the aerodynamic forces and moments
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.
319
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
　
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