Multi-dimensional damage resistance prediction method and device for aero-engine thrust bearing
By constructing a three-dimensional digital prototype of the thrust bearing and conducting multi-dimensional analysis, the problem of damage prediction of the thrust bearing of aero-engine under complex conditions was solved, and more accurate design optimization and life assessment were achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2023-02-17
- Publication Date
- 2026-04-28
AI Technical Summary
Existing technologies are insufficient to effectively predict the damage to aero-engine thrust bearings under high temperature, high speed and complex load conditions, and lack cross-disciplinary simulation analysis methods to support design optimization.
A multi-dimensional evaluation method was adopted, including constructing a three-dimensional digital prototype of the thrust bearing, performing finite element analysis, modal analysis, fluid analysis and dynamic analysis, and combining the load spectrum under working conditions and material properties to predict the damage resistance of the thrust bearing.
It enables damage prediction of thrust bearings under multiple physical phenomena, supports engineering design optimization, and improves the reliability and life prediction accuracy of bearings.
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Figure CN116306112B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of aero-engine damage prediction technology, specifically to a multi-dimensional damage prediction method and device for aero-engine thrust bearings. Background Technology
[0002] Unlike traditional mechanical component simulation, aero-engine thrust bearings, used in the main shaft support position, are characterized by high temperature, high speed, and complex loads. They involve high-speed rotating bodies with multiple physical phenomena such as dynamics, fluid dynamics, and thermal dynamics, and the envelope load spectrum of the aircraft engine during flight must be considered. The core objective of the multi-dimensional damage prediction method is to predict whether the designed aero-engine thrust bearing will suffer damage under typical service envelope load spectrum environments.
[0003] With the increasing design requirements for high reliability of bearings in aero engines, simulation analysis methods need to be combined with engineering design applications to optimize the entire design process while effectively predicting potential bearing damage. Summary of the Invention
[0004] To overcome at least one deficiency in the prior art, this application provides a method and apparatus for predicting multi-dimensional damage resistance of aero-engine thrust bearings.
[0005] Firstly, a multi-dimensional damage prediction method for thrust bearings in aero-engines is provided, including:
[0006] Determine the structural design information of the thrust bearing for aero-engines;
[0007] A three-dimensional digital prototype of a thrust bearing with geometric features was constructed. The three-dimensional digital prototype of the thrust bearing includes an inner ring, rotating rolling elements, and a cage.
[0008] Based on a three-dimensional digital prototype of a thrust bearing, damage prediction is performed on the thrust bearing of an aero-engine according to structural design information.
[0009] In one embodiment, based on a three-dimensional digital prototype of a thrust bearing, damage prediction for aero-engine thrust bearings is performed according to structural design information, including:
[0010] Based on the three-dimensional digital prototype of the thrust bearing, a finite element model of the cage was established according to the structural design information. Modal analysis was used to predict the resonance characteristics of the cage during high-speed operation.
[0011] Based on the resonance characteristics, determine whether the rotational speed corresponding to the resonance intersection of the cage coincides with the rotational speed of the working load spectrum. If they coincide, the cage meets the design requirements; otherwise, adjust the structural design of the cage.
[0012] In one embodiment, based on a three-dimensional digital prototype of a thrust bearing, damage prediction for aero-engine thrust bearings is performed according to structural design information, including:
[0013] Based on the three-dimensional digital prototype of the thrust bearing, a finite element model of the inner ring component was established according to the structural design information, and the fluid domain of the lubrication part under the ring was extracted based on the finite element model of the inner ring component.
[0014] Fluid analysis software was used to analyze the oil flow capacity of the lubrication zone under the inner ring. If the oil flow rate of the oil hole in the inner ring is greater than the oil supply of the thrust bearing, then the inner ring meets the design requirements; otherwise, the structural design of the inner ring is adjusted.
[0015] In one embodiment, based on a three-dimensional digital prototype of a thrust bearing, damage prediction for aero-engine thrust bearings is performed according to structural design information, including:
[0016] Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, the impact force of the rotating rolling elements of the thrust bearing on the cage is determined using explicit dynamic analysis or rigid-flexible coupling dynamic analysis methods.
