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(2.29)
The last terms in equations (2.29) express the effects of the gyroscopic couples:
p˙0 = qh0
z − rh0
y
q˙0 = rh0
x − ph0
z (2.30)
˙ r0 = ph0
y − qh0
x
Ppp, Ppq, . . . , Rn are inertia coefficients, which are derived from the matrix multiplications
involving the inertia tensor I; they have been listed in table 2.2.
Equations (2.28) and (2.29) describe the motions of any rigid body relatively to the
Earth if the following four restrictive assumptions are made:
1. the body is assumed to be rigid during the motions considered (attached spinning
rotors are allowed, provided these are accounted for in the moment equations),
2. the mass of the body is assumed to be constant during the time-interval in
which its motions are studied,
2.1. General rigid-body equations of motion 17
3. the Earth is assumed to be fixed in space, i.e. its rotation is neglected, and
4. the curvature of the Earth is neglected.
The latter two assumptions allow us to assume that the inertial reference frame in
which the motions of the rigid body are considered is fixed to the Earth. If the equations
are to be applied to a moving vehicle, the description of the vehicle motion under
assumptions 3 and 4 are accurate for relatively short-term guidance and control
analysis purposes only. The assumptions do have practical limitations when very
long term navigation or extra-atmospheric operations are of interest [26]. Ref.[33]
contains a more elaborate model for around-the-Earth navigation.
In order to simplify the notations for the remainder this report, the velocity vector
Vc.g. will from this point onwards shortly be denoted as V. The body-axes components
of this vector are u, v, and w, respectively, and the length of this vector is
denoted as V. Likewise, the subscript c.g. will be omitted from the moment vector
Mc.g. in the remainder of this report.
In the next chapter, the dynamics of airplanes will be described in terms of the
rigid-body equations of motion which we just derived. Figure 2.1 gives a graphical
representation of the external forces and moments (Fx, Fy, Fz, L, M, and N), and
the linear and rotational velocity components of the airplane (u, v, w, p, q, and r)
in relation to its body-fixed reference frame. The orientation of the body-axes in
this figure conforms to the definition from section A.7.1 of appendix A. The figure
also shows the graphical representation of the airspeed vector V, the angle of attack
a, and the sideslip angle b, which define the orientation of the flight-path axes in
relation to the aircraft’s body-axes.
M, q
N, p
F , w
F , v
F , u
L, r
Y -axis
c.g.
X -axis
Z -axis
x
y
z
B
B
B
a
b
V
Figure 2.1: Orientation of the linear and angular velocity components, external forces and
moments, angle of attack, and sideslip angle in relation to the body-fixed reference
frame of the aircraft.
18 Chapter 2. Rigid body equations of motion
2.2 Expressing translational motions in flight-path axes
Although it seems logical to express translational velocities in terms of the body-axes
velocity components u, v, and w, it is often more convenient to use the true airspeed
V, angle of attack a, and sideslip angle b instead when considering aerodynamic
problems. The latter variables express the translational motions in relation to the
flight-path axes (the airspeed vector V coincides with the flight-path or wind-axis
XW, while a and b define the orientation of the flight-path reference frame FW in
relation to the body-fixed reference frame FB; see section A.7.1 of appendix A).
2.2.1 Advantages of relating translations to flight-path axes
Since V, a, and b can be expressed in terms of u, v, and w, and vice-versa, it is possible
to use either set of variables for the solution of the equations of motion. However, V,
a, and b are usually better suited for simulation tasks, for two reasons:
1. From a physical point of view it is logical to express the aerodynamic forces
and moments in terms of V, a, and b. For simulation purposes, we want the
linear force equations to be re-written as a set of explicit ordinary differential
equations (ODEs), moving all time-derivatives to one side of the equations and
all other terms to the other side. This may be difficult to achieve, since the aerodynamic
forces and moments may depend on the time-derivatives of the angle
of attack and sideslip angle, while ˙a and ˙b themselves will not not available until
after the force equations have been evaluated. In practice it is often possible
to assume a linear relationship between these time-derivatives and the aerodynamic
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(13)
