Convergence of Thin Vibrating Rods to a Linear Beam Equation

TL;DR

This paper demonstrates that solutions of a scaled nonlinear wave equation in elasticity converge to those of a linear beam equation, using asymptotic expansions and initial data scaling.

Contribution

It introduces a rigorous asymptotic analysis showing convergence from nonlinear elasticity to a linear beam model for thin vibrating rods.

Findings

Solutions converge to the linear beam equation as the scaling parameter tends to zero.

Constructed an approximation using asymptotic expansion based on the beam equation.

Bounded the difference between the nonlinear solution and the approximation.

Abstract

We show that solutions for a specifically scaled nonlinear wave equation of nonlinear elasticity converge to solutions of a linear Euler-Bernoulli beam system. We construct an approximation of the solution, using a suitable asymptotic expansion ansatz based upon solutions to the one-dimensional beam equation. Following this, we derive the existence of appropriately scaled initial data and can bound the difference between the analytical solution and the approximating sequence.