132 W.K. Ao and P. Reynolds Large scale classical tuned mass dampers (TMD) can be employed to provide an effective solution to mitigate vibration problems in civil engineering structures. Usually, viscous dampers are utilised to achieve the required damping in the TMDs. Whilst the design of such dampers tends to be simple, they have several disadvantages. These include oil leakage and the fact that the magnitude of damping cannot usually be adjusted after installation, leading to sub-optimal TMD performance. Therefore, an ECD design was investigated [7] to provide Eddy current damping to an optimally designed large scale TMD. In [8], the basic theoretical and experimental analysis of a magnetically damped TMD was presented and a laboratory scale ECD was implemented to perform vibration suppression on a small mock-up bridge structure. This paper describes firstly the fundamental electromagnetic induction theory, which can be utilised to calculate the analytical damping force of an ECD involving the classical infinite and finite (method of image) boundary of a conductor. Next, a case study of a footbridge structure is presented, which uses a TMD design with Eddy current damping element. This work includes a finite element (FE) model that can be compared to the analytical calculation. 17.2 Fundamental Electromagnetic Induction Theory The basic law of electromagnetism is Faraday’s law or electromagnetic induction. The relative motion between a magnetic field and a conductor can induce an electromotive force (emf) and hence a current within the conductor (Eddy current). This represents an electric generator converting mechanical energy into electrical energy. In this study, this induced energy is being utilised to provide the non-contact Eddy current damping. 17.2.1 Modelling Eddy Currents and Electromagnetic Force of ECD Figure 17.1 shows a rectangular conductive plate which is assumed to have infinite dimension size [9] and [10]. On the conductive plate, there is a finite sized permanent magnet. The dimension of the magnet is 2a in length, 2b in width and t in thickness. Note that the magnet is rectangular in shape. The magnetic flux density is directed into the page as shown in Fig. 17.1. Therefore, the magnetic flux density is in the negative z direction. The vector of magnetic flux density Bcan then be expressed as BDBx i CBy j C. Bz/k (17.1) where Bx, By andBz are the individual components of magnetic flux densityBin the x, y andz directions, respectively. Note that in this case the magnetic flux density of the x and y components is zero, and hence only the z direction component contributes to the electromagnetic force. In Fig. 17.1c, the conductive plate is assumed to move in the positive horizontal direction. The velocity vector v can be written as v Dvx i Cvy j Cvz k (17.2) (a) (b) (c) (d) Fig. 17.1 Concept of generation of electromagnetic force via moving plate in a magnetic field. (a) 3Dviewof vertically moving conductive plate. (b) Front viewwith electrical charge moving in Vy. (c) 3D viewof horizontally moving conductive plate. (d) Front viewwith electrical charge moving inVx
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