eqmotion.hlp

MATLAB在飞行动力学和控制中应用的工具

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System BEAVER - subsystem 'Aircraft Equations of Motion (Beaver)'
=================================================================
The dynamics of a rigid aircraft can be expressed by twelve Ordinary
Differential Equations (ODEs). The equations which determine the mag-
nitude of the time-derivatives of the twelve state variables as a
function of the external forces and moments, external disturbances,
and the states themselves, have been gathered in the subsystem 'Air-
craft Equations of Motion (Beaver)'. This subsystem also contains the
integrator which obtains the states from their time-derivatives.
   The twelve ODEs themselves are independent of the aircraft con-
sidered. However, some of the aerodynamic forces or moments may
depend upon the time-derivatives of the state variables, in which
case the differential equations become implicit and thence unsol-
vable for Simulink (in general). Often, it is not difficult to
rewrite the implicit differential equation(s) back into generalized
explicit ODEs and an aircraft-dependent correction term. In the case
of the system BEAVER, such an aircraft-dependent correction to the
general state-equations proved to be necessary, and has been included
in one of the masked subsystem blocks in the subsystem 'Aircraft
Equations of Motion (Beaver)'. This explains the reason for putting
the extension '(Beaver)' into the name of the subsystem.

The subsystem contains two masked subsystem blocks: uvw, and
xdotcorr (Beaver). The latter block contains the aircraft-dependent
corrections to the state equations for the DHC-2 'Beaver' aircraft,
which were induced when converting the original implicit differen-
tial equations into a set of explicit ODEs (see XDOTCORR.HLP for
more details). The masked subsystem uvw contains the transformation
equations which make it possible to obtain the components of the true
airspeed along the body-axes of the aircraft, given the true air-
speed, angle of attack, and sideslip angle.
    An Integrator block is used to obtain the time-trajectories of
the state variables themselves from their derivatives with respect to
time. The Gain block makes it possible to artificially fix states, by
performing an element-by-element multiplication of the vector xdot
with the vector xfix. All elements of the latter vector either equal
one or zero, meaning that the corresponding states are free to vary
or are artificially kept constant, respectively. Type HELP FIXSTATE
for more info! See also XFIX.HLP.
     Finally, the subsystem 'Aircraft Equations of Motion (Beaver)'
also contains another subsystem, '12 ODEs', in which the twelve ODEs
themselves have been grouped together. See ODES.HLP for more details
about the internal structure of this subsystem.

Computation order:
==================
The time-derivatives of the state variables are now computed as
follows. First, the velocity components u, v, and w are computed,
using the current value of the state vector x, and the outputvector
from the masked subsystem HLPFCN, which contains frequently used
sines and cosines of angular state variables, yhlp. The velocity
components u, v, and w are needed in order to be able to solve the
ODEs for the aircraft-coordinates xe and ye. In order to account for
contributions of non-steady atmosphere, the velocity components due
to wind and/or atmospheric turbulence, uw, vw, and ww, are extracted
from the vector uwind, and added to u, v, and w. Next, the time-deri-
vatives of the states are computed in the subsystem '12 ODEs', which
uses the current state vector x, the helpvector yhlp, the vectors
Ftot and Mtot with total forces and moments, and the vector
[u+uw, v+vw, w+ww]' as inputs.
     The time-derivatives of the states are gathered in the vector
xdot = dx/dt. As said before, the vector xdot that leaves the subsystem
'12 ODEs' needs to be corrected for aircraft-dependent terms. This is
done in the subsystem xdotcorr (Beaver), which needs the current state
vector x, and, in the case of the 'Beaver' aircraft, the outputvector
from the masked subsystem Atmosph in the subsystem 'Airdata Group' (to
be more specific: xdotcorr needs the airdensity for the correction of
the beta-dot equation). The outputvector from xdotcorr, the corrected
version of xdot, is send to the Gain xfix, which is used to artifi-
cially fix states if required by the user (see XFIX.HLP). Finally,
the time-derivatives are sent to the Integrator block, where they are
integrated to obtain the updated value of the state vector x. Note: the
Integrator block uses the vector xinco as initial condition (see below).

The state vector x is fed back to the other subsystems in BEAVER, and
internally in the subsystem 'Aircraft Equations of Motion', because the
time-derivatives of the state variables depend upon the state variables
themselves. The second level of BEAVER contains a masked subsystem
block Hlpfcn, which computes the sines and cosines of the angles alpha,
beta, psi, theta, and phi. These 'helpvariables' are also fed back to
other subsystems in BEAVER, including the system 'Aircraft Equations of
Motion'. The resulting feedback loops distinguish themselves from the
other signals by a magenta foreground color.

