A mathematical model that can predict the forming shape of spherical bearings
By combining geometric relationships and the theory of material elasticity and springback into a mathematical model, the problem of efficient and accurate prediction of the forming shape of spherical bearings was solved, and the reliability of process design and resource optimization were achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- YANSHAN UNIV
- Filing Date
- 2026-01-26
- Publication Date
- 2026-06-02
AI Technical Summary
Existing technologies lack efficient and accurate mathematical models for predicting the forming shape of spherical plain bearings, leading to reliance on experimentation and simulation for process design, which increases uncertainty and development cycle.
A mathematical model is constructed that combines the geometric relationship of the compression process with the elastic rebound theory of materials. By establishing a set of geometric equations and static equilibrium conditions, the explicit relationship before and after rebound is derived, thereby achieving accurate prediction of the outer ring shape of the spherical bearing.
It significantly improves the prediction accuracy of the formed shape, reduces the number of experiments and simulations, shortens the development cycle, saves resources, and provides a reliable basis for process design.
Smart Images
Figure CN122133269A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the fields of computational mechanics and computer-aided engineering technology, specifically to a mathematical model that can predict the forming shape of spherical bearings. Background Technology
[0002] Spherical plain bearings are key basic components in mechanical equipment, and the forming accuracy of their outer ring is crucial to ensuring the overall performance and service life of the bearing. In the field of precision plastic forming of spherical plain bearings, especially in the process of pressing with tapered molds, how to accurately predict the final shape has always been the core issue of process design and optimization.
[0003] Currently, prediction methods in this field are mainly based on physical experiments, numerical simulations, and empirical formulas. These methods provide important basis for process understanding and parameter determination. With the development of digital design and intelligent manufacturing technologies, the industry has an increasing need to establish a prediction model that can efficiently and accurately integrate geometric deformation and material mechanical behavior and facilitate engineering applications, so as to further improve the reliability of process design, shorten the development cycle, and optimize resource allocation.
[0004] Therefore, developing a mathematical model for predicting forming shapes that has a clear mechanism and high computational efficiency is of great theoretical significance and engineering application value. Summary of the Invention
[0005] The purpose of this invention is to address the problems existing in the background art by proposing a mathematical model that can predict the forming shape of spherical bearings.
[0006] The technical solution of this invention: a mathematical model capable of predicting the forming shape of a spherical plain bearing, the specific construction steps of which include: S1. Obtain the geometric dimensions of the outer ring blank of the spherical bearing, the geometric and process parameters of the tapered die, and the elastic modulus of the material; S2. Based on the geometric relationship of the compression process, establish a set of equations, solve them simultaneously to derive the core relationship between the compression amount and the theoretical mean diameter after compression, and calculate the theoretical geometric center layer radius of the outer ring of the spherical bearing without considering the elastic rebound of the material. S3. Introduce the theory of springback under bending, establish the springback geometric constraint equation and the expression of the reverse stress caused by unloading, and derive the explicit relationship between the radius after springback and the radius before springback based on the static equilibrium condition under the pure bending assumption. Make elastic correction to the radius of the theoretical geometric center layer to obtain the final predicted radius of the geometric center layer. S4. Output the final predicted geometric center layer radius as the predicted shape result.
[0007] Preferably, the geometric dimensions of the spherical plain bearing outer ring blank include the blank wall thickness, blank inner diameter, and blank height; The geometric and process parameters of a conical die include the die taper (i.e., the pressing angle), the pressing distance, and the number of pressing cycles. The material parameter is the elastic modulus of the outer ring material of the spherical bearing.
[0008] Preferably, the geometric relationship equations for the compression process are established as follows: Establish a geometric model describing the relationship between the taper of the upper mold and the mean diameter and height of the outer ring of the spherical plain bearing after compression.
