To enhance the performance of the tuned mass damper (TMD), an inerter has recently been incorporated into a conventional TMD system to constitute a new type of an inerter-based passive control system, namely, tuned mass damper inerter (TMDI). The present study proposes a unified design idea for the TMDI based on the Kramers–Kronig relations in the complex effective mass. The TMDI is equivalent to a combination of a static mass and a conventional TMD, and the theoretical links between the TMDI and conventional TMD have been established. Based on the proposed method, the performance of the TMDI is investigated combined with the existing dynamic characteristics of the conventional TMD. It is concluded that the proposed design idea is feasible, and the previously reported dynamic characteristics of the TMDI can be well demonstrated. Moreover, the mass ratio of the equivalent TMD can be used to compare the control performance of the TMDI and TMD, and when it is larger than the physical mass ratio, the inerter enhances the control performance of the TMD.
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