Loom Shed and Warp Tension

What is modelled

On a loom every second warp yarn is lifted and the others lowered by the heddle frames to open a shed, so the weft can pass. A yarn pulled out of line by half the shed height is longer than before, and because the yarn is elastic its tension rises while the shed is open. The weave structure decides how many crossings there are in a repeat, and with thread density and yarn diameter it sets how completely the cloth covers the area.

Equations

• Warp length change with the heddle a distance a from the fell: ε = (√(a² + (s/2)²) − a)/a and, for a shallow shed, ε ≈ s²/(8a²).
• Tension T = T0 + EA ε.
• Cam motion s(φ) = smax (1 − cos φ)/2.
• Cover per direction K = n d; combined cover 1 − (1 − K)².
• Mean float length = cells / interlacings in the repeat.

Checks run in the page

• Exact against small-angle shed strain.
• Mean shed height over a cycle by numerical integration.
• Time the shed stays 90 % open, counted and analytic.
• Float lengths of plain, twill and satin weaves by counting.
• Cover at the jamming limit.

Honest simplifications

A straight-line yarn model with a linear elastic law, a fixed back rest, a fixed cloth fell, a sinusoidal cam and no beat-up, crimp interchange, yarn friction or let-off control. Real warp tension also depends on the whole warp sheet path, so the values are illustrative.

• Raise the shed height and watch the exact curve leave the small-angle curve.
• Choose Satin, big shed: a short heddle distance makes the tension jump sharply.
• Switch the weave structure and compare the mean float and the cover.