An analytical calculation method for the natural frequency of the rotor of a disc permanent magnet motor

Abstract

This invention discloses a method for analytically calculating the natural frequency of a disc-type permanent magnet motor rotor, comprising the following steps: S1. Equilibrating the rotor backplate to a circular ring plate and establishing the vibration equation for the equivalent circular ring plate in a polar coordinate system; S2. Obtaining the natural frequency of the equivalent circular ring plate based on the boundary conditions of the rotor backplate's free modes; and S3. Determining the equivalent stiffness of the rotor backplate based on the natural frequency of the equivalent circular ring plate to obtain the rotor's natural frequency. This method enables rapid and accurate calculation of the natural frequency of a disc-type permanent magnet motor rotor, providing a theoretical basis for high-performance motor design.

Description

Technical Field

[0001] The invention belongs to the technical field of natural frequency analytical calculation, and in particular relates to a natural frequency analytical calculation method for a disc-type permanent magnet motor rotor. Background Art

[0002] Disc permanent magnet motors offer advantages such as high power density, compact size, and large moment of inertia, making them suitable for applications requiring high torque density and compact space, such as electric vehicles, renewable energy systems, flywheel energy storage systems, and industrial equipment. However, during actual operation, the motor's inherent vibration-induced noise is a source of pollution. Excessive noise can not only damage the motor itself and surrounding equipment, but can also adversely affect human health. Therefore, vibration noise is a key consideration in the design of disc permanent magnet motors. The primary source of motor noise is electromagnetic noise, which is primarily determined by the axial electromagnetic force in the air gap, the dynamic response of the motor structure, and the radiation characteristics of the motor surface. Calculating the natural frequency of the motor structure is a prerequisite for implementing electromagnetic noise calculation and suppression techniques. For disc permanent magnet motors, whose rotor is the primary noise radiation source, calculating the rotor natural frequency is a crucial step in this process. Natural frequency and mode shapes can characterize the vibration characteristics of the motor, facilitating the study of motor vibration noise.

[0003] Numerical and analytical models for the natural frequency of radial flux permanent magnet motors (PMMs) have been extensively studied. Researchers typically use an electromechanical analogy, comparing the stator or rotor to a thin cylindrical shell for analytical calculations. However, because the magnetic flux path of a disk-type PMM is axial, the electromagnetic forces and vibration directions acting on the stator and rotor are all axial, making the analytical calculation method based on the thin cylindrical shell unsuitable for disk-type PMMs. Therefore, researchers generally use the finite element method (FEM) to calculate the natural frequency of disk-type PMMs. The FEM method for the natural frequency of disk-type PMMs primarily combines FEM with experimental results. Experimental results are used to modify the FEM model. Once the model is modified, all motors with the same structure and constraints can be calculated using the same FEM. However, the FEM method has long computation times and consumes significant computer resources. Furthermore, most research focuses on the stator, with little research on the rotor's natural frequency. Currently, the lack of a fast and effective analytical method for calculating the natural frequency of disk-type PMM rotors has created significant challenges for researchers in optimizing the noise design of disk-type PMMs. Summary of the Invention

[0004] Technical Problem Solved: In response to the above technical problems, the present invention provides a method for analytically calculating the natural frequency of a disc-type permanent magnet motor rotor, which can quickly and accurately calculate the natural frequency of the disc-type permanent magnet motor rotor.

[0005] Technical solution: A method for analytically calculating the natural frequency of a disc-type permanent magnet motor rotor, comprising the following steps:

[0006] S1. Equivalently transform the rotor backplate into a circular ring plate and establish the equivalent circular ring plate vibration equation in a polar coordinate system.

[0007] S2. Obtain the natural frequency of the equivalent annular plate based on the boundary conditions of the rotor backplate free mode;

[0008] S3. Determine the equivalent stiffness of the rotor backplate based on the natural frequency of the equivalent annular plate, and obtain the natural frequency of the rotor.

