[TheForge] Re: helical railing calculations

[Mike Spencer]

[Bruce's full explanation copied infra for reference.]

Okay, I think I understand that.

> 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....
>
> 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.

Yow! Does Harbor Freight sell the Infinitesimal Cutoff Saw and the
Infinitely Integrating Welder?   Do they have 0" flux-core wire,
too? :-)

So, I think what you want to do is:

    Take a certain length of flat bar (such as 1/4"x1" or 1/2"x2") and
    bend it the hard way (edge-ways only) to some as yet unknown
    radius, so that it will still lie flat on the floor but will form
    a portion of a circular curve on the floor.

    Then you want to just tilt it up to the angle of the stair rail
    and have it be in the right place.  Or close enough to the right
    place that the eye won't detect the difference when several such
    portions are joined.

Instead of doing algebra, try to visualize it.

    Make the bend: circular, best guess for the radius.

    Tilt it up into position.

    Climb up on staging high enough to see it in plan.  The "hole",
    the cylindrical space inside your rail segment will be elliptical.
    That is, the ends of the workpiece will be closer to the axis than
    the middle of the workpiece. [1] You can't get where you're going
    from where you started without further deforming the metal.  This
    is true whether the cross section is round or flat.

> 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 don't think that kind of "pre-twist" gets it.  Lets visualize again.

     Draw a line from corner to corner of a piece of 8-1/2x11
     typewriter paper.

     Roll it into a tube with the long edge forming the end of the
     tube and the drawn line on the outside where you can see it.

The line is the "spiral stair rail" -- the circular helical path of
the rail.

     Un-roll, cut along the line, roll again.

The cut edge is the rail now.  The paper only bends in one direction
-- the one that would be edge-ways for the flat bar.  No twist.
Repeat the experiment with something soft and thick, say, 1" thick
rubber foam sheet.  There may be some ripples in the cut edge (that
represents the rail surface) resulting from compressing the inner
surface while constraining it with the outer surface.  But no twist
around the long axis of the surface that reprsents rail stock.

> 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.

So I think that's wrong, based just on visualizing the
transformations.  I also think that trying to bend a flat bar with a
slight but carefully uniform twist to a uniform circular curve without
trashing the twist would be as difficult and painful (or more so) in
practice than any of the other ways of doing this.

- Mike


[1] Depending on just how you rotate the workpiece in three-space.
    But no matter how you rotate it to tilt it up into position, parts
    of it will be closer to the helix axis than other parts of it
    because you're projecting a circle (tilted with respect to the
    floor plane) onto a plane parallel to the floor plane and that
    results in an ellipse.

    I'm still thinking....

    So if you did your initial flat bend to form part an ellipse, then
    when you tilted it up into position, and if you tilted it just the
    right way with respect to the axes of the ellipse, it's projection
    on the circular plan would be a circle.

    Proposed: Lose the "twist" idea.  Try to determine if there exists
    some segment of a planar ellipse of some dimension which, when
    tilted in a suitable way, will project on the floor as a circle
    and lie uniformly in the path of a circular helix of the desired
    pitch.

    Mumble...mutter....

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

-- 


--- Begin full quote of Bruce's post ---

> Date: Wed, 22 Sep 2004 13:48:35 -0400
> From: "Bruce Freeman" <FREEMAB at pt.fdah.com>
>
> 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
>