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and moments by closely examining the aerodynamic and propulsion models. The
disadvantage of this approach is that it involves working within the aircraft model
itself, which would either restrict our flexibility to implement forces and moments
in whatever format we prefer (for instance, compare the polynomial structure of the
Beaver aerodynamics model with the table-lookup methods often used in industry),
or require the creation of customized trimming tools for every individual aircraft
model.
We will therefore use a more generic method that will deal with the aircraft model
only through its input and output signals, separating the software from the model
as visualized in figure 6.1. The trim algorithm must solve a set of nonlinear simultaneous
equations, derived from the state model. The very complex functional dependence
of the aerodynamic data makes it virtually impossible to solve these equations
analytically, so a numerical algorithm must be used to iteratively adjust the independent
variables until some solution criterion is met.
The numerical solution will be approximate, but can be made arbitrarily close to
the exact solution by tightening up the criterion. It should be realized that solutions
do not have to be unique. For example, for a given engine power, there may be two
different airspeed and angle of attack combinations that result in steady level flight.
Our knowledge of aircraft behaviour makes it possible to specify the steady-state
condition so that the trim algorithm will converge on an appropriate solution.
6.3.2 Specification of the flight condition
We must first decide how to specify the steady-state flight condition, how many of
the control variables may be chosen independently, and what constraints exist on
the remaining variables. We can then develop a numerical algorithm that adjusts the
remaining independent variables and evaluates the constraint equations.
If we neglect the change in air density with altitude, the state equations for the
aircraft coordinates xe, ye, and the altitude H can be disregarded in our search for a
steady-state flight condition, as they no longer couple back into the other equations
of motion. Steady-state flight conditions that are important for flight control system
design can then be defined in terms of the remaining nine state variables of the flat-
Earth equations:
˙ p, ˙ q, ˙ r, ˙V , ˙a, ˙b (or u˙, v˙, w˙ )  0 and u = constant (6.40)
Additional constraints have to be made to define the exact flight condition. Here
we will consider steady wings-level flight, steady turning flight, steady pull-up or
push-over, and steady rolls, which are defined by the following constraints:
steady wings-level flight: j, ˙j, ˙q, ˙y  0 (i.e. p, q, r  0)
steady turning flight: ˙j, ˙q  0, ˙y = turn rate
steady pull-up or push-over: j, ˙j, ˙y  0, ˙q = pull-up rate
steady roll: ˙q, ˙y  0, ˙j = roll rate
To satisfy the conditions p˙, q˙, r˙  0, the angular rates must be zero (as in level flight)
or constant (as in steady turns). The conditions ˙V, ˙a, ˙b  0 require the airspeed,
6.3. Steady-state trimmed flight 89
angle of attack, and sideslip angle to be constant. As a consequence, the aerodynamic
and thrust forces and moments must be zero or constant.1
First of all, we will assume that the aircraft configuration (such as the position of
the landing gear and the aircraft loading) and secondary flight controls (flaps, slats,
speedbrakes) are pre-specified. In addition, we must specify the ‘secondary engine
controls’ (target RPM for a constant-speed propeller, mixture control for a piston engine)
and configuration settings that affect engine power (bleed air requirements for
jet engines, thrust limit settings for specific phases of flight). For the Beaver model,
we will pre-specify the flap position df and the engine RPM n.
For steady-state flight, we expect to be able to specify the (initial) altitude and
the airspeed. The latter can be controlled by changing either the engine power or the
pitch attitude of the airplane (or both), so it is useful to pre-specify at least one of
those variables:
1. We can fix the flight-path angle, within the boundaries imposed by the engine
power, and let the numerical algorithm search for a matching value of the engine
power. This situation is sometimes referred to as ‘speed on thrust’: for
a given flight-path angle, changes in airspeed are achieved by increasing or
decreasing engine power.
2. Alternatively, we can fix the engine thrust, without putting any constraints on
the flight-path angle. This can be called ‘speed on pitch’: for a given power
setting, the airspeed can be increased by lowering the nose of the airplane, and
　
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