12odes.hlp

MATLAB在飞行动力学和控制中应用的工具

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System BEAVER - 4th-level subsystem '12 ODEs'
=============================================
The dynamics of a rigid aircraft have been written in terms of
12 Ordinary Differential Equations (ODEs). The subsystem '12 ODEs',
which is contained in the 3rd level subsystem 'Aircraft Equations
of Motion (Beaver)' of the system BEAVER, contains the general
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:

1 - the airframe is assumed to be a rigid body in the motion under
    consideration
2 - the airplane's mass is assumed to be constant during the time
    interval in which its motions are studied
3 - the Earth is assumed to be fixed in space, i.e. its rotation
    is neglected
4 - the curvature of the Earth is neglected.

Structure of the subsystem '12 ODEs'
====================================
In the subsystem '12 ODEs', these twelve state equations have been
separated in four blocks:

1 - ODEs for the true airspeed V, the angle of attack alpha, and the
    sideslip angle beta, which have been derived from Newton's law
    for translational motions
2 - ODEs for the rotational speeds along the aircraft's body-axes, p,
    q, and r, which have been derived from Newton's law for rotational
    motions
3 - ODEs for the Euler angles psi, theta, and phi, which determine the
    attitude of the aircraft relatively to the Earth (derived from
    basic kinematic relations)
4 - ODEs for the coordinates xe and ye, and the altitude H, all
    measured relatively to the (flat, non-rotating) Earth (derived
    from basic kinematic relations)

The four groups of state equations have been implemented as sepa-
rate masked subsystems, called Vabdot, pqrdot, Eulerdot, and xyHdot,
respectively.

These four masked subsystems all use [x' Ftot' Mtot' yhlp']' as input-
vector. In '12 ODEs', the different subvectors are first Muxed to-
gether into one large inputvector for Vabdot, pqrdot, Eulerdot, and
xyHdot. 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 surroun-
ding air, and the contributions due to atmospheric disturbances (wind +
turbulence). Therefore, xyHdot has one additional input line.

The outputs from Vabdot, pqrdot, Eulerdot, and xyHdot are Muxed
together, to build one large vector xdot (time-derivative of the state
vector). Note: the equations in the subsystem '12 ODEs' 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 system BEAVER, this problem has been solved
by neglecting these terms in the subsystem '12 ODEs', and making
corrections to the vector xdot afterwards in the masked subsystem
xdotcorr.

Inputs to the subsystem '12 ODEs':
==================================
  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 FMSORT)

  Mtot = [L M N]'            (total external moments along body-axes,
                                        computed in the block FMSORT)

  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 HLPFCN)

  [u+uw v+vw w+ww]'        (body-axes velocity components, including
                          contributions due to non-steady atmosphere)
where:

  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]

The inputvariables which are not used by any of the blocks Vabdot,
pqrdot, Eulerdot, or xyHdot, are displayed between accolades. The
angles alpha, beta, psi, theta, and phi do play an important role in
the ODEs, but for reasons of efficiency, they have been extracted
from the helpvector yhlp, which contains the sines and cosines of
these angles. The subsystem Hlpfcn in the 2nd level of BEAVER is used
to create this helpvector. Of yhlp, all elements are used in the
subsystem '12 ODEs'.

Output from the subsystem '12 ODEs':
====================================
  xdot = dx/dt

Note: the vector xdot that leaves the subsystem '12 ODEs' does not yet
take into account the aircraft-dependent influences which are induced
when writing out the implicit differential equations as a set of ex-
plicit ODEs. For the 'Beaver', this means that the influence of the
beta-dot term in the aerodynamic sideforce is neglected up to the sub-
system '12 ODEs'. In the subsystem 'Aircraft Equations of Motion
(Beaver)', the masked subsystem XDOTCORR (Beaver) takes care of the
appropriate corrections to xdot. This subsystem also contains a gain
block XFIX, which makes it possible to fix some of the states artifi-
cially to their initial values.

Parameters which must be set in the Matlab workspace:
=====================================================
The subsystem '12 ODEs' needs the following parameters to be loaded
into the Matlab workspace:

 GM1: matrix with the basic geometric properties and the mass of the
      aircraft
 GM2: matrix with inertia parameters of the aircraft

You may use the load routine LOADER.M to load these parameters from
file; type HELP LOADER for more info. Use MODBUILD.M to create such
datafiles; type HELP MODBUILD for more info.

More info:
==========
See VABDOT.HLP, PQRDOT.HLP, EULERDOT.HLP, and XYHDOT.HLP for more info
about the masked subsystem blocks from the subsystem '12 ODEs'. See
EQMOTION.HLP for more info about the subsystem 'Aircraft Equations of
Motion (Beaver)'. For more info about the parameters which must be set
in the Matlab workspace: type HELP LOADER and/or HELP MODBUILD.

More details about the inputvectors to this subsystem can be found
by examining HLPFCN.HLP and FMSORT.HLP. See XDOTCORR.HLP and XFIX.HLP
for more info about the corrections which can or will be made to the
vector xdot after it leaves the subsystem '12 ODEs'.

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The FDC toolbox, Copyright Marc Rauw 1994-1998.