The effect of unbalanced magnetic pull (UMP) caused 6 by air gap eccentricity on the vibration of a permanent magnet 7 synchronous motor (PMSM) is investigated. The force model is 8 established analytically by the Maxwell stress method. For accu-9 rate consideration of the eccentricity condition, mixed eccentricity, 10 axial-varying eccentricity, and eccentricity caused by motor frame 11 vibration are modeled and combined. The model of the rotor-12 bearing system, which includes the UMP model, is developed with 13 two different methods. In the first method, UMP is added as a 14 linear negative spring to the rotor model, whereas in the second 15 method, UMP is included as an external force. The rotor system of 16 a centrifugal pump driven by an integrated PMSM is modeled using 17 beam elements, and the two distinct modeling approaches for UMP 18 are applied. From the results, the UMP effect on vibration and the 19 difference between the two modeling methods are investigated. To 20 verify the results of the analysis, experimental work is done with a 21 pump test rig, and results of frequency spectra are obtained. Based 22 on the analyses and experimental work, the negative stiffness effect 23 and additional vibration excitations caused by UMP are examined. 24 Index Terms-Axial-varying eccentricity, Eccentricity by frame 25 vibration, Permanent magnet synchronous motor, Mixed eccentric-26 ity, Unbalanced magnetic pull. 27 I. INTRODUCTION 28 E LECTROMECHANICAL interaction in rotating electrical 29 machines is a significant factor in the generation of nonlin-30 ear dynamic behavior of a system. In machines with a small air 31 gap, such nonlinear dynamic behavior can be dangerous for the 32 rotor system, and therefore, many studies have been conducted 33 on unbalanced magnetic pull (UMP) caused by electromechan-34 ical interaction. 35 The topic of UMP has been addressed in numerous studies 36 covering various factors such as asymmetry of rotor and stator, 37 rotor eccentricity, and magnetic saturation. Ortega et al.

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