## Subsystem xdotcorr (Beaver)

xdotcorr (Beaver) converts implicit state equations of the 'Beaver' dynamic model into explicit state equations. It is contained in the subsystem Aircraft Equations of Motion (Beaver) in the Beaver model; being the only aircraft-dependent block of this subsystem. In this case, xdotcorr depends upon the aerodynamic model of the DHC-2 'Beaver' from ref.[1].

The implicitness of the original state equations is caused by a contribution of the time-derivative of beta-dot to the aerodynamic sideforce. Since Simulink can not always cope with implicit ODEs, the ODEs were divided in an aircraft-independent part and a separate correction term. It is this correction term which has been implemented in xdotcorr (Beaver).

xdotcorr can be adapted for other aircraft models, in cases where it is necessary to bring into account contributions of time-derivatives of states to the aerodynamic forces and/or moments, which would otherwise result in implicit state equations. Such implicit state equations most often involve alpha-dot and beta-dot terms.

### Inputvectors for xdotcorr (Beaver)

     xdot = dx/dt = [Vdot alphadot betadot pdot qdot rdot ...
                     psidot thetadot phidot xedot yedot Hdot]'
           (state derivatives, without corrections which make impli-
              cit equations explicit, coming from subsystem 12 ODEs)

     yhlp = [cos(alpha) sin(alpha) cos(beta) sin(beta) ...
             ... tan(beta) sin(psi) cos(psi) sin(theta) ...
             ... cos(theta) sin(phi) cos(phi)]'
               (frequently used sines & cosines, coming from Hlpfcn)

     yatm = [rho ps T mu g]',  (atmospheric properties, computed in
                                                  the block Atmosph)

    {Vdot    : time-derivative of airspeed [m/s^2]                 }
    {alphadot: time-derivative of angle of attack [rad/s]          }
     betadot : time-derivative of sideslip angle [rad/s]
                                                   (uncorrected!)
    {psidot  : time-derivative of yaw angle [rad/s]                }
    {thetadot: time-derivative of pitch angle [rad/s]              }
    {phidot  : time-derivative of roll angle [rad/s]               }
    {pdot    : time-derivative of roll rate [rad/s^2]              }
    {qdot    : time-derivative of pitch rate [rad/s^2]             }
    {rdot    : time-derivative of yaw rate [rad/s^2]               }
    {xedot   : time-derivative of x-coordinate in Earth-fixed
                                            reference frame [m/s]  }
    {yedot   : time-derivative of y-coordinate in Earth-fixed
                                            reference frame [m/s]  }
    {Hdot    : time-derivative of altitude above sea-level [m/s]   }

     rho  : airdensity [kg/m^3]
    {ps   : static pressure [N/m^2]                                }
    {T    : temperature [K]                                        }
    {mu   : dynamic viscosity [kg/(m*s)]                           }
    {g    : acceleration of gravity [m/s^2]                        }

The inputvariables which are not used by xdotcorr have been put between curly braces. Of the vector yhlp, only cos(beta) is used.

### Outputvector of xdotcorr

     xdot = [Vdot alphadot betadot pdot qdot rdot psidot ...
             ... thetadot phidot xedot yedot Hdot]'

     (xdot with correction for beta-dot influence upon sideforce Ya)

Here, betadot has been corrected for the beta-dot influence upon the aerodynamic sideforce Ya!

### Parameters which must be set in the Matlab Workspace

- AM: matrix with stability and control derivatives (currently valid for the nonlinear aerodynamic model of the DHC-2 'Beaver', see ref.[1]). The stability derivative CYbdot is extracted from this matrix to determine the influence of beta-dot upon aerodynamic sideforce Ya.
- GM1: vector with some important geometrical properties of the 'Beaver' aircraft, and the mass of the aircraft (which is assumed to be constant during the motions of interest). In xdotcorr, the wing span b, wing area S, and mass m are extracted from GM1.

The vector GM1 and the matrix AM can be loaded into the workspace from file by applying the routine DATLOAD. Run MODBUILD first if this datafile does not yet exist.

### References

1. R.T.H. Tjee and J.A. Mulder: Stability and Control Derivatives of the De Havilland DHC-2 "Beaver" aircraft. Report LR-556, Delft University of Technology, Delft, The Netherlands, 1988.