Closed-form modal analysis of flexural beam resonators ballasted by a rigid mass

The work deals with the study of free flexural vibrations of constant cross-section elastic beams ballasted by a rigid mass with rotary inertia at any longitudinal position. We analyze five sets of boundary conditions of the beam (fixed-free, fixed-fixed, fixed-pinned, pinned-pinned, and free-free) and hypothesize that the structure is perfectly rigid, where the rigid mass is applied. By employing the Euler-Bernoulli beam theory, a single parametric matrix is obtained, which provides the characteristic equation of motion of the structure. When applied to specific configurations, the proposed analytical model predicts the eigenfrequencies and eigenmodes of the beam as accurately as ad-hoc analytical models available in the literature. The accuracy of the results is also confirmed by comparison with detailed two- and three-dimensional finite element analyses of a test case. By means of a 3D finite element model, the applicability of the rigid mass hypothesis to continuous beams with a composite thickened portion is finally assessed.