## 5 MTMD Applications on Taizhou Bridge
### Step-Updating Framework for the Determination of Parameters  
In order to quantify the control efficiency of the MTMD, the attenuation factor is defined as  
\[\eta=\frac{\beta-\alpha}{\beta}\times 100\% \tag{19}\]  
where \(\alpha\) and \(\beta\) are the RMS (Root Mean Square) of the buffeting displacements with and without MTMD, respectively. This attenuation factor is a performance-oriented index that directly presents the effect of the MTMD system [27].  
As shown in Figure 14, a step-updating framework is presented to analyze the parameter sensitivity of the control efficiency for each MTMD, and determine the preferred values for the main parameters with the expected attenuation factor. The frequency ratio is set as 1 in the analysis, but it is also changed within \(\pm 10\%\) to investigate the robustness of the MTMD system. The step-updating framework gets the main parameters updated one by one via the parametric analysis. A feedback in each step will be included for the next round of updating, which will save time for the parameter determination. By the experience obtained in testing the framework, the prescribed attenuation factor can be satisfied within 2-3 rounds.  
There are seven steps included in the whole procedure, which are detailed as  
**Step 1**: Set the initial values for the four main parameters, including the number of TMDs \(\mathbf{N_{0}}\), the mass ratio \(\mathbf{\mu_{0}}\), the damping ratio \(\mathbf{\zeta_{0}}\), and the frequency bandwidth ratio \(\mathbf{\delta_{0}}\), by experience.  
Figure 14: Step-updating framework for the determination of the main parameters.  
* **Step 2**: Conduct the parametric analysis on the number of TMDs \(\mathbf{N}\) with the other parameters unchanged, and determine a preferred value for \(\mathbf{N}\) in this round. Then, include the positive range between \(\mathbf{N}\) and the attenuation factor in the next round.
* **Step 3**: Conduct the parametric analysis on the mass ratio \(\mathbf{\mu}\) with the updated \(\mathbf{N}\) and two initial other parameters, and determine a preferred value for \(\mathbf{\mu}\) in this round. Then, include the positive range between \(\mathbf{\mu}\) and the attenuation factor in the next round.
* **Step 4**: Conduct the parametric analysis on the damping ratio \(\mathbf{\zeta}\) with the updated \(\mathbf{N}\) and \(\mathbf{\mu}\) as well as \(\delta_{\mathbf{0}}\), and determine a preferred value for \(\mathbf{\zeta}\) in this round. Then, include the positive range between \(\mathbf{\zeta}\) and the attenuation factor in the next round.
* **Step 5**: Conduct the parametric analysis on the frequency bandwidth ratio \(\delta\) with the updated \(\mathbf{N}\), \(\mathbf{\mu}\), and \(\mathbf{\zeta}_{\mathbf{0}}\), and determine a preferred value for \(\delta\) in this round. Then, include the positive range between \(\delta\) and the attenuation factor in the next round.
* **Step 6**: Compare the damping ratio \(\mathbf{\eta}\) of the MTMD with the updated parameters to the prescribed threshold \(\mathbf{\eta}_{0}\). If \(\mathbf{\eta}\geq\mathbf{\eta}_{0}\), go to Step 7; Otherwise, go to Step 2.
* **Step 7**: The final values of the four main parameters are determined as the ones in Step 6 and are renamed as \(\mathbf{N_{d}}\), \(\mathbf{\mu_{d}}\), \(\mathbf{\zeta_{d}}\), and \(\mathbf{\eta_{d}}\), respectively.