Inter-shaft bearings, critical components within aero-engine dual-rotor systems, are susceptible to localized faults due to their unique installation locations. This research delves into a distinctive dynamic phenomenon termed “paroxysmal impulse vibration”, which is subjected to localized faults in the outer ring of inter-shaft bearings. The vibration behavior of this phenomenon differ significantly from those of traditional continuous impulse vibrations. The mechanical mechanisms and analytic requirements governing the emergence of paroxysmal impulse vibrations are systematically studied. Three analytic prediction formulas are derived to anticipate these vibrations. A systematic prediction process is summarized, offering standardized guidance for forecasting paroxysmal impulse vibrations under various working conditions. Furthermore, the established prediction formulas and process forecast all specific frequency components of paroxysmal impulse vibrations. Experimental vibration tests on defective inter-shaft bearings align well with simulation outcomes, confirming the efficacy of the proposed prediction methods for paroxysmal impulse vibration. Additionally, a significant correlation between the frequency components in paroxysmal and continuous impulse vibration is revealed. This identified correlation provides a novel method for elucidating the presentation laws of all specific frequency components within traditional continuous impulse vibrations under different parameter conditions. The research of paroxysmal impulse vibration introduces novel perspectives on inter-shaft bearing vibration mechanisms, offering promising implications for the inter-shaft bearing failure diagnosis.