Modal shape and frequency identification method for motor stator system with inner gear ring

Abstract

The invention discloses a modal shape and frequency identification method for a motor stator system with an inner gear ring, and the method comprises the steps: building a shell unit and beam unit mixed analysis model of the motor stator system provided with the inner gear ring, and obtaining a total stiffness matrix and a total mass matrix under a local rectangular coordinate system through a matrix displacement method, obtaining a structural stiffness matrix and a structural mass matrix under the overall cylindrical coordinate system by using coordinate transformation; and establishing a discrete free vibration model, inputting the structural stiffness matrix and the structural mass matrix into the model, and outputting modal shapes of different orders and corresponding inherent frequencies of the motor stator system, thereby completing identification. The method comprehensively considers the influence of the stator teeth and the tooth part of the inner gear ring, helps to rapidly identify the modal shape and inherent frequency of the stator system, provides guidance for subsequent vibration acquisition and optimization, and has the advantages of simple modeling, rapid identification and high precision.

Description

Modal mode and frequency identification method for motor stator system with internal gear ring Technical Field

[0001] This invention belongs to the field of motor technology, specifically relating to a method for identifying the mode shape and frequency of a motor stator system with an internal gear ring. Background Technology

[0002] With the rapid development of intelligent manufacturing, new energy vehicles, high-end machine tools, and aerospace technology, highly integrated motor and reducer units are widely used in industrial robots, new energy vehicles, high-end CNC machine tools, and aerospace fields, gradually replacing the traditional structure of a motor combined with an external reducer. These integrated units are typically frameless torque motors with an external rotor and internal stator, integrating brushless motors, planetary reducers, and controllers into a single housing, achieving high torque density and high integration within a limited space.

[0003] To achieve a larger reduction ratio and output torque, the internal gear ring of a planetary reducer is typically mounted directly inside the motor. However, in the fields of new energy vehicle drive systems, high-end machine tool equipment, and aerospace precision servo mechanisms, higher requirements are placed on the high torque density and low vibration performance of the integrated motor-reducer drive unit. But under high reduction ratio and high torque density operating conditions, it is subject to the combined excitation of periodic electromagnetic force and gear meshing force, which easily induces its intrinsic modes, causing resonance in the integrated unit and generating severe noise. Currently, most modal characteristic calculation methods focus on the motor stator body, often simplifying or even completely ignoring the reducer. In reality, after mounting the internal gear ring inside the stator, traditional methods that only consider stator modal characteristics (such as the literature [Stator Vibration Modal Analysis of Switched Reluctance Motor. Proceedings of the CSEE, 2025, Vol. 25, No. 22, pp. 148-152]) can no longer reflect the true dynamic behavior of the system. Therefore, a method is needed to accurately calculate the stator modal characteristics considering the internal gear ring, providing a theoretical basis for the structural design and vibration noise of the integrated unit of motor reducer.

[0004] The modal characteristics of the stator-internal gear ring combined structure largely depend on the stator body structure and the geometric parameters of the internal gear ring. The stator core is typically formed by laminating and solidifying silicon steel sheets. The internal gear ring can be integrally formed with the stator core or assembled inside the stator via an interference fit. Its short, thick teeth act as a distributed tooth beam, significantly altering the mass distribution and local stiffness of the stator system. Furthermore, in the integrated structure, the internal gear ring accounts for a relatively high proportion of mass and stiffness, affecting the stator's natural frequencies and mode shapes. Therefore, during the design phase, the modal characteristics of the stator system can be specifically optimized by adjusting structural parameters such as the stator yoke thickness, the internal gear ring thickness, and the number of teeth. This allows for improved vibration characteristics while maintaining high torque density, achieving the goal of motor vibration reduction and noise reduction.

[0005] For stator modal calculations considering the internal gear ring, the finite element method (FEM) is still the most commonly used method in engineering. This involves establishing an overall model of the stator core and the internal gear ring and solving for their modal characteristics. While this method offers high accuracy, its physical analysis process is not sufficiently clear and intuitive. In practical designs, to change the stator and internal gear ring dimensions to meet the comprehensive performance requirements of the prototype, such as output performance, it is often necessary to repeatedly modify structural parameters, establish multiple finite element models for solving, and consume a significant amount of computation time. This is detrimental to preliminary design and parameter sensitivity analysis. The literature [Accurate Calculation Method of Natural Frequency of Radial Flux Slotted Motor Considering End Covers. IEEE Transactions on Industrial Electronics, June 2023, Vol. 70, No. 6, pp. 5516-5526] proposes an energy method, which can consider the connection between the stator and the internal gear ring during the solution process. However, the modeling and solution process is complex, and the derivation workload is substantial. Summary of the Invention

[0006] In view of the above, the present invention provides a method for identifying the mode shape and frequency of a motor stator system with an internal gear ring. By establishing a hybrid analytical model of the stator system's shell elements and beam elements, the mode shape and corresponding natural frequency of the stator system can be identified. This method comprehensively considers the influence of the stator teeth and the internal gear ring teeth, and has both theoretical basis and universality, accuracy and speed.

[0007] A method for identifying the mode shapes and frequencies of a motor stator system with an internal gear ring includes the following steps: (1) For a motor stator system equipped with an internal gear ring, a hybrid analytical model of its shell elements and beam elements is established; (2) Based on the hybrid analytical model, the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system are calculated using the matrix displacement method; (3) Based on the total stiffness matrix and total mass matrix, the structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system are calculated using coordinate transformation; (4) A discrete free vibration model of the motor stator system is established, and the structural stiffness matrix and structural mass matrix are input into the model, and the model automatically identifies the different modes and natural frequencies of the motor stator system.

