Analytical calculation method for strength of cylindrical permanent magnet rotor of high-speed motor

Abstract

The application belongs to the field of motor, and relates to a high-speed motor cylindrical permanent magnet rotor strength analytical calculation method. The stress and strain balance equation is obtained by analyzing the rotor stress and deformation; the mechanical differential equation group under the fixed structure and material is obtained; the rotor stress distribution is obtained by applying boundary conditions in different situations to the equation group; the radial strain and radial deformation are obtained by extracting the stress distribution during rotation; the axial deformation and axial strain are obtained through the radial deformation; the actual axial strain and the final axial stress are obtained by introducing a correction coefficient K; the axial stress distribution is obtained by using the static assembly and the radial and tangential stress under the condition of only considering the temperature; the axial stress is introduced into the stress distribution under the two conditions; the axial stress is introduced into the stress distribution during rotation; finally, the stress distribution under three conditions is accumulated to obtain the rotor stress distribution during motor operation. The application is used for high-speed motor cylindrical permanent magnet rotor strength analytical calculation.

Description

Technical Field

[0001] This invention belongs to the field of electric motors, specifically relating to an analytical calculation method for the strength of a cylindrical permanent magnet rotor in a high-speed motor. Background Technology

[0002] High-speed motors (permanent magnet motors) generate significant centrifugal force during operation due to the high-speed rotation of the rotor. This centrifugal force can potentially damage the permanent magnets. Therefore, a protective sleeve is needed to protect the permanent magnets on the outside of the high-speed motor rotor. This is often achieved using an interference fit assembly. Determining the interference fit and calculating the strength of the rotor components under a given interference fit constitutes the rotor strength problem. There are generally two methods for assessing motor rotor strength: finite element simulation and analytical calculation. Finite element simulation is accurate but time-consuming; while analytical calculation can achieve results close to those of finite element simulation while performing rapid calculations. However, traditional analytical calculation methods do not consider axial stress and strain, leading to significant errors for special rotors with cylindrical permanent magnets. This renders analytical calculation methods unsuitable for motor rotors with solid cylindrical permanent magnets. Summary of the Invention

[0003] The purpose of this invention is to solve the problem of large errors in analytical calculations of motor rotors with solid cylindrical permanent magnet structures, and to provide an analytical calculation method for the strength of cylindrical permanent magnet rotors in high-speed motors.

[0004] This invention, through stress and strain analysis of the motor rotor, builds upon traditional methods by analyzing the influence of radial and tangential stresses in the axial direction, calculating the degree of axial deformation, and further obtaining axial strain and stress. It proposes an analytical calculation method for solid cylindrical permanent magnets with results closer to finite element simulation.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for analytically calculating the strength of a cylindrical permanent magnet rotor in a high-speed motor, the method comprising the following steps:

[0007] Step 1: Determine the rotor model, analyze the forces and deformations of the rotor, and obtain the stress and strain equilibrium equations under the rotor model structure; combine the constitutive equations of the rotor material itself to obtain the mechanical differential equations under a fixed structure and fixed material.

[0008] Step 2: By applying the boundary conditions of force and displacement in equation (4) to the mechanical differential equation system in Step 1, solve the mechanical differential equation system and obtain the rotor stress distribution under different boundary conditions, including three cases: static assembly, considering only rotation, and considering only temperature.

[0009]

[0010] Wherein: S o S is the boundary surface subjected to external forces; u Let be the surface with a given displacement; u is the given displacement along the radial direction on the rotor in the polar coordinate system.

[0011] Step 3: Extract the stress distribution considering only rotation in Step 2, apply the obtained relationship between tangential stress and radial strain at this time to obtain the radial strain considering only rotation, and further obtain the radial deformation considering only rotation.

[0012] Step 4: Assuming the rotor volume remains constant, we go from considering only the radial deformation under rotation in Step 3 to considering only the axial deformation under rotation, and further obtain the axial strain under rotation. We introduce a correction factor K due to the volume change into the axial strain under rotation to obtain the final actual axial strain under rotation, and further obtain the final axial stress under rotation.

