📄 12odes.htm
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FDC help: 12 ODEs
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<h2>
Subsystem <i>12 ODEs</i>
</h2>
<p>The dynamics of a rigid aircraft have been written in terms of 12 Ordinary
Differential Equations (ODEs). The subsystem <i>12 ODEs</i>, which is
contained in the 3<sup>rd</sup> level subsystem <i><a href=
"eqmotion.htm">Aircraft Equations of Motion (Beaver)</a></i> of the <i><a href="beaver.htm">Beaver</a></i> model, contains the generic state
equations for rigid aircraft, which are fully independent of the aircraft
under consideration. The twelve state equations are valid only when four
restrictive assumptions are made: </p>
<ol>
<li>
the airframe is assumed to be a rigid body in the motion under
consideration
</li>
<li>
the airplane's mass is assumed to be constant during the time
interval in which its motions are studied
</li>
<li>
the Earth is assumed to be fixed in space, i.e. its rotation is
neglected
</li>
<li>
the curvature of the Earth is neglected.
</li>
</ol>
<h3>
Structure of the subsystem <i>12 ODEs</i>
</h3>
<p>In the subsystem <i>12 ODEs</i>, these twelve state equations have been
separated in four masked subsystem blocks: </p>
<ol>
<li>
<i><a href="vabdot.htm">Vabdot</a></i> contains ODEs for the true
airspeed <i>V</i>, the angle of attack <i>alpha</i>, and the sideslip
angle <i>beta</i>, which have been derived from Newton's law for
translational motions
</li>
<li>
<i><a href="pqrdot.htm">pqrdot</a></i> contains ODEs for the rotational
speeds along the aircraft's body-axes, <i>p</i>, <i>q</i>, and
<i>r</i>, which have been derived from Newton's law for rotational
motions
</li>
<li>
<i><a href="eulerdot.htm">Eulerdot</a></i> contains ODEs for the Euler
angles <i>psi</i>, <i>theta</i>, and <i>phi</i>, which determine the
attitude of the aircraft relatively to the Earth; derived from basic
kinematic relations
</li>
<li>
<i><a href="xyhdot.htm">xyHdot</a></i> contains ODEs for the
coordinates <i>x<sub>e</sub></i> and <i>y<sub>e</sub></i>, and the
altitude <i>H</i>, all measured relatively to the (flat, non-rotating)
Earth; derived from basic kinematic relations
</li>
</ol>
<p>These four masked subsystems all use the inputvector [<i>x<sup>T</sup></i>
<i>Ftot<sup>T</sup></i> <i>Mtot<sup>T</sup></i>
<i>yhlp<sup>T</sup></i>]<i><sup>T</sup></i>. In <i>12 ODEs</i>, the
different subvectors are first 'Muxed' together into a single large
inputvector for <i>Vabdot</i>, <i>pqrdot</i>, <i>Eulerdot</i>, and
<i>xyHdot</i>. The position of the aircraft relative to the Earth depends
upon the velocity components along the aircraft's body-axes, taking
into account both the velocity of the aircraft with respect to the
surrounding air, and the contributions due to atmospheric disturbances
(wind and turbulence). This explains why <i>xyHdot</i> has an additional
input line.</p>
<p>The outputs from <i>Vabdot</i>, <i>pqrdot</i>, <i>Eulerdot</i>, and
<i>xyHdot</i> are 'Muxed' together, to build one large vector
<i>xdot</i> (time-derivative of the state vector). Note: the equations in
the subsystem <i>12 ODEs</i> are fully independent of the aircraft under
consideration. In practice, the equations of motion may become implicit,
for instance if alpha-dot or beta-dot terms enter the force-equations of
the aerodynamic model. If these equations are written out as explicit
equations (in practice, this will often be possible), the twelve ODEs will
contain aircraft-dependent terms. In the <i><a href="beaver.htm">Beaver</a></i> model, this problem has been solved by neglecting
these terms in the subsystem <i>12 ODEs</i>, and making corrections to the
vector <i>xdot</i> afterwards in the masked subsystem <i><a href=
"xdotcorr.htm">xdotcorr</a></i>. </p>
<h3>
Inputs to the subsystem <i>12 ODEs</i>
</h3>
<pre>
x = [V alpha beta p q r psi theta phi xe ye H]' (states)
Ftot = [Fx Fy Fz]' (total external forces along body-axes,
computed in the block <a href=
"fmsort.htm">FMsort</a>)
Mtot = [L M N]' (total external moments along body-axes,
computed in the block <a href=
"fmsort.htm">FMsort</a>)
yhlp = [cos(alpha) sin(alpha) cos(beta) sin(beta) tan(beta) ...
