17 Analytical and Experimental Study of Eddy Current Damper for Vibration Suppression in a Footbridge Structure 133 which can be re-written as v Dvx i C. vy/j Cvz k (17.3) Figure 17.1a shows the conductive plate vy moving in the negative y direction, whilst the other directional components have no contribution. In addition, all of these values are time-invariant. In general, several basic electromagnetic theories can now be used to calculate the electromagnetic damping force. The magnetic field Bis defined in terms of the force FB acting on a test particle with charge q moving through the field with velocity v. Therefore, the magnetic force or Lorentz’s force on the charge q can be obtained as follows: Fem Dqv BDI idl BDI i A A dl BDZ i A dV BDZ J BdV (17.4) where i is the electric current, Ais an outward normal unit area vector, V is a volume of magnetic pole projection, and J can be defined as the current density, which stands for current per area. However, the current density can also be defined by multiplying the conductivity of the material and the electric field, as follows: J D E (17.5) The total electric field intensity induced on the conductive plate can be calculated as EDEind CEv DEind Cv B (17.6) where Eis total electric field intensity, Eind is the induced electric field intensity of the electric charged particle, andEv is an electric field intensity caused by the magnetic field in addition to that of the electric charged particle, which is also the cross product of velocity and magnetic flux density. Equation (17.5) can now be rewritten as follows: J D .Eind CEv/ (17.7) 17.3 The Method of Image Current The previous section was developed based on a conductive plate with infinite x andy dimensions. In this section, the method of image ([11] and [12]) is presented to calculate the ECD electromagnetic force, using finite x and y dimensions, which cause the Eddy currents to be zero at the edges of the conductive plate. The solid line in Fig. 17.2 shows the infinite boundary condition of Eddy current density J.p/. The two tails of the curve extend infinitely in the positive and negative directions. Due to the finite actual dimensions of the plate, the curve does not exactly express the real current flowing, which potentially might cause a calculation error. To solve this problem, the tails of the profile of J.P/ are reflected about axes of symmetry corresponding with the edges of the conductive plate. In this case, the image current density on the left-hand edge (Jileft ) is reflected to the right and the right-hand edge Jiright is reflected to the left. Hence, the edge effect can be eliminated. From this point of view, when the conductive plate moves in the y direction, the mathematical expression of the primary and image current density shows the following relationship J0x DJ0xi DJ .P/ x i J .iright/ x i J .ileft/ x i (17.8) The image Eddy current density is symmetrical to the primary Eddy current density. Looking at the y movement of the conductive plate, Jileft can be expressed in terms of J.P/ using a coordinate transformation, as given by J .iright/ x i DJ .P/ x .2B x; y/i D vyBz 2 tan 1 y a 2B x b tan 1 y Ca 2B x b Ctan 1 yCa 2B x Cb tan 1 y a 2B x Cb i (17.9)
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