Hello all,(adsbygoogle = window.adsbygoogle || []).push({});

I have yet another mathematical quandary that is robbing me sleep, so I return to gather ideas.

This problem (fuelled from self-study) involves the equation of a catenary.

The profile of a catenary can be expressed as y(x) = T/w*(cosh(w/T*x) – 1). My inputs into this equation are:

y(L/2): Sag at mid-span

w: distributed load

x: position along the cable with respect to the origin. The origin is located at the mid-span of the sagging catenary profile.

Given these inputs I am looking to solve for the minimum tension on the cable, denoted T in the above equation. So far I have been using excel to solve this by “trial-and-error” using the solver add-on. My approach, thus far, has been to rearrange the above equation to the following form: y(L/2)*w/T = cosh(w/T*x) – 1. This has allowed me to set a cell with the RHS – LHS and hence use solver to find a solution.

Some interesting outcomes I have found so far are:

I have compared the results to a parabolic cable profile and seem to be achieving correct results (for the most part)

This method does not always work

For a given set of inputs (listed above) the solution may not converge because of the starting value used for T in the iteration. I have found that tweaking my initial guess for T may sometimes lead to a solution but I was hoping to be able to solve for the correct solution without helping solver all the time.

To the point: is there a fundamentally better way of being able to solve for the minimum cable tension in this equation? I am looking to do this explicitly in excel so this means by iteration (as I have attempted above) or solving for an expression for T (this route doesn’t seem too friendly).

Any thoughts on how to best manipulate this equation?

Cheers

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# Tension force in catenary

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