A generalized 2-DOF model for optimal design of MDOF structures controlled by Tuned Mass Damper Inerter (TMDI)

The paper illustrates a methodology to perform the optimal design of a Tuned Mass Damper Inerter (TMDI) equipped on a multi degree of freedom (MDOF) structure through a generalized model. The TMDI is considered in both configurations grounded and ungrounded. A generalized 2-DOF model is obtained: the primary oscillator, representing the original MDOF structure and the secondary oscillator, representing the control system. A Gaussian zero mean white noise random process is adopted for the excitation. The TMDI design is carried out adopting an energy objective function, maximized over the space of the design parameters which are: the mass and inertance ratios, the frequency ratio, the damping factor and a parameter that accounts for the TMDI location on the MDOF structure. Synthetic performance-based design maps of the TMDI parameters are carried out. It is shown that is possible to have different design configurations to achieve a target performance level. The maps furnish, as limit cases, the design of a Tuned Inerter Damper (TID) and a Tuned Mass Damper (TMD), when the mass or the inertance ratios are respectively assumed null. It is shown straightforwardly the regions of the design parameters where the inerter enhances the performances achievable by classical TMD. Furthermore, an analytical expression which links the optimal design variables necessary to obtain a desired target performance is proposed. Frequency response functions and modal parameters of certain 2-DOF models with optimally designed control system are reported and considerations on how the design parameters influence the system dynamics and the damping capabilities are highlighted. Comparisons of the TMDI optimally designed with cases of no control and same mass TMD are evidenced throughout the paper. Finally, the main responses of the primary structure and the TMDI are estimated in the space of variation of the design parameters in order to assess the effectiveness of the control system.