# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 4 Structural Design of 2 DOFs TMD
### Graphical Approach  
The screw theory as the theoretical foundation of graphical approach have been widely applied to design and analysis the compliant mechanisms [37; 38]. For object behavior design, adding constraints is the most important step to reach the specific motion. Thus, finding the relationship between the constraints and movements is indispensable in TMD design. Currently, the screw theory is the most popular way to describe this relationship. In the screw theory, a unit screw \(\mathbf{\upxi}\) is defined by a straight line with an associated pitch and is represented as a pair of three-dimensional vectors:  
\[\mathbf{\upxi}=\left[ \begin{array}{c}\mathbf{s}\\ \mathbf{s}_{0}+h\mathbf{s}\end{array} \right]=\left[ \begin{array}{c}\mathbf{s}\\ \mathbf{r}\mathbf{\times}\mathbf{s}+h\mathbf{s}\end{array} \right], \tag{23}\]  
Figure 4: Numerical results: FRFs of the deck respect to different measuring position (**a–d**). The excitation point is fixed. Figure 5: Influence of the damping of TMD1 on the FRF of the system. \(\xi=10\%\)\(\xi_{opt}\), \(25\%\)\(\xi_{opt}\), \(50\%\)\(\xi_{opt}\) and \(\xi_{opt}\), respectively. where \(s_{0}=r\times s\) defines the moment of the screw axis about the origin of a coordinate system, \(s\) is a unit vector parallel the screw axis, \(r\) is the position vector of any point on the screw axis with respect to the origin of a coordinate system, and \(h\) is the pitch of the screw. If \(h\) is equal to zero, the screw reduces to a line quantity (Figure 6a):  
\[\mathbf{\xi}_{t}=\left[ \begin{array}{c}\mathbf{s}\\ \mathbf{s}_{0}\end{array} \right]=\left[ \begin{array}{c}\mathbf{s}\\ \mathbf{r\times s}\end{array} \right], \tag{24}\]  
If \(h\) is infinite, the screw reduces to:  
\[\mathbf{\xi}_{w}=\left[ \begin{array}{c}0\\ \mathbf{s}\end{array} \right], \tag{25}\]  
In addition, an infinite-pitch screw can be considered as a line located at infinity, as shown in Figure 6b. For a better understanding and applicability, the two special cases of unit screw (\(\mathbf{\xi}_{t}\), \(\mathbf{\xi}_{w}\)) are visualized by geometric patterns in Figure 7. The unit screw of zero pitch (\(\mathbf{\xi}_{t}\)) stands for a pure rotation in freedom space (rotational freedom line) or a unit pure force in static along the line in constraint space (constraint force line). A unit screw of infinite pitch represents a pure translation in freedom space (translational freedom line) or a pure couple in constraint space (constraint couple line). It is worth noting that the rotational freedom line represents the axis of rotational movement, and the constraint couple line stands for the axis of couple imposed on a rigid body. Based on [PERSON]'s principles of constraints, the freedoms and constraints in a mechanical system can be defined as:  
\[N=6-n, \tag{26}\]  
where \(N\) is the number of DOFs, \(n\) is the number of non-redundant constraints. When a rigid body (e.g., TMD) is constrained by several mechanical connections providing \(n\) constraints, while \(N\) DOFs of  
Figure 6: A unit screw. Figure 7: Geometric patterns representing screws. the body will remains. In this regard, based on the reciprocal screw theory, \(n\) non-redundant constraints form a wrench \(\mathbf{\xi}_{1}\) in constraint space, and the remained DOFs constitute a twist \(\mathbf{\xi}_{2}\) in freedom space [39]. Based on the definition, the reciprocity of these two screw systems is expressed as:  
\[ \begin{array}{ll}\mathbf{\xi}_{1}^{T}\Delta\mathbf{\xi}_{2}&=\mathbf{s}_{1}\cdot(\mathbf{r}_{ 2}\times\mathbf{s}_{2}+h_{2}\mathbf{s}_{2})+\mathbf{s}_{2}\cdot(\mathbf{r}_{1}\times\mathbf{s}_{1} +h_{1}\mathbf{s}_{1})\\ &=(h_{1}+h_{2})(\mathbf{s}_{1}\cdot\mathbf{s}_{2})+(\mathbf{r}_{2}-\mathbf{r}_{1})\cdot(\mathbf{s}_ {2}\times\mathbf{s}_{1})\\ &=(h_{1}+h_{2})\cos\alpha_{12}-a_{12}\sin\alpha_{12}\\ &=0\end{array}  \tag{27}\]  
where \(\Delta=\left( \begin{array}{cc}0&\mathbf{I}\\ \mathbf{I}&0\end{array}