Exotic Ratio of some Noncircular Solids subjected to Torsion Loads using Lagrange Undetermined Multipliers

Opeyemi David Alli¹ , Tajudeen Kolawole Ajiboye²

Section:Research Paper, Product Type: Journal-Paper
Vol.8 , Issue.3 , pp.11-14, Sep-2021

Online published on Sep 30, 2021

Copyright © Opeyemi David Alli, Tajudeen Kolawole Ajiboye . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
A parameter to aid machine designers was established using a mathematical analysis based on the theory of a single Lagrange multiplier for the maximum shear stresses developed during the torsion loading of non-circular elements. The maximum shear stress functions were based on formulas developed by the normal torsion theory for circular shafts, the Prandtl stress function and the Prandtl membrane analogy for four non-circular cross-sections and was used to develop a mathematical ratio known as the exotic ratio for elliptical, rectangular, thin walled tubes and hexagonal cross sections. The exotic ratio depends on two envelope points (x, y) for a minimum and a graph is plotted to show the cross sections with more exotic ratios. Also, from a radar chart, it is observed that the exotic ratio decreases in the order from a thin walled tube to a hexagon, then a rectangle and finally the elliptical solid respectively.

Key-Words / Index Term :
Lagrange multiplier; torsion; geometry; maximum shear stress; exotic ratio

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