# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 2 Formulation of the Bridge-TMD System  
Our goal is to use two 2 DOFs TMDs to control the first four vibration modes of the suspension bridge simultaneously. The 2 DOFs TMD is decoupled in the physical coordinates, their mode shapes follow the physical coordinate of the mock-up, and the corresponding resonance frequencies can be tuned independently to match the desired design. The suspension bridge mock-up and its finite element modelling are detailed described in our previous studies [27, 33, 34]. Here, this paper only lists the vibration characteristics of the bridge mock-up, as shown in Table 1.  
Since the tuning TMDs becomes increasingly complex, we cast the parameter optimization of the 2 DOFs TMDs as a control problem with decentralized static output feedback for minimizing the response of the bridge system. This method has been used successfully for a single mode vibration control of a MDOFs TMD by [PERSON] and [PERSON] [28]. The concept model of the 2 DOFs TMDs is to take the springs as local feedback elements of relative displacements and the dampers as local feedback elements of relative velocities, as shown in Figure 1.  
In this way, the role of the springs and dampers can be replaced by a control force vector, where the control gain is composed of the spring stiffness and damping coefficients (\(k_{i}\) and \(c_{i}\) for \(i=1,\,2,\,\ldots\,\), 6). The mass matrix \(\mathbf{M}_{n\times n}\), stiffness matrix \(\mathbf{K}_{n\times n}\) and viscous damping matrix \(\mathbf{C}_{n\times n}\) is extracted from the numerical model of the suspension bridge mock up, respectively. The 2 DOFs TMD has two planar degrees of freedom, translation \(x_{1}\) (\(x_{2}\)) and rotation \(\theta_{1}\) (\(\theta_{2}\)). Its mass is \(m_{d1}\) (\(m_{d2}\)) and the rotational inertia about its center of mass is \(I_{d}=m_{d}\rho^{2}\), where \(\rho\) is the radius of gyration. The 2 DOFs TMD is connected to the primary system at distances \(d_{1}\) (\(d_{2}\)) from its center of mass via dashpots and springs. Therefore, the control force vector [\(u_{1}\), \(u_{2}\), \(\ldots\,\), \(u_{6}\)] in this case are given by:  
\[u_{1} =k_{1}\Big{(}x_{1}-\mathbf{B}_{2}^{\mathrm{T}}\mathbf{X}-\theta_{1}d_{1} \Big{)}+c_{1}\Big{(}\dot{x}_{1}-\mathbf{B}_{2}^{\mathrm{T}}\dot{\mathbf{X}}-\dot{ \theta}_{1}d_{1}\Big{)}, \tag{1}\] \[u_{2} =k_{2}\Big{(}x_{1}-\mathbf{B}_{2}^{\mathrm{T}}\mathbf{X}+\theta_{1}d_{1} \Big{)}+c_{2}\Big{(}\dot{x}_{1}-\mathbf{B}_{2}^{\mathrm{T}}\dot{\mathbf{X}}+\dot{ \theta}_{1}d_{1}\Big{)},\] (2) \[u_{3} =k_{3}\Big{(}x_{1}-\mathbf{B}_{2}^{\mathrm{T}}\mathbf{X}\Big{)}+c_{3} \Big{(}\dot{x}_{1}-\mathbf{B}_{2}^{\mathrm{T}}\dot{\mathbf{X}}\Big{)},\] (3) \[u_{4} =k_{4}\Big{(}x_{2}-\mathbf{B}_{3}^{\mathrm{T}}\mathbf{X}-\theta_{2}d_{2} \Big{)}+c_{4}\Big{(}\dot{x}_{2}-\mathbf{B}_{3}^{\mathrm{T}}\dot{\mathbf{X}}-\dot{ \theta}_{2}d_{2}\Big{)},\] (4) \[u_{5} =k_{5}\Big{(}x_{2}-\mathbf{B}_{3}^{\mathrm{T}}\mathbf{X}+\theta_{2}d_{2} \Big{)}+c_{5}\Big{(}\dot{x}_{2}-\mathbf{B}_{3}^{\mathrm{T}}\dot{\mathbf{X}}+\dot{ \theta}_{2}d_{2}\Big{)}, \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode** & **Numerical** & **Experimental** & **Experimental** & **Numerical** & **Experimental** \\
**[Hz]** & **[Hz]** & **Damping** & **Mode Shape** & **Mode Shape** \\ \hline
1 st B & 4.5 & 4.4 & 0.14\% & \\
2 nd B & 6.9 & 6.4 & 2.28\% & \\
1 st T & 9.3 & 9.4 & 0.62\% & \\
2 nd T & 10.2 & 10.2 & 0.23\% & \\
3 rd B & 11.2 & 12.2 & 0.49\% & \\ \hline \hline \end{tabular} B stands for bending mode, T stands for torsional mode.  
\end{table} 
Table 1: The numerical and experimental natural frequencies and mode shapes of the bridge mock-up. Figure 1: Concept model of the 2 DOF TMDs: the bridge system equipped with two TMDs.