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m
(−Fx sin a + Fz cos a) + pv cos a + qu cos a + qwsin a − rv sin a

(2.42)
Using equations (2.31) for u, v, and w, we find:
˙a
= 1
V cos b

1
m
(−Fx sin a + Fz cos a)

+ q − (p cos a + r sin a) tan b (2.43)
2.3. Equations of motion in nonsteady atmosphere 21
2.2.5 Derivation of the ˙b-equation
Differentiating equation (2.34) with respect to the time yields:
˙b
=
v˙(u2 + v2) − v(uu˙ + ww˙ )
V2
p
u2 + w2
(2.44)
From equations (2.31) the following relations can be derived:
u2 + w2 = V2 cos2 b
uv = V2 sin b cos b cos a
vw = V2 sin b cos b sin a (2.45)
These values substituted in equation (2.44) yield:
˙b
= 1
V
(−u˙ cos a sin b + v˙ cos b − w˙ sin a sin b) (2.46)
Substituting equations (2.28) for u˙ and w˙ yields:
˙b
= 1
V

1
m
(−Fx cos a sin b + Fy cos b − Fz sin a sin b) + qwcos a sin b +
− rv cos a sin b + pwcos b − ru cos b + pv sin a sin b − qu sin a sin b

(2.47)
If we substitute equations (2.31), many terms can be cancelled and we find:
˙b
= 1
V

1
m
(−Fx cos a sin b + Fy cos b − Fz sin a sin b)

+ p sin a − r cos a (2.48)
2.3 Equations of motion in nonsteady atmosphere
The equations of motion are valid only if the body-axes velocity components are measured
with respect to a non-rotating system of reference axes having a constant translational
speed in inertial space. Under the assumptions 3 and 4 from section 2.1.4,
it is possible to select a reference frame that is fixed to the surrounding atmosphere
as long as the wind velocity vector Vw is constant. In that case, the components u,
v, and w of the velocity vector V express the aircraft’s velocity with respect to the
surrounding atmosphere.
If the wind velocity vector Vw is not constant during the time-interval over which
the motions of the aircraft are studied it is not possible to fix the reference frame to
the surrounding atmosphere. This happens for instance during the approach and
landing of an aircraft, because the wind velocity changes with altitude. Again using
assumptions 3 and 4 of section 2.1.4, the most obvious choice of the reference frame
in this case turns out to be the Earth-fixed reference frame FE [19].
Let Ve be the velocity with respect to the Earth, Va the velocity with respect to the
surrounding atmosphere, and Vw the wind velocity with respect to the Earth. Then
we can write:
Ve = Va + Vw (2.49)
22 Chapter 2. Rigid body equations of motion
or:
ue = ua + uw
ve = va + vw (2.50)
we = wa + ww
where ue, ve, and we are the components of V, ua, va, and wa are the components of
Va, and uw, vw, and ww are the components of Vw, all measured along the body-axes
of the aircraft. The force equations now become:
F = m

¶Ve
¶t
+ W × Ve

(2.51)
Rewriting equation (2.51) yields:
¶Ve
¶t
= F
m
− W × Ve (2.52)
For the individual components along the body-axes we thus find:
u˙e = Fx
m
− qwe + rve
v˙e =
Fy
m
+ pwe − rue (2.53)
w˙ e = Fz
m
− pve + que
In order to compute the aerodynamic forces and moments, it is necessary to know the
values of Va (the true airspeed), a, and b.1 In a manner analogous to the derivation
of ˙V, ˙a, and ˙b in section 2.2, we can find expressions for the time-derivatives of Va,
aa, and ba:
˙V
a = 1
m
􀀀
Fx cos a cos b + Fy sin b + Fz sin a cos b


+
−(qww − rvw + u˙w) cos a cos b + (pww − ruw − v˙w) sin b +
−(pvw − quw + w˙ w) sin a cos b (2.54)
˙a
a = 1
V cos b

1
m
(−Fx sin a + Fz cos a) +
−(pvw − quw + w˙ w) cos a + (qww − rvw − u˙w) sin a

+
+q − (p cos a + r sin a) tan b (2.55)
˙b
a = 1
V

1
m
(−Fx cos a sin b + Fy cos b − Fz sin a sin b) +
+(qww − rvw + u˙w) cos a sin b + (pww − ruw − v˙w) cos b +
+(pvw − quw + w˙ w) sin a sin b

+ p sin a − r cos a (2.56)
The subscript a has been used here for reasons of clarity only, allowing us to make
a clear distinction between the equations for a steady atmosphere and the equations
1Notice that expressions (2.31) to (2.34) remain valid if Va is substituted for V!
　
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