The capacity to change the velocity vector is called manoeuvrability.
Quantifying the manoeuvrability of an airplane involves documenting the acceleration, deceleration, and turning characteristics.
In manoeuvring, the forces of lift, weight, thrust, and drag are altered to generate linear or radial accelerations. The radial acceleration causes a turn in the horizontal, in the vertical, or in an oblique plane. Forces which cause a radial acceleration include: weight, side-force, lift, and thrust. Each of these forces can curve the flight path, turning the airplane. The most common force used to turn, however, is the lift force. Lift variations at zero bank-angle can cause the flight path to curve up or down, but most turns are performed by tilting the lift vector from the vertical. Side-force can cause the flight path to curve, allowing level turns to be performed at zero bank-angle. Generally, a turn has vertical and horizontal components, although the one easiest to analyze is the level turn.
In a stabilized level turn (horizontal plane turn), it can be seen that
The lift must be significantly more than that is required for level flight.(Because the lift vector is moved out of the pure vertical)
The greater the bank angle, the greater the increase in lift required to maintain height.
An increase in lift produces an increase in induced drag. The increase in induced drag requires an increase in thrust to maintain the aircraft in a constant airspeed, constant altitude turn.
The vertical component of lift is still required to offset the aircraft weight, and the horizontal component of lift is the force which is offset by the centrifugal force in the turn. The aircraft experiences a centripetal acceleration toward the center of the turn.
From the force balance equations and trigonometry, it can be found that:
Radius of turn,
$R = {V^2 \over g\sqrt{n^2-1} }$
Turn rate,
$ω = {g\sqrt{n^2-1} \over V}$
Radius of turn as well as turn rate are seen to be functions of velocity and load factor. Minimum turn radius will be a function of the sustained g capability of the aircraft and the velocity.
For a pure vertical plane manoeuvre,
Radius of Turn, $R = {V^2 \over g (n -cos Ɵ)} $, Where Ɵ is the angle measured from the upward vertical to the lift vector (the pitch angle)
Turn Rate, $ω= {g (n -cos Ɵ) \over V} $
Irrespective of the type of turn, we see that both turn rate and turn radius are related to normal acceleration.
Learnings:
- Normal acceleration can be taken as a measure of merit for turning performance.
- Bank angle (ϕ) and pitch angle (Ɵ) will have an effect upon, and therefore need to be accounted for, in the determination of aircraft acceleration.
Concept of ‘Radial g‘
Radial g, nr, is the force which make you turn.
For a turn in the horizontal plane, Radial g is the horizontal component of lift. In this case it can be found that
$n_r = {\sqrt{n^2-1} }$
In the case of a vertical plane turn , the ‘Radial g’ is the vertical component of lift, which can be found as
$n_r= n – cos Ɵ$
similarly, for a oblique plane turn the ‘Radial g’ can be deduced as
$n_r = {\sqrt{n^2+1-2n cos ϕ} }$
The identities for radius of turn and rate of turn are hence
$R = {V^2 \over g n_r}$
$ω = {g n_r \over V}$
The larger the radial g vector, the better the turn performance.
If you pull harder in a turn, i.e, you increase load factor n, which in turn increases nr, it gives you a better turn rate and lesser radius of turn.
Note: You will seldom manoeuvre in either the pure vertical or the pure horizontal; rather, you will be trading airspeed for altitude in an infinite variety of oblique planes. The acceleration of gravity can be used to “tighten up” (decrease the instantaneous radius and increase the rate of a turn) by manoeuvring in the vertical plane.
The Vertical Loop ‘Egg‘ explained through Radial g concept
When the aircraft lift vector is above the horizon (at the bottom of the egg), nr decreases because gravity opposes the load factor of the aircraft, resulting in a larger turn radius and a lower turn rate. When the lift vector is below the horizon (at the top of the egg, when the fighter is inverted), nr increases because gravity assists the load factor and lift, resulting in a smaller turn radius and faster turn rate. When the aircraft is pure vertical (side of the egg), the load factor is parallel to the horizon and therefore equal to nr, which results in an intermediate turn performance.
In the next part we shall explore the manoeuvring limits before we dwell further into flight testing aspects
This is the first part of a four part tutorial.…
PART-2: Introduction to Limits
PART-3: Flight Testing for Manoeuvre Performance
PART-4: Flight Testing for Lift & Thrust Boundaries
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