# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 6 Conclusions  
This study proposed a synthetic approach to design and implement 2 DOFs TMDs for damping the bending and torsional modes of suspension bridges. For verifying the effectiveness of the concept  
Figure 21: Experimental results: frequency response functions of the deck: blue line stands for without the TMDs; red line stands for with two TMDs of 2 DOF each, targeted for the damping of the mode pairs (2B,1T) and (1B,2T).  
Figure 20: Experimental results: acceleration responses of the deck: blue line stands for without the TMDs; red line stands for with two TMDs.  
model, we cast the parameter optimization of the 2 DOFs TMDs conceptual model as a control problem with decentralized static output feedback for minimizing the response of the bridge deck. The optimized frequency ratios play a significant role in the TMD structural parameter design. For designing the expected modes of the 2 DOFs TMDs, the graphical approach is introduced to arrange flexible beams properly according to the exact constraints. Based on the optimized frequency ratios, the dimensions of TMDs are determined by the compliance matrix method. The proposed design has been simulated and implemented successfully on a suspension bridge mock-up. Based on the presented results and interpretations, the main findings are summarized as follows:  
* The synthetic approach based on both the graphical approach and parameterized compliance is an effective way to design the TMD with the expected DOFs (i.e., 1, 2,..., 6). It is also an effective complement to the empirical design for the multi-DOFs TMD. Comparing with the empirical design, this synthetic approach can design the expected multi-DOFs TMD without much design experience, which can save time and cost. The disadvantage of this method is that the influence of prestress is not taken into account in the theoretical model. Thus, the adjuster is an indispensable part of TMDs.
* This study verifies the feasibility of the two 2 DOFs TMDs in vibration reduction of suspension bridges by numerically and experimentally; comparing with the classical configuration of the TMD, the two 2 DOFs TMDs can reduce the weight penalty. The experiment demonstrates the ability of the TMDs for suppressing several vibration modes under laboratory conditions. However, their implementation in a full-scale bridge still needs further research.  
All authors discussed and agreed upon the idea and made scientific contributions. The original idea of the article was provided by [PERSON] and he also designed the methodology, realized its MATLAB program, and wrote the manuscript; [PERSON], [PERSON] and [PERSON] provided with the support of the entire study and revised the manuscript; [PERSON] gave suggestions in the theoretical analyses and tests. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science and Technology Research Project of State Grid (Research on Key Technologies for Improving Impact Resistance of Suspension Spanning Frame, Grant No. 5442 GC180003).  
The authors thank [PERSON] of Active Structures Laboratory from Universite Libre de Bruxelles, Brussels, Belgium for his kind guidance and suggestions in the whole research process; and thank the colleagues of FMRL lab from Beihang University, Beijing, China for the realization of the experimental set-up. The reviewers are very much acknowledged.  
The authors declare no conflict of interest.