Analytical solutions of H₂ control and efficiency-based design of structural systems equipped with a tuned viscous mass damper

In this study, an optimal design of a tuned viscous mass damper (TVMD) incorporated into a single-degree-of-freedom structure system subjected to earthquake-induced random vibrations is studied. Closed-form design formulas of the optimal frequency ratio and damping ratio are proposed for the control of structural displacement and acceleration responses based on H₂ norm optimization. The optimal solutions exist when the mass ratio is smaller than 0.11. The calculated optimal parameters for displacement control and acceleration control are close to each other, which indicates that the displacement and acceleration responses can be controlled simultaneously with the TVMDs designed by the easy-to-use formulas proposed in this study. To expand the use of the proposed formulas for cases with large mass ratios, the alternative suggested design (ASD) is proposed to explicitly represent the recommended relationships between the design parameters. This paper also examines the control effect of a TVMD over linear viscous dampers, defining an efficiency index as a ratio of the response yielded by a TVMD system to that of a viscous damper system. The optimally designed TVMD is generally more efficient than directly adding damping with viscous dampers when the mass ratio is small, while this advantage is insignificant when the mass ratio grows. Based on this outcome, a new method termed efficiency-based design is proposed. This user-friendly method can assist practicing engineers in determining the mass ratio, the frequency ratio, and the damping ratio of TVMDs.