A bearing equivalent stiffness calculation method for rotor dynamics modal analysis

Abstract

This invention discloses a method for calculating the equivalent stiffness of bearings for rotor dynamics modal analysis, belonging to the field of rotor dynamics finite element analysis technology. The method includes: extracting the transient load spectrum obtained from rigid body dynamics analysis as the bearing force boundary; calculating the roller composite contact stiffness coefficient using the Houpert analytical method; constructing the bearing spatial deformation compatibility equation; constructing a nonlinear force balance equation with the transient load spectrum as the target and deriving the transient Jacobian stiffness matrix; constructing a Newton-Raphson numerical iteration and introducing a multiple numerical robust solution mechanism including damping regularization and displacement step truncation to solve the bearing spatial deformation; obtaining the time-averaged equivalent stiffness through time-domain equivalent integration and inputting it as a real constant into the rotor dynamics modal analysis model. This invention improves the problems of simulation distortion, long cycles, and low efficiency caused by relying on experience or experiments to obtain bearing stiffness, and enhances the accuracy of rotor dynamics modal analysis.

Description

Technical Field

[0001] This invention belongs to the field of rotor dynamics finite element analysis technology, specifically relating to a bearing equivalent stiffness calculation method for rotor dynamics modal analysis. Background Technology

[0002] The rotor system is a core component of rotating machinery such as mechanical variable displacement vacuum pumps, aero engines, compressors, and high-speed machine tool spindles. Its dynamic characteristics directly determine the operational reliability and performance boundaries of the entire machine. In rotor dynamics design, critical speed margin verification, unbalanced response prediction, stability analysis, and eddy current characteristic evaluation all rely on the accurate description of the support boundary conditions. As the main supporting element of the rotor, the bearing's stiffness characteristics have a decisive influence on the natural frequency, mode shape, and stability of the rotor system.

[0003] In engineering practice, rolling bearings involve complex micromechanical behaviors such as the rotation and revolution of rolling elements, cage motion, and nonlinear Hertz contact between the inner and outer raceways. If a full three-dimensional solid model containing all rolling elements and contact details were created in general-purpose finite element software for transient dynamics solutions, the enormous number of degrees of freedom and extremely small time integration steps would lead to a drastic increase in computational complexity, which is impractical in engineering. Therefore, a reduced-order equivalent approach is commonly used, simplifying the bearing into macroscopic equivalent spring-damping elements to characterize the support boundary. For example, in commercial finite element software, spring elements are used to replace the bearing's supporting function.

[0004] However, accurately obtaining the equivalent stiffness of bearings has always been a challenge in engineering. In existing technologies, designers typically rely on bearing manufacturers' catalogs to set constant static stiffness values, or simplify the stiffness to an empirical curve related to rotational speed. In reality, rotors are subjected to complex loads such as unbalanced centrifugal force, gyroscopic torque, and transient kinematic overloads during actual operation. The bearing's contact angle, the number of bearing rolling elements, and the load distribution undergo strong nonlinear evolution with the spatiotemporal changes of the external load. Using constant or empirical stiffness to replace the actual time-varying stiffness inevitably leads to distortion in the prediction of higher-order modal frequencies, affecting the reliability and safety of the rotor system design.

[0005] To obtain more accurate bearing stiffness, some studies have constructed dedicated rotor-bearing test benches and used vibration methods or transfer function methods for physical calibration. However, the development of experimental methods is time-consuming and costly, and is limited by loading conditions and sensor placement. They can usually only cover a limited number of operating points, making it difficult to obtain continuous stiffness data for all operating conditions, and even more difficult to provide effective stiffness input for rotor dynamics analysis during the design phase.

[0006] Therefore, there is an urgent need for a method that can efficiently obtain the equivalent stiffness of bearings using digital means. By combining macroscopic rotor dynamics analysis with the microscopic contact mechanics model of the bearing, the accurate inversion and time-domain equivalence of the time-varying stiffness of the bearing can be achieved without physical experiments, thereby providing high-fidelity support boundary parameters for the modal analysis of the rotor system. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides a method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis. The method obtains the time-varying load spectrum of the bearing through rigid body dynamics transient simulation, drives the inversion of the bearing micro-deformation based on Hertz contact mechanics and multiple robust solution mechanisms, and then obtains the high-fidelity equivalent stiffness constant for rotor modal analysis through time-domain equivalent integration, thereby realizing a fully digital closed-loop mapping from macroscopic system loads to underlying support parameters.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis includes:

[0010] Step S1: Establish a macroscopic rotor dynamics model and run transient simulation to extract the three-dimensional follow-up reaction force sequence at the bearing support node to be tested, and generate a transient time-varying external load spectrum as the target external load vector;

[0011] Step S2: Based on the Houpert analytical method, establish the curvature equation of the contact interface between the bearing rolling element and the inner and outer rings, and calculate the comprehensive contact stiffness coefficient of the bearing rolling element.

