Steady-state linear harmonic vibrations of multiple-stepped Euler-Bernoulli beams under arbitrarily distributed loads carrying any number of concentrated elements

Abstract

In this paper, general beam vibration problems with several attachments under arbitrarily distributed harmonic loading are solved. A multiple-stepped beam is modelled by the Euler-Bernoulli beam theory and an extension of an efficient numerical method called Numerical Assembly Technique (NAT) is used to calculate the steady-state harmonic response of the beam to an arbitrarily distributed force or moment loading. All classical boundary conditions are considered and several types of concentrated elements (springs, dampers, lumped masses and rotatory inertias) are included. Analytical solutions for point forces and moments and polynomially distributed loads are presented. The Fourier extension method is used to approximate generally distributed loads, which is very efficient for non-periodic loadings, since the method is not suffering from the Gibbs phenomenon compared to a Fourier series expansion. The Numerical Assembly Technique is extended to include distributed external loadings and a modified formulation of the solution functions is used to enhance the stability of the method at higher frequencies. The method can take distributed loads into account without the need for a modal expansion of the load, which increases the computational efficiency. A numerical example shows the efficiency and accuracy of the proposed method in comparison to the Finite Element Method.