[0009] Preferably, the geometric model is constructed by establishing six basic geometric relation equations: ; ; ; ; ; ; And finally, the relation is obtained: ; in, This represents the distance between point d and point g; This represents the distance between point d and point e; This represents the distance between point c and point d; The distance between point c and point f is represented by h; the height of the outer ring of the spherical bearing is represented by h1; and the distance the upper mold moves downward from the moment it contacts the outer ring of the spherical bearing is represented by h1. r2 represents the pressing angle of the upper mold; r2 represents the outer diameter of the outer ring of the spherical bearing after pressing. The angle between the upper and lower surfaces of the spherical plain bearing after compression is half of the angle; r0 represents the mean diameter of the outer ring of the spherical plain bearing after compression; t represents the thickness of the outer ring of the spherical plain bearing.
[0010] Preferably, the modification by incorporating the theory of springback during bending specifically includes: Establish the rebound geometric constraint equation to describe the relationship between the radius of the strain neutral layer and the geometric center layer before and after rebound: ; ; Establish the expression for the reverse stress caused by unloading: ; ; Based on the static condition that the resultant force at the cross section is zero under pure bending conditions and that the resultant moment is balanced with the loading bending moment, the explicit relationship between the radius after springback and the radius before springback can be obtained by solving the equations simultaneously: ; ; in, This indicates the radius of the strain neutral layer when the outer ring of the spherical bearing is subjected to bending load. This indicates the radius of the geometric center layer after the outer ring of the spherical bearing is bent; This indicates the radius of the geometric center layer of the outer ring of the spherical bearing after springback; This indicates the radius of the neutral layer after the outer ring of the spherical bearing is bent. This indicates the bending radius of the neutral layer after the residual strain of the outer ring of the spherical bearing is released from the bending process and springback. E represents the bending moment during the extrusion deformation process; E represents the elastic modulus of the outer ring material of the spherical bearing. The value represents the reverse bending stress generated in the cross section of the outer ring of the spherical plain bearing during the unloading and springback process; u represents the vertical distance from the point under study to the strain neutral layer on the cross section of the outer ring of the spherical plain bearing, and its range is constrained by the cross section size. Indicates a reference position of the cross section; and These represent the reverse stress caused by unloading. The resultant force and resultant moment generated across the entire cross section; This represents the axial force acting on the cross section during the loading process; This represents the bending moment acting on the cross section during the loading process.
[0011] Preferably, the explicit relationship between the radius after springback and the radius before springback is as follows: ; in, This represents the radius of the geometric center layer of the outer ring of the spherical bearing after springback correction, which is the final shape and size predicted by the model. E represents the moment of inertia of the cross section of the outer ring of the spherical plain bearing about the bending neutral axis when extrusion is completed; E represents the elastic modulus of the material of the outer ring of the spherical plain bearing.
[0012] Compared with the prior art, the above-mentioned technical solution of the present invention has the following beneficial technical effects: This invention presents a mathematical model capable of predicting the forming shape of spherical plain bearings. By organically combining the geometric relationship of the pressing process with the theory of material elastic rebound, this model achieves high-precision prediction of the outer ring shape of the bearing after forming, significantly reducing the uncertainty caused by traditional methods relying on trial and error or extensive finite element simulations. Its core advantage lies in its ability to quickly output a predicted shape corrected for springback based on input key process parameters and material properties, thus providing a reliable theoretical basis and efficient pre-evaluation tool for process design and parameter optimization. This model helps to screen reasonable process windows before physical prototyping, effectively reducing the number of repeated mold modifications and experiments, shortening the product development cycle, and saving related manufacturing costs and computing resources. Furthermore, the model possesses a clear physical mechanism and an adjustable parameter system, making it not only suitable for evaluating specific working conditions but also possessing the potential to be extended to similar forming processes. Attached Figure Description