[0008] Further, in step (1), for the motor stator system equipped with an internal gear ring, the hybrid analytical model of the system shell element and beam element is composed of multiple tooth beam elements, multiple yoke shell elements and multiple internal gear ring beam elements. The tooth beam elements and internal gear ring beam elements adopt Timoshenko beam elements, and the yoke shell elements adopt curved shell elements. The tooth beam element is modeled as a rectangular strip based on the stator tooth height in the motor stator system. The tooth beam element has two nodes located at the tooth tip and tooth root of the stator tooth and the connection between the tooth and the yoke, respectively. The internal gear ring beam element is modeled as a rectangular strip based on the tooth part of the internal gear ring along the tooth height direction. The internal gear ring beam element has two nodes located at the tooth tip and tooth root of the internal gear ring, respectively. The yoke shell element is modeled as a curved shell based on the arc-shaped yoke between two adjacent teeth in the motor stator system. The yoke shell element has four nodes located at the four vertices of the neutral plane rectangle. The geometric position of the neutral plane in space corresponds to the neutral circle radius of the stator yoke and is used to describe the neutral layer in the shell thickness direction.

[0009] Furthermore, the total stiffness matrix in step (2) is an N-dimensional diagonal matrix, specifically expressed as: in: Let be the overall stiffness matrix of the motor stator system in the local Cartesian coordinate system. , , These are the stiffness matrices for the yoke shell element, the internal gear ring beam element, and the toothed beam element, respectively. The first N1 diagonal elements are The last N3 diagonal elements are The remaining diagonal elements are N = N1 + N2 + N3, where N1, N2, and N3 are the number of yoke shell elements, internal gear ring beam elements, and gear beam elements, respectively.

[0010] Furthermore, the stiffness matrix The expression is as follows: in: Let be the translational displacement stiffness matrix of the yoke shell element. Let be the rotational displacement stiffness matrix of the yoke shell element. Let be the translational stiffness interpolation matrix for the yoke shell element. Let be the rotational stiffness interpolation matrix of the yoke shell element, D be the Hooke matrix of the yoke shell element (material-dependent), h be the thickness of the yoke shell element, and J be the Jacobian matrix. This expression calculates the determinant of a matrix, where r and s are the x and y coordinates in the natural coordinate system, respectively. The superscript indicates the matrix. T Indicates transpose. Let be the translational stiffness interpolation matrix for the i-th node in the yoke shell element. Let be the rotational stiffness interpolation matrix for the i-th node in the yoke shell element, where i = 1, 2, 3, 4. and Let x and y be the partial derivatives of the shape function of the i-th node in the yoke shell element with respect to the local physical coordinates x and y at any point on the neutral surface. Let represent the shape function of the i-th node in the yoke shell element in the natural coordinate system. The radius of the neutral circle of the stator yoke. Let be the physical coordinates of the i-th node in the yoke shell element in the global coordinate system.

[0011] Furthermore, the stiffness matrix and The calculation expressions are the same, that is, their respective parameters are substituted into the stiffness matrix of the Timoshenko beam element. Calculated in the middle; in: , , , , , As an intermediate variable, denoted as , where E is the shear influence factor of the internal gear ring beam element or toothed beam element, G is the Young's modulus of the material of the internal gear ring beam element or toothed beam element, A is the equivalent cross-sectional area of ​​the internal gear ring beam element or toothed beam element, L is the equivalent length of the internal gear ring beam element or toothed beam element, I is the moment of inertia of the section of the internal gear ring beam element or toothed beam element, J is the moment of inertia of the internal gear ring beam element or toothed beam element, and k is the shear coefficient of the Timoshenko beam element.

[0012] Furthermore, the total mass matrix in step (2) is an N-dimensional diagonal matrix, specifically expressed as: in: Let be the total mass matrix of the motor stator system in the local Cartesian coordinate system. , , These are the mass matrices for the yoke shell element, the internal gear ring beam element, and the gear beam element, respectively. The first N1 diagonal elements are The last N3 diagonal elements are The remaining diagonal elements are N = N1 + N2 + N3, where N1, N2, and N3 are the number of yoke shell elements, internal gear ring beam elements, and gear beam elements, respectively.

[0013] Furthermore, the mass matrix The expression is as follows: in: Let be the translational displacement-mass matrix of the yoke shell element. Let be the rotational displacement mass matrix of the yoke shell element. Let be the translational mass interpolation matrix for the yoke shell element. Let be the rotational mass interpolation matrix for the yoke shell element. Let be the material density of the yoke shell element, h be the thickness of the yoke shell element, and J be the Jacobian matrix. This expression calculates the determinant of a matrix, where r and s are the x and y coordinates in the natural coordinate system, respectively. The superscript indicates the matrix. T Indicates transpose. Let be the translational mass interpolation matrix for the i-th node in the yoke shell element. Let be the rotational mass interpolation matrix for the i-th node in the yoke shell element, where i = 1, 2, 3, 4. Let represent the shape function of the i-th node in the yoke shell element in the natural coordinate system. Let be the physical coordinates of the i-th node in the yoke shell element in the global coordinate system.

[0014] Furthermore, the mass matrix and The calculation expressions are the same, that is, each parameter is substituted into the mass matrix of the Timoshenko beam element. Calculated in the middle; in: , , , , , , , As an intermediate variable, This refers to the shear influence factor of the internal gear ring beam element or gear beam element. The material density of the internal gear ring beam element or gear beam element. Let be the radius of gyration of the internal gear ring beam element or toothed beam element, E be the Young's modulus of the material of the internal gear ring beam element or toothed beam element, G be the shear modulus of the material of the internal gear ring beam element or toothed beam element, A be the equivalent cross-sectional area of ​​the internal gear ring beam element or toothed beam element, L be the equivalent length of the internal gear ring beam element or toothed beam element, I be the moment of inertia of the section of the internal gear ring beam element or toothed beam element, J be the moment of inertia of the internal gear ring beam element or toothed beam element, and k be the shear coefficient of the Timoshenko beam element.

[0015] Furthermore, the calculation expressions for the structural stiffness matrix and structural mass matrix in step (3) are as follows: Where M and K are the structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system, respectively. and Let be the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system, respectively, and T be the coordinate transformation matrix from the local rectangular coordinate system to the global cylindrical coordinate system. The superscript _____ T This indicates transpose.

