Before moving on to the next topic of the macrocosmic wheel and its application to transporter related constructs fixed to the earth, I feel compelled to backtrack a little and address some nagging concerns with regard to torque as it pertains to the free and dynamic cubodal wheel.
In wrestling with concepts threatening ideas conveyed here previously, I have been left with a fascination toward torque. Torque is typically named the rotational equivalent of force, yet the units of torque are that of energy. As torque ultimately transpires at a point, that energy can be viewed as analogous to the energy content of a point mass like an electron, and may even echo the singularity (s) of the big bang.
Meanwhile, torque holds plenty of interest in the realm of common experience. In a previous post pertaining to torque and the cubodal wheel, the idea of the hub and rim guided by hexagonal geometry was somewhat quantified.
Although there was nothing particularly wrong in the assessment as far as it went, there is more to the story than tracing forces trigonometrically through components, i.e., spokes, hexagon and circular rim.
As it stood, my analysis disagreed with that of the conventional approach, embodied in 2 simple related ways for torque’s calculation: force applied to a point times the distance from it to a pivot times the sine of the angle between them or force times a perpendicular from the pivot to an extended line of force.
In searching for an explanation for the deviation, I found no approaches that followed forces along the actual entities that transmitted the forces. Nevertheless I refused to believe that this 3rd way didn’t have some validity, while at the same time suspecting the deviation had something to do with a lack of accounting for the distance between radial spokes and forces applied parallel to them.
When I finally saw the light after much trial and error, I found the solution had been quietly awaiting recognition all along. In addition to the component of force directed along the side of the hexagon, there was the component of force perpendicular to that side. With this new revelation, I still had to surrender to the standard determination of torque by extending the alternate line of force past the central pivot. Because this other line of force landed on the pivot’s opposite side, its contribution to the torque is directed oppositely and thus needs to be subtracted from the determination made previously. By doing so the combined answer agreed with the conventional analysis employed from the git-go. Although there proved to be a limit to the validity of the 3rd approach, I think it posed some interesting questions.
In taking a fresh look at the picture, it would seem reasonable that the same vertical force component working against the hexagonal side force component should also work on another pivot point – an instaneous one – between the wheel itself and the surface rolled on, and in the same direction of intended rotation as the wheel’s axis.
The above right depiction is the same as the left rotated to make the state of surface contact clear while retaining the same relationship to forces. So, if there are 2 pivot points, interesting questions posed are: Which pivot point receives the force? If shared, how? Does any diminishment of the negative torque on the axis infer more positive torque freed to work there from the other force component?
Intuitively, it seems reasonable that most of the force should be expended to counter the greater rotational inertia around the wheel/surface contact point. To get a precise expression for how much, I followed an approach analogous to a static equilibrium situation (right) by assigning factors (k & 1-k) to the force component acting at each of the 2 pivots. Proceeding with the above situation, angular accelerations about the axis and pivots were equated so that their equations could be solved for k. The expressions arrived at rightly involved hub and wheel radius (r and R), as well as B (the rotational inertia factor), but the expressions seemed too complex and even then did not seem to account for all relevant forces. So I tried the force couplings approach.
No matter what part of the wheel is in contact with the surface at any given moment in relation to the angles of the couplings’ alignments, the difference between perpendiculars extended to the far and near force of the coupling is constant and therefore the net torque exerted on the continuously shifting instantaneous pivot point is always the same.
Because of this rule, I was able to align the wheel to the most easily calculable configuration. With a wheel structure using cubodal geometry, the rotational inertia factor (B) could be somewhat complex, but otherwise, after trying a few simple fractions for B, answers seemed reasonable with proportion of force (K) appropriated by the contact point growing as B declined, inducing a bigger role for the parallel axis theorem’s extra term.
Thus is the virtue of parallel spokes. Previously, the opposing hexagonal rim sides could have served as the force couplings themselves. Otherwise, with lines of force, it doesn’t matter if spokes stop at an inscribed hexagonal rim or run to the (circular) rim without it. That being so, is there any advantage to the hexagonally inscribed rim?
Spokes extending form opposing sides of the hexagonal hub is a big plus, but any even-numbered polygon could theoretically effect the same benefit. The virtue of the hexagonal rim in conjunction with the hexagonal hub is that it -
- bolsters the wheel’s structural integrity
- transmits forces more efficiently and economically
- clarifies analysis of forces with relative simplicity
- supplies a sound connecting intermediary between spoke and rim.
Whether any or all of this is worth the extra rotational inertia is debatable and dependent on the required design criteria for a specific application. The most sublimely unique attribute of the hexagonally structured wheel in general is that the instantaneous pivot between wheel and ground regularly attunes to, by receiving in perfect harmony upon the dynamic of rolling, an extension of the axle centered geometry by reason of the infinite hexagonal potentiality in which the rim of the latter becomes the spoke of the former.
Thus does torque pertain to the cubodal wheel. To deal with it, it was necessary to diverge from the loose analogy of force and torque and come to terms with their antithetical aspects of parallelism and perpendicularity, abstractions seeming to reflect the opposing existential metaphysics of karma and irony. So engaged I have found myself relating to this character from the film in 2001: A Space Odyssey.