Vibrations of Rods and Beams

Synonyms

Vibrations of small amplitude of thin bodies regarded as one-dimensional continua

Definitions

This paper derives the equations of small vibrations of rods and beams, regarded as the one-dimensional continua. It provides also the variational-asymptotic analysis of the three-dimensional equations of elasticity for thin rods and beams that yields the one-dimensional approximate theory.

Introduction

The problem of vibrations of rods and beams is one of the oldest problems in continuum mechanics. Euler (1744) and Bernoulli (1751) have been the first to derive the one-dimensional differential equations of the flexural vibrations of beams by Hamilton’s variational principle of least action. They determined the eigenfunctions and the eigenfrequencies of a beam in the six cases of boundary conditions corresponding to the free, clamped, or fixed edges. The classical Euler-Bernoulli beam theory preceded the exact three-dimensional linear elasticity discovered by Navier, Cauchy, and...