[0017] Based on the impact force of the rotating rolling elements of the thrust bearing on the cage, the finite element analysis method is used to determine the strength of the cage and predict its impact resistance.
[0018] In one embodiment, the method further includes:
[0019] Determine the load spectrum of the thrust bearing of an aero-engine;
[0020] Determine the installation and fitting conditions, material properties, and lubricant properties of the thrust bearing for the aero-engine;
[0021] Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, working condition load spectrum, installation and fitting conditions, material property information, and lubricant property parameters of the aero-engine thrust bearing, a fully dynamic analysis method is used to predict the trajectory of the cage's center of mass.
[0022] In one embodiment, the method further includes:
[0023] Determine the installation and fitting conditions for the thrust bearing of the aero-engine;
[0024] The installation adaptability of thrust bearings is predicted based on the installation fit conditions. The installation adaptability of thrust bearings includes the actual interference fit and circumferential tensile stress of the thrust bearings under working conditions.
[0025] In one embodiment, the method further includes:
[0026] When the actual interference fit is greater than zero and the circumferential tensile stress does not exceed the limit value of the ring material, the design requirements are met.
[0027] In one embodiment, the method further includes:
[0028] Determine the load spectrum of the thrust bearing of an aero-engine;
[0029] Based on the load spectrum, the quasi-static or quasi-dynamic analysis methods are used to predict four evaluation indicators of the thrust bearing under the envelope condition: working clearance, contact stress, oil film lubrication parameters, and life. It is then determined whether the four evaluation indicators meet the design requirements of the thrust bearing application environment.
[0030] In one embodiment, the design requirements for the application environment of thrust bearings include:
[0031] The radial working clearance of the thrust bearing is greater than zero; the bearing contact stress under envelope steady-state conditions is not greater than 2500MPa; the oil film lubrication parameter is greater than 1; the comprehensive cycle life of the bearing is not less than the development life required by the engine; under working conditions, the gap between the non-load-bearing half-circle of the bearing and the rotating rolling element is not less than zero, and the bearing avoids three-point contact, that is, the non-load-bearing half-circle of the thrust bearing does not bear the load; under working conditions, the rotating rolling element cannot climb out of the raceway edge.
[0032] In one embodiment, the method further includes:
[0033] Calculate the power loss of the thrust bearing under different load spectra; transfer the power loss to the thermal field analysis of the bearing cavity in the main thrust bearing application to predict the internal and external temperatures of the thrust bearing; when the internal and external temperatures of the thrust bearing do not exceed the allowable temperature of the bearing material, the design requirements are met.
[0034] Secondly, a multi-dimensional damage prediction device for aero-engine thrust bearings is provided, comprising:
[0035] The structural design information determination module is used to determine the structural design information of the thrust bearing of an aero-engine.
[0036] The 3D digital prototype construction module is used to construct a 3D digital prototype of a thrust bearing with geometric features. The 3D digital prototype of the thrust bearing includes the outer ring, inner ring, rotating rolling elements, and cage.
[0037] The damage prediction module is used to predict the damage resistance of aero-engine thrust bearings based on a three-dimensional digital prototype of the thrust bearing and according to structural design information.
[0038] Compared with the prior art, this application has the following advantages: The multi-dimensional damage prediction method and device for aero-engine thrust bearings of this application adopts a multi-dimensional evaluation of bearing design schemes, is oriented towards engineering design applications, and the constructed simulation environment is based on bearing-specific software and commercial software, which can realize cross-professional integration. Attached Figure Description
[0039] This application can be better understood by referring to the description given below in conjunction with the accompanying drawings, which, together with the detailed description below, are incorporated in and form part of this specification. In the drawings:
[0040] Figure 1 A schematic diagram illustrating the multi-dimensional damage resistance prediction of aero-engine thrust bearings is shown.
