Vibration modes of the Euler–Bernoulli beam equation with singularities

Abstract

We consider the time dependent Euler–Bernoulli beam equation with discontinuous and singular coeffi-cients. Using an extension of the Hörmander product of distributions with non-intersecting singular supports (L. Hörmander, 1983 [25]), we obtain an explicit formulation of the differential problem which is strictly defined within the space of Schwartz distributions. We determine the general structure of its separable solu-tions and prove existence, uniqueness and regularity results under quite general conditions. This formalism is used to study the dynamics of an Euler–Bernoulli beam model with discontinuous flexural stiffness and structural cracks. We consider the cases of simply supported and clamped-clamped boundary conditions and study the relation between the characteristic frequencies of the beam and the position, magnitude and struc-ture of the singularities in the flexural stiffness. Our results are compared with some recent formulations of the same problem.