Abstract <p>The initial model boundary value problem regarding vibrations of a viscoelastic beam with damping of Voigt type is considered. A classical formulation of a mixed problem for a fifth-order linear partial differential equation is adopted. The equation includes a term with a mixed derivative: first-order with respect to the time and fourth-order with respect to the spatial variable. On the basis of integral assessment, the energy identity is established, and the presence of damping vibrations is qualitatively demonstrated. By Galerkin projection, an exact solution of this problem is obtained. An algorithm is determined for calculation of the exact solution as a functional series. An explicit resonant term is obtained in general form.</p>