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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.
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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
Oxbybzb is the angle of attack.It may be noted that,if p = 0, the drag is directed
opposite to the Ox.y axis and lift is directed opposite to the Ozs axis.
Wind axes system (OxwYwzw). Another special case of the body axes sys-
tem is the wind axes system (Fig. 4.2). Here, the Oxw axis points in the direction
opposite to the relative wind and Ozw lies in the plane of symmetry. The OYw
axis is normal to the Oxwzw plane and points towards the right side ofthe vehicle.
Note that, if p #:O, the Oxwz,.o plane will not coincide with the vehicle plane of
symmetry.
The angles locating the wind axes system OxrnYwzu, with respect to the body
axes system Oxb ybzb are sideslip P and angle of attack or. Here, the drag and lift
are always directed opposite to the Oxrn and Ozw axes, respectively. .
4-2-5 Axes Transformation
There are several methods of performing axes transformations so that a vector
givenin one reference system can be expressed with respect to another axes system.
The methods that are most commonly used in flight dynamics are 1) Euler angles,
2) d:irection cosine matrices, and 3) quatemions or the Euler fourparameter method.
In the following, we will discuss each of these three methods.
Eulerang/es. In this method, the orientation ofa given reference frame rela-
tive to another reference system is specified by three angles yr, O, and 4 which are
called Euler angles. These three angles y, 0, and 4 form three consecutive rotations
in that order so that one coordinate axes system is made to coincide with another
system. Note that the order of these rotations is extremely important because any
other order of rotation would normally result in a different orientation.
Suppose we want to describe the orientation ofthe Ox2Y2z2 system with respect
to the OxiYiZi system. We have to perform three consecutive rotations ~, O, and
@ to take the OxiYizi system from the given orientation and makeit coincide with
the Ox2Y2z2 system (Fig. 4.3). J[f the Ox2Y222 system happens to be a body-fixed
system and OxiYizi an Earth-fixed system, then the three Euler angles p, O, and
4 give the orientation of the aircraft in space with respect to the Earth. Usually,
vJ is called the heading or azimuth angle, Q the inclination or pitch angle, and 4
the bank angle. A continuous display of these three Euler angles on the cockpit
instrumentation enables the pilot to have a knowledge of the orientation of the
aircraft with respect to the Earth.
The sequence of the three Euler rotations is as follows:
1) A rotation V about the Ozi axis taking Oxi to Ox; and OYi t0 Oyj such
that Oxj falls in the plane Ox2z2 and Oy; falls in the Oy222 plane.
2) A second rotation 8 about Oyj taking Ox{ to Ox;' and Oz'i to Oz'i' so that
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