[0013] Figure 1 This is a flowchart illustrating the construction of a mathematical model for predicting the forming shape of a spherical bearing, as proposed in this invention. Figure 2 This is a schematic diagram of the spherical bearing compression method proposed in this invention; Figure 3 This invention presents the variation trends of the experimental fitting radius, simulation fitting radius, and numerical calculation radius of the outer ring of the spherical plain bearing after secondary compression with the amount of compression. Detailed Implementation
[0014] Example 1: The present invention proposes a mathematical model that can predict the forming shape of a spherical bearing, such as... Figure 1 As shown, the specific construction steps are as follows: S1. Obtain and set the model input parameters, including the geometric dimensions of the spherical bearing outer ring blank, the parameters of the tapered die, and the process parameters. Simultaneously, specify the material's elastic modulus, specifically: Obtain the geometric parameters of the outer ring blank of the spherical plain bearing, including but not limited to: the wall thickness t of the blank, the inner diameter of the blank, and the height h of the blank; Obtain the geometric and process parameters of the conical die, including but not limited to: the taper of the die, i.e., the pressing angle. The maximum inner diameter of the mold (this diameter must be greater than the outer diameter of the spherical bearing outer ring blank to ensure that the mold can be smoothly inserted into the blank for extrusion and avoid interference), the downward pressing distance h1 of the mold (defined as the vertical distance from when the upper mold just contacts the top of the spherical bearing outer ring to when it moves downward), and the number of times the mold is pressed down. This model is applicable to single-stage extrusion molding with a double-cone die (i.e., completed in a single press) and multiple-stage extrusion molding with a double-cone die (e.g., after the first press, the outer ring is rotated 180° for the second press). It should be noted that the relevant dimensions of the spherical plain bearing outer ring blank are larger than the finished product dimensions, the maximum inner diameter of the conical die is larger than the outer diameter of the spherical plain bearing outer ring blank, and the downward pressing distance of the conical die is greater than zero and less than half the height of the spherical plain bearing outer ring blank. Obtain the material parameters, specifically the elastic modulus E of the outer ring material of the spherical bearing. This parameter is used for subsequent springback correction calculations.
[0015] S2. Based on the geometric relationship of the compression process, establish a system of equations, and derive the core relationship between the compression amount h1 and the theoretical mean diameter r0 after compression by solving the system simultaneously; with known input parameters, solve this relationship numerically to obtain the deformation half-angle. Then, the theoretical geometric center layer radius r0 of the outer ring of the spherical bearing is calculated without considering the elastic rebound of the material, thus completing the shape prediction of pure geometric deformation, specifically: By establishing an accurate geometric model of the compression process, the theoretical shape that the outer ring should achieve under ideal rigid-plastic deformation (i.e., neglecting elastic recovery) is calculated. The core is to solve for the geometric center layer radius r0 (i.e., the mean diameter) of the outer ring after compression. Figure 2 As shown, the following six basic geometric equations are established: ; ; ; ; ; ; The final relationship between the taper of the upper mold, the mean diameter of the outer ring of the spherical plain bearing after pressing, and the height of the outer ring is: ; in, This represents the distance between point d and point g; This represents the distance between point d and point e; This represents the distance between point c and point d; The distance between point c and point f is represented by h; the height of the outer ring of the spherical bearing is represented by h1; and the distance the upper mold moves downward from the moment it contacts the outer ring of the spherical bearing is represented by h1. r2 represents the pressing angle of the upper mold; r2 represents the outer diameter of the outer ring of the spherical bearing after pressing. The angle between the upper and lower surfaces of the spherical plain bearing after compression is half of the angle; r0 represents the mean diameter of the outer ring of the spherical plain bearing after compression; t represents the thickness of the outer ring of the spherical plain bearing.
