The ultimate goal of this paper is to propose a procedure for the optimal design of a Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control the vibration of any possible linear or nonlinear main systems. To this end, the activation characteristic of VI NES at a range of displacement amplitude of a main system is generalized from linear systems to nonlinear systems. It is theoretically proved and experimentally observed that this activation characteristic is almost independent of frequency, which provides direct proof for the effectiveness of VI NES in a broad frequency bandwidth. In terms of vibration control, this feature is very attractive and builds a bridge between linear and nonlinear systems. Then it is applied for the design of VI NES attached to nonlinear systems. In this way, the design of VI NES for a nonlinear system is simplified to the optimal design for a linear system, which is designed to be similar to this target nonlinear system. Because the latter can be analytically calculated, the proposed method is feasible from a quantitative perspective. Finally, this activation characteristic and a proposed design method are applied to control chatter in a turning process, and results prove its feasibility.