Inputs to the subsystem 'Aircraft Equations of Motion (Beaver)':
================================================================
  Ftot  = [Fx Fy Fz]' (total forces along body-axes, coming from FMSORT)
  Mtot  = [L M N]'   (total moments along body-axes, coming from FMSORT)
  uwind = [uw vw ww uwdot vwdot wwdot]'       (atmospheric disturbances)
  yatm  = [rho ps T mu g]'     (atmosphere variables, computed in the
                                               masked subsystem ATMOSPH)
  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, coming from HLPFCN)

  Ftot and Mtot originate from the masked subsystem block FMsort, uwind
  is an external inputvector to the system BEAVER, yatm originates from
  the masked subsystem Atmosph (in the subsystem 'Airdata Group'), and
  yhlp is the outputvector of the masked subsystem Hlpfcn.

  Fx, Fy, Fz: total external forces along body-axes [N]
  L, M, N   : total external moments along body-axes [Nm]
  uw, vw, ww: body-axes components of velocity component due to
              non-steady atmosphere (wind + turbulence) [m/s]
 {uwdot     : d(uw)/dt [m/s^2]                                   }
 {vwdot     : d(vw)/dt [m/s^2]                                   }
 {wwdot     : d(ww)/dt [m/s^2]                                   }
  rho       : air density [kg/m^3]
 {ps        : static pressure [N/m^2]                            }
 {T         : air temperature [K]                                }
 {mu        : dynamic viscosity [kg/(m*s)]                       }
 {g         : acceleration of gravity [m/s^2]                    }

The variables which are not used by any of the (masked) subsystem
blocks from the subsystem 'Aircraft Equations of Motion (Beaver)' have
been put between accolades. Of yhlp, all elements are used.

Outputs from subsystem 'Aircraft Equations of Motion (Beaver)':
===============================================================
  x     = [V alpha beta p q r psi theta phi xe ye H]'    (states)
  xdot  = dx/dt                      (time-derivatives of states)
  ybvel = [u v w]'        (body-axes components of true airspeed)

  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]
  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]

Variables which must be present in Matlab workspace in order for sub-
system 'Aircraft Equations of Motion (Beaver)' to function properly:
=====================================================================
  xinco: Initial value of the state vector x. Usually, a steady-state
         trimmed-flight condition is used as initial condition for the
         aircraft ('condition' = combination of input variables and
         state vector, so xinco does not specify a complete initial
         flight condition by itself!), for which reason a trim-routine
         ACTRIM has been included to the FDC package. Type HELP ACTRIM
         for more info, or examine the source code of ACTRIM.M, con-
         tained in the subdirectory TOOLS. If you double-click the
         block Integrator, you will find that xinco is specified as
         initial value of the state vector.
  xfix : Vector of length twelve (length(xfix) = length(x)), of which
         all elements are equal to either one or zero. If the n'th
         element of xfix is equal to one, the corresponding element of x
         (= the corresponding state) may vary freely, according to the
         'real' solution of the state equations. If the n'th element of
         xfix is equal to zero, the corresponding state is ARTIFICIALLY
         fixed to its initial value, even if the 'real' solution of the
         state equations would give different results. For the analysis
         of some types of aircraft responses, artificially fixing states
         to their initial values may be useful. Type HELP FIXSTATE for
         more info. The vector xfix can be defined manually, or by run-
         ning FIXSTATE.M before starting a new simulation. Usually,
         xfix = 1 or xfix = ones(1,12) will be used for simulations.
         This default value will be set automatically by running
         LOADER.M before starting a new simulation.
  GM1  : Matrix with some basic geometric properties of the aircraft,
         and its mass. GM1 is needed by the subsystem '12 ODEs', see
         its helpfile for more info.
  GM2  : Matrix with inertia parameters of the aircraft under conside-
         ration. GM2 is needed by the subsystem '12 ODEs', see its
         helpfile for more info.

GM1, GM2, and the default value of xfix can be loaded in the Matlab
workspace by applying the routine LOADER.M. Type HELP LOADER or HELP
MODBUILD for more info about the parameter matrices GM1 and GM2. Type
HELP FIXSTATE for more info about the gain xfix. The initial value of
the state vector, xinco, can be loaded from file by applying the load
routine INCOLOAD.M. Type HELP INCOLOAD for more info. Also type HELP
ACTRIM for more info about the determination of steady-state trimmed-
flight conditions.

More info.
==========
See 12ODES.HLP for more info about the subsystem '12 ODEs'. Other rele-
vant helpfiles are UVW.HLP, XDOTCORR.HLP, and XFIX.HLP. Type HELP
FIXSTATE for more info about the option to artificially fix states to
their initial values. More info about the parameters that need to be
defined in the Matlab workspace can be found by typing HELP FIXSTATE,
HELP LOADER, HELP MODBUILD, HELP INCOLOAD, or HELP ACTRIM.
See FMSORT.HLP, ATMOSPH.HLP, and HLPFCN.HLP for more info about the
inputvectors to the subsystem 'Aircraft Equations of Motion'.

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The FDC toolbox, Copyright Marc Rauw 1994-1998.