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(L2 + v2) = 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!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhv4nOqFnAaqB-y0d-g
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-hyperbolic-functions-asymmetric-catenaries-determining-parameters
Hyperbolic Functions: Asymmetric Catenaries: https://youtu.be/vDsLu3DCgyw
Hyperbolic Functions: Catenary: Formula and Proof: https://youtu.be/EYb1p9r1fnM
Hyperbolic Functions: Catenary: Example 4: Arc Length: https://youtu.be/mnBLG_D1nHg
Hyperbolic Trigonometric Identity: cosh(2x) Corollary 1 & 2 Formulas: https://youtu.be/A_juL30FU5k
Hyperbolic Trigonometric Identity: cosh(x-y): https://youtu.be/GMGaQTym4d4
Hyperbolic Functions - tanh(x), sinh(x), cosh(x) - Introduction: http://youtu.be/EmJKuQBEdlc
Hyperbolic Trigonometry Identity Proof: cosh2(x) - sinh2(x) = 1: http://youtu.be/-UXUqIWRNEA .
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