Timoshenko beam with tuned mass dampers and its design curves

The structural analysis and design of a Timoshenko beam with tuned mass dampers (or TMDs) under a harmonic excitation are presented. The dynamic-stiffness matrix of a Timoshenko-beam element is employed to study the dynamic responses for the whole frequency range. A proposed simplified two-degree-of-freedom system and Hartogʹs method are employed to study the dynamic characteristics of TMDs. Some important formulas for the design parameters (such as the mass ratio and stiffness ratio at the tuned frequency, the optimal damping ratio of TMDs, and their upper limits) and a series of the design charts are presented for practical applications, if the beam-own damping is absent and all TMDs are identical.