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the nonlinear aircraft model.
Equations
The equations used by the block Vabdot have been discussed in some more detail in section
2.2.
• Time-derivative of the true airspeed V, [ms−2]:
˙V
= 1
m
(Fx cos a cos b + Fy sin b + Fz sin a cos b)
• Time-derivative of the angle of attack a, [rad s−1]:
˙a
= 1
V cos b

1
m
(−Fx sin a + Fz cos a)

+ q − (p cos a + r sin a) tan b
• Time-derivative of the sideslip angle b, [rad s−1]:
˙b
= 1
V

1
m
(−Fx cos a sin b + Fy cos b − Fz sin a sin b)

+ p sin a − r cos a
Inputs
uVab = [ xT Ftot
T Mtot
T yhlp
T ]T input vector to Vabdot, uVab
where:
x = [ V a b p q r y q j xe ye H ]T state vector, x
Ftot = [ Fx Fy Fz ]T total external forces, Ftot
Mtot = [ L M N ]T total external moments, Mtot
yhlp = [ cos a sin a cos b sin b tan b sin y cos y sin q cos q sin j cos j ]T
frequently used sines and cosines, yhlp
Outputs
yVab = [ ˙V
˙a
˙b
]T time-derivatives of the airspeed, angle of attack, and
sideslip angle, yVab (part of ˙x)
Parameters
Vabdot needs the parameter vector GM1 to be present the MATLAB workspace, in order
to extract the mass m of the aircraft (the mass has been implemented as a parameter, i.e. it
is assumed to be constant during the relative short time intervals considered). The definition
of GM1 can be found in appendix C. GM1 can be loaded from the file AIRCRAFT.DAT,
using the utility DATLOAD (see section 12.4.2). If this datafile has somehow inadvertently
been deleted, it can be re-created by running the program MODBUILD (see section 12.6.1).
Connections
in: x comes from the block Integrator; Ftot and Mtot come from the block FMsort; yhlp
comes from the block Hlpfcn.
out: yVab is muxed together with the time-derivatives of the other state variables into a
single vector ˙x (not corrected for the implicit nature of the ˙b-equation). This timederivative
of the state vector x is then connected to the block xdotcorr.
Type browse Vabdot at the command-line for on-line help.
160 Chapter 8. Aircraft model block reference
xdotcorr (Beaver) Beaver level 1 / Beaver level 2 / Aircraft Equations of Motion / xdotcorr (Beaver)
Main FDC library / Equations of motion
Type
Aircraft-dependent masked subsystem block, essential for solving the equations of motion.
Description
As discussed in section 3.4, the aerodynamic model of the Beaver aircraft includes an implicit
differential equation of the sideslip angle: the time-derivative of the sideslip angle
appears on both sides of the equation (˙b is dependent on the sideforce, while the aerodynamic
component of the sideforce itself is directly dependent on ˙b). To avoid this from resulting
in an algebraic loop (see section 6.4.1), which would slow down the simulations or
make the system too complicated for SIMULINK to solve, the equation has been re-written
as an explicit differential equation. In order to keep the resulting model as generic as possible,
the resulting equation was split into an aircraft-independent part, which neglected
the direct contribution of ˙b to the aerodynamic sideforce, and an aircraft-dependent part,
which took care of this dependency.
The latter task is performed by the block xdotcorr (Beaver), which is the only aircraftdependent
block in the subsystem Aircraft Equations of Motion (Beaver) (hence the suffix
‘Beaver’). Please notice that xdotcorr is in fact part of the aerodynamic model; this
block should be updated if the aircraft model is adapted for a different aircraft type. This
method can be applied for any implicit state equation, as long as the dependencies are linear.
For nonlinear implicit relations, it will be necessary to use a different solution to this
problem, e.g. artificially breaking an implicit loop by introducing a delay in the internal
˙x-feedback loop.
Equations
The ˙b-equation for the Beaver can be written as:
˙b
= 1
Vm

− Fx cos a sin b + Fy
 cos b − Fz sin a sin b + 12
rV2S CY˙b
˙b
b
2V cos b

+
+ p sin a − r cos a
where Fy
 is the side-force without the contribution of ˙b. The ˙b -term on the right-hand
side of this equation can easily be moved to the left-hand side:
˙b
  ˙b

1 −
rSb
4m
CY˙b
cos b

=
= 1
Vm
􀀀
−Fx cos a sin b + Fy
 cos b − Fz sin a sin b
　
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