### AIRCRAFT PERFORMANCE

### Fixed Wing Aircraft

### MANOEUVRES

In order to manoeuvre or turn an aircraft, additional energy must be expended. For the case of a steady level turn, an equilibrium of forces can be used to analyse the situation and determine relevant turn parameters.

If the aircraft is in a balanced turn at a constant airspeed then the following forces are applied to the aircraft. Lift will act at right angles to the fuselage reference line, weight will act vertically down. So to maintain a level turn the lift will need to be increased so that is vertical component balances the weight.

thus

where** n** is called the load factor and**
1/n = cos(φ)**

The unbalanced horizontal component of the Lift (** L
cos(φ)** ) will cause an acceleration of the aircraft in a
direction at right angles to the flight path. As the flight path
will be tangent to a circle based on the turn radius ( **R**
), this acceleration will be the angular acceleration of the turn.

where (**m**) is the mass of the aircraft, ( **W = mg**
).

Hence the bank angle ( **φ**)
and the aircraft velocity ( **V** ) will determine the
rate and size of turn, since **V = ω.R**

thus since **W=mg**

and

While these predictions are relatively simple, there are some
hidden limiting conditions that must be accounted for. As a lift
increase is required for a turn, without change in speed or
altitude, the only way to do this is to increase the angle of
attack (**α**). Hence **C**_{L}
increases but $C_L={nW}/{1/2 ρV^2S} $
must be kept less than **C _{L}(max)**.

As lift increases, lift induced drag (** K.C**_{L}^{2}**
**) will increase, so the excess power balance must be
maintained with **T=D (Ps = 0)**. Thrust will also
need to be increased compared to level flight and required thrust
must not exceed the maximum available at the altitude, otherwise a
minimum** Ps = 0** will not be able to be maintained.
If the maximum thrust requirement is exceeded then **Ps**
will become negative and only descending or lower rate turns will
be possible.

The load factor information can be evaluated along with specific
excess power to determine the maximum load factor possible at a
given speed and altitude whilst still maintaining at least **Ps=0**.
This is shown in the following graph.

Separately it is possible to determine the turn rate available at these load factors. By mapping, contours of maximum turn rate for different speeds and altitudes can be obtained.

Note that the stall speed becomes significantly larger at higher load factors and this will produce significant limitation for the aircraft's operation at lower speed.

These are not simple calculations. For each altitude and
velocity point, different load factors produce different values of
**Ps** and turn rates. An interpolation technique as
shown in the MATLAB
script shown here will be required.