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40 Chapter 3. The aircraft model
minus the gravity terms:
Ax = ax,k + sin q = (Fx − Xgr) /W
Ay = ay,k − cos q sin j = (Fy −Ygr) /W
Az = az,k + cos q cos j = (Fz − Zgr) /W
(3.44)
Ax, Ay, and Az are measured in units of g. These accelerations represent what would
actually be felt by a pilot when he would be located at the aircraft’s center of gravity;
they are usually called specific forces.
3.6.3 Flight-path related variables
To determine the flight-path of the aircraft, it is useful to introduce some additional
observation variables. First of all, the flight-path angle g is computed, using the
following expression:
g = arcsin
˙H
V
(3.45)
This angle is, for instance, useful during approach simulations where it determines
how much the aircraft deviates from the standard glide-path. The acceleration in the
direction of the velocity vector V, measured in units of [g], is called the flight-path
acceleration fpa. It is equal to:
fpa =
p
u˙2 + v˙2 + w˙ 2
g0
=
˙V
g0
(3.46)
Other flight-path related variables are the azimuth angle c and the bank angle F,
which are obtained with the following equations [30]:
c = b + y (3.47)
F = arcsin
sin j sin(90 − q)
= arcsin
sin j cos q
(3.48)
See also the description of the flight-path or wind reference frame FW in appendix A,
section A.7.3.
3.7 Summary
To recapitulate: we have completed the state-space model of the rigid body dynamics,
described in section 2.5, by developing models for the aerodynamic, propulsive,
and gravitational forces and moments, correcting the force factors for nonsteady atmosphere,
and determining some atmosphere and airdata variables that are required
to compute these forces and moments.
All elements combined result in the mathematical model from figure 3.2. This model
was enhanced with several useful output equations, including additional airdata
parameters, acceleration quantities, and flight-path variables. The resulting system
of equations completes the block ‘Aircraft dynamics’ from figure 3.1; the other elements
from that figure will be explored in the next chapters.
The model comprises twelve states and 77 additional output variables (including
some interim results from the computations, such as force and moment coefficients,
and time-derivatives of the states). These variables have been listed in table D.2,
3.7. Summary 41
which can be found appendix E. The input variables to this model are the control
surface deflections, which affect the aerodynamic forces and moments, and the engine
inputs, which affect the propulsive forces and moments. In addition, wind velocity
components and rates enter the model as external disturbances; these signals
are used to determine the force-corrections for nonsteady atmosphere. An overview
of these input variables can be found in table D.3 in appendix E.
Chapter 4
External atmospheric disturbances
In this chapter, we will take a closer look at the unsteady nature of the atmosphere,
which affects the motions of the airplane and its flight-path in relation to the ground.
Some basic models of wind and turbulence will be presented in a format that will
allow easy adaptation in the SIMULINK environment. The wind velocity will be represented
by deterministic functions, while the atmospheric turbulence will be modelled
by means of stochastic functions.
It is not intended to provide a comprehensive description of the entire atmosphere
here, nor will this report provide complete and detailed derivations of the
wind and turbulence models. Instead, the focus of this chapter is to present some
useful models aimed at practical applications, such as the analysis and fine-tuning
of flight control system performance under nonsteady atmospheric conditions. Examples
of such tasks are the simulation of automatic approaches under varying wind
conditions, the assessment of attitude controllers in turbulent weather, and the development
of gust alleviation control laws to improve the ride quality or reduce structural
loads in the airframe.
4.1 Deterministic disturbances
The velocity and direction of the mean wind with respect to the ground usually is not
constant along the flight-path. This variation of the mean wind along the flight-path
is called windshear.1 The influence of windshear upon the motions of the aircraft is
of particular importance during the final approach and landing, and during take-off
and initial climb.
An idealized profile of the mean wind as a function of height above the ground
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