[0041] Figure 2 A flowchart of a multi-dimensional damage prediction method for aero-engine thrust bearings according to an embodiment of this application is shown;
[0042] Figure 3 A structural block diagram of a multi-dimensional damage prediction device for aero-engine thrust bearings according to an embodiment of this application is shown. Detailed Implementation
[0043] Exemplary embodiments of the present application will be described below with reference to the accompanying drawings. For clarity and brevity, not all features of the actual embodiments are described in the specification. However, it should be understood that many embodiment-specific decisions can be made in the development of any such actual embodiment to achieve the developer’s specific objectives, and these decisions may vary as the embodiments differ.
[0044] It should also be noted that, in order to avoid obscuring this application with unnecessary details, only the device structure closely related to the solution according to this application is shown in the accompanying drawings, while other details that are not closely related to this application are omitted.
[0045] It should be understood that this application is not limited to the described embodiments by virtue of the following description with reference to the accompanying drawings. In this document, embodiments may be combined with each other, features may be substituted or borrowed between different embodiments, and one or more features may be omitted in one embodiment, where feasible.
[0046] This application provides a multi-dimensional damage resistance prediction method for aero-engine thrust bearings. Specifically, this method predicts the damage resistance of thrust bearings from three dimensions. Figure 1A schematic diagram illustrating multi-dimensional damage prediction for thrust bearings is shown, including installation and usage evaluation dimensions, quasi-static / quasi-dynamic dimensions, and dynamic and fluid dimensions. This application employs a multi-dimensional evaluation scheme for bearing design, geared towards engineering design applications. The constructed simulation environment is based on bearing-specific software and commercial software, enabling cross-disciplinary integration.
[0047] Figure 2 A flowchart illustrating a multi-dimensional damage prediction method for aero-engine thrust bearings according to an embodiment of this application is shown. See also... Figure 2 The methods include:
[0048] Step S1: Determine the structural design information of the thrust bearing for the aero-engine;
[0049] In this step, based on the detailed design scheme of the aero-engine thrust bearing, the structural design information is determined. Here, the structural design information may include the inner diameter, outer diameter, pitch circle diameter, inner groove bottom diameter, outer groove bottom diameter, rolling element diameter, number of rolling elements, inner flange diameter, outer flange diameter, guiding method, cage outer diameter, cage inner diameter, pocket size, etc.
[0050] Step S2: Construct a three-dimensional digital prototype of a thrust bearing with geometric features. The three-dimensional digital prototype of the thrust bearing includes an inner ring, rotating rolling elements, and a cage.
[0051] In this step, a three-dimensional digital prototype of a bearing with geometric features is built based on the design scheme by designing a collaborative management system and combining parametric methods. It includes an inner ring, rotating rolling elements, and a cage, and may also include an outer ring.
[0052] Step S3: Based on the three-dimensional digital prototype of the thrust bearing, damage resistance prediction is performed on the aero-engine thrust bearing according to the structural design information.
[0053] In this embodiment, damage prediction of the aero-engine thrust bearing from the perspectives of dynamics and fluidity can include damage prediction of the inner ring, rotating rolling elements, and cage.
[0054] In one embodiment, based on a three-dimensional digital prototype of a thrust bearing and according to structural design information, damage prediction for aero-engine thrust bearings may include:
[0055] Based on the three-dimensional digital prototype of the thrust bearing, a finite element model of the cage was established according to the structural design information. Modal analysis was used to predict the resonance characteristics of the cage during high-speed operation.
[0056] Based on the resonance characteristics, determine whether the rotational speed corresponding to the resonance intersection of the cage coincides with the rotational speed of the working load spectrum. If they coincide, the cage meets the design requirements; otherwise, adjust the structural design of the cage.
[0057] In one embodiment, based on a three-dimensional digital prototype of the thrust bearing and according to structural design information, damage prediction for the aero-engine thrust bearing may further include:
[0058] Based on the three-dimensional digital prototype of the thrust bearing, a finite element model of the inner ring component was established according to the structural design information, and the fluid domain of the lubrication part under the ring was extracted based on the finite element model of the inner ring component.
[0059] Fluid analysis software was used to analyze the oil flow capacity of the lubrication zone under the inner ring. If the oil flow rate of the oil hole in the inner ring is greater than the oil supply of the thrust bearing, then the inner ring meets the design requirements; otherwise, the structural design of the inner ring is adjusted.
