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ODEs. This will be outlined for the Beaver aircraft in section 3.4. Table B.5 in appendix
B presents the inertia data on which the aerodynamic model was been based;
corrections to the body-axes moments will be necessary if a different position of the
center of gravity is used, see ref.[34].
3.3.2 Engine Forces & Moments
The propulsive forces and moments also strongly depend upon the type of aircraft
under consideration. For a piston-engined aircraft like the Beaver, the primary engine
control inputs are the engine speed n, and the manifold pressure pz, which
directly affect the engine power P. The engine power also varies with altitude due
to changes in air-density. If the propeller is represented as an ideal pulling disc, it is
possible to express changes in engine power and airspeed in terms of variations of
the non-dimensional pressure increase in the propeller slipstream dpt [34]:
dpt
Dpt
12
rV2
= C1 + C2
P
12
rV3
!
(3.4)
The constants C1 and C2 have been given in ref [34]: C1 = 0.08696, and C2 = 191.18.
P
12
rV3 is measured in [kW kg−1 s3]. For the Beaver aircraft, the engine power in [kW]
can be calculated with the following expression1:
P = 0.7355
− 326.5 + (0.00412 (pz + 7.4)(n + 2010) +
+ (n + 2010) + (408.0 − 0.0965 n)
1.0 −
r
r0
(3.5)
where:
pz = manifold pressure [00Hg],
n = engine speed [RPM],
r = air-density [kgm−3],
r0 = air-density at sea level = 1.225 [kgm−3].
The engine forces and moments, which include propeller slipstream effects, are written
as polynomial functions of x and dpt in a similar way as the aerodynamic model
[34]:
Fprop = d · p2 (x, dpt) (3.6)
where the subscript prop denotes propulsive effects. The vector function p2 contains
the polynomials for the non-dimensional propulsive force and moment coefficients;
1The factor 0.7355 converts from continental horse-power to kilowatt; this factor was used in several
other software packages from the Faculty of Aerospace Engineering of Delft University of Technology
(DUT), including the software that was used to control the engineering flight simulator of the Faculty
during the 1980’s and early 1990’s. However, according to ref.[34], the conversion should be from
brake horse power to kilowatt, which would require a factor of 0.7457 instead. Since it is not known
whether ref.[34] or the software is correct, it was decided not to correct the discrepancy, in order to
ensure that simulations created with the FDC toolbox could be directly compared to those obtained by
other programs at DUT. If the conversion factor in equation (3.5) is incorrect, the engine power is in
fact underestimated by 1.4%, which is deemed acceptable.
32 Chapter 3. The aircraft model
the actual forces and moments are again obtained by pre-multiplication with the
vector d, see equation (3.2). The polynomial functions gathered in p2, which describe
the propulsive force and moment coefficients in the body-fixed reference frame are:
CXp = CXdptdpt + CXa dpt2 a dpt2
CYp = 0
CZp = CZdptdpt
Clp = Cla2dpt
a2dpt
Cmp = Cmdptdpt
Cnp = Cndpt3 dpt3 (3.7)
Table B.4 in appendix B lists the values of the stability and control coefficients from
these polynomial equations. These coefficients are constant over the entire flight
envelope of the Beaver.
3.3.3 Gravitational forces
The weight of the aircraft equals the aircraft’s mass m times the local acceleration of
gravity g. It is directed along the positive ZV axis (see the definition of the vehiclecentered
vertical reference system FV in section A.7.1 of appendix A):
FV
grav = WV = mgV = mg ·
2
4
0
0
1
3
5
V
(3.8)
In this equation, the superscript V has been used to denote the applied reference
frame FV. Using the transformation:
FB
grav = TV!B FV
grav (3.9)
where the transformation matrix is TV!B defined according to equation (A.4) from
appendix A, we find the following equation for the gravity force components along
the aircraft’s body-axes:
Fgrav =
2
4
Xgr
Ygr
Zgr
3
5 = mg ·
2
4
−sin q
cos q sin j
cos q cos j
3
5 (3.10)
In this equation, g is the magnitude of the local acceleration of gravity, q is the pitch
angle of the vehicle, and j is the roll angle of the vehicle. The superscript B has been
omitted here for reasons of brevity. Obviously, the attitude of the vehicle, expressed
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