The tuned mass damper (TMD) is a simple and efficient device, but it is only effective when it is precisely tuned to the frequency of a vibration mode. Because nonlinear vibrating structures have resonant frequencies that vary with the amount of total energy in the system, the efficiency of a TMD is questionable in this case. In the present study, the performance of an essentially nonlinear attachment, termed a nonlinear energy sink (NES), is assessed. It is shown that, unlike the TMD, an NES has no preferential resonant frequency, which makes it a good candidate for vibration mitigation of nonlinear vibrating structures.