Wheels Within Wheels

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.

Force Coupling FormulaAlthough 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.

Greater Cubodal Wheel Force Couplings

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.

Quarter arc torque components

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.

arc force component dispersals

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.

Rotational Inertia and Force Sharing Example

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.

Mechanical and Hydraulic Cubodal Wheels within Wheels

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.

 

 

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Torque Considerations

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.

hexagonal force tracing

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.

Conventional Torque DeterminationAs 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.

Opposing Torque ForceWhen 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.

Instantaneous Rim Pivot Points

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?

Force Sharing Static EquilbriumIntuitively, 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.

Cubodal Wheel Force Couplings

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.

Hexagonal Wheel

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.

 

 

 

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Rounding Serendipity of The Monolithic Cuboda

With the goal of determining the area of spheres left uncovered by cylinders fitted to them diametrically, one of the shell example’s convergence types received detailed graphical treatment in the last post.

transport shell and cuboda convergence

As noted then, the convergence type in question is essentially the same as the basic unaltered cuboda where 2 opposing squares converge on each of the form’s 12 vertices, and where 2 opposing equilateral triangles fill the spaces between the squares.

In the shell example’s convergence, the remaining sphere area – turned out to be PI r-squared / 3 – one twelveth of the spheres complete area – 4 PI r-squared.

Cubodal Sphere Remainders

Similarly, if cylinders are fitted to spheres on either end of lines joining the cuboda’s 12 vertices where spheres were centered, the remaining area of each vertex centered sphere – multiplied by the 12 vertices – equals the area of one fully uncovered sphere!

That this should be so may be obvious to an astute mathematician, but to a simple minded design scientist such as myself who is easily entertained, I don’t see how this should necessarily be the case.

Monolithic CubodaDoes the equality hold for other forms? Yes, at least for the easily pictured cube with spheres centered on each of its 8 vertices, spheres that are then halved and quartered and halved again by cylinders converging identically from 3 directions. Other than the cube I’m guessing that the equality holds for all convex polyhedra with identical vertices, but beyond that, I’m not so sure. Whatever the range of validity, the cuboda holds a very unique claim pertaining to the attribute: It is the only form innately possessing one such full sphere – in its very center.

However much the range of validity, it would seem reasonable to speculate that this geometric relationship has some potential application in physical law and phenomena, much as parabolas delineate focal points and projectile paths; or ellipses, the orbital characteristics of celestial bodies.

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Template Round-offs

This post is a continuation of the last one, which catalogued and applied mathematical expressions of the various geometric forms intrinsic to the transport template. This continuation will specifically address areas and volumes resulting from the rounding techniques described a few posts back.

In doing so, the simplified transporter shell example used in the previous post will again be used. It was determined to have a surface area of approximately 136 sq ft and a volume of about 100 cu ft. To round the hard angles of the shell’s flat converging planes, spheres of 10″ radius are centered on the shell’s vertices where 3 or 4 planes converge. Cylinders are then placed to join spheres at either end of edges common to 2 adjacent planes.

Rounded Transporter Shell

The cross section of such a cylinder viewed directly (above right) shows how the angle of convergence is related to the dihedral angles of the intrinsic forms involved. Six types of convergence manifest in the shell example and are detailed below. Regardless of type, the perpendicular plane translation corresponds to the cylinder radius which multiplied by the translated plane’s plane area (identical to the base plane) yields the volume increase:

Planar translation calculations

The next task is to derive expressions for each of the 6 convergence types, that is, the angles of the cylindrical cross section arcs. With these, the areas and volumes of cylindrical portions between the planar translations may be determined.

Cylindrical Wedges from Converging Template Planes

The cross-sections of cylindrical wedges reflecting convergence types 1, 2, and 3 are depicted (in direct views) because they appear multiple times in the shell example and will likely be the most common in any transporter shell design.

Shell convergence calculations

With derived expressions applied to the shell example in the calculations totaled above, one last component is yet to be accounted for – the portion of the spheres not covered by the cylinders adjoined diametrically to them.

Spherical Remainders

 

The vertex convergence of cylinders depicted above is actually the same as rounding the cuboda with its alternating planes. After the convergence is viewed directly to get a sense of the angular separation of its 2 planes, the convergence is turned to view the (60 degree angles made by the 2 lines comprising one plane. The wedge remaining from the 2 cylindrical enclosures is then turned again to view the remaining area shrunk further by the bisection of the other 2 cylindrical planes. This picture is next set below in an x, y, z coordinate system where equations of the cylindrical end planes are derived.

cylindrical end plane equation

So aligned with the equations transformed to spherical coordinates, key angles (theta) are yielded for use with spherical trigonometry’s the sets of equations, i.e., the laws of sines and cosines.

vertex spherical triangle

squared x the sum of the triangle’s angles minus pi) was obtained from Wolfram’s webpage on spherical triangles. In the shell example with r = 10″, this particular vertex convergence occurs twice (counting the other side) for an areal contribution of  1.5 sq ft and a volume increase of 0.4 cu ft. Of the vertex convergence types found on the transporter shell example, this is type 1.

Tranporter Shell Convergence Types

Convergence types 2 and 3 occur multiple times. In the diagrams of them below, the “sphere patches” are disparately shaped but turn out to occupy to the same area (and thus the same volume).

rounded vertex areas and volumes

 

The sharper shell angles of types vertex convergence types 4 and 5 mean more exposed sphere area and thus volume, although there are only 2 of each type.

Concave Convergence Sphere Remainders

That concludes the assessment of shell areas and volumes, which are listed below for the shell example.

transporter shell total areas and volumes

 

 

 

 

Note that the sphere remainders – by far the most difficult to determine – contribute little to this example. But if spheres centered on vertices are large relative to the shell’s dimensions, their remainders can become quite significant.

Large sphere shell rounding

The concavity of the shell’s backend has not been factored into the compilation of areas and volumes, which invariably will have to be subtracted from the total according to how the concavity is finished off, so for readers up to that challenge, have fun!

 

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