[0060] In one embodiment, based on a three-dimensional digital prototype of the thrust bearing and according to structural design information, damage prediction for the aero-engine thrust bearing may further include:
[0061] Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, the impact force of the rotating rolling elements of the thrust bearing on the cage is determined using explicit dynamic analysis or rigid-flexible coupling dynamic analysis methods.
[0062] Based on the impact force of the rotating rolling elements of the thrust bearing on the cage, the finite element analysis method is used to determine the strength of the cage and predict its impact resistance.
[0063] In one embodiment, the multi-dimensional damage resistance prediction method for aero-engine thrust bearings further includes:
[0064] First, determine the load spectrum of the thrust bearing for aero-engines. Here, based on the different operating conditions of the thrust bearing, the load spectrum is defined, including the ambient temperature, radial load, axial load, and speed requirements under various states such as cruise, idle, climb, and descent.
[0065] Then, determine the installation and fitting conditions, material property information, and lubricant property parameters of the aero-engine thrust bearing. Here, the installation and fitting conditions include the installation and fitting relationship between the shaft and the bearing, and between the housing and the bearing. The material property information includes at least the elastic modulus, Poisson's ratio, coefficient of thermal expansion, and density. The lubricant property parameters include at least the lubricant density, kinematic viscosity, and thermal conductivity.
[0066] Finally, based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, working condition load spectrum, installation and fitting conditions, material property information, and lubricant property parameters of the aero-engine thrust bearing, a complete dynamic analysis method is used to predict the trajectory of the cage's center of mass and to determine the risk of misalignment between the bearing cage and the guide ring.
[0067] In other embodiments, the method of this application also predicts damage resistance from the perspective of installation and use evaluation, and the method further includes:
[0068] Determine the installation fit conditions for the thrust bearing of the aero-engine; here, the installation fit conditions include the installation fit relationships between the shaft and the bearing, and between the housing and the bearing.
[0069] The installation adaptability of thrust bearings is predicted based on the installation fit conditions. The installation adaptability of thrust bearings includes the actual interference fit and circumferential tensile stress of the thrust bearings under working conditions.
[0070] Specifically, in this embodiment, the design requirements are met when the actual interference fit is greater than zero and the circumferential tensile stress does not exceed the limit value of the ring material.
[0071] In another embodiment, the method of this application also performs damage resistance prediction from a quasi-static / quasi-dynamic dimension, and the method further includes:
[0072] Determine the load spectrum of the thrust bearing for aero-engines; here, based on different operating conditions of the thrust bearing, the load spectrum is defined, including the ambient temperature, radial load, axial load, and speed requirements under various states such as cruise, idle, climb, and descent.
[0073] Based on the load spectrum, the quasi-static or quasi-dynamic analysis methods are used to predict four evaluation indicators of the thrust bearing under the envelope condition: working clearance, contact stress, oil film lubrication parameters, and life. It is then determined whether the four evaluation indicators meet the design requirements of the thrust bearing application environment.
[0074] Specifically, in the above embodiments, the design requirements for the application environment of the thrust bearing include:
[0075] The radial working clearance of the thrust bearing is greater than zero; the bearing contact stress under envelope steady-state conditions is not greater than 2500MPa; the oil film lubrication parameter is greater than 1; the comprehensive cycle life of the bearing is not less than the development life required by the engine; under working conditions, the gap between the non-load-bearing half-circle of the bearing and the rotating rolling element is not less than zero, and the bearing avoids three-point contact, that is, the non-load-bearing half-circle of the thrust bearing does not bear the load; under working conditions, the rotating rolling element cannot climb out of the raceway edge.
[0076] Furthermore, the method also includes:
[0077] Calculate the power loss of the thrust bearing under different load spectra; transfer the power loss to the thermal field analysis of the bearing cavity in the main thrust bearing application to predict the internal and external temperatures of the thrust bearing; when the internal and external temperatures of the thrust bearing do not exceed the allowable temperature of the bearing material, the design requirements are met.
