Finding the strongest stable weightless column with a follower load and relocatable concentrated masses

TL;DR

This paper investigates optimal placement of masses on a weightless elastic column under a follower load to maximize stability range, providing analytical insights and computational validation for stability boundaries.

Contribution

It introduces a detailed analytical approach for two masses and conjectures a general formula for multiple masses, supported by extensive computational results using GRANSO.

Findings

Optimal mass placement approaches stability boundaries.

Conjectured formula for maximum load interval for any number of masses.

Validation of the approach through computational experiments.

Abstract

We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. The stability constraint is nonconvex and nonsmooth, making the optimization problem quite challenging. We give a detailed analytical treatment for the case of two masses, arguing that the optimal parameter configuration approaches the flutter and divergence boundaries of the stability region simultaneously. Furthermore, we conjecture that this property holds for any number of masses, which in turn suggests a simple formula for the maximal load interval for $n$ masses. This conjecture is strongly supported by extensive computational results, obtained using the recently developed open-source software package GRANSO (GRadient-based Algorithm for Non-Smooth Optimization) to maximize the load interval subject to an appropriate formulation of the nonsmooth stability constraint. We hope that our work will provide a foundation for new approaches to classical long-standing problems of stability optimization for nonconservative elastic systems arising in civil and mechanical engineering.