[0016] Furthermore, the discrete free vibration model in step (4) calculates the natural angular frequency ω of the motor stator system and the nodal displacement vector at the natural angular frequency ω using the following expression. Furthermore, based on the nodal displacement vector Identify different mode shapes of the motor stator system; Where M and K are the structural stiffness matrix and structural mass matrix of the motor stator system in the overall cylindrical coordinate system, respectively.

[0017] A computer device includes a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the above-described method for identifying the mode shape and frequency of a motor stator system with an internal gear ring.

[0018] A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the above-described method for identifying the mode shapes and frequencies of a motor stator system with an internal gear ring.

[0019] Based on the above technical solution, the present invention has the following beneficial technical effects: 1. The method of the present invention can quickly and accurately complete the modeling of the motor stator system, and fully analyze the influence of the stator teeth and internal gear ring and other structures on the natural frequency of the stator core, and has strong versatility; compared with the existing design methods, the method of the present invention is based on analytical identification, avoiding large-scale finite element simulation and blind scanning optimization, and has the advantages of fast identification speed, clear directionality, small amount of calculation and high accuracy.

[0020] 2. The modal shape identification results of the method of the present invention can be presented intuitively through the processed image, and the modal shape to be considered can be directly identified. Combined with the natural frequency, it can guide subsequent vibration optimization or structural design.

[0021] 3. The method of the present invention takes into account the process influence of the motor stator system equipped with an internal gear ring, and can obtain an intuitive modal shape diagram, which helps to quickly identify the modal shape and natural frequency of the stator system, and provides guidance for subsequent vibration acquisition and optimization. It has the advantages of simple modeling, fast identification and high accuracy. Attached Figure Description

[0022] Figure 1 is a schematic flowchart of the method for identifying the mode shape and frequency of the motor stator system of the present invention.

[0023] Figure 2 is a schematic diagram of the stator system of a motor equipped with an internal gear ring.

[0024] Figure 3 is a schematic diagram of the geometric structure of the shell unit.

[0025] Figure 4 is a schematic diagram of hybrid interpolation node selection.

[0026] Figure 5 is a schematic diagram of the geometric structure of the Timoshenko beam unit.

[0027] Figure 6 is a schematic diagram of the structure of the hybrid unit analytical model.

[0028] Figure 7 is a schematic diagram of the analytical model of the hybrid element in the local coordinate system.

[0029] Figure 8 is a schematic diagram of the analytical model of the hybrid unit in the global coordinate system.

[0030] Figure 9 is a schematic diagram of the overlapping area between the beam element and shell element models.

[0031] Figure 10 is a schematic diagram of node merging in the hybrid unit analytical model.

[0032] Figure 11 is a schematic diagram of the different circumferential mode vibration results of the motor stator core. In the figure, Mode represents the radial mode order of the stator system.

[0033] Figure 12 is a schematic diagram of the effect of the number of stator teeth on the natural frequency of the stator system.

[0034] Figure 13 is a schematic diagram of the influence of the mid-surface radius and shell thickness on the natural frequency of the motor stator system. Figures (a) to (f) show the distribution of the natural frequencies of the 0th, 2nd, 3rd, 4th, 5th and 6th modes of the motor stator system as a function of the mid-surface radius and shell thickness. Detailed Implementation

[0035] To describe the present invention in more detail, the technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0036] The modal vibration mode and frequency identification method of the motor stator system with internal gear ring of the present invention is shown in Figure 1. (1) Establish a mixed analytical model of shell element and beam element of the motor stator system equipped with internal gear ring.

[0037] The hybrid analytical model of the stator system of a motor equipped with an internal gear ring consists of gear-beam elements, yoke-shell elements, and internal gear ring beam elements. These elements are modeled using Timoshenko equivalent beam elements and shell elements, respectively. Specifically, both the gear-beam and internal gear ring beam elements are Timoshenko beam elements, while the yoke-shell elements are curved shell elements. The gear-beam elements are constructed by modeling the stator tooth height in the stator system as rectangular strips, with the two nodes located at the tooth tip and root of the stator tooth, respectively, where they connect to the yoke. The yoke-shell elements are constructed by modeling the arc-shaped yoke between two adjacent teeth in the stator system as curved shells. The internal gear ring beam elements are constructed by modeling the teeth of the internal gear ring along the tooth height direction as rectangular strips, with the two nodes located at the tooth tip and root of the internal gear ring, respectively. In reality, because the neutral axis of the toothed beam element is offset radially relative to the neutral plane of the yoke shell element, the corresponding element nodes are not completely coincident in geometric position. When assembling the element structure stiffness matrix and structural mass matrix, this invention introduces a coordinate transformation matrix to map the two element degrees of freedom to the shell element reference plane, thereby realizing the coordinated constraint of displacement and rotation between offset nodes, accurately characterizing the offset relationship in the actual structure, and finally establishing a hybrid analytical model of the motor stator system.

[0038] (2) Based on the hybrid analytical model of shell element and beam element, the nodal displacement of the structure is used as the basic unknown quantity, and the matrix displacement method is used to obtain the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system.

[0039] In the local rectangular coordinate system of the motor stator system, the overall stiffness matrix is ​​composed of the element stiffness matrix of each element, and the overall mass matrix is ​​composed of the element mass matrix of each element. The element stiffness matrix and element mass matrix are as follows: 2.1 Solving the element mass matrix and stiffness matrix of the shell element: In the local coordinate system of the shell element, the neutral surface is set at z=0. The geometric position of this neutral surface in space corresponds to the radius of the neutral circle of the stator yoke. At the neutral plane, the neutral layer in the thickness direction of the shell is described, as shown in the quadrilateral formed by nodes A1 to A4 in Figure 3. Assuming the displacement of the shell along the thickness direction is linearly distributed, the displacement of the entire shell can be described by the translation and bending at the nodes of the neutral plane. Each node contains three translational degrees of freedom and two rotational degrees of freedom about the x and y axes, while the rotational degrees of freedom about the normal are usually ignored. The shell element model is shown in Figure 3. Its degrees of freedom are described based on multiple coordinate systems, where X, Y, and Z represent the global physical coordinate system, used to characterize the geometric position of the shell element's neutral plane in space; x, y, and z represent the local physical coordinate system at the nodes, used to describe the translational and rotational degrees of freedom of the nodes; in the local physical coordinate system, x is along the stator axis, y is along the stator circumference, and z is along the stator radial direction. Furthermore, to construct the shape functions and numerical integration of the shell element, a natural coordinate system (r, s) is introduced, as shown on the right side of Figure 3. This coordinate system is defined on the neutral plane of the shell element, and the four nodes are located in the natural coordinate system at... At this point, through the geometric mapping function, any point in the natural coordinate system can be mapped to the corresponding position in the physical coordinate system.

