The Jeffcott rotor is the most widely used theoretical model in the analysis of rotor systems. This model satisfies the condition of 1: (-1) internal resonance, and we clarified the effect of this internal resonance on nonlinear steady-state oscillations at the major, twice the major and three times the major critical speeds in the previous papers. In this study, we investigate nonstationary oscillations during acceleration through the major critical speed in the Jeffcott rotor and other rotor systems which almost but not exactly satisfy 1: (-1) internal resonance relation. Especially, we pay attention to the influence of internal resonance. The foilowing are clarified theoretically and experimentally: (a) Nonstationary oscillations of the Jeffcott rotor and other rotor systems whose natural frequencies almost satisfy 1: (-1) internal resonance relation are influenced by the internal resonance. (b) It is easier to pass the major critical speed of a rotor system with internal resonance than that of a rotor system with no internal resonance. (c) When the system has slight discrepancies among critical speeds, that is, almost but not exactly satisfies the relation of internal resonance, the rotor can pass the major critical speed very easily with small amplitude even if the acceleration is small.