[0009] Preferably, in step S1, the vibration equation of the equivalent annular plate is:

[0010]

[0011] Among them, C1, C2, C3, and C4 are coefficients determined by boundary conditions, and J n , I n 、Y n , K n They are the first kind Bessel function, the first kind modified Bessel function, the second kind Bessel function, and the second kind modified Bessel function, n is the modal order of the annular plate, λ is the frequency coefficient, A and θ are constants, is the axial displacement The magnitude of r is the radial distance coordinate, is the angle coordinate, and t is the time.

[0012] Furthermore, the and axial displacement The relationship is as follows:

[0013]

[0014] Where e is a natural constant, i is an imaginary unit, and ω is the natural frequency of the equivalent annular plate.

[0015] Furthermore, the axial displacement is calculated as follows:

[0016]

[0017] Where D is the bending stiffness of the annular plate, which is related to the Young's modulus E, Poisson's ratio v, and thickness h of the annular plate; is a biharmonic operator, is the Laplace operator; ρ is the density of the annular plate.

[0018] Preferably, the boundary conditions of the free mode of the rotor back plate in step S2 are that the inner boundary is free and the outer boundary is free, and the free boundary conditions are:

[0019]

[0020] in, is the axial displacement The magnitude of r is the radial distance coordinate, is the angle coordinate, t is the time, v is the Poisson's ratio of the annular plate, is the Laplace operator.

[0021] Preferably, the natural frequency of the equivalent annular plate in step S2 is calculated as follows:

[0022]

[0023] Where ω is the natural frequency of the equivalent annular plate; λ is the frequency coefficient; D is the bending stiffness of the annular plate, which is related to the Young's modulus E, Poisson's ratio v, and thickness h of the annular plate; and ρ is the density of the annular plate.

[0024] Furthermore, the matrix equation for the frequency coefficient λ is calculated as follows:

[0025]

[0026] Among them, C1, C2, C3, and C4 are coefficients determined by the boundary conditions, M1 and Q1, M2 and Q2, M3 and Q3, and M4 and Q4 are coefficients of the first-kind Bessel function, the first-kind modified Bessel function, the second-kind Bessel function, and the second-kind modified Bessel function in the boundary conditions, respectively. r is the radial distance coordinate, r=a represents the inner boundary condition of inner radius a, and r=b represents the outer boundary condition of outer radius b.

[0027] Preferably, the equivalent stiffness of the rotor backplate in step S3 is calculated as follows:

[0028] K bp =(2πω) 2 M bp

[0029] Among them, K bp is the equivalent stiffness of the rotor back plate, π is the circumference of the circle, ω is the natural frequency of the equivalent annular plate, M bp is the mass of the rotor backing plate.

[0030] Preferably, the natural frequency of the rotor in step S3 is calculated as follows:

[0031]

[0032] Among them, f rn is the natural frequency of the rotor, π is the circumference of the circle, K bpis the equivalent stiffness of the rotor back plate, M rotor is the mass of the rotor.

[0033] Beneficial effects: The present invention equates the rotor backplate to a circular ring plate of the same thickness, and incorporates the permanent magnets and glue into the rotor backplate as additional mass. Taking the natural frequency of the rotor's free mode as an example, the relative error between the analytical calculation result and the finite element result is within |5%|, providing a fast and accurate analytical calculation method for researchers of disc permanent magnet motors to analyze vibration noise; the present invention lays the foundation for further research on the electromagnetic noise calculation and noise optimization method of disc permanent magnet motors. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 Schematic diagram of the rotor structure of the disc-type permanent magnet motor in this embodiment, wherein 1 is the rotor back plate and 2 is the permanent magnet;

[0035] Figure 2 4 is a flow chart of the analytical calculation method for the natural frequency of the disc-type permanent magnet motor rotor in this embodiment. DETAILED DESCRIPTION

[0036] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0037] Example 1

[0038] The implementation test of the present invention was carried out on the rotor of a certain type of disc-type permanent magnet motor.