[0040] The degrees of freedom of the shell element in Figure 3 are defined as follows: Where: u N Let u be the total degrees of freedom of the plane formed by the four nodes A1, A2, A3, and A4 in the shell element. Ak Let u be the total degrees of freedom of the k-th node in the local coordinate system of the shell element. Ak_x u Ak_y u Ak_z Let x, y, and z be the translational degrees of freedom of the k-th node in the local coordinate system, respectively. , These represent the rotational degrees of freedom of the k-th node about the x and y axes, respectively.

[0041] Assuming the displacement of the shell along its thickness is linearly distributed, the displacement field of the entire shell cross-section can be represented by the translational displacement and bending rotation of each node on the neutral plane. As shown in Figure 3, point E is any point inside the shell, and its geometric position is determined by its projection point F on the neutral plane and its radial offset distance relative to point F. The displacement of point E can be expressed as: in: Let E be the displacement of any point E within the shell. Let F be the translational displacement of point F in its local coordinate system. The displacement is caused by the rotation along the shell thickness direction. The subscript ~ represents the displacement of each point on the shell neutral surface, and h is the shell element thickness. Therefore, in order to solve for the displacement of point E, it is first necessary to determine the displacement of point F, which is the projection of point E onto the neutral surface.

[0042] To correlate displacements on the neutral surface with nodal degrees of freedom, shape functions are needed for interpolation. In a 4-node shell element, the bilinear shape function defined in the natural coordinate system is expressed as: in: Let represent the shape function corresponding to the i-th node of the shell neutral surface in the natural coordinate system, which is simplified and denoted as in the subsequent derivation. r and s are the x-coordinate and y-coordinate defined in the natural coordinate system, respectively, with values ​​ranging from -1 to 1.

[0043] Therefore, the displacement of any point F on the neutral surface can be obtained by interpolating the nodal displacements using shape functions: in: .

[0044] According to the small deformation assumption, strain is obtained from the derivative of displacement with respect to physical coordinates. Therefore, a Jacobian matrix is ​​needed to map the derivative in the natural coordinate system to the physical coordinate system. The Jacobian matrix is ​​defined as follows: in: Let F be the physical coordinates of any point F on the neutral surface in the global coordinate system. These are the physical coordinates of the four nodes on the neutral surface in the global coordinate system.

[0045] By using the Jacobian matrix to perform coordinate transformation, we can obtain: in: and Let F and F be the partial derivatives of the shape function of the i-th node with respect to the physical coordinates x and y, respectively, describing the displacement interpolation contribution of the node.

[0046] Substituting the results into the strain calculation formula and expanding along the shell thickness direction, we obtain the normal strain and shear strain: in: Let E be the strain at any point E in the shell. Let F be the strain of point E projected onto the neutral plane of the shell. This refers to the additional strain caused by the rotation in the shell thickness direction.

[0047] Based on the traditional strain expression for shell elements, this invention also introduces a normal strain term in the thickness direction. Without increasing the nodal degrees of freedom, the accuracy of breathing mode frequency prediction is improved by performing numerical integration in the shell thickness direction and weighted summation of the total energy. At the same time, complex high-order elements can be ignored in the stator yoke.

[0048] To uniformly describe the mapping relationship between strain and nodal displacement, we introduce... and Describe the in-plane membrane bending coupling strain and shear strain respectively: The mid-surface radius of the shell appears in the formula It is a stiffness coupling term affected by shell curvature. However, since the thickness dimension of the stator yoke of the prototype used in this invention is small, shear self-locking may occur in the shear strain solution, resulting in excessive stiffness. Therefore, a hybrid interpolation method is used to decouple the shear strain from the standard displacement field, and the original matrix is ​​replaced by the shear-corrected matrix. The selection of the hybrid interpolation point is shown in Figure 4.

[0049] The four nodes corresponding to the yoke shell element and Each constitutes a global translational stiffness interpolation matrix for the shell element. and rotational stiffness interpolation matrix Its matrix is ​​represented by a block matrix as follows: Furthermore, since the mass distributions of translation and rotation along the thickness direction differ, the weighting functions along the thickness direction differ during integration. Therefore, different interpolation matrices are needed to construct the shell element stiffness interpolation matrix and the translational stiffness interpolation matrix. and rotational stiffness interpolation matrix : in: Let be the shape function of the i-th node in the natural coordinate system.

[0050] The four nodes corresponding to the yoke shell element and Each constitutes the overall translational mass interpolation matrix of the shell element. and rotational mass interpolation matrix Its matrix is ​​represented by a block matrix as follows: The obtained translational stiffness interpolation matrix and rotational stiffness interpolation matrix Substitute into the stiffness element solution; use the obtained translational mass interpolation matrix and rotational mass interpolation matrix Substituting into the solution formula for the mass element, we obtain: in: Let be the stiffness matrix of the shell element, B be the interpolation function for the shell element stiffness matrix, N be the interpolation function for the shell element mass matrix, and D be the material-dependent Hooke matrix. Let be the translational displacement stiffness matrix of the yoke shell element. Let B be the rotational displacement stiffness matrix of the yoke shell element. c and B v These are the translational stiffness interpolation matrix and the rotational stiffness interpolation matrix, respectively. Here is the mass matrix of the shell element. The material density of the shell element. Let be the translational displacement-mass matrix of the yoke shell element. Let N be the rotational displacement mass matrix of the yoke shell element. c and N v These are the translational mass interpolation matrix and the rotational mass interpolation matrix, respectively.

