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2.4. Kinematic relations 23
for a nonsteady atmosphere. Since V, a, and b are always determined relatively to
the surrounding atmosphere, these subscripts will be omitted again in the remainder
of this section.
These expressions differ from equations (2.38), (2.43), and (2.48) in that they include
additional terms that depend on the wind velocity components and their timederivatives.
These terms can be expressed in terms of corrections of the external force
components along the aircraft’s body-axes, Fx, Fy, and Fz, This means that we can apply
equations (2.38), (2.43), and (2.48) for steady as well as nonsteady atmosphere, as
long as the force components are properly corrected for the wind, if necessary. These
corrections will be summarized later in section 3.3.4.
2.4 Kinematic relations
So far we have derived differential equations for the true airspeed, angle of attack,
sideslip angle, and the rotational velocity components. However, to solve the equations
of motion it is also necessary to know the attitude of the aircraft relatively to
the Earth, because some contributions to the external forces and moments depend
upon those quantities. We also need to know the altitude of the aircraft, because
the air pressure, temperature, and density change with altitude, affecting both the
aerodynamic and propulsive forces and moments. And finally, we want to be able to
track the flight path relative to the Earth, e.g. for simulations of navigational tasks.
The orientation of the airplane relative to the Earth is given by a series of three consecutive
rotations, the Euler angles y, q, and j, see figure A.2 from appendix A. As
shown in section A.7.3, the rotations can be expressed by three transformation matrices
TY, TQ, and TF. It is possible to express the total angular velocity of the aircraft
expressed in terms of the derivatives with respect to time of the Euler angles:
2
4
p
qr
3
5 =
2
4
˙j
0
0
3
5 +
z }TF| { 2
4
1 0 0
0 cos j sin j
0 −sin j cos j
3
5
2
4
0˙q
0
3
5 +
+
2
4
1 0 0
0 cos j sin j
0 −sin j cos j
3
5
| {z }
TF
2
4
cos q 0 −sin q
0 1 0
sin q 0 cos q
3
5
| {z }
TQ
2
4
0
0˙y
3
5 (2.57)
This can be written as:
2
4
p
qr
3
5 =
2
4
1 0 −sin q
0 cos j sin j cos q
0 −sin j cos j cos q
3
5
2
4
˙j˙q
˙y
3
5 = TR
2
4
˙j˙q
˙y
3
5 (2.58)
where TR is the matrix that transforms angular velocities in the Earth-fixed axis system
into body-axes angular velocities. Consequently, the time-derivatives of the
Euler angles can be found by pre-multiplying equation (2.58) with TR
−1. This yields
24 Chapter 2. Rigid body equations of motion
the following kinematic relations:
˙y
= q sin j + r cos j
cos q
˙q
= q cos j − r sin j (2.59)
˙j
= p + (q sin j + r cos j) tan q = p + ˙y sin q
The position of the aircraft with respect to the Earth-fixed reference frame is given
by the coordinates xe, ye, and ze. To find these coordinates, we need to resolve the
body-axis velocity vector in the Earth-bound reference system FE:
2
4
x˙e
y˙e
z˙e
3
5 = TB!E ·
2
4
ue
ve
we
3
5 (2.60)
where TB!E = TB!V = TV!B
−1 is the transformation matrix from FB to FE, see the
definition in section A.7.3 of appendix A. This results in the following equations:
x˙e = {ue cos q + (ve sin j + we cos j) sin q} cos y − (ve cos j − we sin j) sin y
y˙e = {ue cos q + (ve sin j + we cos j) sin q} sin y + (ve cos j − we sin j) cos y
z˙e = −ue sin q + (ve sin j + we cos j) cos q (2.61)
In practice, the altitude of the aircraft is a more useful property than the coordinate
ze. The relationship between the time-derivatives of H and ze is simple:
˙H
= −z˙e (2.62)
Notice that the positive ZE-axis points downwards.
The state equations (2.59) have the advantage of using physically meaningful variables,
and they express the airplane’s attitude using the minimum number of three
first-order differential equations. However, it should be noted that these equations
also have some significant disadvantages.
First of all, a division by zero occurs if the pitch angle reaches plus or minus 90.
　
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