Analytic calculation method of generator iron core mode

Abstract

The invention relates to an analytic calculation method for a generator iron core mode, which belongs to the technical field of structural analysis of motors and comprises the following steps: equivalently simplifying a generator stator iron core structure into a planar circular ring structure, and calculating equivalent circular ring parameters according to iron core parameters; a polynomial equation of the modal inherent frequency is established, and variables related to the modal inherent frequency are obtained by solving the equation; and calculating the modal inherent frequency of the iron core according to the obtained variable. The method is high in calculation precision and easy and convenient to operate, and standardization of the calculation method and process is achieved.

Description

Technical Field

[0001] This invention relates to a modal analysis method for the stator core structure of an electric motor, applicable to electric motors, synchronous condensers, and steam turbine generators, and belongs to the field of structural analysis technology for electric motors. Background Technology

[0002] When the generator rotor is connected to the excitation current, it becomes an electromagnet, and its magnetic poles generate a magnetic pull that deforms the stator core. This magnetic pull reaches its maximum value at the center of the magnetic poles and its minimum value between the poles, thus causing the stator core of a two-pole generator to deform into an elliptical shape and the stator core of a four-pole generator to deform into a four-lobed shape. When the natural frequency of the core approaches the excitation frequency, destructive resonance can easily occur. Therefore, modal analysis of the stator core is necessary during the generator design phase to ensure that the natural frequency of the core is far away from the frequency of the excitation force, thereby avoiding resonance.

[0003] Previously, the calculation of the natural frequency of generator stator cores commonly used the formula recommended in the electrical engineering technical guidance document "Calculation Formulas for Large Electrical Machines". This formula, based on the Bernoulli-Euler beam theory, establishes the differential equation of motion for the core, considering only bending deformation during vibration but neglecting the effects of shear deformation and rotational inertia. Theoretically, this model is only applicable to thin-walled circular ring structures. Therefore, it introduces certain errors when used to calculate the natural frequency of core structures with large radial thickness. Especially as the vibration order and mode wavenumber increase, the core is divided into multiple short and thick segments by the mode nodes. At this point, the effects of yoke shear deformation and the rotational inertia of the cross-section about the neutral axis are significantly enhanced, leading to a further increase in the error of the original algorithm. Therefore, there is an urgent need to develop a more accurate calculation method to accurately evaluate the natural frequency of generator stator cores and overcome the limitations of traditional methods. Summary of the Invention

[0004] The purpose of this invention is to provide a modal analysis method applicable to generator stator core structures. This method is simple to operate, computationally efficient, and produces accurate results, while also enabling the standardization of analysis methods and processes.

[0005] To achieve the above objectives, the present invention provides an analytical calculation method for generator core modes, which includes the following steps:

[0006] The generator stator core structure is simplified to an equivalent planar circular ring structure, and the equivalent circular ring parameters are calculated based on the core parameters.

[0007] Establish a polynomial equation for the modal natural frequencies, and obtain the variables related to the modal natural frequencies by solving the equation;

[0008] Calculate the modal natural frequencies of the iron core based on the obtained variables.

[0009] Preferably, the core parameters include the outer diameter D. o , inner diameter D i Number of slots n s , groove depth h s Width w s Elastic modulus E, density ρ0, axial cross-sectional area A of the yoke y Axial cross-sectional area A of the tooth t .

[0010] Preferably, the equivalent toroidal parameters include:

[0011] Average radius of the ring

[0012] Tangential cross-sectional area of ​​the ring

[0013] Moment of inertia of the tangential section of the annulus

[0014] equivalent density of toroid

[0015] Preferably, the polynomial equation for solving the modal natural frequencies is used.

[0016] sr 3 λ 3 -[1+srm 2 +(s+1)r(m 2 +1)]rλ 2

[0017] +[m 2 +1+(s+1)rm 2 (m 2 +1)+r(m 2 -1) 2 ]λ-m 2 (m 2 -1) 2 =0

[0018] in, λ is a variable related to the modal natural frequency; m is the modal order, which is taken as m=2 for a two-pole generator core and m=4 for a four-pole generator core.

[0019] Solving for λ yields three roots, with the smallest root corresponding to the modal natural frequency for which the solution is needed.

[0020] Preferably, the natural frequencies of the core modes are solved.

[0021] The technical solution of the present invention also provides an analytical calculation method for the modes of generator core, comprising the following steps:

[0022] Based on the actual structural characteristics of the generator stator core, a set of equilibrium differential equations for the free vibration of the core is established based on Timoshenko beam theory, taking into account the effects of core shear deformation and rotational inertia.

