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airspeed and altitude of the aircraft. Several related variables are needed to compare
simulations with flight test measurements or windtunnel experiments; these variables
also play an important role for aircraft instrumentation. In this report, these
variables are referred to as ‘atmosphere and airdata variables’, or shortly ‘airdata variables’.
Several airdata equations have been included in the aircraft model to allow
the model to be used for a wide range of applications.
The airdata variables depend upon atmospheric properties such as the air pressure,
density, and temperature. Here we use the ICAO Standard Atmosphere model (see
for instance refs.[5] or [30]) to determine these properties. According to this model,
the air temperature T decreases linearly with increasing altitude in the troposphere,
i.e. at altitudes from zero to 11,000 meters above sea level:
T = T0 + lh (3.16)
where:
h = altitude above sea level [m],
T0 = air temperature at sea level = 288.15 [K],
l = temperature gradient in troposphere = −0.0065 [Km−1].
The air pressure depends upon the altitude, according the basic hydrostatic equation:
dps = −r g dh (3.17)
We assume that the ideal gas law can be applied to the air in the atmosphere:
ps
r
= Ra
Ma
T (3.18)
3.5. Atmosphere and airdata variables 35
where:
Ra = molar gas constant = 8314.32 [JK−1kmol−1],
Ma = molecular weight of the air [kg kmol−1].
Combining these equations and neglecting the altitude-dependency of the gravitational
acceleration g yields:
dps
ps
= −
Ma g0
RaT
dh (3.19)
where:
ps = static air pressure, [Nm−2],
g0 = gravitational acceleration at sea level = 9.80665 [ms−2].
To find ps, equation (3.19) needs to be integrated, yielding:
ln
ps
p0
= −
g
lR
ln
T0 + lh
T0
(3.20)
This equation can be written as:
ps
p0
=
1 + lh
T0
− g
lR
=
T0
T
g
lR
(3.21)
R is the specific gas constant, which is defined as:
R
Ra
M0
(3.18) = p0
r0T0
= 287.05 [JK−1kg−1] (3.22)
and:
p0 = air pressure at sea level = 101325 [Nm−2],
r0 = air density at sea level = 1.225 [kgm−3],
M0 = molecular weight of the air at sea level = 28.9644 [kg kmol−1].
Since the gravitational acceleration g was held constant during the integration of
equation (3.19), this actually means that the geometrical altitude h in this equation
must be replaced by the geopotential altitude H.1 In this report the slight distinction
between h and H will be neglected, in view of the relatively low altitudes considered.
In order to remind us of this small inaccuracy, the symbol H will be used in the
remainder of the report to denote the altitude.
Contrary to the pressure equation (3.21), where the acceleration g was assumed
to be equal to g0 for all altitudes, the model does take into account changes in g
with altitude for the computation of the aircraft’s weight. The actual gravitational
acceleration is then computed with the following equation, which is valid for a round
Earth:
g = g0
REarth
REarth + h
2
(3.23)
where:
REarth = radius of the Earth = 6371020 [m]
1
The geopotential altitude H is defined as: H
Z h
0
g
g0
dh
36 Chapter 3. The aircraft model
The air density r (in [kgm−3]) is calculated from ps and T by means of the ideal gas
law (3.18), which yields:
r = psMa
RaT
= ps
RT
(3.24)
The aerodynamic and propulsive forces and moments that act upon the aircraft are
functions of the dynamic pressure qdyn, which takes into account changes in airspeed
and changes in air density:
qdyn 12
rV2 (3.25)
This term originates from Bernoulli’s law, which states that the sum of the static
pressure ps and the dynamic pressure qdyn is constant:
pt ps + qdyn = constant (3.26)
where pt is the ‘total pressure’, which is the pressure in a point where the air is
brought to a complete standstill (this pressure can be measured in the so-called ‘stagnation
point’ of a pitot tube). However, this only applies for incompressible currents,
which in practice means that this law can only be applied for slow aircraft, like the
Beaver. As a guideline, Bernoulli’s law is usually applied only for airspeeds up to
100 ms−1.
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(22)
