Induced current has been proven to be an efficient non-contact method for vibration damping. Recent studies in beams have found a coupling between vibration modes when the external magnetic field acts simultaneously in multiple directions, which was further studied via a two degree of freedom model. That work suggested that existing models may prove insufficient to fully capture this phenomenon and found that, when magnetic fields in various directions are applied simultaneously, the initially excited degree of freedom exhibited a vibration dampening, while the vibrations in the coupled degree of freedom were amplified. A methodology is presented to derive a damping matrix from the motion-induced vibrations, allowing the phenomenon to be studied through a system of equations of motion for damped free vibration in beams. For this purpose, the structure is discretized into nodes, each with three degrees of freedom: the bending displacement, the bending rotation and the torsion rotation. This multiple degree of freedom formulation then allows the computation of both the frequency- and time-domain responses for the bending displacement, the bending rotation and the torsion rotation of each node. Using mass and stiffness matrices available in the literature, and by validating the resulting natural frequencies for bending and torsional modes from existing theoretical expressions, the frequency response demonstrates vibration dampening in the excited mode (bending or torsion), and the amplification of the coupled mode, each in the excited mode’s natural frequencies. Additionally, the proposed methodology also allows to study the nature of the corresponding dissipative forces when this coupling occurs through time-domain analysis.