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