## 5 MTMD Applications on Taizhou Bridge
### Design Parameters of the MTMD  
The MTMD is consisted of groups of TMDs, whose frequencies are uniformly distributed around the frequency of the dominated mode. The MTMD in Figure 13a is equipped with two groups of TMDs with a shared TMD on the central line. For the TMDs from the central one to the last one on each side in one MTMD, they are named from TMD\({}_{1}\) to TMD\({}_{n}\) using the subscript \(1\), \(2\), \(\ldots\), \(n\). For convenience, the mass of the block as well as the damping ratio of each TMD in the same MTMD is assumed to be the same. The basic frequencies of TMDs from TMD\({}_{1}\) to TMD\({}_{n}f_{1}f_{2}\), \(\ldots\), \(f_{n}\), are assumed to increase or decrease with a constant difference. The frequency of the TMD is adjusted by changing the stiffness of the spring component. In the application of MTMD, the mechanical parameter of each TMD, including the mass of the block, the stiffness of the spring, and the damping ratio, must first be known. This raises the determination of limited key parameters of MTMD to facilitate the damper design. In this study, five key parameters are utilized to characterize the structure of the MTMD. They are the number of TMDs \(N\), the mass ratio \(\mu\), the damping ratio \(\zeta\), the frequency bandwidth ratio \(\delta\), and the frequency ratio \(\lambda\), respectively. Once the five parameters are given, the MTMD will naturally be determined. The number of TMDs \(N\) refers to the number of TMDs included in one MTMD. In this study, this number will be odd considering the distribution in Figure 13a. The relationship between \(N\) and subscript \(n\) can be expressed as Equation (9). When the other parameters are given, more TMDs will lead to more intensive natural frequencies. This means the frequency difference between two adjacent TMDs will be smaller with the increment of \(N\). \[\mathbf{N}=2n-1 \tag{9}\]  
The mass ratio \(\mu\) is a key parameter reported to heavily influence the control effectiveness of the MTMD system. \(\mu\) denotes the ratio of the total mass of TMDs in a MTMD to the equivalent mass of the dominated structural mode, which is defined as  
\[\mu=\frac{(2n-1)m}{M_{equ}} \tag{10}\]where \(m\) is the mass of the block for each TMD; \(M_{equ}\) is the equivalent mass of the dominated mode. To facilitate the design of the MTMD system, the damping ratios for all the TMDs in the same MTMD are assumed to be identical. \(\zeta\) is defined as Equation (11) in detail. \[\zeta=\frac{c_{i}}{4\pi mf_{i}} \tag{11}\]  
where \(c_{i}\) and \(f_{i}\) are the damping coefficient and the natural frequency of the \(i\)th TMD, respectively. Frequency bandwidth ratio \(\delta\) is a measurement of the range of frequencies that can be controlled by the MTMD. It is a vital parameter indicating the robustness of the MTMD and makes the MTMD insensitive to frequency variations of the main structure. It is defined as the ratio of the maximum frequency difference in the TMDs to the central frequency, which is detailed as  
\[\delta=\frac{|f_{n}-f_{1}|}{f_{c}} \tag{12}\]  
where \(f_{1}\) and \(f_{n}\) are natural frequencies of TMD\({}_{1}\) and TMD\({}_{n}\), respectively. \(f_{c}\) is the central frequency which is the average of frequencies from TMD\({}_{1}\) to TMD\({}_{n}\). The frequency ratio \(\lambda\) denotes the ratio of central frequency to the dominated frequency of the main structure, as expressed in Equation (13). In engineering applications, this ratio is often determined as 1 for the MTMD case. Moreover, the modal frequencies of such a long-span bridge will not vary too much after the installation of the MTMD with a not very large mass ratio. When the natural frequency of the main structure varies with the performance deteriorations or maintenance, this parameter is utilized to evaluate the frequency shift of the installed MTMD system and for further performance evaluation of the damper. \[\lambda=\frac{f_{c}}{f_{s}} \tag{13}\]  
Once the five parameters above are given, the mechanical parameter for each TMD can be determined.