  We study a mathematical model of a hinged flexible beam with piezoelectric
actuators and electromagnetic shaker in this paper. The shaker is modelled as a
mass and spring system attached to the beam. To analyze free vibrations of this
mechanical system, we consider the corresponding spectral problem for a
fourth-order differential operator with interface conditions that characterize
the shaker dynamics. The characteristic equation is studied analytically, and
asymptotic estimates of eigenvalues are obtained. The eigenvalue distribution
is also illustrated by numerical simulations under a realistic choice of
mechanical parameters.