[0051] When the degree of freedom (DOF) of one type of element completely encompasses that of another, the two can directly share the same node during assembly without the need for additional coupling equations. To improve the model's versatility, the DDF of the shell element is aligned with that of the Timoshenko beam element. Based on shell element theory, the nodal DDF of the shell element is expanded from 5 to 6, and additional stiffness and mass terms are established on this basis, enabling the expanded shell element to fully match the beam element at the degree of freedom level, achieving unconstrained connection. The nodal stiffness matrix of the original shell element is denoted as... After expansion, we get: in: The equivalent torsional stiffness that adds a degree of freedom to the shell element.

[0052] Where: α is the flexible connection coefficient, which is generally a flexible connection at the transition connection of the shell-beam element, and is taken as α=0.2; G is the shear modulus.

[0053] Similarly, the expanded mass matrix is ​​expressed as: in: The equivalent inertial mass that adds a degree of freedom to the shell element.

[0054] Where: β is the mass expansion factor. For a stator core with a stacking factor of 0.95, considering the constraint of interlaminar friction, β = 1.2 is taken, and A is the equivalent cross-sectional area.

[0055] At this point, the nodal mass matrix and stiffness matrix of the shell element are expanded to order 6. Thus, after sharing the same node with the beam element, the torsional stiffness of the beam element will be directly coupled to the free moment of the shell element node, thereby accurately transmitting the torsional effect to the shell. In this way, the merging and mechanical coupling of rigid nodes of the shell and beam can be completed without modifying the continuous medium derivation of the shell element.

[0056] 2.2 Solving for the mass and stiffness matrices of the Timoshenko beam element: As shown in Figure 5, a local rectangular coordinate system and nodal degrees of freedom are established for the Timoshenko beam element, where x, y, and z are the radial, circumferential, and axial coordinate axes of the beam element, respectively. The length of the beam element is denoted as L. Each beam element has two nodes, and each node has six degrees of freedom. The translational degrees of freedom of each node are denoted as... , , The rotational degrees of freedom are denoted as follows: , , , where i = 1 or 2.

[0057] The cross-sectional geometry of the stator teeth and the internal gear ring teeth is not a regular shape: the irregularity of the stator teeth usually comes from the widening at the tooth shoe, which is usually ignored in the analysis and is usually replaced by the groove depth; the structure of the internal gear ring teeth is more complex, involving variable cross-sections caused by involute cutting. Under the premise of ensuring that the root stress and tip deflection of the internal gear ring teeth are consistent, Timoshenko beam elements are used to replace the actual tooth shape for analytical calculation.

[0058] Since the tooth surface contains involute and circular arc segments, each tooth profile is discretized into 8-12 sections along the tooth height direction. For each discrete section, equivalent section parameters are established: tooth width b (which can be taken as a constant in the initial modeling) and tooth thickness varying along the height direction. The cross-sectional area is obtained by multiplying the tooth width and tooth thickness, and the moment of inertia of the cross section is... From tooth width b and tooth thickness Determined as: Therefore, the true compliance of a tooth can be expressed by integrating the bending compliance and shear compliance along the tooth height direction as follows: in: and Let E be the bending moment and shear force acting on the tooth, respectively, and let E be the Young's modulus and k be the shear coefficient.

[0059] To reasonably represent the internal gear teeth as beam elements, stiffness and mass equivalence must be followed during the equivalence process. Stiffness equivalence requires a proper match between bending stiffness, torsional stiffness, and axial stiffness to ensure the actual flexibility of the tooth matches that of the equivalent beam. Simultaneously, the mass and moment of inertia conservation equations must be simultaneously established to keep the moment of inertia and linear density about the tooth root constant. Therefore: in: and These are the equivalent height and equivalent area of ​​the beam, respectively, with the equivalent area equal to the equivalent height. With equivalent width The product of This is the distance from the centroid of the cross section to the end of the beam.

[0060] Substituting the actual internal gear parameters, we obtain the actual equivalent height and equivalent width. Substituting these into the beam element mass matrix and stiffness matrix, we get: Shear Influence Factor for: Where: K e Let E be the element stiffness matrix of the Timoshenko beam element, G be the Young's modulus of the Timoshenko beam element material, A be the shear modulus of the Timoshenko beam element material, L be the cross-sectional area of ​​the Timoshenko beam element, and I be the moment of inertia of the cross section.

[0061] Radius of gyration is: Where: M e Here is the element mass matrix of the Timoshenko beam element. The density of the Timoshenko beam element is given.

[0062] (3) Considering the relationship between displacement and force at different nodes, the structural stiffness matrix and structural mass matrix of the motor stator system in the overall cylindrical coordinate system are obtained by coordinate transformation based on the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system.

[0063] This invention, based on the local coordinate systems of each element in the shell element and Timoshenko beam element models, uses coordinate transformation to convert the force, displacement, element stiffness matrix, and element mass matrix in the local rectangular coordinate system of each element into the force, displacement, element stiffness matrix, and element mass matrix in the global cylindrical coordinate system of the hybrid analytical model.

[0064] (4) Establish a discrete free vibration model of the motor stator system. Input the structural stiffness matrix and structural mass matrix of the motor stator system in the overall cylindrical coordinate system into the discrete free vibration model. The discrete free vibration model outputs the different modes and natural frequencies of the motor stator system, and completes the identification of the modes and frequencies of the motor stator system equipped with the internal gear ring.

[0065] For a motor stator system, the damping matrix can often be neglected or approximated as a linear combination of the mass and stiffness matrices, and the external force is zero when considering the free vibration of the stator system. Assuming the stator system undergoes simple harmonic motion, a discrete free vibration model of the system in the frequency domain can be obtained: Where: ω is the eigenvalue of the hybrid analytical model, i.e., the natural angular frequency of the motor stator system; M and K are the structural mass matrix and structural stiffness matrix of the motor stator system in the global cylindrical coordinate system, respectively. Let be the displacement vector of the nodes in the hybrid analytical model at the natural angular frequency ω of the motor stator system. Different mode shapes of the motor stator system are identified based on these node displacement vectors. T is the transformation matrix from the local Cartesian coordinate system to the global cylindrical coordinate system. The total stiffness matrix of the motor stator system in the local Cartesian coordinate system, with subscripts... xyz The local rectangular coordinate system is used as a reference; F is the force vector in the overall cylindrical coordinate system of the motor stator system, and Δ is the displacement vector of all nodes in the overall cylindrical coordinate system of the motor stator system.

