Wednesday, October 4, 2017

Tighter synthetic weaving joins and marking by bend type

The tightest synthetic weaving join occurs when a double-bump bend (follow upper black wire) accommodates a single-bump bend (follow lower black wire.) (The edges shown have not yet been locked by the overlaying of third strands.)
Making the tightest possible join at a synthetic weave's 'crossings' adds a slight complexity since complementary portions (as opposed to congruent portions) must be brought into alignment. On the other hand, these two complementary types of bends can be exploited to indicate 'open' or 'close', without any other sort of marking needed on the scaffold strand.

The basket will still contain edges of both even and odd lengths (as measured in helical half wavelengths.)

A scaffold strand for a synthetic weave can be described by a four letter code (u, n, d, p). This is an 'undip' word as described earlier in this blog, only now the 'photon' edges have shrunk to zero length. Two sequential undip letters can be 'alike' or 'unlike' in two comparisons: up/down and open/closed.

Edges bounded by letters that compare as alike/alike or unlike/unlike will be even in length.

Edges bounded by letters that compare as alike/unlike or unlike/alike will be odd in length.

More concisely, call letter pairs {u, p} and {d, n} opposites. Then an edge has even length if it is bounded by identical letters or by opposite letters. Otherwise, it has odd length.

Same join as above after overlaying third strands.

Thursday, August 31, 2017

Example of a square 2-ply Z with congruent wires

A square 2-ply Z with congruent wires

Wires arranged anti-parallel as they would be upon unwinding.

Wires arranged parallel to show congruency.
This Z is 1.75 mm diameter ABS with a wavelength of 15 mm, and an outside, peak-to-peak height of 5.2 mm in the coiled ribbon orientation. In terms of diameter (d): wavelength is 8.6 d; p-to-p is 3.0 d. Measured flat, outside p-to-p is 6 mm = 3.4 d.

Coiled-ribbon appearance of square 2-ply

Square 2-ply viewed against a lightbox.
Properly-made square 2-ply can be rotated to an angle where its silhouette resembles coiled ribbon. This is the correct rotation for bending Z's.

Schematic of the "coiled-ribbon" appearance of a square 2-ply. 

A common manufacturing fault that prevents such a resemblance is twist.

Z's from square 2-plies with congruent wires

End and side views of a square 2-ply. Lines mark balance points. Bue dot marks the midpoint of a Z that could be formed by making bends around the red pegs.
It is possible to form a square 2-ply Z from congruent wires. In the figure, the blue dot marks the Z's midpoint, and the red circles mark the bending locations for the shortest possible Z. The wires are still congruent after bending.

Longer Z's can be realized by incrementing both bend locations farther out, two balance points at a time (i.e., symmetrically adding a full wavelength to the interbend distance.)

Wednesday, August 30, 2017

Finding balance points

When a square compound 2-ply lies on a horizontal surface, a vertical view will reveal some of the balance points (half of them to be precise) as the places where silhouettes of the two wires cross. These are also points of minimum apparent width.

Cross-section of a square 2-ply pressed against a horizontal surface.

Balance points marked on a square 2-ply

Balance points and congruent wires in square compound 2-ply

For a cut length of square, compound, 2-ply to be composed of two congruent wires, the midpoint of the length must lie at what I call a balance point. Any length will do, so long as the midpoint is at a balance point.

At a balance point in a square 2-ply compound helix, the line joining the centers of the two plies is perpendicular to a radius of the major helix. For example, this 2-ply has been cut exactly at a balance point
For comparison, this 2-ply has been cut just short of a balance point.
When a length of 2-ply is centered on a balance point, diagonally opposite half segments are congruent, and therefore also, the two wires are congruent over their entire length.
As these end-views show, at a balance point the two cut faces of the wires lie parallel to a side of the 'square'. There are four balance points per wavelength.

Tuesday, August 29, 2017

Wire paths in a 3x2 square wire rope

Wire paths in a 3x2 square wire rope. (Actual cross-sections do not remain circular.)
In this animation contacts at the same level can be seen (intra-strand and inter-strand.) In a packed configuration there are additional contacts between strands crossing above/below.

A 2-ply compound helix (where twist and writhe have the same counter-rotating wavelength) always has a crossed-ellipse appearance in cross-section. These 2-plies can "stack" to accommodate any number of plies, e.g., 2x2, 3x2, 4x2, etc.