  We consider the time dependent Euler--Bernoulli beam equation with
discontinuous and singular coefficients. Using an extension of the H\"ormander
product of distributions with non-intersecting singular supports [L.
H\"ormander, The Analysis of Linear Partial Diffe\-rential Operators I,
Springer-Verlag, 1983], 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 solutions 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 structure of the singularities in the flexural
stiffness. Our results are compared with some recent formulations of the same
problem.
