Analytical Identification of Magnetic Faults Induced by Material Properties and Mechanical Tolerance in PM Synchronous Motor

This paper presents an analytical model suitable for analyzing Permanent Magnet Synchronous motors. The difference between mass eccentricity and magnet eccentricity is described. The full derived vibration mathematical model incorporates both the electrical domain and mechanical domain. The shaft bending force, mass eccentricity exciting force, aerodynamic exciting force, and unbalanced-Magnet-Pull (UMP) are introduced in this three-dimensional motor model. The relative displacement between rotor and stator is calculated in order to avoid rotor to stator rub. The experiments were conducted with Laser Vibrometer for verifying the analysis results. The measurement results prove the effectiveness of the proposed analytic method.


Introduction
Because many tools and machines used today are derived by motors, vibrations can be devastating. Vibration shortens the life of the motor and damages all the components around them. In order to fully understand motor vibration behavior, a thorough theoretical derivation of motor dynamics should be carried out as they can disclose clearly the global performance of the motor. Generally, four types of excitation force such as mass eccentricity force, shaft bending force, UMP force, and fan aerodynamic force can generate the motor lateral and axial vibration and produce motor noise. Mechanical unbalance in a rotating machine is a condition of unequal mass distribution at each section of the rotor [1]. When the unbalanced machine is rotating, the rotor mass center does not coin-cide with the rotating axis and the eccentric force is generated on the rotor. Vibration and stress are induced in the rotor itself and in its supporting structure, which may gradually lead to excessive wear in the joints: bearings, bushings, shafts and spindles. Eventually, the whole system may be broken down. This is a very common malfunction in rotating machines. Vibrations due to one machine's mechanical unbalance may also be transmitted through the floor to adjacent machinery and seriously impair its accuracy or proper functioning, or be transmitted through either the machine structure or the air to generate structure bone or air bone acoustic noise, which will decrease the machine's performance and the quality of the working environment.
Several researchers have studied rotor mass eccentricity in rotating machines in recent years. Rajagopalan et al. [2] proposed a methodology to distinguish mechanically unbalanced rotors by monitoring signals at two side bands of the fundamental frequency of the Brushless DC (BLDC) motor phase current under no stationary conditions. Huang [3] studied the characteristics of torsion vibrations of an unbalanced shaft using the numerical simulation method, and the simulation results are agreeable with the experimental results. Concari et al. [4] proposed a method to discern mechanical torque unbalance in the induction motor. Their method uses the motor phase current sideband component to estimate the unbalance. Kr. Jalan and Mohanty [5] used a motor-based fault diagnosis technology to diagnose the misalignment or unbalanced motor by building the methodical model and calculating residual vibration on the healthy motor.
Other than Mechanical Unbalance (MU), Unbalance Magnetic Pull (UMP) is another big concern that demands a thorough study and understanding of motor design and diagnosis. In fact, UMP generated by rotor eccentricity faults has been an active research topic for more than a hundred years [6]. A switched reluctance motor by Garrigan et al. [7] considered only static rotor eccentricity faults, but also dynamic rotor eccentricity faults. The authors show that UMP can be quickly predicted by their developed Magnetic Equivalent Circuit (MEC) approach when the relative rotor eccentricity is less than 25% of the normal air-gap. Based on the motor shaft movement orbit from a mechanical point of view, Werner [8] uses shaft vibration signals to study induction motor static UMP due to the static eccentricity fault. On the other hand, Rajagopalan et al. [9] use a field reconstruction method to study static UMP in a Permanent Magnet Synchronous Motor (PMSM) Bi et al. [10] analyzed and calculated dynamic UMP within one motor revolution. The lowest order of the extrinsic UMP harmonic is one-in-one motor revolution. Even though there is no eccentricity distance between the rotor center and stator center, the UMP is only related to EM structures such as the Pole pair number, and the slot number still exists.
This type of UMP is called intrinsic UMP, which is formed in even harmonics and can be eliminated by an even slot number. In this paper, the BLDC mathematic model has introduced and validated by oriental motor experimental case study.

Mathematical Model
Due to machining and assemble accuracy limitations, an electrical motor could have a few types of eccentricities. The most important eccentricities in a motor are rotor mass eccentricities, magnetic eccentricities, and rotor bending deflection. In this work, we consider a horizontal mounting motor for analysis. Its dynamic model can be illustrated as in Figure 1. The PM synchronous motor model consists of two lumping mass: rotor mass, Mr, and stator mass, Ms. The rotor is supported by an oil film sleeve bearing at the two ends of the shaft. The oil film sleeve bearing can be modeled by stiffness and damping matrix. For a rotating electrical machine, an additional magnetic stiffness matrix Cn between the rotor and stator exists, which describes the electromagnetic coupling between them. Due to the sleeve bearing housing, damping effect is very low compared to its oil film. Therefore, the bearing house stiffness matrix is the only one to be considered. The motor normally is mounted on the soft base to isolate the vibration from the ground. Compared to the stator of a PM synchronous motor, the mounting media is more flexible, so stiffness and damping matrix is taken into account.
In order to simplify the analysis, the dynamic model can be separated into three sub-models: rotor model, bearing housing model, and stator model. Rotor model includes magnet, rotor yoke, and shaft; bearing house model includes bearing components; stator model includes armature winding, stator core, motor base, left-side and right bearing covers.

