In this paper, we prove the existence and uniqueness of the weak solution of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. Also, we construct a finite element method, based on piecewise cubic Hermitian shape functions. Next, we derive error estimates for the semi-discrete Galerkin approximations. The results are derived from \cite{BS}. Finally, we implement the results of numerical schemes developed.

In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed at the two ends, the resulting model being called hybrid system. The beam is hybrid in the sense that it holds both rigid and elastic motions. Our main result is to show the existence and uniqueness of the weak solution of close-loop system. The closed-loop system stability is shown through Lyapunov-based analysis.