This paper is dedicated to exploit the same optimization mechanism of targeted energy transfer under different types of excitation. Specifically, a linear oscillator (LO) coupled with a vibro-impact (VI) nonlinear energy sink (NES) is analytically studied with an asymptotical method. The optimization mechanism under periodic excitation with a single frequency and under transient excitation is numerically obtained and experimentally validated for the first time. For periodic excitation, the boundary between the regime with two impacts per cycle and that of strongly modulated response (SMR) is proved to be optimal rather than SMR. The chaotic SMR is experimentally observed from the viewpoint of displacement of LO. The above observed mechanism is further applied to explain the optimization mechanism under transient excitation and that under periodic excitation with a range of frequency. It is experimentally verified that the optimization of the latter can be simplified to the optimization under an excitation with a single resonance frequency. For transient excitation, the efficiency of different transient response regimes is experimentally compared, which agrees with the periodic results. Moreover, the efficiency comparison of different lengths of cavity is also experimentally validated. In short, the close relation of optimization under different excitations is clearly demonstrated.