- #1
Engineering01
- 11
- 0
Hello all,
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:
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:
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
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