[0013] Step 5: Using the static assembly from Step 2 and considering only temperature, calculate the radial and rotor tangential stresses σ. θ The axial stress distribution under static assembly and temperature-only conditions is obtained, and the axial stress is introduced into the stress distribution under these two conditions. The axial stress obtained in step four under rotation-only conditions is introduced into the stress distribution under rotation-only conditions. Finally, the stress distributions under the three conditions are summed to obtain the rotor stress distribution under motor operation.

[0014] Furthermore, step one specifically involves:

[0015] The rotor model consists of an outer sheath and a permanent magnet disposed inside the sheath and interference-fitted with it. By analyzing the forces and deformations of the rotor, a set of mechanical differential equations under the rotor model structure is obtained, including stress balance equations and strain balance equations, as shown in equations (1) and (2) respectively:

[0016]

[0017] Where: σ r For rotor radial stress, σ θ Let ρ be the rotor tangential stress, r be the radial position of the analysis point, ρ be the rotor material density, ω be the rotor rotational angular velocity, and ε be the rotor tangential stress. r For rotor radial strain, ε θ This refers to the rotor tangential strain;

[0018] The rotor material is an elastic isotropic material with the same Poisson's ratio ν and Young's modulus E in each direction, resulting in the constitutive equation, as shown in equation (3):

[0019]

[0020] Furthermore, in step three, the process of determining the radial deformation considering only rotation is as follows:

[0021] Solve the mechanical differential equations in step two, namely equations (1) and (2), to obtain the radial stress, tangential stress and radial deformation under three working states: static assembly, rotation only, and temperature only.

[0022] The radial deformation u of the sheath considering only rotation is obtained through equation (5). v,t Equation (6) yields the radial deformation u of the permanent magnet considering only rotation. v,m ;

[0023]

[0024]

[0025] Where: v1 is the Poisson's ratio of the sheath, r is the radial position of the analysis point, ρ1 is the density of the sheath, E1 is the Young's modulus of the sheath, and R... t1 R is the radius of the outer circumference of the sheath. t2 The radius of the inner circumference of the sheath.

[0026] v2 is the Poisson's ratio of the permanent magnet, ρ2 is the density of the permanent magnet, E2 is the Young's modulus of the permanent magnet, and R... m1 ω is the outer diameter of the permanent magnet's outer circular surface, and ω is the rotor's angular velocity.

[0027] Radial deformation u of permanent magnet m for:

[0028]

[0029] Furthermore, in step four, the final process for determining the axial stress considering only rotation is as follows:

[0030] Assuming the permanent magnet has a constant volume during deformation, the corrected axial strain ε can be obtained from equation (9). z0 Equation (10) is obtained by combining equations (8) and (9);

[0031] πR m1 2 l=π(R m1 +u m ) 2 l'(8)

[0032] Where: l is the axial length of the permanent magnet, l' is the axial length of the permanent magnet after deformation, and R m1 U is the diameter of the outer circular surface of the permanent magnet. m The radial deformation of the permanent magnet;

[0033]

[0034]

[0035] A correction coefficient K is introduced into equation (10) to compensate for the reduction in strain caused by volume change. K is a value less than 1, which is related to the ratio of the diameter and axial length of the permanent magnet. The ratio of the half-arc length of the permanent magnet to the axial length is taken to obtain the value of K as shown in equation (11). Then equation (10) becomes equation (12).

[0036]

[0037]

[0038] Where: ε z For the axial strain, the axial strain obtained from equation (12) and the radial and tangential stresses obtained from solving the mechanical differential equations in step two, considering only rotation, are substituted into the material constitutive equation (13) considering the radial, tangential, and axial directions. This yields a system of three linear equations. Solving this system of equations yields the axial stress σ considering only rotation. z As shown in equation (14), the axial stress σ z By combining the radial stress and tangential stress that only consider rotation without considering axial strain, we obtain the axial stress that only considers rotation and takes axial strain into account.

[0039]

[0040]

[0041] In the formula, ε z For axial strain, E r For radial Young's modulus, E θ For tangential Young's modulus, E z For axial Young's modulus, v rθ For radial and tangential Poisson ratios, v rz For radial and axial Poisson's ratios.

