Symmetric Cylindrical Coordinates

[Douglas Goncz A.A.S. M.E.T. 1990]

[[Mod. note -- I'm sorry for the delay in processing this article, which
arrived in my s.p.r inbox on 2023-02-08. -- jt]]

Cylindrical coordinates are defined usually by a first variable of Radius,
a second variable of Height, and a third variable of Angular displacement
from a reference Ray. I will maintain this order here and will refer to R,
H, and A as variables for every point in the system I am trying to
conceptualize this morning.

I posted recently on chat GPT with a full transcript, thanks to moderation
inserting with double carets the omitted notes as to the source of each
block of text. There have been 28 reads of that submission so I will
continue on about homopolar bicycle hubs.

Let's leave off the chain drive for now and consider a rear homopolar hub
for a human electric hybrid bicycle. A homopolar hub accepts and receives
direct current and in an electric hybrid bicycle, that flow of current
would come from and go to an ultra capacitor, most likely. A suitable piece
of hardware, and ultra capacitor rated 16 volts DC, 500 farads, with
balancing resistors, and an internal effective series resistance of
approximately 1,000 of an OHM, with current capacity of order 1,000
amperes, is available from Amazon and other sellers, and represents about
90% the state of the art. This ultra capacitor stores 64 kilojoules, which
is theoretically enough to accelerate a bicycle and Rider weighing about
250 lb to a terminal speed of 190 mph, or to climb a 166-ft hill if I did
my calculations and conversions from English to metric units correctly.
Kinetic energy is 1/2 MV squared. Potential energy is MGH. Ultra capacitor
energy is 1/2 c e squared.

So let's think of a rear axle homopolar human electric hybrid rear hub in
its most symmetrical configuration. The axis which is variable H, is
roughly plus and minus 67 mm for an overlock nut spacing of about 135 mm,
the most common bicycle rear hub overlock nut spacing. Let's say we're
using a 700c rear wheel and say that the radius, that is variable R, goes
from zero up to about 350 mm which if you remember is about the radius of a
standard 700c type bicycle wheel. And of course variable A goes from
including zero to not including 360=C2=B0 or if you prefer 2pi radians.

Now on this rear axle let us establish the symmetries. Almost verything we
consider from this point on will have axial symmetry on variable H, which
is to say if there is a magnet at some location R,H,A, an infinitesimally
small bit of a magnet to be clear, there will also be an identical
infinitesimal bit of matter in this example of magnet at variable R, -H, A.

There is no radial symmetry. Everything starts at zero and goes outward
from the origin for the radius variable R.

There is an angular symmetry for variable A. If there is a bit of matter be
it a bearing, a coil, a magnet, a backing plate, a spoke interface,
accepting the spokes which violate angular symmetry, or a bit of rim or a
bit of inner tube or tire at coordinates R,H,A, then with angular symmetry
there is an identical bit of matter of the same type for all coordinates
and I'll use the wild card * here, all coordinates of value R,H,*
remembering that star can have any value from 0=C2=B0, let's say at the
reference Ray including the tire valve up to and not including 360=C2=B0 or=
if
you prefer up to and not including 2 pi radians.

This concludes the establishment of the symmetries. As I've mentioned the
spokes are asymmetric and this has to do with transmitting torque to the
rim. I would like to introduce first the other asymmetric part of the
system.

Let us consider a symmetric torus of rectangular cross section centered at
the origin extending one quarter of an inch in each direction H, that is
having a thickness of half an inch. And let us establish that the outside
diameter of this rectangular cross section torus is five inches or 127 mm
precisely. And further let us establish a 2 inch inner diameter for this
coil.

I would make this coil form out of beryllium oxide because the wire in this
homopolar Hub is going to, by my calculations overheat drastically when an
attempt is made to capture bicycle potential energy and put it into the
ultra capacitor so I have asked the manufacturer to quote me a price for
this exotic laser machine ceramic and I will let you know if I can get one.

Now let's cut a slot in it at the origin Ray about a 32nd of an inch thick
allowing for winding, and wind it with some 288 turns of 20 gauge magnet
wire in copper and Tuck the ends into a 12 mm axle on which it is mounted,
after machining about an 1/8 inch slot in the axle about a half inch in
length, and let us use an axle made of good quality Steel having an inner
diameter of about an eighth of an inch which is plenty of room for those
two magnet wire leads. I have sourced and purchased this material from
online metals.com and I'm currently working on sawing it to exact lengths
in machining the slot.

The winding is fixed to the axle and drives the Hub by inducing through
Faraday's law, motion of two large magnets connected to the wheel through
spokes. Let me describe the large magnets.

Neodymium magnets of 5-in outer diameter, 2 inch inner diameter, and 1/2-in
thickness are available at around $150 and are used in metalworking for
grabbing and or holding material and moving it or keeping it from moving
during manufacturing and the machining operations. My source for these is
mscdirect.com otherwise known as Manhattan supply company, and I think they
had something to do with the Manhattan project as an aside, and I have an
account there so I can get those magnets. I would need two of them.

Between the coil and the magnets there has to be a gap of about a 32nd of
an inch. But remember the coil breaks the symmetry the way the spokes break
the symmetry. It's helicity is constant all the way around almost a full
360=C2=B0. It's winding is what is called poloidal. It's internal field is =
what
is called toroidal and almost has no external field.

So let's position the magnets on suitable ball bearings which I have
sourced from vxb.com at about a 30 second of an inch away from the coil and
compassing a fold width of slightly more than 2 in, which is a common
spacing for the flanges of an electric bicycle hub.

Further let us arrange the poles of these magnets north to North, repelling
each other strongly, press them into position with screws or a machine
press, and fix them into position with flanges and screws or some means.

The axial load on the bearings will be high and there is a type of bearing
called angular contact that deals with that well. I've also sourced these
from vxb.com.

So all of this is preface to the interesting question to follow.

Is it a motor?

I opened the discussion to the learned readers of the sigh.physics.research
newsgroup, reserving my comments for after each of you who are interested
has written as much as you care to write about what is to me a very
interesting configuration of matter.


Cheers from Douglas
Replikon Research