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The nonlinear vibration of such a system subjects to the base periodic displacement is investigated by numerical simulation and theoretical analysis. The occurrence of the auto-parametric resonance at the resonance point of the translational motion is observed when the ratio between the natural frequencies of the inclinational motion and the translational motion is close to 1:2. Also, the occurrence of an almost periodic motion is observed when this ratio is slightly different from 1:2. The occurrence regions of the auto parametric resonance and the almost periodic motion are discussed. Besides, the influences of discrepancies of the center of mass which produce slight linear coupling terms between the translational motion and the inclinational motion are investigated. 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