[0012] Step S3: Analyze the initial assembly preload deformation of each row of bearings using the assembly axial preload and the comprehensive contact stiffness coefficient; construct the spatial deformation coordination equation under the three-degree-of-freedom system, introduce the axial projection direction coefficient and the unilateral contact constraint control operator, and calculate the effective normal deformation of each bearing rolling element under different spatial displacements based on the initial assembly preload deformation.

[0013] Step S4: Based on the nonlinear constitutive relation, calculate the normal contact force of the bearing rolling element by multiplying the comprehensive contact stiffness coefficient and the effective normal deformation and project it to the global coordinate system. Construct a nonlinear residual force balance equation with the target external load vector as the target value and derive the transient Jacobian tangent stiffness matrix.

[0014] Step S5: Establish the Newton-Raphson numerical algorithm, and construct a multi-robust solution mechanism consisting of damping regularization, generalized pseudo-inverse calculation and space physical domain step size truncation. Use the transient Jacobian tangent stiffness matrix to solve the iterative nonlinear residual force balance equation to update the spatial displacement vector, and recalculate the effective normal deformation with the updated spatial displacement vector until the nonlinear residual force balance equation converges.

[0015] Step S6: After the spatial displacement vector converges, the transient radial principal stiffness and transient axial principal stiffness of each time step are extracted based on the transient Jacobian tangent stiffness matrix.

[0016] Step S7: Perform time-domain equivalent integration on the transient radial principal stiffness and the transient axial principal stiffness, output the radial equivalent stiffness constant and the axial equivalent stiffness constant, and map them to the radial equivalent support element and the axial equivalent support element in the rotor dynamic modal analysis model, respectively.

[0017] Furthermore, in step S1, the transient time-varying external load spectrum is obtained by solving the macroscopic rotor dynamics model based on rigid body dynamics simulation.

[0018] Furthermore, in step S2:

[0019] Define the ratio of the structural geometric parameters of the bearing under test, and introduce the cross-sectional radius of curvature coefficient of the inner ring raceway and the cross-sectional radius of curvature coefficient of the outer ring raceway respectively;

[0020] The first curvature and the coefficient of difference between the first curvature and the elliptical interface where the bearing rolling element contacts the bearing inner ring are analyzed and derived, as well as the second curvature and the coefficient of difference between the second curvature and the elliptical interface where the bearing rolling element contacts the bearing outer ring.

[0021] The ratio of the major and minor axes of the first and second contact ellipses is calculated using the Houpert closed analytical formula based on the first curvature difference coefficient and the second curvature difference coefficient, respectively. Based on Hertz's elastic contact theory, the first analytical contact stiffness is calculated from the first curvature and its ratio to the major and minor axes of the first contact ellipse, and the second analytical contact stiffness is calculated from the second curvature and its ratio to the major and minor axes of the second contact ellipse. The first analytical contact stiffness and the second analytical contact stiffness are then combined using a series of mechanical equations to obtain the comprehensive contact stiffness coefficient.

[0022] Furthermore, in step S3:

[0023] In the aforementioned three-degree-of-freedom system, the spatial displacement vector of the bearing inner ring relative to the bearing outer ring is defined;

[0024] Define the total number of rows of bearing rolling elements of the bearing under test. For each row of bearing rolling elements, introduce the axial projection direction coefficient. When the positive axial displacement causes the row of bearing rolling elements to tend to be pressed together, the axial projection direction coefficient is 1. When it tends to be stretched and unloaded, the axial projection direction coefficient is -1.

[0025] For each row of bearing rolling elements, set the assembly axial preload, and calculate the initial assembly preload deformation of that row based on Hertzian contact theory and static equilibrium analysis.

[0026] Based on the initial assembly preload deformation amount in this column, combined with the circumferential azimuth angle of each bearing rolling element in this column, the radial projection component of the spatial displacement vector, and the product of the axial projection direction coefficient and the axial displacement, the theoretical normal deformation amount of each bearing rolling element is calculated.

