Up and Over: Evaluating Flight Parameters in Pull-Up and Vertical Loop Manoeuvres

Introduction

In this following blog we will analyse an aircraft performing a loop manoeuvre. Loops will be analysed using our previous definition of load factor and the basics of circular motion. If we are able to analyse the forces on the aircraft in such a manoeuvre we can also apply this to an aircraft performing a general pulling up manoeuvre.

If the pilot pulls back on the stick, provided there is sufficient thrust, the angle of attack will be increased and the lift will also increase. This increases the load factor. If the aircraft enters a constant radius loop, that is one with constant radius and constant airspeed we can consider, the forces are shown below..

If the angular displacement is denoted by γ, with γ = 0 being the bottom of the loop, increasing clockwise, then the equations of motion are, in the aircraft longitudinal direction:

T -d-W sin γ = 0

L – W cos γ = ^WV2 / _gr

Where, F_c =^WV2 /_gr 

is the force generated from the centripetal acceleration in circular motion. 

At different point in the loop, we can summarise:

For a perfect loop the normal acceleration is constant, the load factor will vary according to:

 L – W cos γ = ^WV2 / _gr 

L / W – cos γ = ^V2 / _gr

Since n = L / W , then 

n = cos γ + ^v2 / _gr

At points A to D in the loop above, we have the load factor as:

Point A:  1 + ^V2/_gr

Point B : ^V2/_gr

Point C: ^V2/_gr -1

Point D: ^V2/_gr 

Hence it is the load factor required to initiate the loop, point A, that will set the minimum turn radius, and hence the minimum turn radius is given by:

r_min = ^V2 /_g(n-1)

Therefore the minimum value of the equation above will give the minimum radius for a constant radius loop. 

The graph below demonstrates how the minimum loop radius changes with speed.

Some key points that can be made from this graph are:

• The minimum turn radius speed is the same as the manoeuvre speed, V_A  The lowest point on the graph above is approximately 450m with a speed of 155 m/s.
• At speeds below V_A the turns are limited by the onset of stall.
• At speeds above V_A  the turns are limited by structural limitations.

In our example illustrated above it is therefore apparent that the aircraft will need to vary the thrust and lift produced throughout this manoeuvre in order to produce a constant centripetal force and keep the radius of the loop constant. With this in mind we will analyse a loop with a constant load factor and determine the shape of the loop produced by aircraft making this manoeuvre. The equation for the load factor in a constant radius loop can be rearranged for the radius:

r  = ^V2 / _{g(n-cos γ)}

The radius used in this equation could be confusing. In the equation the radius refers to the radius of a theoretical circle that would be flown if, at any point in the loop, the instantaneous value of the pitching velocity were maintained. Effectively, the equation shows that the distance flown during a given section of the loop is inversely proportional to the load factor – so the aircraft flies further during the parts where cos γ  is closer to zero as the radius of the turn increases at these points. This means that the shape flown is elongated vertically to give a tighter turn at the top and bottom of the loop.

Flightpath in loop manoeuvre with constant load factor and speed

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