Rotor Model
where w m is rotor mass. xx c , yy c , zz c is the bearing stiffness coefficient in X, Y and Z directions; d xx , d yy , d zz is the bearing damping coefficient in X, Y and Z directions.

Stator Model
It is assumed that the motor is flexibly mounted on the ground through horizontal and vertical spring with damper. The spring stiffness and damper coefficient in X and Y direction are same. The stator mass is located at its geometry center. The dynamic action forces on the X S , Y S , Z S is two end bearing house action forces, rotor magnet action force and supporting spring and damper action forces. In Figure 4, the individual forces on the stator in X, Y and Z directions are: ( ) The following assumptions are made for the geometrical relationship of the  rotor mass magnet system: • The distance between fluid bearing orbit and pre-bent shaft orbit is ˆr e ; • The distance between rotor orbit and rotor mass center is ˆm e ; • The distance between rotor orbit and rotor magnet center is ˆn e .
Thus from Figure 2 the equations can be derived as: ( ) ( ) Given by motor whole system mass matrix M, damping matrix D, stiffness Matrix C, coordinate vector and excitation force vector, the linear inhomogeneous differential equation can be derived: q is coordinate vector and can be present as: System mass matrix M can be expressed as: and M 11 is 6 × 6 mass matrix which includes mass of stator and that of rotor as: System damping matrix D also can be given by: and D 11 is damping of oil film in bearing between stator and ground in xyz directions and as: D 22 is damping of oil film in bearing between rotor and stator in xyz directions.
System stiffness matrix C also can be represented as: and it includes four sub matrix which are present as: which n C is the stiffness of rotor in radial direction respectively; C zz is that of rotor in axial direction. C by and C gy , C bz and C gz are the stiffness of bearing and ground in y and z directions respectively.
C bx of C 12 is the stiffness of bearing in x direction. Moreover, C 12 = C 21 .
In Equation (16) where is the rotor rotating speed and m ϕ is the rotor rotating phase angle. Assuming that the radius of rotor is R, then, the bending moment I of rotor can be described as: The following assumptions are made: the load which makes the rotor bend is uniformly distributed, the rotor distance between two sliding bearing is L, the rotor material modulus is E, then, the deflection of rotor can be expressed as: Based on Equation (29), f r can be calculated as: Magnet eccentricity forces are also referred as Unbalanced Magnetic Pulls If the rotor center is offset from the stator center and it also rotates around the stator center, the dynamic UMP will be generated. The dynamic UMP excitation force vector is: Regarding forced vibration, the standard harmonic excitation is considered, and the equation of motion in frequency domain takes the form, where ω is the forcing frequency, { } F is the complex force vector, and { }   Table 1 shows that only number of pole order vibration has in the static eccentricity faulty motor. There are first order and number of pole plus one order vibration in the dynamic eccentricity faulty motor. These analytical results will be verification through experimental results later.

Results and Discussions
The experimental specimen design and vibration measurement are the same as in [11]. The real time vibration results of the healthy motor and the faulty ones with the static and dynamic eccentricities are shown in Figures 5(a)-(c) respectively.
It can be seen that the faulty condition is difficult to be identified in time domain even though the low pass filter is used.
The rotor y-z orbits of the healthy motor, the faulty motor with static eccentricity, and that with the dynamic eccentricity are shown in Figures   6(a)-(c) respectively. It can be observed that the faulty motor and the healthy one can be easily differentiated; the vibration pattern of the healthy motor is much more regular than that of faulty motor. Moreover, the vibration amplitude of the healthy motor is much smaller than that of the faulty motor. Therefore, the technology can be adopted to distinguish faulty motor from healthy motor.
However, the type of fault cannot be distinguished, whether it is due to static or dynamic eccentricity. So the orbit of the rotor cannot be used as a tool to diagnose the motor running fault condition. Therefore, in this work, FFT signal processing algorithm is developed to obtain the vibration amplitude with different faulty grades and the results in each frequency in normalized percentage are shown in Table 2. From the Table 2, we find that the vibration speed amplitude changes of static eccentricity is the largest in 10th orders of the running speed compared to others types of motor in two directions, we therefore can decide that the motor has a static eccentricity. On the other hand, we find that the vibration speed amplitude changes of dynamic eccentricity is the largest in 11th orders running speed compared to others types of motor in two directions. Moreover, there are second significant changes in 1st order running speed in the case of dynamic eccentricity.

Conclusion
In this paper, we have introduced a mathematical model which takes into ac-