[0027] By applying the unilateral contact constraint control operator, the values ​​of the theoretical normal deformation of each bearing rolling element that are less than zero are set to zero, thus obtaining the effective normal deformation.

[0028] Furthermore, in step S4, the transient Jacobian tangential stiffness matrix is ​​obtained by taking the partial derivative of the transient bearing total support reaction vector with respect to each component of the spatial displacement vector, and its main diagonal elements constitute the radial principal stiffness and axial principal stiffness of the bearing; the radial principal stiffness is obtained by summing the local contact stiffness of each effective bearing rolling element according to the projection weights of their respective contact angles and azimuth angles, characterizing the load-displacement response capability of the bearing in the radial plane; the axial principal stiffness is obtained by summing the local contact stiffness of each effective bearing rolling element according to the sinusoidal projection weights of the contact angles, characterizing the load-displacement response capability of the bearing in the axial direction; each local contact stiffness is determined by the nonlinear relationship between the comprehensive contact stiffness coefficient and the corresponding effective normal deformation.

[0029] Furthermore, in step S5:

[0030] The damping regularization is implemented, the maximum value of the diagonal of the transient Jacobian tangent stiffness matrix is ​​extracted, and the damping factor is injected to construct the correction matrix;

[0031] The generalized pseudo-inverse calculation is performed by using singular value decomposition technology and directly solving the displacement increment vector numerically using the generalized Moore-Ponous pseudo-inverse operator of the correction matrix and the nonlinear residual force balance equation of the current iteration step.

[0032] Implement the spatial physical domain step size truncation to limit the displacement modulus of a single iteration update to no more than a preset threshold;

[0033] The numerical value of the nonlinear residual force balance equation for the current iteration step is determined based on the difference between the transient bearing total support reaction vector and the target external load vector.

[0034] Furthermore, in step S6: when the nonlinear residual force balance equation satisfies the set convergence judgment condition, it is determined that the spatial displacement vector of the current transient time step has converged. The spatial displacement vector in the converged state is substituted into the explicit differential analytical expression of the transient Jacobian tangent stiffness matrix, and the elements on the main diagonal of the matrix are extracted as the transient radial principal stiffness and the transient axial principal stiffness under that time step, respectively.

[0035] Furthermore, in step S7, the time-domain equivalent integration is as follows: the transient radial principal stiffness and the transient axial principal stiffness of each time step are integrated along the time-domain interval of the set working condition, and divided by the total integration time to obtain the radial equivalent stiffness constant and the axial equivalent stiffness constant.

[0036] Specifically, the mapping input involves mapping the radial equivalent stiffness constant to the radial spring damping support unit in the modal analysis model, and mapping the axial equivalent stiffness constant to the axial spring damping support unit in the modal analysis model.

[0037] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis.

[0038] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis.

[0039] The beneficial effects of this invention are as follows:

[0040] First, it achieves a fully digital replacement for obtaining bearing stiffness. This invention, without requiring any physical prototypes or dedicated experimental platforms, can faithfully invert the dynamic stiffness characteristics of bearings under arbitrarily complex time-varying loads using only conventional rotor dynamics simulation. This significantly shortens the product design iteration cycle, greatly reduces reliance on physical testing, and achieves cost reduction and efficiency improvement.

[0041] Secondly, it significantly improves the accuracy of modal analysis of rotor systems. This invention directly maps the radial and axial equivalent stiffness constants obtained from time-domain equivalent integration to the support elements in general-purpose finite element software, making the simulation boundary conditions highly consistent with the actual mechanical state of the bearing. This effectively avoids the prediction deviation of natural frequencies and mode shapes caused by empirical stiffness values ​​being too hard or too soft, thus ensuring the reliability of rotor dynamics design.

[0042] Third, it possesses good numerical robustness and engineering applicability. This invention introduces a multi-robust mechanism in the nonlinear solver, consisting of damped regularization, generalized pseudo-inverse calculation, and step size truncation in the spatial physical domain. This mechanism can smoothly handle the stiffness matrix singularity problem caused by large-area unloading of rolling elements under extreme conditions such as large loads and large oscillations of bearings, ensuring stable and efficient calculations in continuous time steps.

[0043] Fourth, it has a wide range of applications and strong compatibility. Through a unified system of generalized analytical equations, this invention can be compatible with single-row bearings, back-to-back (DB), face-to-face (DF), tandem (DT), and any combination of multiple rows of bearings. It can be widely applied to rotor dynamics analysis of various rotating machinery such as aero-engines, gas turbines, centrifugal compressors, high-speed machine tool spindles, and vacuum pumps, and has good versatility. Attached Figure Description

[0044] Figure 1 This is a flowchart illustrating a bearing equivalent stiffness calculation method for rotor dynamic modal analysis according to the present invention.

