# A Multi-Degree of Freedom Tuned Mass Damper Design for Vibration Mitigation of a Suspension Bridge
## 1 Introduction  
With the tendency to use longer spans, the damping of modern suspension bridges is seriously reduced. Complex vibration problems follow, such as wind-induced vibration, human-induced vibration, cable-structure interactions and flutter instabilities [1; 2; 3; 4]. It is admitted that the oversensitivity to dynamic excitation of suspension bridges is associated with the very low structural damping in the global bridge modes [5; 6]. Therefore, the dissipation of the vibration energy generated by the dynamic loadings is a central issue in their design. At present, the use of damping systems such as tuned mass damper (TMD) [7], viscous dampers [8; 9], or active tendon control [10] is a classical way to alleviate the vibrations in structures. This study aims at the design of multi-degree of freedom TMD for vibration damping of a suspension bridge deck. Considering their simplicity and effectiveness, tuned mass dampers have been widely used in bridges such as the London Millenium bridge, for damping both lateral and vertical vibrations of the deck. Since [PERSON] et al. proposed the fundamental theory, TMDs have seen numerous applications in civil engineering [11; 12; 13]. Thus, [PERSON] and [PERSON] [14; 15] proposed a new semi-active electromagnetic TMD of pendulum type to damp the torsional mode of an evolving bridge mock-up. Their studies showed that the tuned pendulum damper (TPD) is very effective in vibration damping,qualitatively and quantitatively confirming the theoretical predictions. However, the TMDs can only control single mode and easy detune, which limits their further development. For the robustness of vibration control and targeting several vibration modes, the multiple tuned mass dampers (MTMDs) has been proposed, and its performance is more effective as compared to the single TMD. The superior effectiveness of the MTMDs is able to control almost any type of vibration in civil structures [16]. The MTMDs is used to damp suspension bridges for several purposes. In some studies, the MTMDs are used to the suppression of buffeting, flutter or increasing the critical flutter wind speed [17, 18]. Other studies consider MTMDs for alleviating pedestrian- and jogger-induced vibration [19, 20, 21, 22] or traffic-induced vibration [23, 24, 25, 26]. Generally, the weight of a TMD is limited to 1-3% of the structure weight. Hence, as the number of targeted modes increases, a large number of TMDs will increase the burden on the primary system and limit the damping performance (called weight penalty). To avoid such a penalty, in our previous study [27], we proposed to design a two-degree of freedom TMD, where the original mass of TMD is redistributed in such a way that the TMD has a bending mode and a torsional mode. In this design the resonance frequencies and the modal damping of the two modes can be tuned independently. In addition, [PERSON] and [PERSON] have proposed a multiple degrees of freedom TMD (MDOFs TMD), and experimentally demonstrated that the MDOFs TMD can damp six modes of the primary structure. They also showed that a MDOFs TMD can be used to attain better vibration suppression for single mode vibration of a primary structure [28]. [PERSON] et al. described a novel method for selecting the parameters of a 2 DOFs TMD with translational and rotational degrees of freedom [29]. [PERSON] and [PERSON] et al. presented a design of a multi-DOFs TMD to alleviate the dominant mode of the work piece/fixture assembly in milling [30, 31, 32]. As the DOF increases, by only selecting the appropriate DOFs and tuned frequencies, the TMDs can reach the best vibration control of the primary structure. Therefore, designing a TMD with expected DOFs and natural frequencies becomes an urgent problem. Unfortunately, due to its complicated structure and easily detuning, the further study on the implementation of MDOFs TMD is rare.