  We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected
to one another and forming a particular network which is a chain beginning with
a string. We study two stabilization problems on the same network and the
spectrum of the corresponding conservative system: the characteristic equation
as well as its asymptotic behavior are given. We prove that the energy of the
solutions of the first dissipative system tends to zero when the time tends to
infinity under some irrationality assumptions of the length of the strings and
beams. On another hand we prove a polynomial decay result of the energy of the
second system, independently of the length of the strings and beams, for all
regular initial data. Our technique is based on a frequency domain method and
combines a contradiction argument with the multiplier technique to carry out a
special analysis for the resolvent.
