This paper proposes the design criteria for optimally tuned Vibro-Impact (VI) Nonlinear Energy Sink (NES) to control vibration under periodic and transient excitation. Firstly, a generalized dimensionlessmodel of a two degrees of freedom(DOF) system comprising a harmonically excited linear oscillator (LO) strongly coupled to a VI NES is investigated. Bifurcation analysis and efficiency of Targeted Energy Transfer (TET) around the Slow Invariant Manifold (SIM) are studied with the variation of clearance. As a result, the optimal clearances for periodic and transient excitation are calculated fromtwo transition points of the SIM, respectively. Then the procedure is extended to the case of multiple VI NESs in parallel with the LO. Two principles of additivity and separate activities of VI NESs are verified theoretically. Finally, experiments involving the whole system embedded on an electrodynamic shaker are performed. The results show that the design criteria can not only predict the efficient TET at resonance frequency, but can also achieve an optimal performance in a range of frequencies. Furthermore, the criteria can be straightforward for the application of multiple VI NESs, so as to make VI NESs work robustly under different types of excitation.