Vibration mode analysis of multi-degree-of-freedom permanent magnet synchronous motor

2.1. Composition of structure

The liquid suspension hybrid drive multi-degree-of-freedom permanent magnet synchronous motor mainly includes the rotor, stator coil, spherical shell and the liquid oil film between two layers of spherical shells. In the multi-degree-of-freedom motor, the mechanical bearing between the stator and rotor of the motor is different from the traditional motor structure. The motor is filled with oil film between the stator shell and the rotor, and the rotor is suspended in the liquid. In the way, the friction wear of the motor can be greatly reduced, which is more flexible and applicable to be controlled.

According to different requirements, the motor can be divided into output shaft structure and non-output shaft structure. The non-output shaft is also called the rotor embedded structure. The difference between them is that the oil film structure of the latter is a complete hollow spherical structure, as shown in Figs. 1 and 2.

Fig. 1Model with non-output shaft

Fig. 2Model with the output shaft

A large range adjustable permanent magnet is embedded in the vertical direction of the rotor shell, and a single stage fine tuned permanent magnet is embedded in the tail. The output shaft is fixedly connected with the permanent magnet. There is a fine motion coil system composed of 5 coils and cores at the end of the spherical shell [14]. The hybrid drive mode is adopted by multi-degree-of-freedom permanent magnet synchronous motor. By controlling the magnitude and direction of the coil currents, the large range motion and fine deflection motion of the rotor can be achieved. In the control process, a large range motion control method is used. If the control accuracy cannot be achieved, then the fine control system is used to fine tune the rotor to achieve the control accuracy. The control accuracy and efficiency can be improved to the greatest extent through this control mode. The rotation and deflection of the motor do not affect each other. The detailed arrangement of structure can be shown in Figs. 3 and 4.

Fig. 3Structure of the large motion adjustment

Fig. 4Structure of the small motion adjustment

2.2. Modeling of the motor

In the established model, the embedded shaft structure is selected. The stator shell and the rotor are built into two complete spherical structures. According to the structure which is mentioned above, the assumption of a modeling process can be shown as follows: 1) ignore the magnetic circuit saturation; 2) during the research of magnetic field, ignore the eddy current effects caused by magnetic field changes; 3) the magnetic field created by the energized coils only has effects on the rotor magnetic poles which are nearby the coils. The magnetic field has less effects on rotor magnetic poles which is not adjacent to the coils so that the leakage magnetic field can be ignored; 4) the magnetic material is isotropic medium. Detailed parameters are listed in Table 1.

3.1. Principle of resonance

Resonance is a case where a physical system vibrates at a given frequency in the greater amplitude than other frequencies, which is called the resonance frequency. During the operation of multi-degree-of-freedom permanent magnet motor, the coupling calculation of electromagnetic field, solid mechanics and solid heat transfer are involved, and the coupling and vibration noise produced by operation are more complex [15].

Before investigating the resonance law and coupling, the correct model is set up and the electromechanical coupling is realized. The interaction law and engineering application of electromechanical coupling analysis are studied, and the corresponding material is selected to view the intrinsic properties.

When the multi-degree-of-freedom PM motor in operation, the electromagnetic resonance will be occurred. Under certain frequencies, the multiple modal resonances are excited by electromagnetic resonance at the same time. To stimulate multiple resonances, it needs to satisfy two aspects: 1) it is necessary to satisfy its mathematical relation; 2) the modal of the system is required to have a proper coupling relationship [16]. Therefore, the coupled vibration study of the motor is essential.

3.2. Experimental principles

The sensors are used to monitor the modal changes of the stator shell. The modal analysis experiment is carried out on the stator shell of the multi-degree-of-freedom motor and the natural frequency of the stator is studied by the modal experiment. Compared with the data of the stator modal displacement which measured by FEM, the accuracy is verified. Because the stator of the motor is spherical, it chooses the 2nd order mode of the stator shell as the object of experiment and calculation.

In the experimental platform, the single point excitation of hammer method is applied. The force sensor is mounted on the hammer head to measure the excitation signal [17]. This experiment is used to detect the force deformation of the stator shell under the two order modes. The signals of displacement and deformation are returned to the computer through the sensor, and the analysis is carried out, which is compared with the data of the modal deformation of the stator shell calculated by FEM. The data is imported into the MATLAB, which directly presents the 2nd order modal changes of the stator.

3.3. Coupling of magnetic field and solid

After the main coil of the motor and the fine coil system are energized, since the coils are fixed on the iron cores, the magnetic domain of the magnetic material rotates along the magnetic field under the action of the external magnetic field. The direction of the magnetization tends to be consistent, which leads to the elongation or shortening of the core length along the direction of the magnetic force line, thus producing the magnetostrictive stress within the core [18, 19]. The magnetostriction stress of the stator core of the motor is calculated by the elastic mechanics method. As the motor is energized, the coils generate heat to make the coils and cores for further deformation. The phenomenon of magnetic structure coupling is the main phenomenon of the resonance of the motor. Therefore, in the research of motor structure system, it is also an important content to discuss the deformation caused by the force produced by the iron core after the coils are energized. Therefore, the magnetic field of the motor should be analyzed firstly, which is different from the calculation principle of two-dimensional static magnetic field. The method of three-dimensional static magnetic field calculation is the edge method, that is, the edges of meshing elements are the degrees of freedom for the solution of amount of field parameters.

The magnetic field equations of the motor windings are shown in Eq. (1):

where $A$($A_x$, $A_y$, $A_z$) is the magnetic vector and the $J$($J_x$, $J_y$, $J_z$) is the current density vector; $B$($B_x$, $B_y$, $B_z$) is the magnetic induction vector; $E$ is the electric field intensity; ${\nabla }^2$ is Laplace Operator; $\nabla$ is Hamiltonian operator. By using the method of separation of variables, the internal and external windings are taken into account, and the magnetic induction intensity is expressed as Eq. (2):

where $G_i^k$ is the expression of the coordinate function and the current density, $i=$ 1, 2, ..., 6.