[0066] The specific implementation process of this invention is as follows: First, a hybrid analytical model of shell elements and beam elements of a motor stator system equipped with an internal gear ring is established. The structure of the motor stator system equipped with an internal gear ring is shown in Figure 2. Each shell element has four nodes, and each node contains five independent degrees of freedom: three translational degrees of freedom along the x, y, and y directions, and two rotational degrees of freedom about x and y. Each Timoshenko beam element has two nodes, and each node contains six degrees of freedom. The axial displacement, lateral displacement, shear angle, and torsional angle of any point within the element can be expressed as functions of the nodal degrees of freedom. The stiffness equations of each element are listed according to the matrix displacement method. By combining the stiffness equations into a matrix form, the stiffness matrix of each element can be obtained. Similarly, considering the nodal acceleration force equations, the mass matrix of each element can be obtained.

[0067] Using the nodal displacements of the structure in the model as the basic unknowns, the matrix displacement method is used to obtain the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system. The element stiffness matrix in the total stiffness matrix is ​​a matrix that reflects the relationship between the displacements and forces at both ends of the element. This relationship can be derived through two approaches: one is by using the static method, and the other is by using the energy principle or the virtual work principle.

[0068] The analytical model of the hybrid element is shown in Figure 6. The model includes three basic elements: (1) yoke shell element, with four nodes represented by A, B, C, and D, where A1, A2, and A... n These are the nodes at one end of the first, second, and nth shell elements along the circumferential direction, respectively. A and D are the shell element nodes in the same cross section. (2) Toothed beam element, the two end nodes are represented by J and K, where q in the subscript is the tooth number in the circumferential direction, and m in the subscript is the m-th segment of the beam element after discretization of a single tooth. Specifically, J 111 J 1q1 and J 21m These are, respectively, a node at one end of the first beam element on the first tooth of the first layer, a node at one end of the first beam element on the q-th tooth of the first layer, and a node at one end of the m-th beam element on the first tooth of the second layer. K 111 K 1q1 and K 21m These are, respectively, a node at the other end of the first beam element on the first tooth in the first layer, a node at the other end of the first beam element on the qth tooth in the first layer, and a node at the other end of the mth beam element on the first tooth in the second layer. (3) Internal tooth ring beam element, the two end nodes are represented by P and Q, where p in the subscript is the tooth number in the circumferential direction, specifically, P 11 and P 1pQ represents one node at one end of the beam element on the first tooth of the first layer and the other node at one end of the beam element on the p-th tooth of the first layer. 11 and Q 1p These represent a node at one end of the beam element on the first tooth of the first layer and a node at the other end of the beam element on the p-th tooth of the first layer, respectively. Since the connections between the elements are not considered at this point, the matrix form of the global stiffness equation of the entire stator system in the local coordinate system is as follows: in: This is the force vector in the local Cartesian coordinate system of the motor stator system. , … , … , …represent the force vectors of nodes A1, B1…P1, Q1…J1, K1…in the local Cartesian coordinate system; , , These are the stiffness matrices for the shell element, the internal gear ring teeth, and the stator teeth, respectively. Let be the displacement vector of all nodes in the local Cartesian coordinate system of the motor stator system. , … , … , …represent the nodal displacement vectors of nodes A1, B1…P1, Q1…J1, K1… in the local Cartesian coordinate system.

[0069] Similarly, the total mass matrix in the local coordinate system of the motor stator core as follows: in: , and These are the mass matrices for the shell unit, the internal gear ring teeth, and the stator teeth, respectively.

[0070] Considering the relationship between displacements and forces at different nodes, the structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system are obtained by coordinate transformation based on the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system.

[0071] For the overall stiffness matrix and overall mass matrix of the motor stator system in the local Cartesian coordinate system, the orientations of the local coordinate systems of each element are different. To establish the equilibrium equations of the nodes, forces and displacements must have a unified positive direction, i.e., the positive direction of the global coordinate system. This requires converting the force, displacement, element stiffness matrix, and element mass matrix in the local coordinate system into the force, displacement, element stiffness matrix, and element mass matrix in the global coordinate system. As shown in Figures 7 and 8, considering that the stator motor system is largely a cylindrical shell, a global cylindrical coordinate system is adopted. The transformation matrices between the coordinate systems of the element nodes in the two coordinate systems are as follows: in: This is the coordinate transformation matrix for transforming a local relative coordinate system to a global cylindrical coordinate system. y is the angle between the y-axis of the local relative coordinate system and the t-axis of the global polar coordinate system.

[0072] After completing the coordinate system transformation, it is necessary to further consider the actual connection method of the elements in geometric space to merge the nodes. As shown in Figure 9, the dashed line represents the stator yoke represented by the shell element, and the blue box represents the tooth represented by the beam element. When the beam element node and the shell element node are merged, their solid parts will partially overlap with the stator yoke corresponding to the shell element. The height of the overlapping area is half the height of the stator yoke. For the internal gear ring tooth involved in this invention, its height is smaller than the stator yoke height, and this factor cannot be ignored. Therefore, this embodiment adds a translation transformation at the beam element node to solve the problem of direct connection between the shell element node and the beam element node.

[0073] For structural nodes with 6 degrees of freedom, a coordinate transformation matrix is ​​established using the displacement relationships of rigid body motion to map the stiffness and mass matrices of the beam element onto the nodes on the shell mid-surface. The beam element nodes are offset only in the radial direction. A matrix can be constructed: in: The vector offset from the tooth root to the node on the mid-surface of the shell; the vector offset of the internal tooth. Vector offset of external teeth , This is the radial offset of the beam element node relative to the neutral plane of the shell element, which is numerically equal to the shell element thickness h. The subscript z indicates that the offset is taken in the radial direction. As an antisymmetric matrix, coupled with the rotation angle, it effectively generates an additional bending moment.