[0045] Figure 2 This is a schematic diagram of the structure of the bearing to be tested provided in an embodiment of the present invention;

[0046] Figure 3 The geometric parameters of the bearing under test and the transformation relationship between the global coordinate system of the bearing and the local coordinate system of the bearing rolling elements are provided for embodiments of the present invention.

[0047] Figure 4 The target external load vector of the bearing at each transient time step is provided in the embodiments of the present invention based on transient rigid body dynamics simulation. ;

[0048] Figure 5 The target external load vector provided in the embodiments of the present invention The equivalent stiffness values ​​of the bearing corresponding to the transient radial principal stiffness and the transient axial principal stiffness of the bearing are calculated below.

[0049] Figure label:

[0050] 1. Bearing outer ring; 101. Bearing outer ring raceway; 2. Bearing inner ring; 201. Bearing inner ring raceway; 3. Bearing rolling elements. Detailed Implementation

[0051] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0052] This embodiment provides a method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis. This method relies on computer equipment with data processing capabilities. For example... Figure 1 As shown, the method specifically includes:

[0053] Step S1: Establish a macroscopic rotor dynamics model and run transient simulation. Extract the three-dimensional follow-up reaction force sequence at the support node of the bearing under test under the set working conditions, generate the transient time-varying external load spectrum of the bearing, and use it as the target external load vector of the bearing. ;

[0054] Step S2: Based on the Houpert analytical method, establish the micro curvature equations of the contact interfaces between the bearing rolling elements and the bearing inner ring and the bearing outer ring respectively, and calculate the comprehensive contact stiffness coefficient of the bearing rolling elements.

[0055] Step S3: Based on the static equilibrium equation, the initial assembly preload deformation of each row of bearings is analytically calculated using the assembly axial preload and the comprehensive contact stiffness coefficient. A spatial deformation coordination equation for the bearing under test, which contains at least one row of bearing rolling elements in a three-degree-of-freedom system, is constructed. An axial projection direction coefficient is introduced to characterize the vector projection relationship between the contact angle and the spatial displacement. Using the initial assembly preload deformation as a benchmark, the real-time normal deformation of each bearing rolling element under different azimuth angles is calculated. A single-sided contact constraint control operator is introduced to determine the effective bearing boundary.

[0056] Step S4: Based on the nonlinear mechanical constitutive relation, calculate the normal contact force of the bearing rolling elements by multiplying the comprehensive contact stiffness coefficient and the effective normal deformation, and project it onto the global coordinate system to calculate the transient bearing support reaction vector. A nonlinear residual force balance equation is constructed using the target external load vector as the target value, and the transient Jacobian tangent stiffness matrix is ​​derived simultaneously.

[0057] Step S5: Establish the Newton-Raphson numerical algorithm and set up a multi-numerical robust solution mechanism consisting of damping regularization, generalized pseudo-inverse calculation and space physical domain step size truncation. Solve the iterative nonlinear residual force balance equation to update the spatial displacement vector, and recalculate the effective normal deformation with the updated spatial displacement vector. Then calculate the transient bearing total support reaction vector to calculate the nonlinear residual force balance equation until the nonlinear residual force balance equation converges.

[0058] Step S6: After the nonlinear residual force balance equation converges, the transient Jacobian tangent stiffness matrix corresponding to each target external load vector is calculated based on the converged spatial displacement vector, and the transient radial principal stiffness and transient axial principal stiffness of each time step are extracted.

[0059] Step S7: Perform equivalent numerical integration of the transient radial principal stiffness and the transient axial principal stiffness along the time domain interval of the set working condition, output the radial equivalent stiffness constant and the axial equivalent stiffness constant, and map them into the radial equivalent support unit and the axial equivalent support unit in the rotor dynamic modal analysis model, respectively.

