Investigation of analytical solution of super harmonic resonance of rotor system in permanent magnet synchronous motors considering mixed eccentricity

The unbalanced magnetic pull results in the nonlinearity of the system for the permanent magnet synchronous motor rotor having mixed eccentricity. The mass unbalance and static eccentricity form multi-frequency excitation, which results in the coupling effect of forward and backward whirling motions on the steady-state of super harmonic resonance. The analysis is difficult due to the phases of complex amplitudes of two motions which is no longer time invariant compared with the main resonance. To solve this problem, an analysis method of complex amplitudes is presented in this paper. The rotor is considered a Jeffcott rotor, and multi-scale method is applied to this system. A mathematical analysis method of the complex amplitude phases is employed for their time variant characteristics. The results show that both phases are linear time variant and their variation rates are both equal in magnitude to the difference between twice the excitation frequency and natural frequency of the generating system. The characteristics lead to the frequency shift of the two motions by the difference value from the natural frequency of the generating system. The numerical analysis reveals that the phases are independent of the initial condition and hence the complex amplitudes. The analysis solves two problems: the analytical solution of the rotor system carrying phases information can be expressed, which is in good agreement with the numerical result, and the solution construction is analyzed easily, which indicates the contribution of every component of the analytical solution. The phase characteristics analysis is the prerequisite for the frequency characteristics which can be easily carried out, and the coupling mechanism of the two motions is explained.