Advanced theory of vibration of uniform beams

In classical theory the equation of a dynamic Euler – Lagrange beam is solved by using the composition of the displacements into the sum of harmonic vibrations to obtain the ordinary differential equation. The solution of this equation with prescribed set of boundary conditions is a typical Sturm – Liouville problem with the infinite, discrete eigenvalues and modes of vibration. The purpose of this paper is to reveal that an elastic beam is a limited continuum with limited domain of physically existing, continuous eigenvalues and modes of vibration. In contrast to the classical theory, the advanced theory of free vibration of beams without damping in present investigation is based on the analysis with transversal and angular stiffness, initiated by external transient excitations and inherited by beams in compliance with energy conservation law. The output of this investigation demonstrates the fundamental distinction between the dynamic characteristics of uniformed beams established by classical theory with infinite, discrete eigenvalues and derived characteristics of beams with continuous eigenvalues and modes of vibration in limited domains. The theoretical investigation shows that only few, natural, discrete eigenvalues and normal modes of vibration physically exist in limited domains.

Nomenclature: k⁴ = ω²a^−2, a² = E I g (A γ) ^−1, E − modulus of elasticity, g − gravitational acceleration, A − area of the beam’s cross - section, γ − specific gravity of the beam’s material, ω = 2Ï€f, f − frequency of vibration per second, angular frequency p = a (kl)²/l², I – length of beam. I − moment of inertia of area.