[0042] Furthermore, in step five, the process of obtaining the stress distribution of the motor during actual operation is as follows:

[0043] By solving the mechanical differential equations in step two, the radial and tangential stresses under static assembly and considering only temperature are extracted. The axial stress under these two conditions is obtained through equation (15), and the stress distribution under these two conditions is also obtained. Finally, the stress distribution under the three working conditions is obtained through equation (16): considering only the stress σ during rotation. v The compressive stress σ of static assembly sConsidering only the thermal stress σ at temperature T The summation yields the rotor stress distribution during motor operation;

[0044] σ z =v(σ r +σ θ (15)

[0045] σ=σ s +σ v +σ T (16).

[0046] The beneficial effects of this invention compared to the prior art are as follows: This invention is a method for determining the axial stress of a cylindrical solid permanent magnet motor rotor under different working conditions. When studying the strength problem of solid permanent magnets, this invention introduces axial stress into the stress distribution under various working conditions, thereby realizing the determination of the three-dimensional stresses of axial, tangential and radial directions, making the analytical calculation of equivalent stress more accurate. Attached Figure Description

[0047] Figure 1 This is a flowchart of the analytical calculation method for the strength of the cylindrical permanent magnet rotor of the high-speed motor according to the present invention;

[0048] Figure 2 This is a cross-sectional view of the rotor model;

[0049] Figure 3 This is a schematic diagram showing the radial expansion and axial contraction of the permanent magnet when only rotation is considered.

[0050] Figure 4 This is a comparison diagram of the stress distribution of the corrected permanent magnet considering only rotation; where: Figure 4 (a) is the axial stress distribution of the permanent magnet after correction, considering only centrifugal force. Figure 4 (b) is the equivalent stress distribution of the permanent magnet when only centrifugal force is considered after correction;

[0051] Figure 5 This is a comparison diagram of the stress distribution of the corrected permanent magnet during operation; among which: Figure 5 (a) is a diagram showing the axial stress distribution of the permanent magnet during operation after the correction. Figure 5 (b) is the equivalent stress distribution of the permanent magnet during operation after correction. Detailed Implementation

[0052] Specific implementation method one: as follows Figure 1 , Figure 2 As shown in the figure, this embodiment discloses an analytical calculation method for the strength of a cylindrical permanent magnet rotor in a high-speed motor. The method includes the following steps:

[0053] Step 1: Determine the rotor model, analyze the forces and deformations of the rotor, and obtain the stress and strain equilibrium equations under the rotor model structure; combine the constitutive equations of the rotor material itself to obtain the mechanical differential equations under a fixed structure and fixed material.

[0054] Step 2: By applying the boundary conditions of force and displacement in equation (4) to the mechanical differential equation system in Step 1, solve the mechanical differential equation system and obtain the rotor stress distribution under different boundary conditions, including three cases: static assembly, considering only rotation, and considering only temperature (all three cases are obtained from the mechanical differential equation system).

[0055]

[0056] Wherein: S o S is the boundary surface subjected to external forces; u Let S be the surface with a given displacement; u is the displacement along the radial direction on the rotor in polar coordinates (when solving the mechanical differential equations in step one, it is necessary to combine the boundary conditions, and by applying the boundary conditions of force and displacement, solutions under different working conditions can be obtained. For a surface with a given displacement, S is used). u In the polar coordinate system, if the displacement along the radial direction of the rotor is given as u, then the boundary conditions of force and displacement are given by equation (4).

[0057] Step 3: Extract the stress distribution considering only rotation in Step 2, apply the obtained relationship between tangential stress and radial strain at this time to obtain the radial strain considering only rotation, and further obtain the radial deformation considering only rotation.

[0058] Step 4: Assuming the rotor volume remains constant, we go from considering only the radial deformation under rotation in Step 3 to considering only the axial deformation under rotation, and further obtain the axial strain under rotation. We introduce a correction factor K due to the volume change into the axial strain under rotation to obtain the final actual axial strain under rotation, and further obtain the final axial stress under rotation.

