[TheForge] helical railing calculations

[Bruce Freeman]

Mike,

Imagine taking one full "rotation" of a helix of round stock (eg. a
coil spring) and cutting it in quarters.  My (unproven) assertion is
that each quarter-arc segment will ALMOST lie flat.  

If instead of cutting it into quarters, we cut it into a large number
of segements, each segment would lie flat to ANY degree of accuracy we
chose.  That statement is provable.  

This is really no more than saying that if you divide a curve up into a
large enough number of straight line segments, you will not be able to
distinguish the assemblage of straight lines from the original curve.

So my real assertion is that four segments per revolution around the
helix is sufficient.  SOME number (six, eight, twelve, ...) WILL be
sufficient, but for practical reasons the smaller the number the better.
 (Two or three would be better than four, but I'm not convinced these
would do.)

What I've done - in my preliminary sketches - is to determine the
radius of the arc necessary for each of these flat segments.  (I may
also have to determine the degrees of arc necessary.  I don't THINK so,
but that remains to be seen.)

The above assumes the railing is round in cross-section.  If it were
not round, then a twist would have to be imparted to give keep the "top"
of the railing on top.  I have attempted to calculate the degree of
twist.  (I will have to test my assumptions.)

What I'm shooting for is the following scenario:  Take a flat bar of
the calculated length.  Twist it a the calculated number of degrees. 
Then bend it to the calculated radius.  Do this 4 times the number of
revolutions of the staircase.  Then assemble the quarter-segments
together.  Voila', a completed helical railing to the size needed.

This is untested.

Bruce
NJ

>>> Mike Spencer <mspencer at tallships.ca> 9/22/2004 12:40:40 PM >>>

> played with the geometry of helices, as in helical railings for
> circular staircases....Is this subject of any interest or use to
> anyone on this list?

Yes. Haven't thought about this for some time but I may soon break
down and make a helical staircase for the shop.  Peggy's getting tired
of using a ladder to get to her loom.

> ...feasibility of constructing these railings from four FLAT curves,
> each representing 1/4 rotation.  I think the answer is yes, but I
> haven't tested it yet.  The question I addressed was, what would be
> the radius of such a curved railing.  That, I got an answer for.

I don't quite understand what you mean.  And being a night owl, right
this minute I have to go off an try to see some *normal* people while
*they* think it's still daytime. :-)  But I'll get back to it tonight.

In the meantime, could you give a little more explanation of
"constructing these railings from four FLAT curves"?

I did once try to do some calculations for an elliptical helical stair
and, after some pages of scribbling and grovelling through books,
smacked into something called "elliptical integrals" that I couldn't
hack.  Maybe now that I have Maple V [1] it would be less
impenetrable?

- Mike

[1] [OT]  Anybody have a Maple V Reference Manual they'd part with?

-- 
Michael Spencer                  Nova Scotia, Canada       .~. 
                                                           /V\ 
mspencer at tallships.ca                                     /( )\
http://home.tallships.ca/mspencer/                        ^^-^^

-- 


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