# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 4 Structural Design of 2 DOFs TMD
### Parametric Design of 2 DOFs TMDs  
The conceptual model of TMDs only have two expected DOFs, but in actual design, due to the material and geometric properties, the TMD may have more DOFs than expected. The corresponding redundant modes of TMD may affect the performance of TMD, even lead to TMD failure. This is a key problem that has been perplexing TMD design. In general, the redundant modes which are far from the targeted modes can be ignored, and the empirical design has always been the major tool to achieve this goal. However, as the number of DOF increases, the empirical method gradually fails to realize the complex design of TMDs. Furthermore, the empirical design may cause the increase in time and cost. Therefore, the theoretical guidance has become particularly important in the multi-DOFs TMD design. In this section, we introduce a parameterized compliance approach for parametric design of the 2 DOFs TMD. Figure 10 shows the configuration of the TMD. It can be seen that the TMD is formed by eight slender beams in parallel distributed on mass block, which is transformed from physical models (Figure 9c). Each slender beam is considered as cantilever beam (Figure 10a) with length \(L\), width \(w\)  
Figure 8: Shapes of the targeted modes and the corresponding optimal location of the TMD: the optimal location of the modes (2B,1T) at point A, while it is point B for modes (1B,2T); the freedom space of the TMDs. Figure 9: Conceptual design of 2 DOFs TMDs: (**a**) freedom space of 2 DOFs TMD; (**b**) constraint space of 2 DOFs TMD; (**c**) physical model of 2 DOFs TMD. and thickness \(t\). Then, according to the Bernoulli-Euler model, the compliance matrix \(C\mathbf{c}_{p}\) (\(p=1,\,\ldots,\,n\)) for each slender beam at location coordinate system \(Oxyz\) is given as follows:  
\[C\mathbf{c}_{p}=\frac{L_{p}}{EI_{y}}diag\bigg{\{} \begin{array}{ccc}\alpha&1& \frac{1}{\lambda\gamma}&\frac{L_{p}^{2}}{12}&\frac{L_{p}^{2}}{12}\alpha&\frac{ L_{p}^{2}}{12}\beta\end{array} \bigg{\}}, \tag{28}\]  
where:  
\[\alpha=\bigg{(}\frac{t}{w}\bigg{)}^{2},\beta_{p}=\bigg{(}\frac{t}{L_{p}} \bigg{)}^{2},\chi=\frac{G}{E}=\frac{1}{2(1+\
u)},\gamma=\frac{J}{I_{y}}, \tag{29}\]  
and  
\[I_{x}=\frac{w^{3}t}{12},I_{y}=\frac{wt^{3}}{12},J=I_{x}+I_{y}, \tag{30}\]  
where \(G\), \(E\) is the shear and Young's modulus, respectively. \(\gamma\) is the ratio of torsion constant over moment of inertia, \(\
u\) is the Poisson's ratio. Further, to simplify the following non-dimensionalization, we choose \(t=w\), and according to Equations (29) and (30), \(\gamma=2\). Due to the different length of slender beam (\(L_{1}\), \(L_{2}\) and \(s=L_{1}/L_{2}\)), the compliance matrix of two type slender beams can be written as:  
\[C\mathbf{c}_{A}=\frac{L_{2}}{EI_{y}}diag\bigg{[} \begin{array}{ccccc}s&s&\frac{s} {2\chi}&\frac{s^{3}L_{2}^{2}}{12}&\frac{s^{3}L_{2}^{2}}{12}&\frac{sL_{p}^{2}}{ 12}\beta_{2}\end{array} \bigg{]}, \tag{31}\]  
\[C\mathbf{c}_{B}=\frac{L_{2}}{EI_{y}}diag\bigg{[} \begin{array}{ccccc}1&1&\frac{1 }{2\chi}&\frac{L_{p}^{2}}{12}&\frac{L_{p}^{2}}{12}&\frac{L_{p}^{2}}{12}\beta_{ 2}\end{array} \bigg{]}, \tag{32}\]  
In order to combine the local compliance matrix \(\mathbf{C}\mathbf{c}_{p}\) of the eight slender beams, they should be transformed from the local to global coordinate system. The origin \(O^{\prime}\) of the global coordinate system \(O^{\prime}XYZ\) is defined in the centroid of the mass block (Figure 10b). For the parallel flexure mechanism, the global compliance matrix can be given as:  
\[\mathbf{C}_{s}=\left(\sum_{p=1}^{m}\big{(}\mathbf{A}\mathbf{d}_{p}C\mathbf{c}_{p}\mathbf{A}\mathbf{d} _{p}^{T}\big{)}^{-1}\right)^{-1}, \tag{33}\]  
where \(m\) is the number of slender beams; \(\mathbf{A}\mathbf{d}_{p}\) is the adjoint transformation matrix from the \(p\)th element to the global system:  
\[\mathbf{A}\mathbf{d}_{p}=\left[ \begin{array}{cc}\mathbf{R}_{x,y,z}(\theta)&0\\ \mathbf{T}\mathbf{R}_{x,y,z}(\theta)&\mathbf{R}_{x,y,z}(\theta)\end{array}