[0023] The differential equations for the free vibration of the iron core are transformed into a system of algebraic equations.

[0024] The modal natural frequencies of the iron core are obtained by solving the equations.

[0025] Preferably, based on the balance of the axial force, tangential force and torque of the core element, a set of equilibrium differential equations for the core during free vibration is established in polar coordinates.

[0026] Preferably, based on the physical equations, the internal forces on the micro-element cross section are expressed as normal force, shear force, and bending moment;

[0027] Substituting the internal forces into the equilibrium differential equations, we obtain the motion differential equations for the free vibration of the iron core.

[0028] Preferably, the degrees of freedom and internal forces are expressed in trigonometric function form;

[0029] Substituting the trigonometric function form into the system of differential equations of motion transforms it into a system of algebraic equations.

[0030] Preferably, the system of algebraic equations is represented in matrix form;

[0031] Calculate the determinant of the matrix and set it equal to zero;

[0032] Solve the determinant equation to obtain the modal natural frequencies.

[0033] In summary, the present invention has the following beneficial technical effects:

[0034] The method proposed in this invention effectively overcomes the limitations of insufficient calculation accuracy in existing technologies, enabling precise and rapid analysis of the modal natural frequencies of generator stator cores. The advantages of this method lie in its consideration of the effects of shear deformation and rotational inertia when calculating the modal natural frequencies of the generator stator core, resulting in accurate calculations and simple operation. By promoting the standardization of calculation methods and processes, it significantly improves analysis efficiency.

[0035] This invention, based on Timoshenko beam theory, establishes the kinematic differential equations for the free vibration of a generator stator core structure. By expressing the core vibration response and internal forces in trigonometric function form, the original differential equations are transformed into algebraic equations. Finally, by solving these algebraic equations, the modal natural frequencies of the stator core are calculated. The outstanding advantage of this invention is that it achieves the accurate calculation of the modal natural frequencies of a generator stator core for the first time, filling the gap in existing analytical methods that cannot accurately calculate natural frequencies due to neglecting shear deformation and rotational inertia. Attached Figure Description

[0036] Figure 1 This is the cross-section of the generator stator core;

[0037] Figure 2 This is a schematic diagram of the force analysis when a circular ring is in free vibration. Detailed Implementation

[0038] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0039] Example 1:

[0040] This invention provides a method for modal natural frequency analysis of generator stator cores. This method offers high calculation accuracy and ease of operation, and standardizes the calculation method and process. It includes the following steps:

[0041] Step 1: Based on the actual structural characteristics of the generator stator core, establish a set of equilibrium differential equations for the free vibration of the core. To ensure calculation accuracy, the equations are based on Timoshenko beam theory, taking into account the effects of core shear deformation and rotational inertia. Based on the balance of radial force, tangential force, and moment of inertia of the core element, establish the set of equilibrium differential equations for the core during free vibration in polar coordinates as follows:

[0042]

[0043] Where: θ is the angular coordinate of the infinitesimal element; t is time; N, Q, and M are the normal force, shear force, and bending moment on the cross section of the infinitesimal element, respectively; u, w, and Let R be the radial force, shear force, and angular displacement of the infinitesimal element, respectively; R be the average radius of the infinitesimal element; A be the area of ​​the tangential section of the infinitesimal element; I be the moment of inertia of the tangential section of the infinitesimal element; and ρ be the density.

[0044] According to the physical equations, the internal forces on the infinitesimal cross section can be expressed as:

[0045]

[0046] Where: k is the cross-sectional shape factor, which is 5 / 6 for rectangular cross-sections; E is the elastic modulus; and G is the shear modulus. Equation 2 above uses Timoshenko beam theory to consider the influence of core shear deformation, while equation 3 considers the influence of core rotational inertia. Substituting equation (2) into equation (1), we obtain the set of differential equations for the free vibration of the core.

[0047]

[0048] Step 2: Transform the differential equations of the free vibration of the iron core into an algebraic equation system. The free vibration of the iron core can be regarded as simple harmonic motion. According to the phase relationship between each degree of freedom and the internal force described by equation (2), the degrees of freedom u, w, and The internal forces N, Q, and M are expressed in the following trigonometric function form.

[0049]

[0050] Where: U0, W0 and Φ0 are the amplitudes in the radial, tangential and angular directions, respectively, when the core vibrates freely; N0, Q0 and M0 are the amplitudes of the changes in normal force, shear force and bending moment on the cross section, respectively, when the core vibrates freely; m is the order of free vibration; ω is the natural frequency of free vibration.

