The working part of the last post culminated with a determination of the proportion factor of the force shared between the (2) pivot points of a rolling wheel: the axis of rotation and continually shifting instantaneous point of contact between wheel and surface rolled upon.
Although this factor’s expression indicates the sharing is independent of the size of the separations between force coupling componets, the torques generated by them are vary much dependent on the separation. In theory, force couplings or parallel spoke pairs transmitting them may have separations ranging from zero (radial spokes) to a fully inscribed rim hexagon.
Within those extremes, an interesting arrangement is posed by the greater cubodal wheel’s hexagonally arrayed circles. Those circles’ diameters are each one 3rd of the circle overarching them. Each of the 3 sets of 3-circle alignments pose the settings for force couplings laid tangentially to each alignment.
Examining the force’s application to each pivot in isolation, the torque on the axle is FR/3, with R being the larger wheel’s radius. Multiplied by 6 spokes (assuming an equal force applied to each), the total torque equals 2FR. By the law of force couplings, the same torque can be ascribed to the ever moving wheel/surface contact points. However, with 2 pivot points regarded together as inevitably they must by reason of their motion being linked directly, the forces are shared between the pivots according to the factors k = b + 1 / 2b + 1 and 1 – k = 1 – [b + 1 / 2b + 1]).
As cut and dried as this is, breaking down the forces with vectors along the force-carrying arcs poses some interesting results. To undrstand them, an isolated quarter circle arc is simplified with a single diagonal, and subsequently divided angularly into 30° and 10° segments, and so on. The cosine component multipliers of increasingly smaller angles resulting from more divisions yield a tangential force vector at the rim having no apparent loss in bending it around 90° such that the torque on the axis equals FR.
This would be nice if that was all there was to it. But the other force vector components -acting orthogonally from each change of direction at the discreetly divided arc, and and radially from the continuum of the fully rounded arc – add to pose a force countering the tangentals to the tune of negative 2/3 F. This means a torque is imparted in the opposite direction to yield a net torque of FR – 2FR/3 = FR/3.
It seems logical that the sum of negative component forces would be directed at the 45° average, but notice how each radial line of force (one of an infinity of infinessimals) passes the point of the ground pivot to intersect points along a at a continuum of the rim at all times. Considering any one of these individally, it is joined by 5 parallel other lines of force from the other 5 arcs to pose an infinity of force trifectas.
What does all this mean practically? Aside from the well known strength of arches, the arced wheel so configured poses an optimal dispersal of (torsional) forces, forces that can then be strategically funneled into the hexagonal (and its alternate) lattice for a smooth rolling wheel. To illustrate these ideas, an example with an actual rotational inertia calculalted using the relevant components is instructive.
The trickiest part of the problem is to get the 4 individual rotational inertias (as derived in Intrinsic Wheel Quantification) onto a common basis. To do this uniform linear density () is assumed so that mass (M) is proportional to length with the latter stated in terms of radius (R). Once done, there is still a fair amount of work getting a total rotational inertia factor (B), which, as was stated previously and is evident above, can be pretty complex. What the result ultimate (k) says is that 78% of force applied will be absorbed in the torque at the wheel’s instantaneous point of contact with the surface, while the remaining 22% is applied to the torque working on the wheel’s axis.
The arced rim also suggests a further division of wheels within wheels. As the structure of hexagonal lines suggests an innate accommodation pathway to mechanical advantage with a kind of wholistic semi-passive leveraged gearing, so the arced wheel suggests, by its fluid-like continuum, a hydraulic pathway to same.
I was disappointed that what initially looked like a way to finesse force through the arc to a greater moment arm (for greater torque) also came with intrinsic counterforces. With the discovery of the negative force, it was also a letdown that this component could not simply work on the rim instead – and allow the positive component felt by the axis to remain undiminished. Maybe someone can find a way to do this someday. Until then, I’ll have to be satisfied that cubodal geometry responds to the wheel dilemma in a sound manner.