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in terms of the Euler angles q and j needs to be known to be able to determine the
weight components along the body-axes; see section A.7.3 for the relevant definitions.
3.3.4 Forces and moments due to nonsteady atmosphere
In section 2.3 it was shown that corrections to the external force components along
the aircraft’s body-axes are needed whenever the aircraft is flying in nonsteady at3.4.
Converting the implicit state equations to explicit equations 33
mosphere. These corrections are equal to:
Fwind =
2
4
Xw
Yw
Zw
3
5 = −m ·
2
4
u˙w + qww − rvw
v˙w − pww + ruw
w˙ w + pvw − quw
3
5 (3.11)
3.4 Converting the implicit state equations to explicit equations
As explained in section 2.5, it is possible that the external forces and moments depend
upon time-derivatives of state variables. This causes the general state equations
(2.66) to become implicit, which results in the equation (2.69). In the case of
the Beaver model, the aerodynamic sideforce was shown to be dependent on the
time-derivative of the sideslip angle:
CYa = CY0 + CYb b + CYp
p b
2V
+ CYr
rb
2V
+ CYda da + CYdr dr + CYdradra + CY˙b
˙b
b
2V
(3.12)
Notice the last term in this equation. The time-derivative of the sideslip angle was
given by equation (2.48):
˙b
= 1
Vm
−Fx cos a sin b + Fy cos b − Fz sin a sin b
+ p sin a − r cos a (3.13)
which shows that ˙b depends on the sideforce Fy, which includes the aerodynamic
contribution Ya = 1
2rV2S CYa . As a consequence, the time-derivative ˙b appears on
both sides of equation (3.13).
Using these raw relations in a simulation model would result in a so-called algebraic
loop. Although such loops can be solved numerically by means of an iterative process,
as illustrated in section 6.4.1, this is not desirable in practice, because the increased
number of computations would severely slow down simulations. A better solution
is to search for an algebraic solution instead.
In this particular case, it is easy to re-write the ˙b-equation such that it becomes
explicit. First of all, the contribution of ˙b to the side-force Fy can be written as a
separate term:
˙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 (3.14)
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
+ p sin a − r cos a (3.15)
Based upon these equations the following calculation sequence can be used in the
simulation model:
1. compute the external forces and moments as usual, but neglect the contribution
of ˙b to the aerodynamic side-force for the time being,
34 Chapter 3. The aircraft model
2. substitute the thus obtained forces and moments into the general ˙b equation,
to obtain the value ˙b , and
3. compute the true value of ˙b with the expression ˙b = ˙b
1 − rSb
4m CY˙b cos b
−1
.
Since the last step ‘corrects’ the originally computed value ˙b to obtain the actual
value of ˙b, the multiplication factor from step 3 represents a correction term. The
correction is aircraft-dependent because it contains CY˙b , whereas the equation for ˙b
itself is applicable to all aircraft. Thus, the computation scheme allows us to separate
the aircraft-dependent from the aircraft-independent terms, which is desirable
in view of the intended standardization of aircraft models.
3.5 Atmosphere and airdata variables
So far, the aerodynamic and propulsive forces and moments were expressed in terms
of non-dimensional force coefficients, but to solve the equations of motion, we need
to know their actual values. For the Beaver aircraft, this requires knowledge of the
dynamic pressure. Other aircraft models often also take into account compressibility
effects, which would require additional knowledge of the Mach number, and
sometimes it may be necessary to account for scale-effects (e.g. in case of windtunnelderived
models) which would require knowledge of the Reynolds number as well.
These variables are closely related to the properties of the atmosphere, and the
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(21)