[0074] From geometric relationships, it can be deduced that the coordinate transformation matrix of the stator yoke shell element node can be expressed as: For all nodes labeled A (such as A1, A2, etc.), the coordinate transformation is adopted. The remaining marked nodes all use the corresponding coordinate transformation.

[0075] The nodal transformation matrix of the toothed beam element can be represented as follows: The structural stiffness matrix and structural mass matrix in the global cylindrical coordinate system after coordinate transformation take into account the connections between nodes of the elements, so there are no overlapping nodes. The merged model is shown in Figure 10. In the figure, the yoke node of the stator system is represented by M, the stator tooth node by L, and the internal gear tooth node by N. Specifically: M 11 and M 1n These are the first node and the nth node in the first layer of the yoke after node merging, respectively; L 111 L 11m-1 and L 1qm-1 These are, respectively, the first node on the first tooth of the first layer of the stator teeth after node merging, the (m-1)th node on the first tooth of the first layer, and the (m-1)th node on the (q)th tooth of the first layer; N 11 and N 1p These are the first node in the first layer of the internal gear ring after node merging, and the p-th node in the first layer, respectively.

[0076] Therefore, the transformation matrix T from the local Cartesian coordinate system to the global cylindrical coordinate system can be derived: Similarly, the transformation of the force vector can be expressed as follows: in: It is the force vector of all nodes in the local coordinate system. It is the force vector of all nodes in the global cylindrical coordinate system.

[0077] Finally, a discrete free vibration model of the motor stator system is established. The structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system are input into the discrete free vibration model. The discrete free vibration model outputs the different mode shapes and corresponding natural frequencies of the motor stator system, thus completing the identification of the mode shapes and frequencies of the motor stator core equipped with the internal gear ring.

[0078] This embodiment was experimentally verified on a 36-slot, 42-pole jointed modular motor with 48 teeth on the internal gear ring. The extracted stator yoke node modal amplitude displacement vectors and the corresponding identified natural frequencies, along with the circumferential mode shape results of different orders, are shown in Table 1 and Figure 11. Table 1 lists the natural frequencies and their relative errors using the hybrid shell element and beam element model of this invention. It can be seen that the results have good consistency, indicating that this invention can be used to accurately analyze the influence of the internal gear ring on the natural frequencies of the stator core.

[0079] Table 1 To leverage the wide applicability of the unit hybrid analytical model proposed in this invention, we further investigated the influence of structural parameters on the modal characteristics of the stator system and conducted parameter sensitivity analysis. As shown in Figures 12 and 13, it can be seen that the frequencies of lower-order modes generally decrease gradually with the increase of the number of teeth, while the natural frequencies of the 5th and 6th orders show slight fluctuations. Among all modes, the natural frequencies of each order show a significant decreasing trend with the increase of the shell mid-surface radius. The main reason is that the increase of the shell mid-surface radius is equivalent to increasing the radial flexibility of the structure, which reduces the bending stiffness of the entire shell, thereby leading to a decrease in modal frequencies. This can be theoretically verified in the stiffness coupling term. With the shell mid-surface radius unchanged, the frequencies of each order increase to varying degrees with the increase of the shell thickness. The main reason is that the increase in structural thickness enhances the bending and shear resistance and improves the overall stiffness. Moreover, compared with lower-order modes, higher-order modes show a smoother response surface to changes in shell thickness, indicating that the parameter sensitivity of higher-order modes is lower and is more reflected in local mode shape changes.

[0080] Therefore, the method of this invention can quickly and accurately model the motor stator system, and comprehensively analyze the influence of stator teeth and internal gear rings on the natural frequency of the stator core, demonstrating strong versatility. Compared with existing design methods, this invention's method, based on analytical identification, avoids large-scale finite element simulation and blind scanning optimization, and has advantages such as fast identification speed, clear directionality, low computational load, and high accuracy, which can guide subsequent vibration optimization and structural design.

[0081] The above description of the embodiments is provided to enable those skilled in the art to understand and apply the present invention. Those skilled in the art can readily make various modifications to the above embodiments and apply the general principles described herein to other embodiments without creative effort. Therefore, the present invention is not limited to the above embodiments, and any improvements and modifications made to the present invention by those skilled in the art based on the disclosure thereof should be within the scope of protection of the present invention.

Claims

1. A method for identifying the mode shapes and frequencies of a motor stator system with an internal gear ring, characterized in that, The steps include: (1) Establishing a hybrid analytical model of shell elements and beam elements for a motor stator system equipped with an internal gear ring; (2) Calculating the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system using the matrix displacement method based on the hybrid analytical model; (3) Calculating the structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system using coordinate transformation based on the total stiffness matrix and total mass matrix; (4) Establishing a discrete free vibration model of the motor stator system, inputting the structural stiffness matrix and structural mass matrix into the model, and the model automatically identifies the different mode shapes and natural frequencies of the motor stator system.

2. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 1, characterized in that: In step (1), for the motor stator system equipped with an internal gear ring, the hybrid analytical model of the system shell element and beam element consists of multiple tooth beam elements, multiple yoke shell elements, and multiple internal gear ring beam elements. The tooth beam elements and internal gear ring beam elements adopt Timoshenko beam elements, and the yoke shell elements adopt curved shell elements. The tooth beam element is modeled as a rectangular strip based on the stator tooth height in the motor stator system. The tooth beam element has two nodes located at the tooth tip and tooth root of the stator tooth and the connection between the tooth and the yoke, respectively. The internal gear ring beam element is modeled as a rectangular strip based on the tooth part of the internal gear ring along the tooth height direction. The internal gear ring beam element has two nodes located at the tooth tip and tooth root of the internal gear ring, respectively. The yoke shell element is modeled as a curved shell based on the arc-shaped yoke between two adjacent teeth in the motor stator system. The yoke shell element has four nodes located at the four vertices of the neutral plane rectangle. The geometric position of the neutral plane in space corresponds to the neutral circle radius of the stator yoke and is used to describe the neutral layer in the shell thickness direction.

3. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 1, characterized in that, The total stiffness matrix in step (2) is an N-dimensional diagonal matrix, specifically expressed as: in: Let be the overall stiffness matrix of the motor stator system in the local Cartesian coordinate system. 、 、 These are the stiffness matrices for the yoke shell element, the internal gear ring beam element, and the toothed beam element, respectively. The first N1 diagonal elements are The last N3 diagonal elements are The remaining diagonal elements are N = N1 + N2 + N3, where N1, N2, and N3 are the number of yoke shell elements, internal gear ring beam elements, and gear beam elements, respectively.

4. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 3, characterized in that, The stiffness matrix The expression is as follows: in: Let be the translational displacement stiffness matrix of the yoke shell element. Let be the rotational displacement stiffness matrix of the yoke shell element. Let be the translational stiffness interpolation matrix for the yoke shell element. Let be the rotational stiffness interpolation matrix of the yoke shell element, D be the Hooke matrix of the yoke shell element, h be the thickness of the yoke shell element, and J be the Jacobian matrix. This expression calculates the determinant of a matrix, where r and s are the x and y coordinates in the natural coordinate system, respectively. The superscript indicates the matrix. T Indicates transpose. Let be the translational stiffness interpolation matrix for the i-th node in the yoke shell element. Let be the rotational stiffness interpolation matrix for the i-th node in the yoke shell element, where i = 1, 2, 3, 4. and Let x and y be the partial derivatives of the shape function of the i-th node in the yoke shell element with respect to the local physical coordinates x and y at any point on the neutral surface. Let represent the shape function of the i-th node in the yoke shell element in the natural coordinate system. The radius of the neutral circle of the stator yoke. Let be the physical coordinates of the i-th node in the yoke shell element in the global coordinate system.

5. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 3, characterized in that: The stiffness matrix and The calculation expressions are the same, that is, their respective parameters are substituted into the stiffness matrix of the Timoshenko beam element. Calculated in the middle; in: 、 、 、 、 、 As an intermediate variable, denoted as , where E is the shear influence factor of the internal gear ring beam element or toothed beam element, G is the Young's modulus of the material of the internal gear ring beam element or toothed beam element, A is the equivalent cross-sectional area of ​​the internal gear ring beam element or toothed beam element, L is the equivalent length of the internal gear ring beam element or toothed beam element, I is the moment of inertia of the section of the internal gear ring beam element or toothed beam element, J is the moment of inertia of the internal gear ring beam element or toothed beam element, and k is the shear coefficient of the Timoshenko beam element.

6. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 1, characterized in that, The total mass matrix in step (2) is an N-dimensional diagonal matrix, specifically expressed as follows: in: Let be the total mass matrix of the motor stator system in the local Cartesian coordinate system. 、 、 These are the mass matrices for the yoke shell element, the internal gear ring beam element, and the gear beam element, respectively. The first N1 diagonal elements are The last N3 diagonal elements are The remaining diagonal elements are N = N1 + N2 + N3, where N1, N2, and N3 are the number of yoke shell elements, internal gear ring beam elements, and gear beam elements, respectively.

7. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 6, characterized in that, The mass matrix The expression is as follows: in: Let be the translational displacement-mass matrix of the yoke shell element. Let be the rotational displacement mass matrix of the yoke shell element. Let be the translational mass interpolation matrix for the yoke shell element. Let be the rotational mass interpolation matrix for the yoke shell element. Let be the material density of the yoke shell element, h be the thickness of the yoke shell element, and J be the Jacobian matrix. This expression calculates the determinant of a matrix, where r and s are the x and y coordinates in the natural coordinate system, respectively. The superscript indicates the matrix. T Indicates transpose. Let be the translational mass interpolation matrix for the i-th node in the yoke shell element. Let be the rotational mass interpolation matrix for the i-th node in the yoke shell element, where i = 1, 2, 3, 4. Let represent the shape function of the i-th node in the yoke shell element in the natural coordinate system. Let be the physical coordinates of the i-th node in the yoke shell element in the global coordinate system.

8. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 6, characterized in that: The mass matrix and The calculation expressions are the same, that is, each parameter is substituted into the mass matrix of the Timoshenko beam element. Calculated in the middle; in: 、 、 、 、 、 、 、 As an intermediate variable, This refers to the shear influence factor of the internal gear ring beam element or gear beam element. The material density of the internal gear ring beam element or gear beam element. Let be the radius of gyration of the internal gear ring beam element or toothed beam element, E be the Young's modulus of the material of the internal gear ring beam element or toothed beam element, G be the shear modulus of the material of the internal gear ring beam element or toothed beam element, A be the equivalent cross-sectional area of ​​the internal gear ring beam element or toothed beam element, L be the equivalent length of the internal gear ring beam element or toothed beam element, I be the moment of inertia of the section of the internal gear ring beam element or toothed beam element, J be the moment of inertia of the internal gear ring beam element or toothed beam element, and k be the shear coefficient of the Timoshenko beam element.

9. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 1, characterized in that, The calculation expressions for the structural stiffness matrix and structural mass matrix in step (3) are as follows: Where M and K are the structural stiffness matrix and structural mass matrix of the motor stator system in the global cylindrical coordinate system, respectively. and Let be the total stiffness matrix and total mass matrix of the motor stator system in the local rectangular coordinate system, respectively, and T be the coordinate transformation matrix from the local rectangular coordinate system to the global cylindrical coordinate system. The superscript _____ T This indicates transpose.

10. The method for identifying the mode shape and frequency of a motor stator system with an internal gear ring according to claim 1, characterized in that: The discrete free vibration model in step (4) calculates the natural angular frequency ω of the motor stator system and the nodal displacement vector at the natural angular frequency ω using the following expression. Furthermore, based on the nodal displacement vector Identify different mode shapes of the motor stator system; Where M and K are the structural stiffness matrix and structural mass matrix of the motor stator system in the overall cylindrical coordinate system, respectively.

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