Nonlinear Vibration Characteristics Of Eccentricity Rotor Of Permanent Magnet Synchronous Motor In HEV

The electromechanical transmission system of hybrid electrical vehicle(HEV)is complex,and the eccentricities such as mass eccentricity,static eccentricity(parallel misalignment)and angular eccentricity(angle misalignment)of the rotor of permanent magnet synchronous motor(PMSM)used in the system are caused by the manufacturing tolerances and assembly errors even if a dynamic balancing is performed for the rotor.Therefore,when PMSM runs geometrical eccentricities occur which cause unbalanced magnetic pull(UMP)and additional transverse electromagnetic moment with the nonlinearity resulting in the complex dynamic behaviors.In this paper,we focus on this problem and investigate the dynamic characteristics of the rotor in whole frequency domain due to PMSM operating over wide speed range.First,the radial stiffness characteristics of the perfect alignment rotor system are analyzed.The analysis shows that the system undergoes the pitchfork bifurcation when the system stiffness in the case of the concentric rotor and stator changes from negative number into positive number.The system has multiple equilibrium points which include an isolated equilibrium point and a continuum of equilibrium points.The latter is cyclically symmetric with respect to the former.Using eigenvalue-based stability analysis and center manifold theorem,the stability analysis for the multiple equilibrium points is performed which offers a stiffness condition for the system stability.In free vibration study,amplitude modulation effects are discovered of which the mechanism is explained and the period of modulating signal is carried out by phase analysis and averaging method.The analysis indicates that the effects are caused by the modal coupling of forward and backward whirling motions.In forced vibration study,considering mass eccentricity,frequency characteristics are obtained by harmonic balance method,and the stability of periodic solution is investigated by Routh–Hurwitz criterion.The frequency characteristics analysis indicates that the response amplitude is limited in the range between the amplitudes of the two kinds of equilibrium points.Furthermore,the frequency characteristics curve does not pass through the unstable equilibrium point if the damping is nonzero.In the vicinity of the continuum,the system hardly provides resistance to bending,and hence external disturbances easily cause loss of stability.Second,the radial stiffness characteristics analysis of the parallel misalignment rotor system reveals that the zero points of the restoring force of rotor shaft and UMP do not coincide with each other when the static eccentricity is introduced.Therefore,the cyclic symmetry of the equilibrium points is spoiled and the continuum is degrades into two unstable isolated equilibrium points.Considering the static eccentricity ratio as an unfolding parameter,a generic bifurcation with defect of the system equilibrium points occurs.Therefore,for the parallel misalignment rotor system,in order to obtain the stability,the static eccentricity should be limited in the range of the critical static eccentricity(eccentricity condition)in addition to the stiffness condition.The frequency characteristics of the main resonance and the super-harmonic resonance caused by the mass eccentricity are investigated through multi-scale method.The result exhibits that the modal coupling of the forward and backward whirling motions occurs.In the main resonance,we can observe the jump phenomenon,and the amplitude-frequency curve of the forward whirling motion is similar to that of the perfect alignment rotor system,while the backward whirling motion is very weak.However,this motion increases significantly for the comparatively small damping,large mass eccentricity and static eccentricity.In the super-harmonic resonance,the amplitude-frequency curves of the two whirling motions are both single-valued.However,the influences of the damping,mass eccentricity and static eccentricity on the characteristics are similar to those of main resonance.Third,an additional transverse electromagnetic moment caused by the inclination of the rotor is modeled based on Maxwell stress tensor method and the analytical expression is carried out.The stiffness characteristics of the angle misalignment rotor system are similar to those of the parallel misalignment rotor system.Under the prerequisite of the stiffness condition of the system stability,when the static angular eccentricity does not exist,the nonlinearity of this moment results in the multiple equilibrium points: the isolated equilibrium point and the continuum of equilibrium points.When static angular eccentricity is introduced,the symmetry of the equilibrium points is spoiled and the continuum degrades into two unstable equilibrium points.The static angular eccentricity which passes through the critical values leads to a generic bifurcation with defect.Therefore,the stiffness condition and eccentricity condition are complete.The frequency characteristics of the system are obtained by harmonic balance method.However,comparing with the radial vibration,the frequency response curves generally pass through all the equilibrium points.For the disk-shaped rotor,the frequency characteristics exhibit hardening type and in a same branch of the frequency curves,all of the solutions have the same stabilities as the equilibrium point in this branch.For the cylindrical rotor,the characteristics show softening type and only the solutions running near the equilibrium point have the same stabilities as the equilibrium point in a same branch and there exists the frequency band in which the response is globally unstable.However,for the comparatively large damping,inertia ratio,small stiffness ratio and static angular eccentricity,the frequency characteristics of the cylindrical rotor are similar to those of the disk-shaped rotor.The frequency band with globally unstable response turns smaller and then vanishes.Even the stabilities of the solutions are consistent in a same branch.Last,the approximate analytical expressions of UMP and additional transverse electromagnetic moment caused by comprehensive eccentricity are obtained based on the Maxwell stress tensor method.A four degree-of-freedom motion equation of the rotor system is modeled and then its dynamic behaviors are analyzed and the theoretical analysis is verified by the test.