The paper first gives a rigorous proof of existence and highlights proprieties of the eigenvalues and eigenfunctions for a bounded body with peridynamical Dirichlet boundary conditions. In particular, the mechanical behavior of the body is described by mixed local and nonlocal operators where, for the latter, the regional fractional Laplacian is used. The dynamics of the1-dimensional case is thereafter analyzed. More precisely, the previous results are applied to analyze the evolutionary problem which corresponds to free oscillations of a bar taking also into account the damping effects. A peculiar numerical approach is finally proposed to solve both the eigenvalue problem and the time evolution problem. Comparisons with classical local models and super- and sub-critical behaviors are highlighted.