# Performance of Numerically Optimized Tuned Mass Damper with Inerter (TMDI)
## 4 Results  
TMDI with Inerter in Parallel to Stiffness and Viscous Damper, \(\beta=1\%\), Tuning According to Parameters Described in the Literature  
First, the normalized structural displacement and damper relative motion of the TMDI with an inerter in parallel to the stiffness and viscous damper, \(\beta=1\%\) and parameters set according to [33], were compared to the classical TMD with the same mass ratio for harmonic and random excitations; see Figures 4 and 5. The results demonstrate that the TMDI with tuning according to [33] led to a far greater structural displacement than the classical TMD. The shape of the primary structure response due to the TMDI corresponded well with the results presented in [32]. The response of the TMDI was similar to the response due to an overdamped TMD, which was confirmed by the simulation result due to a TMD with the same mass ratio and a viscous damping coefficient that was 2.3434 times greater than the value according to Den Hartog (Table 2). The observation that the design according to [33] leads to an overdamped TMDI was also seen from the viscous damping coefficient of 121.66 kNs/m (Table 3), which is very close to the value of the overdamped TMD 2.3434 \(\times\) 53.069 kNs/m = 124.36 kNs/m (Table 2). The reason for the extremely suboptimal tuning of the TMDI according to [33] is that the TMDI parameters are derived for a TMD with a mass ratio \(\mu+\beta\) that is not correct, since the inerter force is in proportion to the relative acceleration between the primary structure and the damper mass. Figure 3: Power Spectral Density estimate via [PERSON]’s method of white noise excitation force (close-up of frequency range 0 Hz to 10 Hz). TMDI with Inerter in Parallel to Stiffness and Viscous Damper, \(\beta=1\%\), Numerically Optimized Parameters  
The second case investigated the performance of the TMDI with an inerter in parallel to the stiffness and viscous damper, \(\beta=1\%\) and numerically optimized parameters. The results are depicted in Figures 6 and 7 for harmonic and random excitations and compared to the classical TMD with same mass ratio and Den Hartog parameters and the TMD with \(\mu=0.5\%\) and numerically optimized parameters. It can be seen that the classical TMD still performed significantly better than the TMDI with \(\beta=1\%\) and numerically optimized parameters. Notice that the left peak of the classical TMD is slightly higher than the right peak because \(\zeta_{1}=1\%\) was used for the computations, but the Den Hartog formulae assume \(\zeta_{1}=0\%\). Furthermore, it was observed that mitigation of the primary structure due to the TMDI was approximately equal to that due to a TMD with half mass. Needless  
Figure 4: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu=1\%\), \(\beta=1\%\), parameters as in [33]), TMD (\(\mu=1\%\), Den Hartog parameters) and overdamped TMD (\(\mu=1\%\)) for harmonic excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. Figure 5: Performances of TMDI with inerter in parallel to stiffness and viscous damper (\(\mu=1\%\), \(\beta=1\%\), parameters as in [33]), TMD (\(\mu=1\%\), Den Hartog parameters) and overdamped TMD (\(\mu=1\%\)) for random excitation: (**a**) primary structure normalized displacement response and (**b**) normalized damper relative motion response. to say, the relative motion of the TMD with half mass was significantly greater than that of the TMD with \(\mu\) = 1% and that of the TMDI with \(\mu\) = 1% and \(\beta\) = 1%. TMDI with Inerter in Parallel to Stiffness and Viscous Damper, \(\beta\) = 0.02%, Numerically Optimized Parameters  
The third case investigated how the far smaller inertance ratio of \(\beta\) = 0.02% influenced the performance of the TMDI. For this case, only numerically optimized TMDI parameters were considered, as the first case demonstrated that the TMDI parameters set according to [33] led to a far suboptimal tuning.