*5.1. Experimental Setup*

The laboratory mock-up of the suspension bridge (in Figure 14) is used to investigate the performance of the 2DOFs TMDs. An impact hammer is used to excite the structure. Prior to vibration measurements, the data acquisition system is established, which involves a single-axial accelerometer, positioned to measure vertical accelerations. The position of the accelerometer is illustrated in Figure 14. The output data are obtained by successive hammering all positions on the deck (Figure 15). The modal parameters of the bridge are estimated by hammering method. The natural frequencies and mode shapes of the mock-up are shown in Table 1.

**Figure 14.** Laboratory mock-up of the suspension bridge.

**Figure 15.** Hammer points on the deck.

#### *5.2. Hammer Tests of TMDs*

The structure of 2DOFs TMDs with adjustable natural frequencies is presented in Figure 16. Each TMD consists a mass block, two pedestals, and eight flexible beams with adjusters. For each flexible beam, one side is fixed on the pedestal by bolts and the other side is connected to the mass block. Though changing the positions of the adjuster, the effective length of the flexible beam can be adjusted, then tunable stiffness of the TMD is realized. The adjuster not only improves the tuning ability of TMD, but also compensates for the errors. The errors include the machining error and the calculation error which is caused by ignoring the prestress of flexible beams (Equations (39) and (40)). Thus, the adjuster is an indispensable part of the TMDs.

**Figure 16.** Prototype of the 2 DOFs TMDs.

Based on Equations (39) and (40), the natural frequencies of 2DOFs TMDs are determined by the dimensions of the flexible beams. In order to improve the tuning ability at runtime, the TMDs are designed with tunable *L*1 and *L*2 which is the effective length of the flexible beams (Figure 16).

In hammer tests, the TMDs are excited by an INV9311impact hammer, and the acceleration responses are recorded by a PSV-500-1D scanning laser vibrometer. The experimental results are shown in Figure 17. The figure shows the experimental acceleration responses of the designed 2DOFs TMDs: (a) when *L*1 = 48.5 mm, *L*2 = 55.5 mm, the first three order frequency of TMD1 is 4.37 Hz, 10.08 Hz, and 65.81 Hz; (b) when *L*1 = 37.5 mm, *L*2 = 40 mm, the first three order frequency of TMD2 is 6.21 Hz, 9.47 Hz, and 72.19 Hz. Comparing with the first two order frequency, the third order frequency of TMDs is a relatively large value, which is far beyond the bandwidth (0–15 Hz) we are considering. Therefore, the 2DOFs TMDs meet the design requirements and verify the validity of the theoretical model.

**Figure 17.** Experimental acceleration responses of the designed TMDs. (**a**) TMD1; (**b**) TMD2.

#### *5.3. Vibration Suppression of Bridge with TMDs*

The designed 2DOFs TMDs are mounted on point *A* and point *B*, which has discussed in Section 4.2. Figure 18 shows a close view of the TMD1 (shown in Figure 12); except the different dimension parameters, the two TMDs have the same configuration and can be tuned in the same way. For damping the target modes, the optimum FRF of these two TMDs exhibits two distinct modes at 4.37 Hz, 10.08 Hz, and 6.21 Hz, 9.47 Hz, respectively, as shown in Figure 17.

**Figure 18.** Experimental implementation of the 2DOFs TMDs. The damping is tuned by changing the magnets.

By referring the previous design [27], the damping is introduced by using eddy current damping, where two symmetric powerful magnets are attached on the pedestal below the mass block of the TMD in Figure 19. The damping value can be tuned by manually adjusting the size and number of the magnets. In Figure 19, the replaceable magnets are used to set the translational damping and the rotational damping at the same time. If necessary, another two symmetric magnets which are attached on the pedestal (the position at the red dashed circles in Figure 19) can provide additional damping for the bending mode.

**Figure 19.** Layout of the symmetric magnets.

The experimental results are shown in Figures 20 and 21. The Figure 20 shows the acceleration responses of the deck: (i) without TMD, (ii) when the structure is equipped with two 2DOFs. The result shows that the 2DOFs TMD is very effective in vibration damping of the bridge. In order to quantitatively confirm the performance of TMDs, Figure 21 shows the FRFs of the deck: (i) without TMD, (ii) when the structure is equipped with two 2DOFs TMDs, targeting the mode pairs (2B,1T) and (1B,2T), respectively. One sees that TMD1 can attain 13.8 dB amplitude reduction of the first bending mode (1B) and 8.8 dB amplitude reduction of the second torsional mode (2T); Meanwhile, a 10.7 dB amplitude reduction of the second bending mode (2B) is observed after using TMD2, and 10 dB amplitude reduction of the first torsional mode (1T).

**Figure 20.** Experimental results: acceleration responses of the deck: blue line stands for without the TMDs; red line stands for with two TMDs.

**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).

The experimental results indicate that the 2DOFs TMDs which designed under the kinematic constraint theory guidance can target the bending and the torsion modes at the same time. Comparing with the classical TMD, the 2DOFs TMD not only achieves vibration reduction, but also avoids increasing the load of the bridge. It is worth mentioning that the final frequency ratios are different from the optimized frequency ratios (Table 4). The optimized frequency ratios are the result of theoretical model calculation. However, the error between the theoretical model and the real structure, as well as the machining and calculation errors, may result in the change of the final frequency ratios. But the role of the optimized frequency ratios in the TMD structural parameter design cannot be neglected.
