Approximate analysis of eddy-current force under time-varying velocity motion for structural control

Eddy-current force generated from a relative movement between a magnet and a conductor has been studied for possible application in vibration mitigation of civil engineering structures. This eddy-current force is found to be linearly proportional to the relative velocity between the magnet and the conductor when this relative velocity is low and constant. Under a relative velocity with large value or high frequency, the relationship between the eddy-current force and the velocity becomes nonlinear. This nonlinear behaviour however is less investigated. To address this issue, the behaviour of eddy-current force under time-varying velocity motion is examined in this study. An approximate analysis based on the magnetoquasistatic assumption is derived. Analysis shows that the eddy-current force under time-varying motion can be approximated as the sum of a dissipative force, a pseudo-inertial force and a pseudo-dissipative force that are respectively proportional to the relative velocity, acceleration and jerk between the magnet and the conductor. Under harmonic motion, the eddy-current force behaves as a Kelvin-Voigt model with stiffness coefficient and damping coefficient linearly increases and linearly decreases with the square of vibration frequency, respectively. Experimental studies are conducted to validate the proposed approximate analysis. Results show that the proposed approximate analysis predicts the eddy-current force reasonably well. When a structure equipped with an eddy-current damper is subjected to harmonic or earthquake ground motions, it is shown that ignoring the nonlinearity of eddy-current force in the analysis could underestimate the relative displacement and the absolute acceleration of the structure.