## 2 Optimal Design of the TID
### Optimization Results  
The case that the loads on the structure are white noise excitations is first considered. From Equation (9), it can be identified that the optimal parameter for a specific mass ratio is related to the damping ratio of the isolation structure, but is independent of its natural frequency of vibration. The optimal parameters for TID with different mass ratios are presented in Figure 4, when the damping ratios \(\xi_{b}\) of the base-isolated structure are 0.05, 0.1, 0.15, 0.2, and 0.25, respectively. Through observation, the optimal frequency ratio \(f\) is inversely proportional to the mass ratio \(\mu\) and the damping ratio \(\xi_{b}\) of the base-isolated structure. In addition, it is observed that the optimal damping ratio \(\xi_{d}\) is proportional to the mass ratio \(\mu\) and it is also independent of the damping ratio \(\xi_{b}\) of the base-isolated structure. For convenience, the curve-fitting method based on the least-squares principle was used to fit the optimal parameters curves of the TID. The expressions of the fitting formulas in terms of mass ratio \(\mu\) and damping ratio \(\xi_{b}\) of the base-isolated structure are given in Equation (14). As shown in Figure 4, there is a close agreement between the optimal parameters of the TID given by numerical optimization and the fitting formulas (FF). The errors of the fitting formulas are no greater than 0.5% for all mass ratios, which satisfies the actual design requirements of the TID. \[f^{opt}=\frac{\sqrt{1+\mu/2}}{1+\mu}-0.49\sqrt{\mu}\xi_{b}+0.37\mu\xi_{b},\ \xi_{d}^{opt}=\sqrt{\frac{\mu(1+3\mu/4)}{4(1+\mu)(1+\mu/2)}} \tag{14}\]  
Next, in the case of modeling three kinds of ground motion power spectrum with Clough-Penzien spectrum, the optimal parameters of TID are solved. In this case, the optimal parameters for a specific mass ratio are related to both the damping ratio and the natural vibration frequency of the based-isolated structure, which is different from the above-mentioned case of white noise excitation. Figure 5 shows the connection between the optimal parameters of the TID and the dynamic characteristics (damping ratio and natural vibration period) of the based-isolated structure, respectively, for the far-fault, no pulse near-fault and pulse near-fault ground motions when \(\mu\) is equal to 0.2. From the figure, the following distribution characteristics of the optimal parameters can be  
Figure 4: The optimal frequency ratio and optimal damping ratio of the TID under white-noise excitation when the damping ratios of the base-isolated structure are 0.05, 0.1, etc., respectively. Figure 3: The optimal frequency ratio and optimal damping ratio of the TID under white-noise excitation when the damping ratio of the base-isolated structure is zero. made: (a) the impact of the dynamic characteristics of the based-isolated structure on the optimal parameters is similar for these three kinds of ground motion. Moreover, the optimal frequency ratio and optimal damping ratio are most significantly influenced by the dynamic characteristics of the based-isolated structure for the no pulse near-fault ground motions. (b) The optimal frequency ratio and optimal damping ratio are basically proportional to the natural vibration period. For the far-fault, no pulse near-fault and pulse near-fault ground motions, when the natural vibration period is changed from 1 s to 6 s, the optimal frequency ratio is increased by 11.8%, 29.3%, and 17.5%, the optimal damping ratio is increased by 6.0%, 32.0%, and 15.9%, respectively. (c) The impact of the damping ratio of the based-isolated structure on the optimal parameters is significant with the increase of the natural vibration period. Specifically, the larger the damping ratio of the isolation structure is, the larger the optimal frequency ratio and optimal damping ratio of the TID are.