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Vibration analysis and vibration mitigation are very important to ensure the safe and stable operation of electrical machines applied in flywheel energy storage systems. In traditional integer slot synchronous machines, it is generally believed that there is only the zeroth-mode vibration caused by odd tooth harmonics. However, the homopolar inductor machine (HIM) adopts the end cap excitation structure, which makes its air-gap magnetic field contain the dc component, odd tooth harmonics and even tooth harmonics. The force density distribution on the slotted stator is analyzed by the magnetomotive force-multiplying permeance method. The results show that there are both zeroth- and pth(pole pairs)-mode vibrations caused by tooth harmonics and dc component of the air-gap flux density. Furthermore, to reduce vibration and resonance, and to improve the electromagnetic performance of HIM, it is best to use the stator skew slot, the magnetic slot wedge and the sinusoidal rotor slot at the same time. Finally, a 54-slot/6-pole end cap permanent magnet excitation homopolar inductor machine (EPE-HIM) is tested to validate the analytical and the finite-element method (FEM) results, and the variation trend of the experimental results and FEM results is the same.