[0059] Step 5: Using the static assembly from Step 2 and considering only temperature, calculate the radial and rotor tangential stresses σ. θ (i.e., the mechanical differential equations of equation (2)) yield the axial stress distribution under static assembly and temperature-only conditions, and introduce the axial stress into the stress distribution of these two conditions; introduce the axial stress obtained in step four under rotation-only conditions into the stress distribution under rotation-only conditions; finally, sum the stress distributions under the three conditions (static assembly, rotation, and temperature) to obtain the rotor stress distribution when the motor is running.

[0060] Furthermore, step one specifically involves:

[0061] The rotor model consists of an outer sheath and a permanent magnet disposed inside the sheath and interference-fitted with it. By analyzing the forces and deformations of the rotor, a set of mechanical differential equations under the rotor model structure is obtained, including stress balance equations and strain balance equations, as shown in equations (1) and (2) respectively:

[0062]

[0063]

[0064] Where: σ r For rotor radial stress, σ θ Let ρ be the rotor tangential stress, r be the radial position of the analysis point, ρ be the rotor material density, ω be the rotor rotational angular velocity, and ε be the rotor tangential stress. r For rotor radial strain, ε θ This refers to the rotor tangential strain;

[0065] The rotor material is an elastic isotropic material (both the sheath and the permanent magnet are elastic isotropic materials), and the Poisson's ratio ν in each direction is the same as Young's modulus E. The constitutive equation is obtained as shown in equation (3):

[0066]

[0067] Furthermore, in step three, the process of determining the radial deformation considering only rotation is as follows:

[0068] Solve the mechanical differential equations in step two, namely equations (1) and (2), to obtain the radial stress, tangential stress and radial deformation under three working states: static assembly, rotation only, and temperature only.

[0069] The radial deformation u of the sheath considering only rotation is obtained through equation (5). v,t Equation (6) yields the radial deformation u of the permanent magnet considering only rotation. v,m ;

[0070]

[0071]

[0072] Where: v1 is the Poisson's ratio of the sheath, r is the radial position of the analysis point, ρ1 is the density of the sheath, E1 is the Young's modulus of the sheath, and R... t1 R is the radius of the outer circumference of the sheath. t2 The radius of the inner circumference of the sheath.

[0073] v2 is the Poisson's ratio of the permanent magnet, ρ2 is the density of the permanent magnet, E2 is the Young's modulus of the permanent magnet, and R... m1 ω is the outer diameter of the permanent magnet's outer circular surface, and ω is the rotor's angular velocity.

[0074] Radial deformation u of permanent magnet m for:

[0075]

[0076] Furthermore, in step four, the final process for determining the axial stress considering only rotation is as follows:

[0077] like Figure 3 When considering only the rotational case, the permanent magnet expands radially and contracts axially. It is easy to see that the axial contraction does not change with the radial position, therefore the axial strain is a constant in the radial position. Assuming the volume of the permanent magnet remains constant during deformation, the corrected axial strain ε is obtained from equation (9). z0 Equation (10) is obtained by combining equations (8) and (9);

[0078] πR m1 2 l=π(R m1 +u m ) 2 l'(8)

[0079] Where: l is the axial length of the permanent magnet, l' is the axial length of the permanent magnet after deformation, and R m1 U is the diameter of the outer circular surface of the permanent magnet. m The radial deformation of the permanent magnet;

[0080]

[0081]

[0082] According to elasticity, an object will inevitably change its volume when subjected to force. Under this condition, the permanent magnet tends to deform outward due to rotation. Its own stress is directed outward, which will cause the volume to expand. Due to the expansion of the volume, part of the radial deformation is caused by the volume expansion. The remaining part will cause the axial inward contraction. This will result in the actual axial strain being lower than that in equation (10). Therefore, a correction coefficient K is introduced into equation (10) to compensate for the reduction in strain caused by the volume change. K is a value less than 1 and is related to the ratio of the diameter and axial length of the permanent magnet. The value of K is obtained as shown in equation (11) by taking the ratio of the semi-arc length of the permanent magnet to the axial length. Then equation (10) becomes equation (12).

[0083]

[0084]

[0085] Where: ε zFor the axial strain, the axial strain obtained from equation (12) and the radial and tangential stresses obtained from solving the mechanical differential equations in step two, considering only rotation, are substituted into the material constitutive equation (13) considering the radial, tangential, and axial directions. This yields a system of three linear equations. Solving this system of equations yields the axial stress σ considering only rotation. z As shown in equation (14), the axial stress σ z By combining the radial stress and tangential stress that only consider rotation without considering axial strain, we obtain the axial stress that only considers rotation and takes axial strain into account.

