In this paper, we investigate a problem of passive mitigation of friction-induced vibrations due to mode coupling instability in breaking systems. For this purpose, the well-known Hultèn’s model is coupled to one Nonlinear Energy Sinks (NES). In previous work by the authors [1], an asymptotic analysis has been performed on the system allowing to explain the steady-state response regimes observed in the numerical simulations and predict some of them. However, because of the appearance of a three-dimensional critical manifold, the prediction of the mitigation limit (i.e. the friction coefficient value which separates harmful situations from harmless situations) has been not performed. This present work completes this previous study using a general dynamics reduction method allowing to obtain one dimensional critical manifolds. The mitigation limit is therefore predicted and compared, for validation purposes, to numerical integration of the system.