The use of a dynamical basis for computing the modes of a beam system with a discontinuous cross-section

In this work we determine the modes and the frequency equation of Euler-Bernoulli beams with discontinuous properties in the transversal section by using a dynamical basis which is generated by a fundamental solution of a fourth-order differential equation [1-4]. Free vibrations of stepped beams have been studied by several authors applying exact and numerical techniques. We can cite Gorman [5], Jang and Bert [6, 7], Nagulseswaran [8-10], De Rosa [11-13], Vu et al. [14], Turhan [15], Korenev and Reznikov [16], Krylov [17], among others. The frequency equation and mode shapes have been formulated in terms of the classical Euler basis involving the roots of the associated characteristic polynomial of a fourth-order differential equation. The use of the dynamical basis allows the identification of factors that frequently appear in the literature, as well as writing the frequency equations and modes in a compact form.