[0086]

[0087]

[0088] In the formula, ε z For axial strain, E r For radial Young's modulus, E θ For tangential Young's modulus, E z For axial Young's modulus, v rθ For radial and tangential Poisson ratios, v rz For radial and axial Poisson's ratios.

[0089] Furthermore, in step five, the process of obtaining the stress distribution of the motor during actual operation is as follows:

[0090] By solving the mechanical differential equations in step two, the radial and tangential stresses under static assembly and considering only temperature are extracted. The axial stress under these two conditions is obtained through equation (15), and the stress distribution under these two conditions is also obtained. Finally, the stress distribution under the three working conditions is obtained through equation (16): considering only the stress σ during rotation. v The compressive stress σ of static assembly s Considering only the thermal stress σ at temperature T The summation yields the rotor stress distribution during motor operation;

[0091] σ z =v(σ r +σ θ (15)

[0092] σ=σ s +σ v +σ T (16).

[0093] The difference between conventional methods and the method of this invention:

[0094] Conventional methods do not consider the influence of axial stress when performing equivalent stress calculations. This has almost no impact on surface-mounted permanent magnet rotor structures. However, when calculating the strength of solid permanent magnets, the order of magnitude of axial stress is the same as that of radial and tangential stress, and cannot be ignored.

[0095] This invention categorizes the stress during rotor operation into three cases: static assembly, considering only rotation, and considering only temperature. The axial stress distribution under static assembly and temperature-only conditions can be directly obtained from radial and tangential stresses. However, the axial stress under rotation-only conditions needs to be indirectly obtained by converting tangential and radial stresses into deformation, and then from deformation into stress, ultimately yielding the overall stress distribution during operation.

[0096] After axial stress correction, the original analytical method and the current analytical method, considering only the rotational effect, are compared with the simulation results. Figure 4 (a) and Figure 4 As shown in (b), the error of the axial stress of the permanent magnet decreased from 8.8 MPa to 1.5 MPa, and the equivalent stress also changed to a curve with the same trend and smaller error.

[0097] like Figure 5 (a) and Figure 5 (b) The graph shows the axial stress and equivalent stress of the permanent magnet during normal operation of the motor after correcting for stresses in all three directions. The error in axial stress decreased from 11.2 MPa to 2.9 MPa, and the equivalent stress curve showed the same trend with small error. This is sufficient to prove the effectiveness of the correction method.

[0098] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for analytical calculation of the strength of a cylindrical permanent magnet rotor in a high-speed motor, characterized in that: The method includes the following steps: Step 1: Determine the rotor model, analyze the forces and deformations of the rotor, and obtain the stress and strain equilibrium equations under the rotor model structure; combine the constitutive equations of the rotor material itself to obtain the mechanical differential equations under a fixed structure and fixed material. Step 2: By applying the boundary conditions of force and displacement in equation (4) to the mechanical differential equation system in Step 1, solve the mechanical differential equation system and obtain the rotor stress distribution under different boundary conditions, including three cases: static assembly, considering only rotation, and considering only temperature. （4） in: The boundary surface subjected to external forces; Let the surface be a given displacement; in the polar coordinate system, let the given displacement along the radial direction on the rotor be... ; This refers to the radial stress of the rotor. Step 3: Extract the stress distribution considering only rotation in Step 2, and apply the obtained relationship between tangential stress and radial strain at this time to obtain the radial strain considering only rotation, and further obtain the radial deformation considering only rotation; the process of obtaining the radial deformation considering only rotation is as follows: Solving the system of mechanical differential equations yields the radial stress, tangential stress, and radial deformation under three working conditions: static assembly, considering only rotation, and considering only temperature. Equation (5) yields the radial deformation of the sheath considering only rotation. Equation (6) yields the radial deformation of the permanent magnet considering only rotation. ; （5） （6） in: For the sheath Poisson's ratio, To determine the radial position of the analysis point, For sheath density, For the Young's modulus of the sheath, The radius of the outer circumference of the sheath. The radius of the inner circumference of the sheath. For permanent magnets, Poisson's ratio Density of permanent magnets This refers to the Young's modulus of a permanent magnet. The radius of the outer circular surface of the permanent magnet is denoted as . The angular velocity of the rotor; Radial deformation u of permanent magnet m for: （7）； Step 4: Assuming the rotor volume remains constant, we go from considering only the radial deformation under rotation in Step 3 to considering only the axial deformation under rotation, and further obtain the axial strain under rotation. We introduce a correction factor K due to the volume change into the axial strain under rotation to obtain the final actual axial strain under rotation, and further obtain the final axial stress under rotation. Step 5: Using the radial and rotor tangential stresses obtained in Step 2 under static assembly and temperature-only conditions, obtain the axial stress distribution under static assembly and temperature-only conditions, and introduce the axial stress into the stress distribution under these two conditions; introduce the axial stress obtained in Step 4 under rotation-only conditions into the stress distribution under rotation-only conditions; finally, sum the stress distributions under these three conditions to obtain the rotor stress distribution under motor operation.

