We consider the response of a linear structural system when coupled to an attachment containing strong or even essential nonlinearities. For this system, the attachment is designed as a nonlinear vibration absorber, serving to dissipate energy from the structural system. Moreover, the attachment not only leads to a reduction in the total energy of the system, but also couples together the vibration modes of the linear structural system, thereby allowing for energy to also be redistributed among these structural modes of the system. The effect of the nonlinear attachment on the linear primary system can be quantified in terms of equivalent measures for the damping and frequency of each mode, derived through consideration of the energy in each mode. The identification of these equivalent measures is illustrated on a two degree-of-freedom primary system. Moreover, this procedure depends only on the time history of the response and is therefore suited to both simulation and experimental results.