Finite element approximation of the vibration problem for a Timoshenko curved rod

the aim of this paper is to analyze a mixed finite element method for computing the vibration modes of a timoshenko curved rod with arbitrary geometry. optimal order error estimates are proved for displacements and rotations of the vibration modes, as well as a double order of convergence for the vibration frequencies. these estimates are essentially independent of the thickness of the rod, which leads to the conclusion that the method is locking free. a numerical test is reported in order to assess the performance of the method.