During the transient start-up process (SUP) of mixed-flow pumps (MFP), pressure pulsation (PP) and principal shaft vibration (PSV) in MFP are two main indicators for analyzing failure types and evaluating operational stability. To investigate the effect of start-up mode (SUM) on the stability of MFP, the PP and PSV of MFP under nonlinear and linear SUM were collected. And the stability of MFP under different SUMs was quantitatively evaluated by the proposed WF-SampEn method which combining two-dimensional (2D) sample entropy and wavelet fusion diagram. Then, the relationship between the SUM and the stability of the MFP were studied. Experimental results indicate that the energy distribution of PSV in the frequency domain can be improved by the concave exponential function (CCEF) SUM. And the CCEF SUM can suppress the shock vibration of the shaft and reduce the duration of high-amplitude (HA) PSV, thereby decreasing the probability of shaft instability failure. Specifically, 2D sample entropy of the PSV and PP are the lowest when the MFP start-up in CCEF SUM, and the stability of MFP in CCEF SUM is 39.9% and 22.8% higher than that in Linear and CVEF SUM. The results provide a quantitative theoretical criterion and a practical diagnostic tool for designing start-up strategy and preventing instability-induced failures in fluid machinery.
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