2. The analytical calculation method for the strength of a cylindrical permanent magnet rotor in a high-speed motor according to claim 1, characterized in that: Step one is as follows: The rotor model consists of an outer sheath and a permanent magnet installed inside the sheath with an interference fit. By analyzing the forces and deformations of the rotor, a set of mechanical differential equations under the rotor model structure is obtained, including stress balance equations and strain balance equations, as shown in equations (1) and (2), respectively: （1） （2） in: For rotor radial stress, For rotor tangential stress, To determine the radial position of the analysis point, For rotor material density, The rotor's angular velocity. For rotor radial strain, This refers to the rotor tangential strain; The rotor material is an elastic isotropic material with a Poisson's ratio in each direction. With Young's modulus Similarly, the constitutive equation is obtained, as shown in equation (3): （3）。 3. The analytical calculation method for the strength of a cylindrical permanent magnet rotor in a high-speed motor according to claim 2, characterized in that: In step four, the final process for determining the axial stress considering only rotation is as follows: Assuming the permanent magnet has a constant volume during deformation, the corrected axial strain can be obtained from equation (9). Equation (10) is obtained by combining equations (8) and (9). （8） in: This is the axial length of the permanent magnet. This represents the axial length of the permanent magnet after deformation. The radius of the outer circular surface of the permanent magnet is denoted as . The radial deformation of the permanent magnet; （9） （10） A correction coefficient K is introduced into equation (10) to compensate for the reduction in strain caused by volume change. K is a value less than 1, which is related to the ratio of the radius and axial length of the permanent magnet. The ratio of the half-arc length of the permanent magnet to the axial length is taken to obtain the value of K as shown in equation (11). Then equation (10) becomes equation (12). （11） （12） In the formula: For the axial strain, the axial strain obtained from equation (12) and the radial and tangential stresses obtained from solving the mechanical differential equations, considering only rotation, are substituted into the material constitutive equation (13) considering the radial, tangential, and axial directions. This yields a system of three linear equations. Solving this system of equations yields the axial stress considering only rotation. As shown in equation (14), the axial stress By combining the radial stress and tangential stress that only consider rotation without considering axial strain, we obtain the axial stress that only considers rotation and takes axial strain into account. （13） （14） In the formula, For axial strain, For radial Young's modulus, For tangential Young's modulus, For axial Young's modulus, For radial and tangential Poisson ratios, For radial and axial Poisson's ratios.

4. The analytical calculation method for the strength of a cylindrical permanent magnet rotor in a high-speed motor according to claim 2, characterized in that: Step five involves obtaining the stress distribution of the motor during actual operation as follows: The radial and tangential stresses under static assembly and considering only temperature are extracted by solving the system of mechanical differential equations. The axial stress under these two conditions is obtained through equation (15), and the stress distribution under these two conditions is also obtained. Finally, the stress distribution under the three working conditions is obtained through equation (16): considering only the stress during rotation. Extrusion stress in static assembly Considering only thermal stress at temperature The summation yields the rotor stress distribution during motor operation; （15） （16）。

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