[0078] Based on the same inventive concept as the multi-dimensional damage resistance prediction method for aero-engine thrust bearings provided in the embodiments of this application, the embodiments of this application also provide a multi-dimensional damage resistance prediction device for aero-engine thrust bearings. Figure 3 A structural block diagram of a multi-dimensional damage prediction device for aero-engine thrust bearings according to an embodiment of this application is shown. The device includes:
[0079] Structural design information determination module 31 is used to determine the structural design information of the thrust bearing of the aero-engine.
[0080] Here, based on the detailed design scheme of the aero-engine thrust bearing, the structural design information is determined. This structural design information may include the inner diameter, outer diameter, pitch circle diameter, inner groove bottom diameter, outer groove bottom diameter, rolling element diameter, number of rolling elements, inner flange diameter, outer flange diameter, guiding method, cage outer diameter, cage inner diameter, pocket size, etc.
[0081] The 3D digital prototype construction module 32 is used to construct a 3D digital prototype of a thrust bearing with geometric features. The 3D digital prototype of the thrust bearing includes an outer ring, an inner ring, rotating rolling elements, and a cage.
[0082] Here, by designing a collaborative management system and combining it with parametric methods, a three-dimensional digital prototype of a bearing with geometric features is built according to the design scheme. It includes an inner ring, rotating rolling elements, a cage, and may also include an outer ring.
[0083] The damage prediction module 33 is used to predict the damage resistance of the aero-engine thrust bearing based on the three-dimensional digital prototype of the thrust bearing and according to the structural design information.
[0084] In this embodiment, damage prediction of the aero-engine thrust bearing from the perspectives of dynamics and fluidity can include damage prediction of the inner ring, rotating rolling elements, and cage.
[0085] In summary, the multi-dimensional damage prediction method and device for aero-engine thrust bearings proposed in this application adopts a multi-dimensional evaluation of bearing design schemes, is geared towards engineering design applications, and the constructed simulation environment is based on bearing-specific software and commercial software, enabling cross-disciplinary integration.
[0086] The above descriptions are merely various embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.
Claims
1. A multi-dimensional damage prediction method for thrust bearings of aero-engines, characterized in that, include: Determine the structural design information of the thrust bearing for aero-engines; the structural design information includes inner diameter, outer diameter, pitch circle diameter, inner groove bottom diameter, outer groove bottom diameter, rolling element diameter, number of rolling elements, inner flange diameter, outer flange diameter, guiding method, cage outer diameter, cage inner diameter, and pocket size; A three-dimensional digital prototype of a thrust bearing with geometric features is constructed. The three-dimensional digital prototype of the thrust bearing includes an inner ring, rotating rolling elements, and a cage. Based on the aforementioned three-dimensional digital prototype of the thrust bearing, and according to the structural design information, damage resistance prediction is performed on the aero-engine thrust bearing; including: Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, a finite element model of the cage is established. Modal analysis is used to predict the resonance characteristics of the cage during high-speed operation. Based on the resonance characteristic law, determine whether the rotational speed corresponding to the resonance intersection of the cage coincides with the rotational speed of the working load spectrum. If they coincide, the cage meets the design requirements; otherwise, adjust the structural design of the cage. Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, a finite element model of the inner ring component is established, and the fluid domain of the lubrication part under the ring is extracted based on the finite element model of the inner ring component. Fluid analysis software was used to analyze the oil flow capacity of the lubrication zone under the ring. If the oil flow rate of the oil hole of the inner ring is greater than the oil supply of the thrust bearing, the inner ring meets the design requirements; otherwise, the structural design of the inner ring is adjusted.
2. The method as described in claim 1, characterized in that, Based on the aforementioned three-dimensional digital prototype of the thrust bearing, and according to the structural design information, damage prediction is performed on the aero-engine thrust bearing, including: Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, the impact force of the rotating rolling elements of the thrust bearing on the cage is determined using explicit dynamic analysis or rigid-flexible coupling dynamic analysis methods. Based on the impact force of the rotating rolling elements of the thrust bearing on the cage, the strength of the cage is determined and the impact resistance of the cage is predicted using the finite element analysis method.
