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+ p sin a − r cos a
Based upon this equation the following calculation sequence has been used in the simulation
model of the Beaver:
1. compute the external forces and moments as usual, but neglect the ˙b-contribution
to the aerodynamic side-force for the time being (Aeromod),
2. substitute the thus obtained forces and moments into the general ˙b-equation, to
obtain the value ˙b (12 ODEs), and
3. compute the actual value of ˙b using the expression ˙b = ˙b
1 − rSb
4m CY˙b
cos b
−1
(xdotcorr).
8.2. The aircraft model block libraries 161
Inputs
˙x = [ ˙V
˙a
˙b
˙ p ˙ q ˙ r ˙y
˙q
˙j
˙ xe ˙ ye ˙H ]T uncorrected time-derivative of state vector, xdot
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
yatm = [ r ps T μ g ]T basic atmospheric properties, yatm
Outputs
˙x = [ ˙V
˙a
˙b
˙ p ˙ q ˙ r ˙y
˙q
˙j
˙ xe ˙ ye ˙H ]T time-derivative of state vector, xdot, corrected for ˙b
Parameters
The block xdotcorr (Beaver) needs the parameter vector GM1 to be present the MATLAB
workspace, in order to extract the wing span, wing surface, and 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). It also needs the parameter matrix AM, to
extract the stability derivative CY˙b
. The definition of AM and GM1 can be found in appendix
C. Both variables 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 (not corrected for the implicit nature of the ˙b-equation) comes from the subsystem
12 ODEs; yhlp comes from the block Hlpfcn; yatm comes from Atmosph.
out: ˙x (with a corrected value of ˙b) is connected to the block xfix.
Type browse xdotcorr at the command-line for on-line help.
162 Chapter 8. Aircraft model block reference
xfix Beaver level 1 / Beaver level 2 / Aircraft Equations of Motion / xfix
Main FDC library / Equations of motion
Type
Aircraft-independent masked Gain block, not necessary for solving the equations of motion
themselves but quite useful for purposes such as autopilot design and analysis.
Description
Sometimes it is useful to artificially fix state variables to their initial values by simply
setting their time-derivatives to zero, thus totally disregarding the values of the timederivatives
resulting from the model equations. Neglecting parts of the system dynamics
can be useful to establish ‘reference’ system responses, against which ‘actual’ responses
can be measured. Examples are the design of a speed-hold autothrottle controller, whose
responses can be compared to the ‘ideal’ situation in which the airspeed is artificially
held constant, or the analysis of lateral-longitudinal cross-coupling effects by comparing
simulations from the full aircraft model with results in which the longitudinal or lateral
aircraft dynamics are artificially neglected.
The block xfix can be used to force state variables from the aircraft model to remain
fixed to their initial value during the simulation. It is in fact a masked Gain block that
multiplies the time-derivatives of all state variables with a value of either one, or zero.
In the first situation, the computed time-derivative is used, in the second situation, the
time-derivative is artificially set to zero, causing the corresponding state variable to stay
constant.
Equations
The block xfix modifies the time-derivative of the state vector by performing an elementby-
element multiplication (in the equation below denoted with the symbol ?) with the
vector xfix. The elements from this vector are equal to one or zero, and the total number
of elements corresponds to the length of the state vector.
• Modified time-derivative of the state vector:
x˙ new = x˙ old ? xfix =
2
6666666666666666664
˙V˙a˙b˙
p˙
q˙
r
˙y˙q
˙j˙
xe
y˙e
˙H
3
7777777777777777775
?
2
6666666666666666664
xfix(1)
xfix(2)
xfix(3)
...............
xfix(12)
3
7777777777777777775
where: xfix(i) =
0
1 i 2 {1, 2, . . . , 12}
Inputs
˙xold = [ ˙V
˙a
˙b
˙ p ˙ q ˙ r ˙y ˙q
˙j
˙ xe ˙ ye ˙H ]T unmodified time-derivative of state vector, xdot
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(79)
