[1] "Pole distance: 150"[1] "Curve length: 157"[1] "Sag: 20"Catenary Explorations
mathematics
Recently, I have been playing around with slide rules, specifically the Pickett N4-ES, which has hyperbolic scales on it. Those can be handy for things such as determining the following values for a catenary1 curve.
\[\begin{equation}\tag{1} y = a \ \cosh{\frac{x}{a}} \end{equation}\]
The scaling factor a is usually interpreted as the ratio between the horizontal component of the tension on the cable and the weight of the cable per unit length. In Cartesian coordinates, the a is the distance of the lowest point above the x-axis. The directrix is the x-axis of the catenary. If a is very large, the curve is very flat. Then y may be described as the vertical displacement at a given horizontal position of x.
Using a reference from Baumeister (1958), the formula above is given as,
\[\begin{equation}\tag{2} y = a \ \cosh{\frac{x}{a}} - 1 \end{equation}\]
But what if we don’t know what the a ratio is, or what the cable weighs?
However, before I get into that, let’s work the above using not the Pickett N4-ES, but the Post Versalog 1460 slide rule. This rule doesn’t have hyperbolic scales, so we will use an alternate method on the above formula. We will use,
\[\begin{equation}\tag{3} \cosh{x} = \frac{e^{x} + e^{-x}}{2} \end{equation}\]
Let’s assume we do know the a value as 11 and the x value as 25. Using normal division for \(\frac{x}{a}\), we must determine,
\[y = 11 \ \cosh{\frac{25}{11}} = 11 \ \cosh{2.273}\]
Using the LL/-LL scales, we set 2.273 on the D scale, and read 9.71 on the LL3 scale, and 0.103 on the -LL3 scale. Adding those together (9.81) and dividing by 2 gives 4.91. Multiplying by a shows y = 53.95.
Now let’s assume we don’t know the value of a. Without getting into Algebra on the slide rule, we turn to the instrument that spelled doom for the slide rule, the calculator.
As an example, let’s say we know several things: the height of two poles and the distance between them. Let the pole height, P = 34 feet, and distance D = 150 feet. Assume both poles are at the same height and we want the cable’s lowest point, a, to be 14 feet above ground. Let’s use the formula where we arbitrarily select a as 65,
\[\begin{equation}\tag{4} \int_{-x}^{x} \cosh{\frac{x}{a}} \ dx \quad \Leftrightarrow \quad \int_{-75}^{75} \cosh{\frac{x}{65}} \ dx \end{equation}\]
We integrate using the above formula which gives a curve length of 185.5 feet. Given that 150 + (34 x 2) = 218, the sag might be too much. If we solve for a in the formula where L = 1/2 of curve length (92.78),
\[\begin{equation}\tag{5} \left(\frac{P + a}{a}\right)^{2} - \left(\frac{L}{a}\right)^{2} = 1 \quad \Leftrightarrow \quad \left(\frac{34 + a}{a}\right)^{2} - \left(\frac{92.78}{a}\right)^{2} = 1 \end{equation}\]
we get a = 109.6. So, plugging that value into formula 4, we see,
\[\begin{equation} \int_{-75}^{75} \cosh{\frac{x}{109.6}} \ dx = 162 feet \end{equation}\]
The lowest point can be determined by using a table for x,y values, and examining the y value where x is 0. We want a maximum sag of about 20. So, if we try a = 145, we see,
That’s kind of a trial-and-error method to massage the values, but it will work. Notice that even a small added length increases sag significantly.
I really enjoy using R Core Team (2023), Allaire et al. (2023) and Xie, Dervieux, and Presmanes Hill (2023) for things like this. It’s great fun to see what I can produce. Have a great day and may God Bless you and yours this holiday season!
References
Allaire, JJ, Yihui Xie, Christophe Dervieux, Jonathan McPherson, Javier Luraschi, Kevin Ushey, Aron Atkins, et al. 2023. Rmarkdown: Dynamic Documents for r. https://CRAN.R-project.org/package=rmarkdown.
Baumeister, Theodore. 1958. Marks’ Mechanical Engineers’ Handbook: 6th Edition. McGraw-Hill, Inc.
R Core Team. 2023. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.
Xie, Yihui, Christophe Dervieux, and Alison Presmanes Hill. 2023. Blogdown: Create Blogs and Websites with r Markdown. https://CRAN.R-project.org/package=blogdown.
Footnotes
The word catenary is from the Latin word catena, meaning chain.↩︎