# Performance of Numerically Optimized Tuned Mass Damper with Inerter (TMDI)
## 5 Conclusions  
This study investigated the mitigation efficiencies of a TMDI and TMD with the same mass ratios for harmonic and random excitations. Two TMDI topologies were considered, i.e., the inerter in parallel to the spring and viscous damper of the damper mass and the serial arrangement of spring, inerter and viscous damper in parallel to the main spring of the damper mass. The TMDI was computed for the mass ratio of 1%, the commonly used inertance ratio of 1% and the far smaller inertance ratio of 0.02%, with analytical parameters set according to values described in the literature and numerically optimized parametric values. The results for harmonic and random excitation demonstrated:  
* That the TMDI with parallel stiffness, damping and inerter and with the typical inertance ratio of 1% led to greater normalized primary structure displacement than the TMD; for harmonic  
Figure 11: Performances of TMDI with serial arrangement of stiffness, inerter and damper (\(\mu=1\%\), \(\beta=0.02\%\), numerically optimized parameters), TMD (\(\mu=1\%\), Den Hartog parameters) and two half TMDs (\(\mu=1\%\), numerically optimized parameters) for harmonic excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response.  
Figure 12: Performances of TMDI with serial arrangement of stiffness, inerter and damper (\(\mu=1\%\), \(\beta=0.02\%\), numerically optimized parameters), TMD (\(\mu=1\%\), Den Hartog parameters) and two half TMDs (\(\mu=1\%\), numerically optimized parameters) for random excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response.  
excitation the maximum displacement response was 27.47% greater than the response due to the classical TMD with Den Hartog parameters;
* That the parallel TMDI with the very small inertance ratio of 0.02% led to approximately the same primary structure displacement response as the classical TMD with numerically optimized parameters;
* That if the inertance ratio of the serial TMDI is too big, then the favorable three-degrees-of-freedom dynamics are not observed and;
* That the TMDI with the serial arrangement of spring, inerter and viscous damper besides the main spring of the damper resulted in an additional degree of freedom. This system, with numerically optimized parameters and the small inertance ratio of of 0.02%, performed better than the TMD; for harmonic excitation, the structural displacement response was additionally reduced by 17.82%. This benefit must be related to the technical efforts that are needed to realize the serial arrangement of a spring, an inerter and a viscous damper without producing undesirable significant friction in the inerter and the connections between these three elements.  
Conceptualization, F.W., F.B., J.D. and C.B.; methodology, F.W., F.B. and J.D.; software, F.W. and F.B.; formal analysis, F.W. and F.B.; writing--original draft preparation, F.W., F.B., J.D. and C.B.; writing--review and editing, F.W. and F.B.; project administration, F.W., F.B., J.D. and C.B. All authors have read and agreed to the published version of the manuscript.  
This research received funding from the Stiftung Maurer Sohne.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors acknowledge the support of Maurer SE, Stiftung Maurer Sohne and OTH Regensburg.  
The authors declare no conflict of interest.