## 6 Optimization of TMDI and Equivalent TMD
### Effect of Mass Ratio and Inertance Ratios on the Performance of TMDI  
Control performance of the TMDI with the inerter connected between TMD at the top and 39 th floor is investigated against varying mass ratio and inertance ratio under filtered white-noise excitations. Four unitless parameters are defined based on the controlled and uncontrolled responses as follows:  
\[ \begin{array}{rl}R_{1}&=\frac{STD\text{ displacement at top floor (controlled)}}{STD\text{ displacement at top floor (uncontrolled)}}\\ R_{2}&=\frac{STD\text{ acceleration at top floor (controlled)}}{STD\text{ acceleration at top floor (uncontrolled)}}\\ R_{3}&=\frac{\text{ Peak displacement at top floor (controlled)}}{\text{ Peak displacement at top floor (uncontrolled)}}\\ R_{4}&=\frac{\text{ Peak acceleration at top floor (controlled)}}{\text{ Peak acceleration at top floor (uncontrolled)}}\end{array}  \tag{9}\]  
As defined by Equation (9), \(R_{1}\) and \(R_{2}\) represent the performance of TMDI for the reduction of root mean square (RMS) displacement and acceleration at the top floor, whereas \(R_{3}\) and \(R_{4}\) represent the performance for the reduction of peak displacement and peak acceleration at the top floor. Figure 9 shows the three-dimensional (3D) plot of normalized structural responses with respect to mass and inertance ratios. The normalized 3D plot of standard deviation (STD) and peak responses are then projected into the \(\mu\)-\(\beta\) plane, as shown in Figure 9.  
Figure 9a shows the variation of STD displacement, and Figure 9b shows the variation of STD acceleration. Figure 9c shows the variation of peak displacement. Figure 9d shows the variation of peak acceleration of the top floor with respect to \(\mu\) and \(\beta\). All the controlled responses are normalized by dividing their corresponding uncontrolled responses as defined by Equation (9). It is crucial to notice that the first line along the \(\mu\) axis is the performance of pure TMD because there is a negligible inertance ratio. Similarly, the first line along the \(\beta\) axis is the pure tuned inerter damper (TID) with no mass attached at the top. Except for the first two lines along each axis, the device acts as TMDI. As discussed in the previous sections, for the sizeable inertance value, the physical mass of the device could be significantly high. For a given mass ratio, a small inertance ratio reduces the device's performance because it damages the optimum frequency and damping ratio, resulting in less energy dissipation. It is observed in Figure 9a,b that there is no significant reduction in the responses when the \(\beta\) increases beyond 0.6. Similarly, peak displacement and acceleration may remain constant when \(\beta\) is increased beyond 0.5, as shown in Figure 9c,d. The optimum \(\beta\) for a given mass ratio cannot be too small (close to zero) or too large (close to 1). This observation is supported by several analytical and experimental studies on TMDI, as described in the previous sections. An enlarged view of Figure 9a, which illustrates the effect of mass ratio and inertance ratio on the STD displacement of the top floor, is shown in Figure 10.  
In Figure 10, point A represents the virtually uncontrolled case where the mass ratio is \(1\times 10^{-5}\), and the inertance ratio is \(1\times 10^{-10}\). Thus, the value of \(R_{1}\) is equal to 1 at point A. The curve that goes through points A, B, and C is the typical TMD with a frequency ratio of 0.99 and a damping ratio of 8.5%. Those parameters (frequency ratio of 0.99 and damping ratio of 8.5%) correspond to a 3% mass ratio. Unless otherwise mentioned, this value (3% mass ratio) has been used entirely in the paper. At points B and C, the mass ratio is nearly 2% and 4%, respectively, but the inertance ratio remains negligible (\(1\times 10^{-10}\)). The response reduction is sharp from point A to B but flatter from point B to C. With the mass ratio increase from 0 to 2%, STD displacement is reduced by nearly 55%.