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easily be included to the model, e.g. ground forces during taxiing, or a ground-effect
model for aircraft that fly close to the ground. Figure 3.2 shows the individual contributions
to the total external forces and moments; the dashed box represents additional
factors that are not taken into account in this report.
3.3.1 Aerodynamic Forces & Moments
The aerodynamic forces and moments depend upon the flight condition, defined by
the state vector x and the aircraft configuration (i.e. the position of movable elements
3.3. External forces and moments 29
Aerodynamics
Propulsion
Gravity
Wind
Corrections
Atmosphere/
Airdata
-
6
-
-
-
-
j -
j
-
-
j -
TW!B
TE!B
- å
-
Equations of Motion
x˙ = f(x, Ftot ,Mtot)
Z dt - x˙ x
FB
tot
MB
tot
FBa
ero
MBa
ero
FWa
ero
MWa
ero
FBp
rop
MBp
rop
FEg
rav
FBg
rav
FBw
ind
uaero
uprop
u
wind
qdyn,
M, ...
-
-
-
-
-
-
- -
Figure 3.2: Block-diagram of the general rigid body dynamics
30 Chapter 3. The aircraft model
on the airplane). For the DHC-2 Beaver aircraft, a sophisticated aerodynamic model
has been determined from flight tests in 1988 [34]. This model expresses the aerodynamic
forces and moments along the aircraft’s body-axes in terms of polynomial
functions of the aircraft states, the time-derivative of the state vector, and the flight
control inputs:
Faero = d · p1 (x, x˙ , uaero) (3.1)
where Faero is a vector of aerodynamic forces and moments, and p1 is a polynomial
vector-function that yields non-dimensional force and moment coefficients. The input
vector uaero gathers the positions of the elevator, ailerons, and rudder (the primary
flight controls for the Beaver), and flaps (the secondary flight controls). For the
Beaver model, the x˙ -term is linear and only takes into account the direct contribution
of ˙b to the aerodynamic side-force Ya. The pre-multiplication with the diagonalmatrix
d converts these non-dimensional coefficients to dimensional forces and moments;
d equals:
d = qdynS · diag ([ 1 1 1 b c b ]) (3.2)
S is the wing-area of the aircraft, b is the wing-span, c is the mean aerodynamic chord,
and qdyn is the dynamic pressure (qdyn = 1
2rV2, see section 3.5).
The polynomial functions from p1, describing the aerodynamic force and moment
coefficients in the body-fixed reference frame are:
CXa = CX0 + CXaa + CXa2 a2 + CXa3 a3 + CXq
qc
V
+ CXdr dr + CXd f
df + CXad f
adf
CYa = CY0 + CYb b + CYp
p b
2V
+ CYr
rb
2V
+ CYda da + CYdr dr + CYdradra + CY˙b
˙b
b
2V
CZa = CZ0 + CZaa + CZa3 a3 + CZq
qc
V
+ CZde de + CZdeb2 deb2 + CZd f
df + CZad f
adf
Cla = Cl0 + Clb b + Clp
p b
2V
+ Clr
rb
2V
+ Clda da + Cldr dr + Cldaadaa
Cma = Cm0 + Cmaa + Cma2 a2 + Cmq
qc
V
+ Cmde de + Cmb2 b2 + Cmr
rb
2V
+ Cmd f
df
Cna = Cn0 + Cnb b + Cnp
p b
2V
+ Cnr
rb
2V
+ Cnda da + Cndr dr + Cnq
qc
V
+ Cnb3 b3
(3.3)
The stability and control coefficients from these polynomial equations are constant
for the entire flight-envelope of the Beaver; table B.3 in appendix B lists their values.
Notice the cross-coupling between lateral motions and longitudinal forces and moments.
For example, the gyroscopical effect of the propeller can be recognized in
the factors Cmr
rb
2V and Cnq
qc
V . According to table B.3, Cmr is negative, which means
that positive yawing (i.e. to the right) also causes a pitch-down moment due to the
gyroscopical precession of the propeller; Cnq is positive, which means that pitchingup
also causes the aircraft to yaw to the right. This is consistent with a clockwiserotating
propeller, which indeed corresponds to the lay-out of the Beaver engine.
Also notice the contribution of ˙b to the aerodynamic side-force Ya, which explains
why the time-derivative ˙x appears on the right-hand side of the general polynomial
equation (3.1). Due to this phenomenon the state equation (2.66) becomes implicit.
3.3. External forces and moments 31
In general such relationships are assumed to be linear, similar to the Beaver model
shown here, which makes it easy to re-write the state equations as a set of explicit
　
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