[0060] like Figure 2 As shown, this embodiment performs equivalent stiffness extraction and mapping on the back-to-back (DB configuration) 7314 type angular contact ball bearing assembly used in the shaft system of a certain type of all-metal scroll vacuum pump. The back-to-back (DB configuration) 7314 type angular contact ball bearing assembly includes a first row of bearings and a second row of bearings. Each row of bearings consists of an outer ring 1, an inner ring 2, and rolling elements 3. An outer ring groove 101 is provided on the inner side of the outer ring 1, and an inner ring groove 201 is provided on the outer side of the inner ring 2. The rolling elements 3 are evenly distributed between the inner ring groove 201 and the outer ring groove 101. (Refer to...) Figure 3 The geometric parameters of the back-to-back (DB configuration) 7314 type angular contact ball bearing assembly, as well as the global coordinate system and the local coordinate system of the bearing rolling elements 3, are further defined. The geometric center of the bearing assembly is the origin O of the global coordinate system, and the global coordinate system is O-XYZ. The radial direction of the bearing assembly corresponds to the XOY plane, and the arrangement direction of the bearing assembly is the Z-axis of the global coordinate system. The bearing rolling elements 3 are evenly arranged between the inner ring 2 and the outer ring 1 of the bearing. The angle between the line connecting the center of the bearing rolling element 3 to the origin O of the global coordinate system and the X-axis of the global coordinate system O-XYZ is [insert angle here]. The center of the bearing rolling element 3 is the origin o of the local coordinate system; the direction of the line connecting the global coordinate origin O and the center of the bearing rolling element 3 is defined as the x-axis of the local coordinate system of the bearing rolling element 3; the direction perpendicular to the line connecting the center of the bearing rolling element 3 and the global coordinate origin O is the y-axis of the local coordinate system; the z-axis of the local coordinate system is the same as the z-axis of the global coordinate system. The diameter of the bearing rolling element 3 is... The contact angles between the bearing rolling element 3 and the bearing inner ring raceway 201 and the bearing outer ring raceway 101 are both... The diameter of the bearing pitch circle formed by the centers of the rolling elements 3 is... .

[0061] Specifically, in step S1:

[0062] A rigid body dynamics model was established using general-purpose multibody dynamics software via computer equipment. This model included the mass distribution of the all-metal vortex vacuum pump rotor and its shaft system, as well as the unbalanced eccentricity and support span. A transient rigid body dynamics simulation under specified operating conditions was then performed. (Refer to...) Figure 4After the simulation converges, the three-dimensional follow-up support reaction force sequence at the support node of the 7314 type double-row angular contact ball bearing assembly under test is extracted, and it is discretized along the time axis to generate a transient time-varying external load spectrum, which is stored in the memory and defined as the target external load vector of the bearing at each transient time step. ,in, They represent The bearing is in time , , Target external load value in the direction, superscript This indicates transpose.

[0063] In step S2:

[0064] The computer equipment reads the internal structural geometric parameters of the 7314 type angular contact ball bearing under test. In this embodiment, the exact engineering parameter extracted from the input is: the diameter of the bearing rolling element 3. The bearing pitch circle diameter formed by the centers of the three rolling elements is 25.4 mm. The radius of curvature coefficient of the bearing inner ring raceway 201 is 110mm. The radius of curvature coefficient of the cross section of the bearing outer ring raceway 101 is 0.52. The contact angle between each bearing rolling element 3 and the bearing inner ring raceway 201 and the bearing outer ring raceway 101 is 0.52. It is 40°.

[0065] First, calculate the ratio of dimensionless structural geometric parameters. Subsequently, based on the aforementioned geometric parameters, the curvature and curvature difference coefficient at the contact interface between the bearing rolling element 3 and the bearing inner ring raceway 201 and the bearing outer ring raceway 101 were rigorously analyzed and derived:

[0066] The curvature of the contact interface of the bearing inner ring raceway 201 and ,

[0067] Curvature difference coefficient of the contact interface of the bearing inner ring raceway 201 ;

[0068] The curvature of the contact interface of the bearing outer ring raceway 101 and ,

[0069] Curvature difference coefficient of the contact interface of the bearing outer ring raceway 101 .

[0070] Furthermore, the ratio of the major and minor axes of the ellipse at the 201 contact interface of the bearing inner ring raceway is directly calculated using the Houpert nonlinear regression closed analytical formula. And the ratio of the major and minor axes of the contact ellipse of the bearing outer ring raceway 101. .

[0071] Finally, the basic mechanical properties (including elastic modulus) of the Inconel 718 high-temperature alloy, the bearing material specially designed in this embodiment, are read. GPa, Poisson's ratio Then, based on the approximate solutions of the first and second kinds of complete elliptic integrals, the analytical contact stiffness at the contact interface between the bearing rolling element 3 and the bearing inner ring groove 201 is obtained. for: Calculate the analytical contact stiffness at the interface between the rolling element 3 of the bearing and the outer ring raceway 101 of the bearing. for: The comprehensive contact stiffness coefficient of the bearing rolling elements is obtained by synthesizing series mechanical equations. N / mm 1.5 .