... sin(psi) cos(psi) sin(theta) cos(theta) ...
... sin(phi) cos(phi) ]'
(sines & cosines, computed in the block <a href=
"hlpfcn.htm">Hlpfcn</a>)
[u+uw v+vw w+ww]' (body-axes velocity components, including
contributions due to non-steady atmosphere)
V : true airspeed [m/s]
{alpha: angle of attack [rad] }
{beta : sideslip angle [rad] }
p : roll-rate [rad/s]
q : pitch-rate [rad/s]
r : yaw-rate [rad/s]
{psi : yaw-angle [rad] }
{theta: pitch-angle [rad] }
{phi : roll-angle [rad] }
{xe : X-coordinate, relative to Earth-axes [m] }
{ye : Y-coordinate, relative to Earth-axes [m] }
{H : altitude above sea level [m] }
Fx, Fy, Fz: total external forces along body-axes [N]
L, M, N : total external moments along body-axes [Nm]
u : component of V along XB-axis [m/s]
v : component of V along YB-axis [m/s]
w : component of V along ZB-axis [m/s]
uw : wind+turbulence velocity component along XB-axis [m/s]
vw : wind+turbulence velocity component along YB-axis [m/s]
ww : wind+turbulence velocity component along ZB-axis [m/s]
</pre>
<p>The inputvariables which are not used by any of the blocks <i><a href=
"vabdot.htm">Vabdot</a></i>, <i><a href="pqrdot.htm">pqrdot</a></i>, <i><a
href="eulerdot.htm">Eulerdot</a></i>, or <i><a href=
"xyhdot.htm">xyHdot</a></i>, are displayed between curly braces. The angles
<i>alpha</i>, <i>beta</i>, <i>psi</i>, <i>theta</i>, and <i>phi</i> do play
an important role in the ODEs, but for reasons of efficiency, they have
been extracted from the helpvector <i>yhlp</i>, which contains the sines
and cosines of these angles. The subsystem <i><a href=
"hlpfcn.htm">Hlpfcn</a></i> in the 2<sup>nd</sup> level of <i><a href=
"beaver.htm">Beaver</a></i> is used to create this helpvector. All elements
from the vector <i>yhlp</i> are used by the blocks from <i>12 ODEs</i>. </p>
<h3>
Output from the subsystem <i>12 ODEs</i>
</h3>
<pre>
xdot = dx/dt
</pre>
<p>Note: the vector <i>xdot</i> which leaves the subsystem <i>12 ODEs</i> does
not yet take into account the aircraft-dependent influences which are
induced when writing out the implicit differential equations as a set of
explicit ODEs. For the 'Beaver', this means that the influence of
the beta-dot term in the aerodynamic sideforce is neglected up to the
subsystem <i>12 ODEs</i>. In the subsystem <i><a href=
"eqmotion.htm">Aircraft Equations of Motion (Beaver)</a></i>, the masked
subsystem <i><a href="xdotcorr.htm">xdotcorr (Beaver)</a></i> takes care of
the appropriate corrections to <i>xdot</i>. This subsystem also contains a
gain block <i><a href="xfix.htm"><i>XFIX</i></a></i>, which makes it
possible to fix some of the states artificially to their initial values. </p>
<h3>
Parameters to be defined in the Matlab workspace
</h3>
<p>The subsystem <i>12 ODEs</i> needs the following parameters to be defined
in the Matlab workspace: </p>
<ul>
<li>
<i>GM1</i>: matrix with the basic geometric properties and the mass of
the aircraft
</li>
<li>
<i>GM2</i>: matrix with inertia parameters of the aircraft
</li>
</ul>
<p>You may use the load routine <i><a href="datload.htm">DATLOAD</a></i> to
load these parameters from file. Use <i><a href=
"modbuild.htm">MODBUILD</a></i> to create such datafiles.</p>
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