# Performance of Numerically Optimized Tuned Mass Damper with Inerter (TMDI)
## 2 Modelling
### Primary Structure with TMDI with Inerter in Parallel to the Spring and Viscous Damper  
The topology of the conventional TMDI is given by the parallel arrangement of the inerter, spring and viscous damper, as depicted in Figure 1b. The idea behind this topology is that the inerter, although producing a force in proportion to the relative acceleration of its two terminals, increases the TMD inertial force without adding real mass. The model of the primary structure with conventional TMDI is given by the following two-degrees-of-freedom system:  
\[\begin{bmatrix}m_{1}+b&-b\\ -b&m_{2}+b\end{bmatrix}\begin{bmatrix}\ddot{x}_{1}\\ \ddot{x}_{2}\end{bmatrix}+\begin{bmatrix}c_{1}+c_{2}&-c_{2}\\ -c_{2}&c_{2}\end{bmatrix}\begin{bmatrix}\ddot{x}_{1}\\ \dot{x}_{2}\end{bmatrix}+\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}f_{ex}\\ 0\end{bmatrix} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model Mass** & **Eigenfrequency** & **Damping Ratio** & **Stiffness Coefficient** & **Viscous Damping Coefficient** \\ _m_**1 (t)** & _f_**1 (Hz)** & \(\xi_{1}\)**(\%)** & \(k_{1}\)**(MN/m)** & \(c_{1}\)**(kNs/m)** \\ \hline
50,000 & 0.14 & 1\% & 38.689 & 879.65 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Assumed primary structure modal parameters.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Damper** & **Mass Ratio** & **Tuning** & **Natural Frequency** & **Damping Ratio** & **Stiffness** & **Viscous Damping Coefficient** \\
**Type** & \(\mu\) = _m\({}_{2}\)/m\({}_{1}\)_ & **Tuning** & _f_**2 (Hz)** & \(\xi_{2}\)**(Hz)** & \(k_{2}\)**(kN/m)** & \(c_{2}\)**(kNs/m)** \\ \hline TMD & 1\% & Den Hartog & 0.1386 & 6.03\% & 379.27 & 53.069 \\ \hline TMD & 1\% & Overdamped & 0.1386 & 2.3434 \(\times\) 6.03\% & 379.27 & 2.3434 \(\times\) 53.069 \\ \hline TMD & 0.5\% & \
\begin{tabular}{c} Numerically \\ optimized \\ \end{tabular} & NA & NA & 191.12 & 19.134 \\ \hline TMD & 1\% &
\begin{tabular}{c} Numerically \\ optimized \\ \end{tabular} & NA & NA & 378.11 & 54.648 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the considered TMDs.  
where \(b\) denotes the so-called inertance with unit (kg) that is determined by the selected inertance ratio \(\beta\) as follows:  
\[b=\beta\ m_{1} \tag{6}\]  
Notice that the inertance \(b\) not only increases the mass \(m_{2}\) of the TMDI but \(b\) occurs at all locations in the modal mass matrix of Equation (5) because the inerter force \(f_{b}\) is in proportion to the relative acceleration of the inerter grounds, which is here the difference between primary structure modal acceleration \(\tilde{x}_{1}\) and acceleration \(\tilde{x}_{2}\) of the TMDI mass \(m_{2}\).  
\[f_{b}=b\ \big{(}\tilde{x}_{1}-\tilde{x}_{2}\big{)} \tag{7}\]  
For the TMDI with conventional topology and the often-adopted inertance ratio \(\beta=1\%\), the parameters \(f_{2}\), \(\zeta_{2}\), \(k_{2}\) and \(c_{2}\) are, on the one hand, designed according to the closed-form solutions (8)-(11), according to [33], and, on the other hand, the parameters \(k_{2}\) and \(c_{2}\) are numerically optimized for minimum structural displacement (Table 3).  
\[f_{2}=f_{1}\frac{\sqrt{1+0.5(\beta+\mu)}}{1+\beta+\mu} \tag{8}\]  
\[k_{2}=(b+m_{2})\ (2\pi f_{2})^{2} \tag{9}\]  
\[\zeta_{2}=\sqrt{\frac{(\beta+\mu)(1+0.75(\beta+\mu))}{4(1+\beta+\mu)(1+0.5( \beta+\mu))}} \tag{10}\]  
\[c_{2}=2\ \zeta_{2}\ (b+m_{2})\ (2\pi f_{2}) \tag{11}\]  
According to [33], the closed-form solutions (8)-(11) do not minimize the structural displacement response but yield a reasonable suboptimal solution as these solutions are derived for a classical TMD with a total mass ratio \(\mu+\beta\). This means that one can expect a good but not optimal performance of the conventional TMDI with parameters (8)-(11). For the inertance ratio \(\beta=0.02\%\), only the numerically optimized TMDI is used for the comparative study because the very small inertance ratio of \(0.02\%\) has not been considered in the literature so far.