In this video I go over the derivation of a very useful formulation that describes catenaries in general and includes parameters such as the length of the cable, the vertical and horizontal distances, and the physical properties of the material being used. Although I derive this formulation with “asymmetric” catenaries in mind, it is applicable to generic symmetric catenaries as well. The formula I derive is sqrt(L^2 + v^2) = 2a*sinh(h/(2a)) where L is the length of the cable, v is the vertical distance, and h the horizontal distance between the two catenary points. This is a very useful formula because it allows for the engineering and design of various lengths of catenaries and of various material characteristics. The formula is a “transcendental” function and thus we can’t solve it using typical mathematical functions but rather must use numerical techniques to solve it. Although the derivation in this video is very extensive, the final result is quite concise, so make sure to watch this video to understand the rigorous mathematics behind seemingly simple equations!
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View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-hyperbolic-functions-asymmetric-catenaries-determining-parameters
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Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction:
Hyperbolic Trigonometry Identity Proof: cosh^2(x) - sinh^2(x) = 1: .
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I don't always design catenaries but when I do I usually first derive a very useful formula that includes all the parameters ;)
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-hyperbolic-functions-asymmetric-catenaries-determining-parameters