After the above Eq. (2) calculates the magnetic field, according to the principle of classical mechanics, the differential equations of vibration in the system of three degrees of freedom linear vibration system, such as Eq. (3) are obtained:

where $\mathbf{M}$ mass matrix is in the matrix and $\mathbf{C}$ is a damping matrix; $\mathbf{K}$ is the stiffness matrix; $\left\{\mathbf{x}\mathrm{\text{'}}\mathrm{\text{'}}\right\}$ is the acceleration vector; $\left\{\mathbf{x}\mathrm{\text{'}}\right\}$ is a velocity vector; $\left\{\mathbf{x}\right\}$ is the displacement vector; $F\left(t\right)$ is static load. In the present structural analysis, the amount of time variation is neglected, and the Eq. (3) can be simplified as Eq. (4):

According to the basic principle of magneto elastic mechanics, the magnetostrictive stress and magnetostrictive strain are linear; the magnetostrictive stress can be expressed as Eq. (5):

where $\sigma$ is the magnetostrictive stress and the $D$ is an elastic tensor; $\epsilon$ is Magnetostrictive strain.

4.1. Coupling of magnetic and solid

When the motor is stationary, the shell of the stator should be selected and the parameters should be obtained. Because the model is a multi-degree-of-freedom permanent magnet synchronous motor, the stator shell approximation can be regarded as a hollow spherical shell, so only a spherical shell model is needed to illustrate the stator. In the modal analysis, the natural frequency of the model is analyzed. The expression of the intrinsic frequency $A$ is as shown in Eq. (6):

where $m$ is the equivalent mass of the object; $k$ is the equivalent stiffness of the model. By establishing finite element model of stator of the motor, the changes of the stator modal parameters are observed. The 1st order and 2nd order modal diagrams of the stator are shown in Figs. 5 and 6.

The modal analysis of the spherical shell of the stator is carried out in the state of complete free condition. In the finite element simulation software, the mode changes of the first six orders of the stator spherical shell are respectively intercepted under without any constraints.

Since the vibration displacement of the stator surface caused by the radial electromagnetic force is inversely proportional to the 4th mode of the spatial order, the contribution of the first four orders model electromagnetic force to the vibration and noise is usually considered.

It can be seen from Figs. 5 to 6 that the modal deformation of the stator shell is matched with the force wave distribution of each order. Here, the 3rd order model is intercepted, and the 3rd order modal parameters of the stator are obtained, as shown in Fig. 7.

Fig. 5The 1st mode of stator

Fig. 6The 2nd order mode of stator

Fig. 7The 3 order mode of the stator

4.2. Influence of electromagnetic force on vibration modes

During the operation of the motor, the constraints in the surface of the stator shell are naturally formed after the coils are energized. The influence of coil winding on the modal frequency of the motor has been a difficult problem in the modal analysis of the motor. The electromagnetic force generated by the coils will also have some influence on the modal deformation of the motor stator [20, 21].

The whole motor is different from the natural frequency of the stator shell, so only the modal analysis during the operation of the motor is needed to see the displacement and deformation of the stator shell and compared with the modal analysis of single stator shell. The vibration displacement of the surface of the spherical shell can be calculated by modal superposition method [22]. The amplitude of $m$-order vibration displacement is generated by radial force wave of the space m-order, which can be expressed as Eq. (7):

where $D_{si}$ is the stator diameter and $M$ is stator quality; $P_{rm}$ is the amplitude of radial force wave in space $m$ order; $f_m$ is the modal frequency of the stator $m$ order mode; $h_m$ is a structure amplification factor, which can be expressed as Eq. (8):

Here, the order ${\phi }_m$ is the $m$-order modal damping, and $f$ is the m-order electromagnetic force frequency.

There are the modal deformation diagrams of the stator shell and the core coils when the motor is running, as shown Figs. 8 and 9. Compared with Figs. 5 and 6, it can be seen that the modal response of the first three orders of the spherical shell is the inherent and overall characteristics of the elastic material under without any constraint. Through the finite element method, the main modal characteristics of its structure material are derived for the stator shell without external force effects. When the coil is electrified, the stator shell will be affected by the electromagnetic force, so it is necessary to study the modal changes of the stator in the operation process.

As shown in Fig. 8, the first order mode of the whole motor is shown. It can be seen that the natural frequency of the motor is about 1760 Hz. From the deformation distribution, it is found that in the 1st order mode of the motor, the stator has a great influence on the modal frequency of the model. The deformation of the stator core and the coils is small, most of which are concentrated in contacting with the stator shell. It also shows that the motor does not have electromagnetic resonance at this frequency.

Fig. 8The 1st order mode in operation

As shown in Fig. 9, the 2nd order mode of the motor under operation is shown. Compared with the 1st order mode, the stator shell and the iron cores have large deformation, and the iron cores will also appear a certain deviation phenomenon. Because the cores and the frequency of the 2nd mode of the stator shell are close to each other, and the 2nd modal frequency of the overall structure of the motor is relatively low and more likely the electromagnetic resonance will occur near that frequency, which will increase the stress and deformation of the motor.

Fig. 9The 2nd order mode in operation

Fig. 10The 3rd order mode in operation

Fig. 10 shows the 3rd order mode change of the motor. As can be seen the electromagnetic resonance is eliminated when the motor runs and the frequency reaches to 4750 Hz. In this frequency range, the force deformation of the motor cores and the coils are obviously greater than the deformation of the stator shell.