A prototype application in development (beta version)
Here comes the plot!
The shape of a hanging slackline is a testimony that, indeed, the book of Nature is written in the language of mathematics. The catenary curve was first described mathematically for a uniform and flexible but non extendible medium by J. Bernoulli, G. Leibniz and C. Huygens in 1691. Though, if the hanging line is elastic, it stretches under its own weight in a non-uniform manner complicating the problem. The theory of elastic catenaries was developed first by Routh (1891) and Feld (1930) and more recently by Irvine and Sinclair (1976) who introduced the Lagrangian co-ordinate approach. Going through this theory is beyond the scope of the present page, though, it has been key in developing the code behind this web application. SciPy routines made this work easier.
On the right side the boxes accept values for a number of variables. More options will be added as the complexity of the application builds up. For the moment there are from top to bottom: D: the horizontal distance from anchor to anchor, accepted range: 10–1000 m. ρ: the linear density/weight of the webbing, accepted range: 1–300 g/m (accounting for additional weight due to a backup line). E: the webbing stretch at 10kN (%), accepted range: 1.5–30%. To: the pretension of the free-hanging line at the anchors, accepted range: the minimum possible depends on D, ρ and E. Lw: the length of unstrained webbing between the anchors, accepted range: 0.5D–1.15D. When To is provided, Lw is computed and vice versa. m: the mass of the slackliner, accepted range: 0–200 kg (accounting for rescue operations). δ: the fractional distance (dimensionless) from left anchor: accepted range: 0–1.
At the bottom right of the plot a number of the computed variables are shown: dLo: the absolute elongation of the free-hanging webbing. So: the midpoint sag of the free-hanging webbing. dL: the total elongation of the webbing supporting the slackliner. S: the line sag at the point of the slackliner. T: the line tension at the left anchor (always larger than To). Tx: the horizontal component of T. Tz: the vertical component of T. \(\phi\): the line angle immediately to the left of the slackliner (as seen in the plot) indicated by the line segment in pale-green. The angle shown for \(\delta\)=0 is not the angle of the free-hanging line, instead it should be interpreted as the angle that the slackliner will experience when reaching or leaving the anchor.
(I) Using this application one can predict the line tension and the midpoint sag with and without the slackliner. One simply needs to determine Lw by knowing the total webbing length and subtracting the length of the tails. This is a handy and very accurate method.
(II) Suppose that one wants to achieve a specific pretension or a given midpoint sag with the slackliner on the line. This application helps: simply determine the required Lw and pull the right amount of webbing until the segment between the anchors has a length equal to Lw (at zero tension). For this kind of operations it would be useful to have the webbing marked at regular distances so as to easily determine how much webbing lies between the weblocks.
(III) Do you want to consider also the weight of the backup line? Simply divide the total weight of the backup line between the anchors with Lw to find the effective linear density of the backup loops and add this to the linear density of the mainline. Use their sum as ρ and voila! As expected, for the same Lw, all tensions and sags increase, do not they?
(IV) The angle \(\phi\) for \(\delta\)=0 is indicative of the walkability of the slackline. If this angle is too steep, it will be hard, or even impossible, to walk the line anchor-to-anchor. Also, in the event of a backup fall, ascending to the anchor may be particularly hard without an ascender or/and without assistance. One can check this by endering the length of the backup line as Lw, accordingly adjusting ρ and E.
(V) What is more? After following III, one can keep the computed sag (S) and enter that in the backup fall simulator.
Built with Python via the Flask framework.
© Panos Athanasiadis, Bologna, Italy, 2018.