[0039] Figure 1 The figure is a schematic diagram of the rotor structure of a certain disc permanent magnet motor. 1 is the rotor back plate and 2 is the permanent magnet. Figure 1 The rotor natural frequency is calculated analytically.

[0040] Figure 2 The process of the analytical calculation method for the rotor natural frequency of a disc permanent magnet motor is provided, including the following specific implementation steps:

[0041] Step 1: Establish an equivalent circular ring plate model based on the prototype rotor, then establish the vibration equation of the equivalent circular ring plate in the polar coordinate system, and solve the general solution of the vibration equation.

[0042] 1) In the polar coordinate system, establish the vibration equation of the equivalent circular ring plate:

[0043]

[0044] Where D is the bending stiffness of the annular plate and is related to the Young's modulus E, Poisson's ratio v, and thickness h of the annular plate. is a biharmonic operator, is the Laplace operator, is the axial displacement, r is the radial distance coordinate, is the angular coordinate, t is the time, and ρ is the density of the annular plate;

[0045] 2) Establish axial displacement The expression:

[0046]

[0047] in, is the axial displacement The amplitude of , e is a natural constant, i is an imaginary unit, ω is the natural frequency of the equivalent annular plate;

[0048] 3) Substitute formula (2) into formula (1) to obtain The expression:

[0049]

[0050] Among them, C1, C2, C3, and C4 are coefficients determined by boundary conditions, and J n , I n 、Y n , K n are the first kind Bessel function, the first kind modified Bessel function, the second kind Bessel function, and the second kind modified Bessel function, n is the modal order of the annular plate, λ is the frequency coefficient, A and θ are constants;

[0051] Step 2: Taking the rotor back plate under the free mode as an example, the boundary conditions are free inside and free outside, and the natural frequency of the equivalent annular plate is obtained.

[0052] 1) The free boundary is:

[0053]

[0054] 2) Substitute equation (3) into equation (4) to obtain the matrix equation for the frequency coefficient:

[0055]

[0056] Wherein, M1 and Q1, M2 and Q2, M3 and Q3, M4 and Q4 are the coefficients of the first-kind Bessel function, the first-kind modified Bessel function, the second-kind Bessel function, and the second-kind modified Bessel function in the boundary conditions, respectively; r = a represents the boundary condition of inner radius a, and r = b represents the boundary condition of outer radius b;

[0057] 3) Obtain the frequency coefficient λ through the coefficient matrix of formula (5), and then obtain the natural frequency of the equivalent annular plate:

[0058]

[0059] Step 3: According to the natural frequency of the equivalent annular plate, the equivalent stiffness of the rotor back plate is determined to obtain the natural frequency of the rotor.

[0060] 1) Determine the equivalent stiffness of the rotor backplate:

[0061] K bp =(2πω) 2 M bp (7)

[0062] Among them, K bp is the equivalent stiffness of the rotor back plate, π is the circumference of the circle, M bp is the mass of the rotor back plate;

[0063] 2) Obtain the natural frequency of the rotor:

[0064]

[0065] Among them, f rn is the natural frequency of the rotor, M rotor is the mass of the rotor.

[0066] Step 4: Extract the analytical calculation results of the second-order, third-order, and fourth-order free modal natural frequencies and compare them with the finite element results, as shown in Table 2.

[0067]

[0068] As can be seen from Table 2, the errors between the analytical results and the finite element results are both within |3%|, which demonstrates the feasibility and accuracy of the analytical calculation method for the rotor natural frequency of the disc permanent magnet motor proposed in this invention and provides an effective theoretical basis for optimizing the noise of the disc permanent magnet motor.

[0069] The analytical calculation method proposed in this invention can quickly and accurately calculate the natural frequency of the rotor of a disk permanent magnet motor and can be used for electromagnetic noise calculation and noise optimization design of disk permanent magnet motors. Taking the rotor of a certain type of disk permanent magnet motor as an example, this invention details the specific implementation process of the method proposed in this invention, analytically calculates the axial free modal natural frequency of the rotor, and compares it with the calculation results of the finite element method to verify the effectiveness of the invention. This invention provides an effective calculation method for disk permanent magnet motor researchers to quickly and accurately calculate the rotor natural frequency, laying the foundation for improving the performance of disk permanent magnet motors.

