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where p0 = 101325 Nm−2 is the air pressure at sea level and R = Ra/M0 = 287.05
JK−1kg−1 is the specific gas constant of the air (Ra = 8314.32JK−1kmol−1 is the molar
gas constant, and M0 = 28.9644 kg kmol−1 is the molecular weight of the air at sea
level).
• Air density r, [kgm−3], according to the gas law for ideal gasses:
r = ps
RT
• Coefficient of the dynamic viscosity μ, [kgm−1s−1], according to Sutherland’s equation
[30]:
μ = 1.458 · 10−6 T32
T + 110.4
• Gravitational acceleration g, [ms−2]:
g = g0
REarth
REarth + h
2
where g0 = 9.80665 ms−2 is the gravitational acceleration at sea level and REarth =
6371020 m is the radius of the Earth. Note: although this equation uses the radius
of the Earth, the state equations in the block 12 ODEs are still based on a flat-Earth model!
This equation only takes into account the altitude-dependency of the gravitational
acceleration.
8.2. The aircraft model block libraries 137
Inputs
x = [ V a b p q r y q j xe ye H ]T state vector, x
Outputs
yatm = [ r ps T μ g ]T basic atmospheric properties, yatm
Parameters
All parameters for Atmosph are defined within the block itself; Atmosph does not use any
parameters from the MATLAB workspace.
Connections
in: x comes from the block Integrator.
out: yatm is connected to the blocks Airdata1, Airdata2, Airdata3, Power (Beaver), Gravity,
and xdotcorr (Beaver)
Type browse atmosph at the command-line for on-line help.
138 Chapter 8. Aircraft model block reference
Beaver level 1 Beaver level 1
Main FDC library / Complete system Beaver
Type
First level of the Beaver dynamics model, which provides connections to other SIMULINK
systems and an interface between the model and the MATLAB workspace.
Description
The first level of the aircraft model (see figure 8.2) takes care of the input/output functions
of the simulation model. It collects all input and output signals in vectors which
are sent to the MATLAB workspace by means of To Workspace blocks. A related timeline,
which can be used for post-simulation analysis and processing of simulation data, is also
made available. Furthermore, Beaver Level 1 contains a set of relevant Inport and Outport
blocks, to allow the aircraft model to be connected to external SIMULINK systems, and
to provide access points for analytical MATLAB functions (most noticably the trimming
functions).
There are twelve Inport blocks: six control inputs and six inputs representing atmospheric
disturbances. On the output-side, only a relevant subset of the system outputs
have been connected to Outport blocks, to prevent the ‘interface level’ from becoming too
cluttered. This subset provides enough information for the autopilot simulation models,
which will be discussed in chapter 15.
The aircraft model comes in three flavours: a stand-alone system, called Beaver, and
two subsystem equivalents of this model. The stand-alone system is used for modelanalysis
from the MATLAB environment (simulations, trimming, linearisation), while the
subsystem equivalents are used as subsystems within larger simulation models. These
variants are virtually identical, except that one of the subsystem equivalents combines
the output signals in two vectors, instead of using 16 individual scalar outputs.
It should be noted that the structure of this ‘interface level’ is actually a leftover from earlier
incarnations of this toolbox, which were designed for earlier SIMULINK versions, that
did not allow the use of vector inputs and outputs in top-levels of a system. Versions 1.0,
1.1, and 1.2 of the FDC toolbox used a graphical ‘S-function’ to implement the aircraft
model, i.e. an entirely separate SIMULINK system, which could be called by other models,
if desired.
The advantage of this approach, compared to using ordinary subsystems, was that
a single model could be called by several larger models, without having to duplicate
any code. However, the feature to treat graphical models as ‘S-functions’ was dropped
from SIMULINK 2 (nowadays, the term ‘S-function’ is used only to describe specially formatted
program functions, written in MATLAB, C, or Fortran). The current version of
the FDC toolbox uses the newer SIMULINK library functionality to maintain a centralized
copy of the aircraft model.
Subsystems and/or blocks
The first level of the aircraft model uses Mux and To Workspace blocks to collect the inputs
and output signals, and send these to the MATLAB workspace. A Clock block provides
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(68)
