## 2 Optimal Design of the TID
### Control Effectiveness  
This section showcases the control effectiveness of the TID for various ground motions. In order to quantitatively evaluate the control effectiveness, the displacement reduction ratio \(J\) of the base-isolated structure is expressed in Equation (15):  
\[J=\frac{\sigma_{w}^{2}}{\sigma_{w}^{2}}=\frac{\int_{-\infty}^{\infty}{|y_{w}( \omega)|}^{2}S(\omega)d\omega}{\int_{-\infty}^{\infty}{|y_{w\omega}(\omega)|}^ {2}S(\omega)d\omega} \tag{15}\]  
where \(\sigma_{w}^{2}\) and \(\sigma_{w}^{2}\) are the mean squared displacement responses of the base-isolated structure with and without TID, respectively. \(y_{w\omega}(\omega)\) and \(y_{w}(\omega)\) are the frequency transfer function between the displacement of the base-isolated structure and the base acceleration excitation, respectively.  
Figure 7 shows the control effectiveness of the TID with mass ratios of 0.2 and 0.5 for the base-isolated structure corresponding to the far-fault, no pulse near-fault and pulse near-fault ground motions, respectively. The results of the three conditions have the similar trends, that is, the larger the damping ratio of the base-isolated structure is, the smaller the displacement reduction ratio is. Conversely, the control effectiveness of the TID exhibits a negative correlation with the isolation period, but the degree of influence is much weaker than the impact of the damping ratio.  
In addition, the mass ratio and control effectiveness also have a strong correlation. For further investigating the impact of the mass ratio, the period and damping ratio of the based-isolated structure are combined in two groups as in the previous section. Figure 8 shows the relationship between the control effectiveness and the mass ratio of the TID. The control effectiveness of the TID under white-noise excitation are also shown in the figure for comparison. Another point to note is that the control effectiveness of the TID under white-noise excitation is only related to the damping ratio of the based-isolated structure, and has nothing to do with the natural vibration period of the based-isolated structure. As illustrated in Figure 8 the effectiveness of TID in reducing structural vibration is very significant. When the mass ratio <0.2, the displacement reduction ratio \(J\) decreases sharply, however, after the mass ratio exceeds 0.2, the benefit of increasing the mass ratio is no longer significant. Another result shows that the control effectiveness of the TID under three types of ground motion is better than that under white-noise excitation, and it increases with the increase of the natural vibration period. However, the difference among the control effectiveness of the TID under the three ground motions is not obvious, especially when the mass ratio is less than 0.2. Therefore, for engineering practice, considering that if the designed mass ratio is too large, the economic cost will be significantly increased, so adopting a suitable mass ratio (<0.2) can often achieve ideal control requirements.  
Figure 7: The relationship between displacement reduction ratio \(J\) and the dynamic characteristics (damping ratio and natural vibration period) of the based-isolated structure (\(\mu=0.2,\,0.5\)).