Claims

1. A method for analytically calculating the natural frequency of a disc-type permanent magnet motor rotor, characterized in that: The steps are as follows: S1. Equivalently transform the rotor backplate into a circular ring plate and establish the equivalent circular ring plate vibration equation in the polar coordinate system. The equivalent circular ring plate vibration equation is: Among them, C1, C2, C3, and C4 are coefficients determined by boundary conditions, and J n , I n 、Y n , K n They are the first kind Bessel function, the first kind modified Bessel function, the second kind Bessel function, and the second kind modified Bessel function, n is the modal order of the annular plate, λ is the frequency coefficient, A and θ are constants, is the axial displacement The magnitude of r is the radial distance coordinate, is the angle coordinate, t is the time; S2. Obtain the natural frequency of the equivalent annular plate based on the boundary conditions of the rotor backplate free mode. The boundary conditions of the rotor backplate free mode are free inner and free outer boundaries. The free boundary conditions are: in, is the axial displacement The magnitude of r is the radial distance coordinate, is the angle coordinate, t is the time, v is the Poisson's ratio of the annular plate, is the Laplace operator; S3. Determine the equivalent stiffness of the rotor backplate based on the natural frequency of the equivalent annular plate and obtain the natural frequency of the rotor. The equivalent stiffness of the rotor backplate is calculated as follows: K bp =(2p) 2 M bp Among them, K bp is the equivalent stiffness of the rotor back plate, π is the circumference of the circle, ω is the natural frequency of the equivalent annular plate, M bp is the mass of the rotor backing plate.

2. The method for calculating the natural frequency of a disc-type permanent magnet motor rotor according to claim 1, characterized in that: described and axial displacement The relationship is as follows: Where e is a natural constant, i is an imaginary unit, and ω is the natural frequency of the equivalent annular plate.

3. The method for analytical calculation of the natural frequency of a disc-type permanent magnet motor rotor according to claim 2, characterized in that: The axial displacement is calculated as follows: Where D is the bending stiffness of the annular plate, which is related to the Young's modulus E, Poisson's ratio v, and thickness h of the annular plate; is a biharmonic operator, is the Laplace operator; ρ is the density of the annular plate.

4. The method for analytical calculation of the natural frequency of a disc-type permanent magnet motor rotor according to claim 1, characterized in that: The natural frequency of the equivalent annular plate in step S2 is calculated as follows: Where ω is the natural frequency of the equivalent annular plate; λ is the frequency coefficient; D is the bending stiffness of the annular plate, which is related to the Young's modulus E, Poisson's ratio v, and thickness h of the annular plate; and ρ is the density of the annular plate.

5. The method for analytical calculation of the natural frequency of a disc-type permanent magnet motor rotor according to claim 4, characterized in that: The matrix equation for the frequency coefficient λ is calculated as follows: Among them, C1, C2, C3, and C4 are coefficients determined by the boundary conditions, M1 and Q1, M2 and Q2, M3 and Q3, and M4 and Q4 are coefficients of the first-kind Bessel function, the first-kind modified Bessel function, the second-kind Bessel function, and the second-kind modified Bessel function in the boundary conditions, respectively. r is the radial distance coordinate, r=a represents the inner boundary condition of inner radius a, and r=b represents the outer boundary condition of outer radius b.

6. The method for analytical calculation of the natural frequency of a disc-type permanent magnet motor rotor according to claim 1, characterized in that: The calculation method of the natural frequency of the rotor in step S3 is as follows: Among them, f rn is the natural frequency of the rotor, π is the circumference of the circle, K bp is the equivalent stiffness of the rotor back plate, M rotor is the mass of the rotor.

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