Theoretical results for finding the damping coefficients of a magnetic damper consisting of a circular magnetic flux and an arbitrarily shaped conductor have been obtained. In the analysis the exact solution in polar coordinates for the governing equation of the electromagnetic fields is utilized. The boundary condition for arbitrarily shaped boundaries of the conductor is satisfied directly by means of the Fourier expansion collocation method. To discuss the accuracy of the present approximate results, the analysis also has been performed on damper consisting of a circular flux and a circular conductor. The comparison between the present results and the exact ones for the typical damper shows very good agreement.

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