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air is negligible.
As explained earlier, it may also be necessary sometimes to take into account scale
effects, e.g. if the aerodynamic model is determined by windtunnel measurements,
using a scale model. In that case, the Reynolds number needs to be known. Often
the Reynolds number is related to the mean aerodynamic chord c, which yields the
non-dimensional value:
Rc = rVc
μ
(3.38)
It is also possible to use the Reynolds number per unit length (in [m−1]), which
equals:
Re = rV
μ
(3.39)
In equations (3.38) and (3.39), μ is the coefficient of the dynamic viscosity, which can
be calculated with the equation of Sutherland:
μ = 1.458 · 10−6 T32
T + 110.4
(3.40)
From all variables mentioned in this paragraph, only p, T, r, and qdyn really must
be calculated in order to be able to solve the equations of motion for the Beaver
airplane. However, the other airdata (-related) variables may be useful for many
analytical purposes and some of them will be needed to solve equations of motion
when a different aerodynamic model that takes into account compressibility and/or
scale effects would be used instead.
Although the current list of atmosphere and airdata equations is quite comprehensive,
it is by no means complete: expanding the flight-envelope beyond the troposphere
(i.e. to altitudes above 11,000 meters) and beyond the speed of sound will
require additional modifications to the equations for pressure, density, and temperature.
For more information, refer to ref. [5] or other books about aircraft instrumentation.
3.6 Additional observation variables
In the previous paragraph we have obtained a list of state variables, time-derivatives
of the state variables, forces and moments, atmospheric variables, and airdata variables.
It is possible to enhance this list with a large number of additional output variables.
Here we will include additional normalized kinematic accelerations, specific
forces, body-axes velocity rates and some flight-path (-related) variables, but the list
can easily be expanded if required.
3.6.1 Body-axes velocity rates
For educational purposes, it may be useful to take a closer look at the components of
the aircraft’s acceleration along its body-axes. These body-axes ‘velocity rates’ u˙, v˙,
3.6. Additional observation variables 39
and w˙ are equal to:
u˙ = Fx
m
− qw + rv
v˙ =
Fy
m
+ pw − ru (3.41)
w˙ = Fz
m
− pv + qu
In section 2.2 it was explained that the true airspeed V, angle of attack a, and sideslip
angle b were better suitable as state variables than the body-axes velocity components
u, v, and w. This is why equations (3.41) have been implemented as additional
output equations, rather than state equations.
3.6.2 Kinematic accelerations and specific forces
It is possible to calculate accelerations and outputs from accelerometers, which may
be useful for post-flight analysis or required for certain flight control tasks. A typical
example of the latter is a turn-coordination system that is based on a feedback-loop of
the acceleration along the YB axis. Also, these signals may be useful for applications
in the field of manoeuvre load limiting. The aircraft model from the FDC-toolbox
considers accelerations in the vehicle’s center of gravity only, but equations for positions
outside the center of gravity can easily be included if necessary. See ref.[14] for
the required transformations.
The body-axis acceleration vector a can be expressed as:
a = ˙V = ¶V
¶t
+ W × V (3.42)
where W is the rotational velocity vector of the aircraft. Expanding this equation
into its components along the body-axes and substituting for u˙, v˙, and w˙ – see equation
(3.41) – yields:
ax,k = 1
g0
(u˙ + qw − rv) = Fx
W
ay,k = 1
g0
(v˙ + ru − pw) =
Fy
W
(3.43)
az,k = 1
g0
(w˙ + pv − qu) = Fz
W
The difference between the kinematic accelerations and u˙, v˙, and w˙ is the inclusion of
additional angular and translational velocity cross-product terms. In addition, contrary
to equations (3.41), the kinematic accelerations have been measured in units
of g, which explains the division by g0. W = m g is the total weight of the aircraft,
measured in N. The index k denotes that these variables represent kinematic accelerations
in the body-fixed reference frame.
The outputs of accelerometers which are oriented along the body-axes, and located
in the vehicle’s center of gravity are equal to the kinematic body accelerations
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(24)