3. The method as described in claim 1, characterized in that, The method further includes: Determine the load spectrum of the thrust bearing of an aero-engine; Determine the installation and fitting conditions, material properties, and lubricant properties of the thrust bearing for the aero-engine; Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, working condition load spectrum, installation and fitting conditions, material property information, and lubricating oil property parameters of the aero-engine thrust bearing, a complete dynamic analysis method is used to predict the trajectory of the center of mass of the cage.
4. The method as described in claim 1, characterized in that, The method further includes: Determine the installation and fitting conditions for the thrust bearing of the aero-engine; The installation adaptability of the thrust bearing is predicted based on the installation fit conditions, wherein the installation adaptability of the thrust bearing includes the actual interference fit and circumferential tensile stress of the thrust bearing under working conditions.
5. The method as described in claim 4, characterized in that, The method further includes: The design requirements are met when the actual interference fit is greater than zero and the circumferential tensile stress does not exceed the limit value of the ring material.
6. The method as described in claim 1, characterized in that, The method further includes: Determine the load spectrum of the thrust bearing of an aero-engine; Based on the load spectrum under the operating conditions, a pseudo-static analysis method or a pseudo-dynamic analysis method is used to predict four evaluation indicators of the thrust bearing under the envelope operating conditions: working clearance, contact stress, oil film lubrication parameters, and life. It is then determined whether the four evaluation indicators meet the design requirements of the thrust bearing application environment.
7. The method as described in claim 6, characterized in that, The design requirements for the application environment of the thrust bearing include: The radial working clearance of the thrust bearing is greater than zero; the bearing contact stress under envelope steady-state conditions is not greater than 2500MPa; the oil film lubrication parameter is greater than 1; the comprehensive cycle life of the bearing is not less than the development life required by the engine; under working conditions, the gap between the non-load-bearing half-circle of the bearing and the rotating rolling element is not less than zero, and the bearing avoids three-point contact, that is, the non-load-bearing half-circle of the thrust bearing does not bear the load; under working conditions, the rotating rolling element cannot climb out of the raceway edge.
8. The method as described in claim 6, characterized in that, The method further includes: Calculate the power loss of the thrust bearing under different load spectra; transfer the power loss to the thermal field analysis of the bearing cavity in the thrust bearing host to predict the internal and external temperatures of the thrust bearing; when the internal and external temperatures of the thrust bearing do not exceed the allowable temperature of the bearing material, the design requirements are met.
9. A multi-dimensional damage prediction device for aero-engine thrust bearings, characterized in that, include: The structural design information determination module is used to determine the structural design information of the thrust bearing for aero-engines; the structural design information includes inner diameter, outer diameter, pitch circle diameter, inner groove bottom diameter, outer groove bottom diameter, rolling element diameter, number of rolling elements, inner flange diameter, outer flange diameter, guiding method, cage outer diameter, cage inner diameter, and pocket size. A three-dimensional digital prototype construction module is used to construct a three-dimensional digital prototype of a thrust bearing with geometric features. The three-dimensional digital prototype of the thrust bearing includes an outer ring, an inner ring, rotating rolling elements, and a cage. The damage prediction module is used to predict the damage resistance of the aero-engine thrust bearing based on the three-dimensional digital prototype of the thrust bearing and according to the structural design information. The damage prediction module is also used for: Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, a finite element model of the cage is established. Modal analysis is used to predict the resonance characteristics of the cage during high-speed operation. Based on the resonance characteristic law, determine whether the rotational speed corresponding to the resonance intersection of the cage coincides with the rotational speed of the working load spectrum. If they coincide, the cage meets the design requirements; otherwise, adjust the structural design of the cage. Based on the three-dimensional digital prototype of the thrust bearing, and according to the structural design information, a finite element model of the inner ring component is established, and the fluid domain of the lubrication part under the ring is extracted based on the finite element model of the inner ring component. Fluid analysis software was used to analyze the oil flow capacity of the lubrication zone under the ring. If the oil flow rate of the oil hole of the inner ring is greater than the oil supply of the thrust bearing, the inner ring meets the design requirements; otherwise, the structural design of the inner ring is adjusted.
Citation Information
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