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vice versa. Notice that because of equation (6.40), it is not possible to achieve
steady-state flight with an engine thrust that exceeds the weight of the airplane,
as even for vertical flight that would result in a linear acceleration.
The first method introduces an additional constraint on the pitch angle, while letting
the engine thrust be determined numerically. In the latter case, the pitch angle will be
adjusted numerically, but the thrust will be pre-specified. Notice that nonzero values
of the flight-path angle will result in ‘instantaneous’ steady-state flight conditions
only, because the air-density will no longer remain constant if the aircraft is climbing
or descending.
6.3.3 Remaining control and state variables
The primary flight controls de, da, and dr (elevator, ailerons, rudder) enter the model
through the aerodynamic data. In general, it is not possible to determine any analytical
constraints on these variables, so these three control inputs will be tuned by
the numerical algorithm. The thrust can either be pre-specified (‘speed on pitch’), or
adjusted by the numerical algorithm (‘speed on thrust’).
For the Beaver aircraft, the primary engine control variable is the manifold pressure
pz. The engine RPM n, which is the target speed for the regulator mechanism
of the constant-speed propeller, is considered a secondary engine control signal that
can be pre-specified.
1Because of this requirement, steady-state pull-up or push-over and steady-state roll conditions can
only exist instantaneously. However, it can still be useful to trim the aircraft dynamics in such flight
conditions and use the resulting operating point values of x and u for aircraft model linearisation,
because flight control systems must operate there too.
90 Chapter 6. Analytical tools
In steady-state translational flight conditions, the state variables j, p, q, and r are
identically zero and y can be selected freely. Since V and H are also specified by the
user, and xe and ye are irrelevant for the steady-state solution, the only remaining
state variables to be determined are a, b, and q.
The angle of attack a must be adjusted by the trim algorithm to generate the
amount of lift needed to support the weight of the aircraft (remember that we chose
to pre-specify the airspeed and altitude, and hence, the dynamic pressure). The sideslip
angle b must be adjusted by the trim algorithm to zero out the sideforce Fy.
For a given value of the flight-path angle g (‘speed on thrust’), the pitch angle q
will be constrained, as we will explain in the next section, which presents a general
rate-of-climb constraint that allows nonzero roll angles. If the thrust is specified by
the user (‘speed on pitch’), the pitch angle can be adjusted by the trim algorithm.
In steady-state turning flight conditions, the state variables j, p, q, and r will differ
from zero. The turn can be specified directly in terms of the yaw rate ˙y or indirectly
through the turn radius R. These variables are interrelated as follows:
˙y
= V
R
(6.41)
The initial heading can still be specified freely. If the attitude angles q and j are
known, it is possible to determine the angular rates p, q, and r from the kinematic
relations given in equation (2.59) of section 2.4.
In principle, it is also possible to define the turn by specifying the roll angle j
directly. However, that will in general introduce a significant sideslip angle b, resulting
in a skidding turn, in which a side-force is experienced by the flight crew and
passengers. To avoid this, an additional constraint for coordinated turns will be derived
in section 6.3.5, in order to compute the roll and sideslip angles such that the
aircraft is banked at an angle with no component of the aerodynamic and propulsive
side-force Ya +Yp.
If the roll angle j is known, the required pitch angle q can be obtained from the
rate-of-climb constraint, which will be derived in the next section. Since the turncoordination
constraint will be shown to involve both q and j, it must be solved
simultaneously with the rate-of-climb constraint.
6.3.4 The rate-of-climb constraint
A constraint for the rate-of-climb ˙H can be found by combining the z˙e part of equation
(2.61), and equations (2.31), (2.62), and (3.45):
sin g =
˙H
V
= −
z˙e
V
=
= −ue sin q + (ve sin j + we cos j) cos q =
= a sin q − b cos q (6.42)
with:
a  cos a cos b
b  sin j sin b + cos j sin a cos b (6.43)
6.3. Steady-state trimmed flight 91
Solving for q, the resulting rate-of-climb constraint is found to be [33]:
　
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