[0016] S3. The theoretical shape is corrected by introducing the theory of springback under bending; the springback geometric constraint equation and the expression for the reverse stress caused by unloading are established. Based on the static equilibrium condition (zero resultant force and equilibrium resultant moment) under the pure bending assumption, the explicit relationship between the radius after springback and the radius before springback is derived; the corrected final predicted shape is calculated using parameters such as the material's elastic modulus E, the moment of inertia of the section, and the bending moment. Specifically: Establish the rebound geometric constraint equation. Before and after rebound, the strain neutral layer and the geometric center layer satisfy the following relationship: ; ; Establish the expression for the reverse stress caused by unloading, that is, unloading is equivalent to applying a reverse bending moment, and the resulting stress is: ; ; The radius after springback is determined based on the static equilibrium condition, i.e., the compression loading process is defined as pure bending, where the section is only subjected to bending moment. The effect is that after unloading, the residual stress in the cross-section self-balances, satisfying: The net force is zero: ; The resultant moment is balanced with the applied bending moment: ; Solving the equations simultaneously, we obtain the relationship between the radius of the outer ring after springback and the radius before springback: ; in, This indicates the radius of the strain neutral layer when the outer ring of the spherical bearing is subjected to bending load. This indicates the radius of the geometric center layer after the outer ring of the spherical bearing is bent; This indicates the radius of the geometric center layer of the outer ring of the spherical bearing after springback; This indicates the radius of the neutral layer after the outer ring of the spherical bearing is bent. This indicates the bending radius of the neutral layer after the residual strain of the outer ring of the spherical bearing is released from the bending process and springback. E represents the bending moment during the extrusion deformation process; E represents the elastic modulus of the outer ring material of the spherical bearing. The value represents the reverse bending stress generated in the cross section of the outer ring of the spherical plain bearing during the unloading and springback process; u represents the vertical distance from the point under study to the strain neutral layer on the cross section of the outer ring of the spherical plain bearing, and its range is constrained by the cross section size. Indicates a reference position of the cross section (such as the distance from the edge of the cross section to the neutral layer); and These represent the reverse stress caused by unloading. The resultant force and resultant moment generated across the entire cross section; This represents the axial force acting on the cross section during the loading process; This represents the bending moment acting on the cross section during the loading process; This represents the radius of the geometric center layer of the outer ring of the spherical bearing after springback correction, which is the final shape and size predicted by the model. The moment of inertia of the outer ring of the spherical plain bearing about the bending neutral axis is represented by E when the extrusion is completed; E represents the elastic modulus of the material of the outer ring of the spherical plain bearing. Based on this, the predicted geometric center layer radius of the outer ring of the spherical bearing, after springback correction, is obtained, which more closely approximates the actual forming result. .
[0017] S4. Output the predicted shape after springback correction as the result, and verify the model accuracy by comparing mathematical models, finite element simulations, and actual experimental data. Analysis shows that the model has a small prediction error (e.g., up to 4.2%) in the stage of large compression (e.g., h1>3.5mm), and is suitable for process prediction in the stable deformation stage. It can effectively reduce the number of experiments and development cycle, specifically: The final predicted radius calculated in step S3 As the core output of the model, this result can be directly used to evaluate the shape of the outer ring of the spherical bearing after compression molding under given parameters; like Figure 3As shown, by systematically comparing the trends of the predicted radius of this mathematical model, the fitted radius of the finite element simulation, and the fitted radius of the actual physical experiment with the downward pressure h1, the prediction accuracy and applicable range of this model can be clearly defined: In the low compression stage (h1<3.5mm): there is a significant gap between the predicted results and the actual results. The main reason is that in this stage, the contact area between the outer ring and the upper mold is small, and the local stress concentration leads to a significant "upsetting" phenomenon on the contact surface, which deviates from the pure bending assumption based on uniform deformation of the model. High-low pressure phase ( The mathematical model's predictions are very close to the actual results. This is because as the pressure increases, the deformation tends to be uniform, and the initial upsetting region no longer dominates the deformation mode. The actual situation is more in line with the geometric and mechanical assumptions of the model. The trend shows that as the downforce h1 increases, the difference between the mathematical model prediction and the actual test results gradually narrows; for example, when the downforce reaches about 5.5 mm, the error between the two can be as low as about 4.2%, showing high engineering prediction value. Therefore, this model is particularly suitable for the design and optimization of compression process parameters when the compression amount is large (e.g., greater than 3.5 mm) and the deformation has entered a stable stage. In practical applications, this model can be used to quickly predict and screen different parameter combinations, effectively reducing the number of physical tests and simulation calculation resources required to determine the process and shortening the development cycle.
[0018] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited thereto. Various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention.
Claims
1. A mathematical model capable of predicting the forming shape of a spherical plain bearing, characterized in that, Its construction steps specifically include: S1. Obtain the geometric dimensions of the outer ring blank of the spherical bearing, the geometric and process parameters of the tapered die, and the elastic modulus of the material; S2. Based on the geometric relationship of the compression process, establish a set of equations, solve them simultaneously to derive the core relationship between the compression amount and the theoretical mean diameter after compression, and calculate the theoretical geometric center layer radius of the outer ring of the spherical bearing without considering the elastic rebound of the material. S3. Introduce the theory of springback under bending, establish the springback geometric constraint equation and the expression of the reverse stress caused by unloading, and derive the explicit relationship between the radius after springback and the radius before springback based on the static equilibrium condition under the pure bending assumption. Make elastic correction to the radius of the theoretical geometric center layer to obtain the final predicted radius of the geometric center layer. S4. Output the final predicted geometric center layer radius as the predicted shape result.