[0072] In step S3:

[0073] Define the total number of columns in the back-to-back bearing assembly to be tested. The total number of rolling elements 3 in a single-row bearing is 2. The instantaneous spatial displacement vector of the bearing inner ring 2 relative to the bearing outer ring 1 is 11; , They represent The instantaneous spatial displacement of the inner ring 2 of the bearing relative to the outer ring 1 in the X, Y, and Z directions. In this embodiment, the axial preload of the bearing is applied by the lock nut. The value is 1500 N. Based on static equilibrium, substituting into the above calculation... The value, obtained by computer parsing, is the first... List( ∈ The initial assembly preload deformation is mm. To accurately represent spatial relationships, an axial projection direction factor is introduced. In a DB back-to-back configuration, the positive axial displacement is specified. The system automatically assigns values ​​to press the rolling elements 3 of the first bearing in the bearing assembly together. Simultaneously, this causes the rolling elements 3 of the second row of bearings in the bearing assembly to tend to loosen, resulting in the assigned value. Therefore, the first The first in the list indivual( ∈ ) Azimuth angle is The theoretical normal deformation of the bearing rolling element 3 is characterized as follows: Subsequently, a one-sided contact constraint control operator is applied to cut off the invalid load, so that the effective normal deformation of the bearing rolling element 3 participating in the final calculation satisfies the condition. .

[0074] In step S4:

[0075] Based on the effective normal deformation obtained in step S3, the computer processor calculates the normal support reaction force of each effective bearing rolling element 3 according to the Hertz nonlinear mechanical constitutive relation. Then, the support reactions of all bearing rolling elements 3 are tensor-projected and superimposed onto the global three-dimensional coordinate system O-XYZ to analytically construct the transient total support reaction vector of the bearing assembly. The explicit algebraic summation formulas for the components of the support reactions (i.e., in the X, Y, and Z directions) in the global three-dimensional global spatial coordinate system O-XYZ are as follows:

[0076] ,

[0077] Furthermore, using the target external load vector in step S1 To optimize the approach to the target, a nonlinear residual force equilibrium equation is constructed. To derive the transient Jacobian tangent stiffness matrix corresponding to the bearing assembly. According to the chain rule of calculus for the system, the total support reaction vector of the bearing assembly is... Regarding spatial displacement vector To perform partial differential equations, we first define the microscopic local tangent stiffness coefficient as... ,pass The 3×3 transient Jacobian tangent stiffness matrix containing fully coupled terms is analytically derived and assembled. The explicit differential analytical expressions of the elements of its main diagonal matrix are defined as follows:

[0078] radial principal stiffness components ;

[0079] radial principal stiffness components ;

[0080] Axial principal stiffness components .

[0081] In step S5:

[0082] The high-speed rotation of the vacuum pump generates a non-constant large load on the bearing assembly, which causes non-linear shedding of the bearing rolling element 3, resulting in the transient Jacobian tangential stiffness matrix generated in step S4. To address potential singularity issues, the computer employs a Newton-Raphson numerical algorithm structure and implements the following robust iterative update mechanism: First, damping regularization is applied to extract the transient Jacobian tangent stiffness matrix. The largest element on the main diagonal, injected with damping factor Construct the correction matrix ,in It is a 3×3 identity matrix; then, the generalized pseudo-inverse calculation is performed on the correction matrix. Singular value decomposition is performed using the generalized Newton's iterative increment formula. Solve for the displacement increment vector of the current iteration step, where Represent the generalized Moore-Penrose pseudo-inverse operator; finally, implement space physics domain step size truncation to control the displacement increment modulus of a single iteration to satisfy... If the value exceeds the limit, it is scaled proportionally along the original direction and truncated to 0.01 mm. Under this mechanism, the computer iteratively updates the spatial displacement vector within each transient time step. In the formula, This represents the spatial displacement vector of the previous iteration step. This represents the updated spatial displacement vector for the current iteration step.

[0083] In step S6:

[0084] During the iterative solution process, when the nonlinear residual force equilibrium equation satisfies the set convergence criterion, that is, the L2 norm of the residual force vector satisfies... At this point, it is determined that the spatial displacement vector of the current transient time step has fully converged. After convergence, the computer processor calculates the value of the spatial displacement vector in the current convergent state. Substituting back into the explicit differential analytical expression of the transient Jacobian tangent stiffness matrix described in step S4, calculate and output the final, accurate transient Jacobian tangent stiffness matrix corresponding to the current time step. The numerical values ​​are then extracted, and the elements on the main diagonal of the matrix are directly used as the transient radial principal stiffness of the bearing assembly at that time step. , With bearing transient axial principal stiffness Please refer to Figure 5 .

