The Optimal Shape of a Javelin

TL;DR

This paper determines the optimal tapering shape of a free javelin to maximize its fundamental vibration frequency, enhancing damping efficiency, by solving complex differential equations with singularities.

Contribution

It introduces a method to find the optimal tapering of a javelin that maximizes vibration frequency using similarity solutions for singular differential equations.

Findings

Optimal tapering increases fundamental frequency nearly fivefold.

Derived differential equations with singularities are solved using similarity solutions.

Optimal shape enhances damping efficiency for free javelins.

Abstract

The problem of finding the optimal tapering of a free (non-supported) javelin is described and solved. For the optimal javelin, the lowest mode of vibration has the highest possible frequency. With this tapering inner damping will lead to the cessation of the vibration at the fastest possible rate. The javelin is modeled as a beam of uniform material. The differential equations governing the vibration and the tapering of the beam are derived. These equations have a difficult singularity at the tips of the beam. A procedure using a similarity solution is used to solve this singular system, and the solution is found. The maximal frequency is found to be almost 5 times larger than the frequency of a cylindrical rod.