Rigorous implementation of the Galerkin method for stepped structures needs generalized functions

In this paper we study free vibrations of stepped structures, specifically for longitudinal bars and flexural rectangular plates, providing two versions the Galerkin method. Specifically, first apply straightforward version method which stipulates employment procedure to be conducted in each subdomain, or step, structure. Second, rigorous realization is presented where structural parameters, like rigidity mass, are treated as generalized functions over entire domain. This latter implementation utilizes unit step functions, well Dirac's delta function, its derivative treat changes parameters across steps. It turns out that leads additional terms do not appear (or “naïve”) Both methods compared with exact solutions considered problems. increase number expansion, rigorous, generalized-functions based tends solution. contrast naïve Galerkin's method, usually utilized literature, does tend demonstrates extreme care must taken when implementing only should employed.