TMD-Inerter for Tall Building Damping: Approximate Closed-Form Solution, Performance and Conclusions

This paper analyzes the performance of the TMD-Inerter for tall building damping. The analysis is performed by simulation to ensure ideal working behaviour of the inerter, i.e., the inerter produces a force in proportion to the relative acceleration of its terminals without any friction of real inerter devices such as fly wheels. For the study, the most realistic TMD-Inerter configuration is considered where the inerter is grounded to the TMD mass and the structural mass next to the TMD mass, i.e., the TMD-Inerter is installed in the top floor room of the structure. Approximate closed-form solutions for the tuning of the TMD-Inerter parameters are derived based on the characteristics of the inerter force. The resulting frequency response functions for different inertance ratios are compared to those of the classical TMD with same mass ratio. The results clearly demonstrate that the TMD-Inerter worsens the tall building damping compared to the classical TMD for the realistic situation that the inerter is grounded to the structural mass next to the TMD. There are two physical reasons why the inerter worsens the efficiency of the TMD. First, the inerter force is per definition in proportion to the relative acceleration of its two terminals, i.e., it is not in proportion to the damper mass (absolute) acceleration whereby it does not increase the damper mass. Second, for harmonic excitation the inerter force characteristics show negative stiffness behaviour which explains why the TMD stiffness must be designed by taking into consideration both the TMD physical mass and the inertance to ensure the correct tuning of the TMD-Inerter natural frequency.