[0085] In step S7:

[0086] The computer equipment reads the bearing transient stiffness data for all time steps within the time domain, along the time domain starting interval of the set operating condition. The numerical trapezoidal integration method is used for smooth equivalent calculations, and the integration formula is explicitly stated as follows:

[0087] ,

[0088] ,

[0089] ,

[0090] Reference Figure 5 Based on the operating condition calculation output of this embodiment, the constant equivalent stiffness value of the all-metal scroll vacuum pump bearing assembly is obtained, wherein the radial equivalent stiffness constant is... 3.254×10 5 N / mm, 3.445×10 5 N / mm, axial equivalent stiffness constant It is 4.717×10 5 N / mm. Finally, the computer equipment will write the obtained constant real constant form of the bearing equivalent stiffness into the corresponding COMBIN214 radial equivalent support element attribute file and COMBIN14 axial equivalent support element attribute file in the modal analysis model of the all-metal scroll vacuum pump rotor shaft system. This allows the high-fidelity drive system solver to perform critical speed prediction and multi-order modal frequency and mode shape extraction analysis of the shaft system.

[0091] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis.

[0092] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis.

[0093] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. 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. A method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis, characterized in that, include: Step S1: Establish a macroscopic rotor dynamics model and run transient simulation to extract the three-dimensional follow-up reaction force sequence at the bearing support node to be tested, and generate a transient time-varying external load spectrum as the target external load vector; Step S2: Based on the Houpert analytical method, establish the curvature equation of the contact interface between the bearing rolling element and the inner and outer rings, and calculate the comprehensive contact stiffness coefficient of the bearing rolling element. Step S3: Analyze the initial assembly preload deformation of each row of bearings using the assembly axial preload and the comprehensive contact stiffness coefficient; construct the spatial deformation coordination equation under the three-degree-of-freedom system, introduce the axial projection direction coefficient and the unilateral contact constraint control operator, and calculate the effective normal deformation of each bearing rolling element under different spatial displacements based on the initial assembly preload deformation. Step S4: Based on the nonlinear constitutive relation, calculate the normal contact force of the bearing rolling element by multiplying the comprehensive contact stiffness coefficient and the effective normal deformation and project it to the global coordinate system. Construct a nonlinear residual force balance equation with the target external load vector as the target value and derive the transient Jacobian tangent stiffness matrix. Step S5: Establish the Newton-Raphson numerical algorithm, and construct a multi-robust solution mechanism consisting of damping regularization, generalized pseudo-inverse calculation and space physical domain step size truncation. Use the transient Jacobian tangent stiffness matrix to solve the iterative nonlinear residual force balance equation to update the spatial displacement vector, and recalculate the effective normal deformation with the updated spatial displacement vector until the nonlinear residual force balance equation converges. Step S6: After the spatial displacement vector converges, the transient radial principal stiffness and transient axial principal stiffness of each time step are extracted based on the transient Jacobian tangent stiffness matrix. Step S7: Perform time-domain equivalent integration on the transient radial principal stiffness and the transient axial principal stiffness, output the radial equivalent stiffness constant and the axial equivalent stiffness constant, and map them to the radial equivalent support element and the axial equivalent support element in the rotor dynamic modal analysis model, respectively.

2. The method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis according to claim 1, characterized in that, In step S1, the transient time-varying external load spectrum is obtained by solving the macroscopic rotor dynamics model based on rigid body dynamics simulation.

3. The method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis according to claim 1, characterized in that, In step S2: Define the ratio of the structural geometric parameters of the bearing under test, and introduce the cross-sectional radius of curvature coefficient of the inner ring raceway and the cross-sectional radius of curvature coefficient of the outer ring raceway respectively; The first curvature and the coefficient of difference between the first curvature and the elliptical interface where the bearing rolling element contacts the bearing inner ring are analyzed and derived, as well as the second curvature and the coefficient of difference between the second curvature and the elliptical interface where the bearing rolling element contacts the bearing outer ring. The ratio of the major and minor axes of the first contact ellipse and the ratio of the major and minor axes of the second contact ellipse are calculated using the Houpert closed analytical formula based on the first curvature difference coefficient and the second curvature difference coefficient, respectively. Based on Hertz's elastic contact theory, the first analytical contact stiffness is calculated by the first curvature and the ratio of the major and minor axes of the first contact ellipse, and the second analytical contact stiffness is calculated by the second curvature and the ratio of the major and minor axes of the second contact ellipse. The first analytical contact stiffness and the second analytical contact stiffness are combined using a series of mechanical equations to obtain the comprehensive contact stiffness coefficient.

