Tether line calculation

[Qacer]

Hi all,

I was looking at some pics of some balloon and started wondering about the length of the tether line. From a 2D view, it seemed like a slowly rising exponential curve at certain angles. How would I come up with a mathematical equation to model the tether line? I figure if I use the equation, then I should be able to guess the real length of the line. The next step would be to add in wind effects on the line, so it would look like a dynamically changing curve.

Is my logic on the right track?

 

[CTho9305]

http://mathworld.wolfram.com/Catenary.html

 

[Qacer]

Very nice. I'm not a mathematician but it is good to know this stuff. ;-)

 

[eLiu]

You're very close--it's a combination of (real) exponentials, a hyperbolic trig function. Should be cosh() in this case.

Remember any differential equations? You need to write the force balance in differential form to get the shape of the rope under gravity only. Then you need to add a RHS (representing a driving force) which can be variant in time & space or whatever to model the wind.

The final shape will always have exponential components, though it may not look that way depending on the wind function.