[0051] Substituting equations (4) and (5) into equation (3), the differential equations for the free vibration of the iron core can be transformed into a set of algebraic equations.

[0052]

[0053] In the above formula, let

[0054]

[0055]

[0056] Equation (6) can then be expressed in the following matrix form.

[0057]

[0058] Step 3: Obtain the natural frequency of the iron core by solving the equation. Equation (10) is a homogeneous linear system of equations, the only condition for which it has a non-zero solution is that the determinant of the coefficient matrix is ​​zero, i.e.

[0059]

[0060] In the above formula: The expression represents the determination of the determinant; s and r are coefficients related to material parameters and structural dimensions, which can be determined by equations (8) and (9) respectively after the core design scheme is determined; m represents the order of the mode shape. For a two-pole generator core, its main vibration mode is elliptical, so m = 2 is taken. For a four-pole generator core, its main vibration mode is four-lobed, so m = 4 is taken. The left side of equation (11) can be specifically expanded into a cubic equation in one variable λ using the matrix determinant formula.

[0061]

[0062] In engineering practice, this equation is usually solved using numerical methods. Currently, several mature numerical computing tools support solving such equations; for example, MATLAB and Python both provide corresponding toolkits that enable convenient and efficient computation.

[0063] After solving, three solutions are obtained, among which the solution with the smallest value is closely related to the mode of interest in the design. According to equation (7), the natural frequency of the iron core can be further calculated, and the specific expression is as follows:

[0064]

[0065] Example 2:

[0066] This invention also provides a method for modal natural frequency analysis of generator stator cores, which includes the following steps:

[0067] Step 1: Simplify the generator stator core structure into a planar circular ring with the same radial thickness as the yoke thickness. The core design parameter is the outer diameter D. o , inner diameter D i Number of slots n s , groove depth h s Width w s When the elastic modulus E and density ρ0 are given, the relevant parameters of the equivalent annulus can be calculated using the following formula:

[0068] Average radius of the ring

[0069] Tangential cross-sectional area of ​​the ring

[0070] Moment of inertia of the tangential section of the annulus

[0071] equivalent density of toroid

[0072] Step 2: Establish and solve the polynomial equation for the modal natural frequencies. Calculate the coefficients using the following formula.

[0073]

[0074] Then, establish the polynomial equation for the modal natural frequencies based on the following formula.

[0075] sr 3 λ 3 -[1+srm 2 +(s+1)r(m 2 +1)]rλ 2

[0076] +[m2 +1+(s+1)rm 2 (m 2 +1)+r(m 2 -1) 2 ]λ-m 2 (m 2 -1) 2 =0

[0077] Where λ is a variable related to the modal natural frequency; m is the modal order. For a two-pole generator core, its main vibration mode is elliptical, so we take m = 2. For a four-pole generator core, its main vibration mode is four-lobed, so we take m = 4.

[0078] Solving this cubic equation yields three roots of λ, with the smallest root corresponding to the modal natural frequency to be solved. The modal natural frequencies of the core can be calculated using the following formula.

[0079]

[0080] Example 3:

[0081] This embodiment, based on Embodiment 2, further illustrates the method in conjunction with the accompanying drawings:

[0082] like Figure 1 The figure shown is a cross-sectional schematic diagram of a generator stator core, with the core outer diameter D. o It is 2896mm in diameter and has an inner diameter D. i It is 1316mm, and the number of inlay slots is n. s The depth of each groove is 36, and the depth of each groove is h. s It is 274.3mm in diameter and has a width of w. s The core is 32.5mm thick. It is constructed from stacked silicon steel sheets, and its circumferential modulus of elasticity E is 1.724 × 10⁻⁶. 5 MPa, density ρ0 is 7.5 × 10 -9 ton / mm 3 This will be used as the object of analysis in this embodiment.

[0083] The method for analyzing the modal natural frequencies of the generator stator core in this embodiment is as follows:

[0084] Step 1: Simplify the calculation model. The rigidity of the core is mainly provided by the yoke, so the influence of the teeth can be ignored, simplifying it into a planar circular ring structure with a radial thickness equal to the thickness of the yoke. The stress state of this structure during free vibration is as follows: Figure 2 As shown. According to Figure 1 Given the core dimensions, the average radius of the simplified planar circular ring structure is calculated as follows:

[0085]

[0086] The tangential cross-sectional area of ​​the ring is

[0087]

[0088] The moment of inertia of the tangential section of the annulus is

[0089]

[0090] Calculate the material parameters of the simplified model. Since the calculation model ignores the tooth structure, its mass needs to be included in the yoke, and the equivalent density needs to be calculated. First, calculate the axial cross-sectional area A of the core yoke. y Axial cross-sectional area A of the teeth t for

[0091]

[0092] The equivalent density of the simplified toroidal model is then...