4. The method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis according to claim 1, characterized in that, In step S3: In the aforementioned three-degree-of-freedom system, the spatial displacement vector of the bearing inner ring relative to the bearing outer ring is defined; Define the total number of rows of bearing rolling elements of the bearing under test. For each row of bearing rolling elements, introduce the axial projection direction coefficient. When the positive axial displacement causes the row of bearing rolling elements to tend to be pressed together, the axial projection direction coefficient is 1. When it tends to be stretched and unloaded, the axial projection direction coefficient is -1. For each row of bearing rolling elements, set the assembly axial preload, and calculate the initial assembly preload deformation of that row based on Hertzian contact theory and static equilibrium analysis. Based on the initial assembly preload deformation amount in this column, combined with the circumferential azimuth angle of each bearing rolling element in this column, the radial projection component of the spatial displacement vector, and the product of the axial projection direction coefficient and the axial displacement, the theoretical normal deformation amount of each bearing rolling element is calculated. By applying the unilateral contact constraint control operator, the values ​​of the theoretical normal deformation of each bearing rolling element that are less than zero are set to zero, thus obtaining the effective normal deformation.

5. The method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis according to claim 1, characterized in that, In step S4, the transient Jacobian tangent stiffness matrix is ​​obtained by taking the partial derivative of the transient bearing total support reaction vector with respect to each component of the spatial displacement vector, and its main diagonal elements constitute the radial principal stiffness and axial principal stiffness of the bearing; the radial principal stiffness is obtained by summing the local contact stiffness of each effective bearing rolling element according to the projection weights of their respective contact angles and azimuth angles, which characterizes the load-displacement response capability of the bearing in the radial plane; The axial principal stiffness is obtained by summing the local contact stiffness of each effective bearing rolling element according to the sinusoidal projection weight of the contact angle, which characterizes the load-displacement response capability of the bearing in the axial direction; each local contact stiffness is determined by the nonlinear relationship between the comprehensive contact stiffness coefficient and the corresponding effective normal deformation.

6. The method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis according to claim 5, characterized in that, In step S5: The damping regularization is implemented, the maximum value of the diagonal of the transient Jacobian tangent stiffness matrix is ​​extracted, and the damping factor is injected to construct the correction matrix; The generalized pseudo-inverse calculation is performed by using singular value decomposition technology and directly solving the displacement increment vector numerically using the generalized Moore-Ponous pseudo-inverse operator of the correction matrix and the nonlinear residual force balance equation of the current iteration step. Implement the spatial physical domain step size truncation to limit the displacement modulus of a single iteration update to no more than a preset threshold; The numerical value of the nonlinear residual force balance equation for the current iteration step is determined based on the difference between the transient bearing total support reaction vector and the target external load vector.

7. The method for calculating the equivalent stiffness of bearings for rotor dynamic modal analysis according to claim 1, characterized in that, In step S6: when the nonlinear residual force balance equation satisfies the set convergence judgment condition, it is determined that the spatial displacement vector of the current transient time step has converged. The spatial displacement vector in the converged state is substituted into the explicit differential analytical expression of the transient Jacobian tangent stiffness matrix, and the elements on the main diagonal of the matrix are extracted and used as the transient radial principal stiffness and the transient axial principal stiffness of the current time step, respectively.

8. The method for calculating the equivalent stiffness of a bearing for rotor dynamic modal analysis according to claim 1, characterized in that, In step S7, the time-domain equivalent integration is as follows: the transient radial principal stiffness and the transient axial principal stiffness of each time step are integrated along the time-domain interval of the set working condition, and divided by the total integration time to obtain the radial equivalent stiffness constant and the axial equivalent stiffness constant. Specifically, the mapping input involves mapping the radial equivalent stiffness constant to the radial spring damping support unit in the modal analysis model, and mapping the axial equivalent stiffness constant to the axial spring damping support unit in the modal analysis model.

9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the bearing equivalent stiffness calculation method for rotor dynamic modal analysis as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement the bearing equivalent stiffness calculation method for rotor dynamic modal analysis as described in any one of claims 1-8.

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