Static deformation of heavy spring due to gravity and centrifugal force

TL;DR

This paper analyzes the static deformation of a heavy spring under gravity and centrifugal force using discrete and continuum models, applying variational calculus to derive deformation equations.

Contribution

It introduces a combined discrete and continuum modeling approach for heavy spring deformation under gravity and rotation, utilizing variational calculus for the continuum case.

Findings

Derived deformation equations for heavy spring under gravity and centrifugal force.

Compared discrete point-mass models with continuum limits.

Provided analytical solutions for static deformation in both scenarios.

Abstract

The static equilibrium deformation of a heavy spring due to its own weight is calculated for two cases. First for a spring hanging in a constant gravitational field, then for a spring which is at rest in a rotating system where it is stretched by the centrifugal force. Two different models are considered. First a discrete model assuming a finite number of point masses connected by springs of negligible weight. Then the continuum limit of this model. In the second case the differential equation for the deformation is obtained by demanding that the potential energy is minimized. In this way a simple application of the variational calculus is obtained.