[0093]

[0094] For isotropic linear elastic materials, with a typical Poisson's ratio μ of 0.3, their shear modulus can be obtained from the relationship between elastic modulus and Poisson's ratio.

[0095]

[0096] Step 2: Solve the polynomial equations for the modal natural frequencies of the iron core. Substitute the R, A, I, and G calculated in Step 1 of this embodiment into equations (8) and (9) to calculate the coefficients.

[0097]

[0098] For a two-pole generator, let m = 2, and substitute the coefficients s and r into equation (12), we can obtain a cubic equation in one variable λ.

[0099] 1.184×10 -5 λ 3 -0.0237λ 2 +6.426λ-36=0

[0100] In this embodiment, the equation is numerically solved using Python's NumPy toolkit, yielding three roots of 5.723, 316.32, and 1679.64. The smallest root, 5.723, corresponds to the elliptic mode. Substituting this into equation (13), the natural frequency of the elliptic mode of the core can be calculated as follows:

[0101]

[0102] This translates to an engineering frequency of 170.12 Hz.

[0103] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An analytical calculation method for the modes of a generator core, characterized in that, Includes the following steps: The generator stator core structure is simplified to an equivalent planar circular ring structure, and the equivalent circular ring parameters are calculated based on the core parameters. Establish a polynomial equation for the modal natural frequencies, and obtain the variables related to the modal natural frequencies by solving the equation; Calculate the modal natural frequencies of the iron core based on the obtained variables.

2. The analytical calculation method for generator core modes according to claim 1, characterized in that, Core parameters include outer diameter D o , inner diameter D i Number of slots n s , groove depth h s Width w s Elastic modulus E, density ρ0, axial cross-sectional area A of the yoke y Axial cross-sectional area A of the tooth t .

3. The analytical calculation method for generator core modes according to claim 2, characterized in that, The equivalent toroidal parameters include: Average radius of the ring Tangential cross-sectional area of ​​the ring Moment of inertia of the tangential section of the annulus equivalent density of toroid 4. The analytical calculation method for generator core modes according to claim 3, characterized in that, Solving the polynomial equations for modal natural frequencies sr 3 λ 3 -[1+yrm 2 +(s+1)r(m 2 +1)]rλ 2 +[m 2 +1+(s+1)rm 2 (m 2 +1)+r(m 2 -1) 2 ]λ-m 2 (m 2 -1) 2 =0 in, λ is a variable related to the modal natural frequency; m is the modal order, which is taken as m=2 for a two-pole generator core and m=4 for a four-pole generator core. Solving for λ yields three roots, with the smallest root corresponding to the modal natural frequency for which the solution is needed.

5. The analytical calculation method for generator core modes according to claim 4, characterized in that, Solve for the natural frequencies of the iron core modes 6. An analytical calculation method for the modal characteristics of a generator core, characterized in that, Includes the following steps: Based on the actual structural characteristics of the generator stator core, a set of equilibrium differential equations for the free vibration of the core is established based on Timoshenko beam theory, taking into account the effects of core shear deformation and rotational inertia. The differential equations for the free vibration of the iron core are transformed into a system of algebraic equations. The modal natural frequencies of the iron core are obtained by solving the equations.

7. The analytical calculation method for generator core modes according to claim 6, characterized in that, Based on the balance of the axial force, tangential force, and torque of the core element, a set of equilibrium differential equations for the core during free vibration is established in polar coordinates.

8. The analytical calculation method for generator core modes according to claim 7, characterized in that, According to the physical equations, the internal forces on the cross section of the infinitesimal element can be expressed as normal force, shear force, and bending moment; Substituting the internal forces into the equilibrium differential equations, we obtain the motion differential equations for the free vibration of the iron core.

9. The analytical calculation method for generator core modes according to claim 8, characterized in that, Express the degrees of freedom and internal forces in trigonometric function form; Substituting the trigonometric function form into the system of differential equations of motion transforms it into a system of algebraic equations.

10. The analytical calculation method for generator core modes according to claim 9, characterized in that, Represent the system of algebraic equations in matrix form; Calculate the determinant of the matrix and set it equal to zero; Solve the determinant equation to obtain the modal natural frequencies.

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