17 Analytical and Experimental Study of Eddy Current Damper for Vibration Suppression in a Footbridge Structure 135 Hence, the total Eddy current density in the x direction can be calculated by combiningJ.P/ y j, J .iright/ y j andJ .ileft/ y j and then the result can be used to calculate the electromagnetic force, as given by Fem;x DFem;x i D Z Jy0 BzdVi D t Z a a Z b b Jy .P/ J y .iright/ J y .ileft/ Bzdxdyi (17.15) 17.4 ECD Application to Tuned Mass Damper Design for a Footbridge In this section a real footbridge is used as a case study. Figure 17.3 shows a footbridge located in the UK. The structure is very slender and has three long spans between two column supports. An ECD is considered as a damping element within a TMD to achieve the required damping for the project. Figure 17.4a shows the results of modal analysis carried out by the bridge designer. It can be seen that the second mode shape of this structure shows significant vertical bending. A vibration serviceability assessment of the bridge was carried out by Full Scale Dynamics Ltd (FSDL). For application of the excitation scenarios from the NA to BS EN 1991-2:2003, Fig. 17.4b and c show responses calculated due to a group of people jogging and a crowd of people walking. For the response analysis, modal damping ratios were assumed in all modes of vibration to be 0.2%. It was assumed that the maximum allowed peak acceleration was 1.0 m/s2 and the walking path was along the centreline of the bridge. For both cases of group jogging and crowd walking, the responses exceeded the allowed limits in the second span of the footbridge, which represented a potential vibration serviceability problem on this structure. Hence, the use of a tuned mass damper (TMD) box (as shown in Fig. 17.5) was considered to overcome the vibration problem. The basic components of a TMD are tuning mass, stiffness element and damping device. An ECD can be chosen to provide a damping element for the TMD. Because the ECD is an non-contact damper, it does not have any additional Coulomb’s friction contribution. The space available for the ECD is 640 260 88 mm. From the TMD design, the required damping coefficient is 2.4 103 Ns/m, the maximum velocity is ˙0.48 m/s and the required stroke is ˙30mm. Section 17.2.1 showed the copper plate moving in the vertical direction. The horizontal magnetic pole projection can be used for ECD design to achieve the required damping. Therefore, the layout shown in Fig. 17.6 can be utilised. The clearance space between the TMD mass block and bridge structure is 88 mm, including the moving stroke ˙30mm. The magnet height is designed to have 28 mm height. Figure 17.6b and c show the relative dimensions of the ECD design. The ECD has a volume of 350 182 56.5 mm and the copper plate has 4 mm thickness with height 30 mm. Figure 17.7a shows the FEM meshing of the ECD. As the copper plate moves with velocity 0.48 m/s, the damping force calculation is shown in Fig. 17.7b. The vertical component of the output force is the target design damping force. The maximum value achieved is 1260 N and the average value is 1150 N, which meets the design requirement. The damping Fig. 17.3 Case study footbridge
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