2. The mathematical model for predicting the forming shape of a spherical bearing according to claim 1, characterized in that, The geometric dimensions of the outer ring blank for a spherical plain bearing include the blank wall thickness, blank inner diameter, and blank height. The geometric and process parameters of a conical die include the die taper (i.e., the pressing angle), the pressing distance, and the number of pressing cycles. The material parameter is the elastic modulus of the outer ring material of the spherical bearing.
3. The mathematical model for predicting the forming shape of a spherical bearing according to claim 2, characterized in that, The geometric relationships during the compression process are established by the following system of equations: Establish a geometric model describing the relationship between the taper of the upper mold and the mean diameter and height of the outer ring of the spherical plain bearing after compression.
4. The mathematical model for predicting the forming shape of a spherical bearing according to claim 3, characterized in that, The geometric model is constructed by establishing six basic geometric relation equations: ； ； ； ； ； ； And finally, the relation is obtained: ； in, This represents the distance between point d and point g; This represents the distance between point d and point e; This represents the distance between point c and point d; The distance between point c and point f is represented by h; the height of the outer ring of the spherical bearing is represented by h1; and the distance the upper mold moves downward from the moment it contacts the outer ring of the spherical bearing is represented by h1. r2 represents the pressing angle of the upper mold; r2 represents the outer diameter of the outer ring of the spherical bearing after pressing. The angle between the upper and lower surfaces of the spherical plain bearing after compression is half of the angle; r0 represents the mean diameter of the outer ring of the spherical plain bearing after compression; t represents the thickness of the outer ring of the spherical plain bearing.
5. A mathematical model for predicting the forming shape of a spherical bearing according to claim 4, characterized in that, The specific modifications include incorporating the theory of springback during bending. Establish the rebound geometric constraint equation to describe the relationship between the radius of the strain neutral layer and the geometric center layer before and after rebound: ; ; Establish the expression for the reverse stress caused by unloading: ; ; Based on the static condition that the resultant force at the cross section is zero under pure bending conditions and that the resultant moment is balanced with the loading bending moment, the explicit relationship between the radius after springback and the radius before springback can be obtained by solving the equations simultaneously: ； ； in, This indicates the radius of the strain neutral layer when the outer ring of the spherical bearing is subjected to bending load. This indicates the radius of the geometric center layer after the outer ring of the spherical bearing is bent; This indicates the radius of the geometric center layer of the outer ring of the spherical bearing after springback; This indicates the radius of the neutral layer after the outer ring of the spherical bearing is bent. This indicates the bending radius of the neutral layer after the residual strain of the outer ring of the spherical bearing is released from the bending process and springback. E represents the bending moment during the extrusion deformation process; E represents the elastic modulus of the outer ring material of the spherical bearing. The value represents the reverse bending stress generated in the cross section of the outer ring of the spherical plain bearing during the unloading and springback process; u represents the vertical distance from the point under study to the strain neutral layer on the cross section of the outer ring of the spherical plain bearing, and its range is constrained by the cross section size. Indicates a reference position of the cross section; and These represent the reverse stress caused by unloading. The resultant force and resultant moment generated across the entire cross section; This represents the axial force acting on the cross section during the loading process; This represents the bending moment acting on the cross section during the loading process.
6. The mathematical model for predicting the forming shape of a spherical bearing according to claim 5, characterized in that, The explicit relationship between the radius after springback and the radius before springback is: ; in, This represents the radius of the geometric center layer of the outer ring of the spherical bearing after springback correction, which is the final shape and size predicted by the model. E represents the moment of inertia of the cross section of the outer ring of the spherical plain bearing about the bending neutral axis when extrusion is completed; E represents the elastic modulus of the material of the outer ring of the spherical plain bearing.