I think that Mathematics gets boring when it tries to prove itself.
Tumble Axis Concept – enhancing the spatial stability of rotating objects
This concept of the tumbling axis, viewing the rotational axis of a body, or better, accepting it as such, stabs deep into the heart of concept ground resonance.
The concept of ground resonance has delivered some tools to mitigate what it effects, however it does not address the underlying issue – that which leads to the phenomena called ground resonance for loss of a better explanation:
The rotational axis itself being a „rotating body“.
Please stop a moment to consider the following: The precessing rotational axis of a rotating body can itself be viewed as a rotating body that has a fix point about which it will (want to) rotate. And as such our visible spinning body is only a very specific case of a body rotating with a body rotating within a body.
For visual clarification I have added some diagrams at the end oft he letter that illustrate the behaviour of a precessing axis in different scenarios.
The ground is not causing nor is it the originator of a resonance travelling up and down the rotational axis of the rotor. The ground is an arrestor of the precession of the rotational axis. Immediately on contact a lever, which is the rotational axis, with a fixpoint and a force moment comes into existence.
The destruction of the aircraft is then the result of leverage-forces that come into play when the precession is halted. The precessing axis itself has a fix point, the ground contact exerts a halting force to its movement and in that instant that you have a lever, that, dependent on the position of the fix-point and ground contact along the rotational axis, will easily lift a heavy aircraft. As the process repeats for each rotational / time increment this can easily lift or flip the aircraft and thus lead to total destruction of the same.
The question is: How can this view of things, poured in to a concept aptly named Tumbling Axis be realised technically for gain, i.e. to achieve, for helicopters
and in extension then for all rotating bodies and axis driven machinery.
Obbservational research of mine over some 20 years that has led to the establishment of this concept:
The word tumbling describes dynamically the change in precession of the rotational axis of a rotating body. It is for this that I coin the term Tumbling Axis. As opposed to a stable precession, which, by definition, is only an order of magnitude in time.
How to realise 1 – 6 above technically:
The principle is to locate a Tumbling Axis Compensator (TAC) on the rotational axis of a body at a distance to the centre of gravity / fixpoint of rotation of that rotating body.
The concept of the Tumbling Axis considers the position that the rotational axis of a rotating body will take up in three dimensional space within the next increment of future time - after all impulses acting on the rotating body during the current increment of time have acted on the rotating body.
The resultant (force) will induce a displacement to the rotational axis and orient it differently in 3D space.
But the individual forces do not have to be known. Current technology allows for the measurement of the resultant. As such this delivers a value that can be used to calculate a moment of force that would deflect the rotational axis back into its stable (desired) position in 3D space during the next or next few rotational / time increments as defined for workability.
Current technology allows only for static balancing. Dynamic balancing, which can be extracted from this concept will be the addition to the future technology we create.
Helicopters with their slow rotational speed lend themself well to this evaluation and realization / development of solutions, which will then impact all rotational problems concerning stability and balancing, adding dynamic balancing to current technical know how.
One design could be to locate, at the extended distal end of the rotating shaft, a tumbling axis compensator (TAC) disc that is composed of two interlaced disc arrangements where each disc, alternately fixed tot he rotating shaft and a fix-cage, contains electromagnets that can apply attractive or repelling forces to the opposite electromagnet to induce a force moment for each rotational / time increment that will induce the precessing axis back into its stable / desired postion in 3D space.
Please see the diagrams for a visual reference.
As a further reference – the measurement of force moments on a rotational axis by static balancing enterprises is capable of and should be able to deliver ample data for the above realization.
This technology is readily available.
Currently it is left to the pilot to, with experience, capture and compensate for the continuously occurring resultants. While in the air he has few arrestors but a number of force moments, even a denser pocket o fair will affect the position the rotational axis will want to take. On landing and take-off, critical, lever inducing moments, he has to get the precessing axis to lie stable in 3D space to effect safe landings or take offs.
The Tumbling Axis is a concept that pre-thinks the orientation of a rotational axis, viewing that as a rotating body itself, in the next increment of rotation / time after all forces acting upon the rotating body during the current increment of rotation / time deliver a resultant force, which induces the rotational axis to re-orient itself in 3D space. It furthermore proposes, following from that a technical realization that will effect a moment of force that will induce the rotational axis to return back to ist original, stable (desired) orientation ins space.
The following diagrams have been included to give visual reference and heighten understanding.
The tip of the spinning top wanders across the ground
The top of the spinning top wanders across the ground
Both the tip and the top of the spinning top wander across the ground.
A helicopter is akin to a spinning top, the same principles are in effect. The hand drawn circles show the movement of the preceding axis, which the helicopter will follow because of the leverage forces in effect. The destruction of the helicopter occurs when an arresting moment like ground contact arrests the precession, introducing a new fix-point for the preceding axis.
Ideal overlap of axes
Misalignment leads to levers coming into effect.
It is when the helicopter touches the ground, which arrests the precession of the rotational axis of the rotor, with misaligned axes RA /\ TA, that leverage forces come into effect which can easily flip or destroy the aircraft. The magnitude of the leverage forces are then determined by the degree of separation of the axes, the location of the fix-points of the Tumbling Axis and Rotational Axis and the point of ground contact.
Example of a possible TAC – Tumbling Axis Compensator to effect dynamic balancing.
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