© 2017 L&H Scientific Publishing, LLC.An oscillator which impacts against a rigid barrier is taken into account. A cycle with zero impact velocity is discussed. The main result of this article concerns stability of the grazing cycle. A significant attention to a model with the variable coefficient of restitution depending on velocity is paid. The mechanical reasons for that are provided as well as new theo- retical advantages have been discovered for the investigation of dynamics near a grazing cycle. The W-map which reduces the system with variable moments of impacts to that with fixed moments and simplifies the analy- sis, is defined. A new type of linearization system with two compounds is applied to investigate the stability of the grazing cycle whose existence is easily examined. A new approach to suppress a singularity, caused by the tangency, in linearization has been developed. Simulations are provided to visualize the stability of the grazing cycle.