Pivotal altitude and reversal height
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Pivotal altitude and reversal height |
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The following discussion explains the cause of a ground reference visual illusion which is considered to be a contributory factor in loss of control situations near the ground. Such illusions can cause no problem on final approach if the pilot confines external scanning to the intended flight path and to the check for conflicting aerial traffic.
Pivotal altitude is a term used by the proponents of ground reference manoeuvres such as 'eights on pylons'. It is a particular height above ground at which, from the pilot's sight line, the extended lateral axis of an aircraft doing a 360° level turn [in nil wind conditions] would appear to be fixed to one ground point, and the aircraft's wingtip thus pivoting on that point.
Imagine an inverted cone with its apex sitting on the ground reference point and the aircraft flying around the periphery of its inverted base. The vertical distance from the reference point to the centre point of the inverted base is the pivotal altitude or height [h] and the distance from the periphery to that centre point is the turn radius [r]. The bank angle [ø] is formed between the outer wall of the cone and the radius line.
(We will just mention here that trigonometrically the tangent of the bank angle equals the height divided by the radius.)
i.e Equation 1: tan ø = h / r
The pivotal height in nil wind conditions is readily calculated by squaring the TAS in knots and dividing by 11.3. So any aircraft circling at a speed of 80 knots would have a pivotal height [80 × 80 / 11.3] around 550 feet, no matter what the bank angle.
In other than still air conditions the pivotal height varies with the ground speed. If the wind was northerly and the aircraft was turning anticlockwise, viewed from above, then groundspeed would be lower on the eastern side of the turn and higher on the western side. When in the northern quadrant the aircraft would be drifting towards the centre point while in the southern quadrant it would drift away. At 70 knots ground speed the pivotal height is reduced to 450 feet, at 90 knots it's about 700 feet.
Thus an exercise requiring a continuous 360° balanced turn around a ground reference point, whilst holding pivotal height, involves constantly changing the height above ground so that the line of pivot around each point is constantly held – rather than maintaining a constant distance from the 'pylon'. The bank angle must also be constantly changed as the wind drifts the aircraft towards or away from the pivot point. It is not an easy exercise to do well requiring an ability to manoeuvre accurately whilst including the ground reference point in the normal scan pattern. Usually two ground reference points, about 5 seconds apart, are included in a figure eight pattern; otherwise known as 'eights on pylons'.
Reversal height Now imagine two cones, the upper is the inverted cone with the aircraft flying around the periphery of its inverted base and below that is a second cone with its base on the ground and its apex connecting with the apex of the upper cone.The vertical distance from the ground through the cone intersection to the centre point of the inverted base is the aircraft height.
So when an aircraft is turning at pivotal height the wingtip appears to be fixed to a single point in the landscape, but when at any height other than pivotal the wing tip will appear to move across the landscape. When an aircraft is turning at a height greater than the pivotal altitude, which is the normal situation in flight, the wingtip appears to move backwards over the landscape – path A in the diagram. However when an aircraft is turning at a height less than pivotal altitude [ thus close to the ground] the wingtip appears to move forward over the landscape – path B in the diagram. Thus when a turning and descending aircraft descends below pivotal altitude there is an apparent reversal of the wingtip movement from backward to forward, which is the reason pivotal altitude is sometimes termed reversal height. There is some thought that the phenomenon may cause problems to inexperienced or unaware pilots during the final turn on approach to landing. If the aircraft is in a banked turn below reversal height, and should the pilot look down over the wingtip, she/he may get the impression that the aircraft is not turning and may then add additional bottom rudder so that the wingtip then appears to move backwards in the turn – the normal movement. This will cause a yaw and the aircraft's nose will move down, the aircraft may then appear to be nose-low and the pilot's reaction is to increase back pressure on the control column. Low speed, excessive bottom rudder and an increasing control column back pressure are the prerequisites for the aircraft to stall and roll toward the lower wing – an incipient spin entry. All pilots should be aware of this illusion and that wind drift may exacerbate them – the final approach turn is probably the most important ground reference manoeuvre that recreational pilots regularly perform. See the notes about descending turns and the stall/spin in the Control section of the Flight Theory Guide. [ The next section in the airmanship and safety sequence is section 13 Operations at non-controlled airfields ] The remainder of this article is just an explanation of the mathematics supporting the pivotal height =V²/11.3 formula . |
When an aircraft turns, in any plane, an additional force must be continuously applied to overcome inertia, particularly its normal tendency to continue in a straight line. This is achieved by applying a force towards the centre of the curve or arc – the centripetal force – which is the product of the aircraft mass and the acceleration required. [Remember that acceleration is the rate of change of velocity, either speed or direction or both.]
The acceleration, as you know from driving a car through a curve, depends on the speed [V] at which the vehicle is moving around the arc and the radius [r] of the turn. Slow speed and a sweeping turn – very little acceleration, but high speed and holding a small radius involves high acceleration with consequent high centripetal force required and difficulty in holding the turn.
(a) Expressed mathematically the acceleration towards the centre of the turn = V²/r metres per second per second and the centripetal force cf required to produce the turn equals m × V²/r newtons where r is the turn radius in metres, m is the aircraft mass in kilograms and V is the velocity in metres per second. Note that we are using aircraft mass not weight.
(b) The force due to gravity, or weight, of an aircraft on the ground, or in flight is expressed as W = m × g The standard gravity acceleration constant g = 9.806 metres per second per second, or 32.16 feet per second per second.
The diagram below shows the relationships between centripetal force [cf], weight [W], lift [L] and bank angle [ø].
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Copyright © 2002 John Brandon [contact information]