  The numerical analysis for the small amplitude motion of an elastic beam with
internal damping is investigated in domain with moving ends. An efficient
numerical method is constructed to solve this moving boundary problem. The
stability and convergence of the method is studied, and the errors of both the
semi-discrete and fully-discrete schemes are derived, using Hermite's
polynomials as a base function, proving that the method has order of quadratic
convergence in space and time.
  Numerical simulations using the finite element method associated with the
finite difference method (Newmark's method) are employed, for one-dimensional
and two-dimensional cases. To validate the theoretical results, tables are
shown comparing approximate and exact solutions. In addition, numerically the
uniform decay rate for energy and the order of convergence of the approximate
solution are also shown.
