During operation, failures in a centrifugal pump’s rotary shaft seal—such as wear, deformation, or thermal cracking—can adversely affect system performance. This study utilizes both theoretical and experimental methods to investigate the vibration characteristics of centrifugal pumps under different rotary-shaft-seal conditions. Vibration signals are collected and processed using empirical mode decomposition (EMD) and autoregressive (AR) modeling to generate an EMD-AR spectrum. The results show that rotary-shaft-seal failure leads to decreases in both the head and efficiency of the centrifugal pump. For improved operation stability, centrifugal pumps should operate at or slightly above their design flow rates (Q_d), while avoiding low-flow conditions. Furthermore, the amplitude of the EMD-AR spectrum increases progressively as rotary-shaft-seal degradation worsens. Therefore, the EMD-AR spectrum provides a reliable diagnostic indicator for detecting rotary-shaft-seal damage.