On Wave-Like Differential Equations in General Hilbert Space. The Functional Analytic Investigation of Euler-Bernoulli Bending Vibrations of a Beam as an Application in Engineering Science

TL;DR

This paper explores wave-like differential equations within Hilbert spaces, focusing on the Euler-Bernoulli beam vibrations, providing a functional analytic framework for understanding these equations in engineering and physics.

Contribution

It introduces a rigorous Hilbert space approach to analyze Euler-Bernoulli beam vibrations, constructing positive selfadjoint operators for various boundary conditions, extending the mathematical understanding of wave-like PDEs.

Findings

Construction of positive selfadjoint operators for beam vibrations

Comparison with wave equations of strings using Laplacian

Framework applicable to various wave-like PDEs in engineering

Abstract

Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a generalization of the primary wave (partial) differential equation. A short overview over three wave-like problems in physics and engineering is presented. The mathematical procedure for achieving positive, selfadjoint differential operators in an $\mathrm{L}^2$-Hilbert space is described, operators which then may be taken for wave-like differential equations. Also some general results from the functional analytic literature are summarized. The main part concerns the investigation of the free Euler--Bernoulli bending vibrations of a slender, straight, elastic beam in one spatial dimension in the $\mathrm{L}^2$-Hilbert space setup. Taking suitable Sobolev spaces we perform the mathematically exact introduction and analysis of the corresponding (spatial) positive, selfadjoint differential operators of $4$-th order, which belong to the different boundary conditions arising as supports in statics. A comparison with free wave swinging of a string is added, using a Laplacian as differential operator.