The Tallest Column -- A Dynamical System Approach using a Similarity Solution

TL;DR

This paper introduces a novel similarity solution approach to the classic tallest column problem, transforming the equations into an autonomous system to efficiently find solutions without iterative methods.

Contribution

The paper presents a new method using similarity solutions to solve the tallest column problem, effectively handling endpoint singularities and simplifying the solution process.

Findings

Successfully applied to clamped and hinged boundary conditions

Eliminates the need for iterative integration methods

Applicable to other endpoint singularity problems

Abstract

A classic problem, the design of the tallest column, is solved again using a different method. By the use of a similarity solution the equations are transformed and the difficult singularity at the endpoint is peeled away. The resulting autonomous system has a critical point and the solution must be on its stable manifold. The solution is found by starting near the critical point in the direction of the stable manifold, and solving backwards numerically. This removes the need for an iterative integration method that was previously used. The method is shown to work for clamped or hinged boundary condition and can also be used for other problems involving singularities at the endpoints.