# Performance of Numerically Optimized Tuned Mass Damper with Inerter (TMDI)
## 4 Results
The results of the TMDI with \(\beta\) = 0.02% and numerically optimized parameters are plotted in Figures 8 and 9 for harmonic and random excitations and compared to the results of the TMD (\(\mu\) = 1%) with Den Hartog parameters and the TMD (\(\mu\) = 1%) with numerically optimized parameters. Figure 6: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu\) = 1%, \(\beta\) = 1%, numerically optimized parameters), TMD (\(\mu\) = 1%, Den Hartog parameters) and TMD (\(\mu\) = 0.5%, numerically optimized parameters) for harmonic excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. Figure 7: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu\) = 1%, \(\beta\) = 1%, numerically optimized parameters), TMD (\(\mu\) = 1%, [PERSON] parameters) and TMD (\(\mu\) = 0.5%, numerically optimized parameters) for random excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. Figure 8 shows that the TMDI with the very small inertance ratio of \(\beta=0.02\%\) and numerically optimized parameters led to a far better performance than the numerically optimized TMDI with \(\beta=1\%\). However, the primary structure displacement was slightly worse (\(X_{1}/(F_{w}/k_{1}))=11.46\)) than for the numerically optimized TMD with the same mass ratio (\(X_{1}/(F_{w}/k_{1}))=11.37\)). A difference in their maximum relative motions could not be observed. TMDI with Serial Arrangement of Stiffness, Inerter and Viscous Damper, \(\beta=1\%\) and 0.02%, Numerically Optimized Parameters  
The fourth case considered the TMDI with a serial arrangement of stiffness, inerter and viscous damper. First, the TMDI with \(\mu=1\%\), \(\beta=1\%\) and numerically optimized parameters was considered. The results for harmonic excitation depicted in Figure 10 demonstrate that this TMDI slightly reduced  
Figure 8: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu=1\%\), \(\beta=0.02\%\), numerically optimized parameters), TMD (\(\mu=1\%\), Den Hartog parameters) and TMD (\(\mu=1\%\), numerically optimized parameters) for harmonic excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. Figure 9: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu=1\%\), \(\beta=0.02\%\), numerically optimized parameters), TMD (\(\mu=1\%\), Den Hartog parameters) and TMD (\(\mu=1\%\), numerically optimized parameters) for random excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. the primary structure displacement by \((11.37\)-\(11.09)/11.37=2.5\%\), while the TMDI relative motion was almost the same. However, it was also observed that this TMDI topology with \(\beta=1\%\) did not show the envisaged three-degrees-of-freedom response characteristics; see Equation (12). This means that the serial arrangement of stiffness, inerter and viscous damper does not make sense for great inertance ratios. Therefore, the performance of the TMDI with serial topology but with the far smaller inertance ratio of \(\beta=0.02\%\) was analyzed next. Figures 11 and 12 depict the normalized primary structure displacements and damper relative motions due to the numerically optimized TMDI with \(\mu=1\%\) and \(\beta=0.02\%\), the classical TMD (\(\mu=1\%\)), Den Hartog tuning and the numerically optimized two half TMDs, each with \(\mu=0.5\%\), for harmonic and random excitations. For the case of harmonic excitation, the theoretical model of the TMDI and the physical model of the fly wheel were computed; for random excitation, the physical model only was computed, which was numerically more robust because it avoids the zeros in the fourth row of the mass matrix of the theoretical TMDI model (12).