CHEN Yijie ¹, XU Guoqian ¹, YAO Kaiwen ², ZHU Hu ³, XU Changyou ³, CHEN Mingxing ¹,

ZHAO Hui ¹, ZHANG Guoliang ¹, LI Zhenming ¹

(1. Nanchang University of Science and Technology, Nanchang 330044, Jiangxi, China; 2. 722 Research Institute of China State Shipbuilding Corporation, Wuhan 430205, Hubei, China; 3. School of Mechanical and Electronic Engineering, Jingdezhen Ceramic University, 

Jingdezhen 333403, Jiangxi, China)

Extended abstract:[Background and purposes] The plastic deformation of ceramic rolling bearings has a significant impact on their service life and motion accuracy. In view of this, a method was proposed for damping vibration based on mass loss spring resonators in view of the force characteristics of ceramic materials, conducting modeling and virtual simulation analysis of ceramic rolling bearings. Tetrahedral ceramic materials were constructed through the spring harmonic oscillator model. A volume spring was introduced to provide volume force to the model, while the deformation process of the ceramic material under force was simulated by the deformation of the spring. In addition, the loss behavior of ceramic bearings under actual working conditions was simulated in combination with the loss model. The ceramic bearing model was constructed based on Chai3D, and the virtual simulation program for damping vibration of the spring harmonic oscillator was written in C++ to ensure that the model can be used to accurately characterize the internal force variation law and pressure-bearing limit characteristics of the ceramic material under different loads. The reliability of the model was verified through the design of experiments, such as fixed-point stretching and extrusion. It was experimentally shown that this model can be used to effectively simulate the physical properties of ceramic materials. The outcome not only provides a new method for the design and optimization of ceramic rolling bearings, but also lays a theoretical foundation for the application of ceramic materials in the field of precision machinery.[Methods] A ceramic bearing model was established by using tetrahedral spring harmonic oscillator elements, where the material was discretized into a set of particles (nodes) and a set of springs (edges). Nodes represent mass points. Spring internal forces were simulated in accordance with Hooke's law, while the elastic deformation was simulated through linear springs. Damping springs dissipate energy. To enhance the volume retention capacity, an innovative volume spring mechanism was introduced. A volume force directed towards the center of gravity was applied between the center and the vertices of the tetrahedron (Formula 3), which strengthened the stability of the model in restoring its initial shape. The Newtonian mechanics framework (Formula 4) is adopted to solve the node motion. The resultant force includes the spring force (Formula 1), damping force (Formula 2), volumetric force and external force. The node position, velocity and acceleration are updated in real time through explicit Euler integral (Formula 5–7) to ensure the real-time interaction with low computational load. A segmented comprehensive model of wear depth (Formula 8) is proposed for the wear behavior of ceramic bearings. This model divides the wear process into the running-in period (exponential decay), the stable period (linear Archard wear), and the accelerated period (fatigue-wear coupling). The influences of pressure, speed and time were comprehensively considered, while parameters, such as material hardness and environmental factors, were introduced to quantify the dynamic process from volume consumption to the failure threshold. Based on the chai3D framework, a virtual simulation platform was developed using C++. The bearing geometry model was constructed through the tetrahedral mesh generator TetGen and visualized in combination with OpenGL rendering.[Results] The performance of the improved spring harmonic oscillator model was verified through four sets of experimental systems. Several experimental results were obtained. Firstly, linear deformation was verified (Fig. 5). Fixed-point tensile tests were conducted on cubic models of alumina, zirconia and silicon carbide. During the small deformation stage (<1 mm), the force and displacement have a strict linear relationship, which conforms to Hooke's Law. When the deformation exceeds the critical threshold (material-related), the model breaks, restoring the brittle characteristics of ceramics. The model accurately characterizes the linear elastic behavior and fracture characteristics of ceramic materials. Secondly, body spring recovery was verified (Fig. 7), comparing the recovery performance of ceramic, rubber and metal ring models in extrusion experiments. When there is no body spring, ceramic has the lowest recovery rate (only 9.7%), while metal (85%) and rubber (98%) have higher elasticity. After the introduction of body springs, the recovery rate of ceramics is increased to 45%, while that of rubber is increased by 53% (reaching nearly full recovery). The volume spring significantly enhances the model's volume retention capacity, especially improving the deformation accuracy of low-elastic materials such as ceramics. Thirdly, wear model was verified (Fig. 9), by comparing the wear depth of the silicon nitride ceramic bearing simulation with that of the real drilling machine experiment. In the running-in period of 0–10,000 turns, the wear depth increases exponentially, with the error between the simulation and the actual measurement to be ≤5.8%. In stable period of 10,000–50,000 cycles, the wear rate remains constant, conforming to the Archard linear model. In the acceleration period (>50,000 cycles), the fatigue-wear coupling effect leads to nonlinear acceleration and the simulation predicted failure threshold (0.02 mm) is consistent with the industrial standard. The segmented loss model accurately simulates the wear law throughout the entire life cycle of ceramic bearings. Lastly, real-time verification was achieved (Fig. 10). Under torque/pressure load, the collision detection rates of the model and the finite element model are compared. With hierarchical bounding box algorithm, the model reaches 281 FPS, which is 26% faster than the finite element model (222 FPS). For continuous collision detection, the performance of the model proposed (198 FPS) was improved by 19%. For discrete collision detection, the model (175 FPS) was improved by 14%. The simplified solution mechanism of particle-spring significantly reduces the computational load and meets the requirements of real-time interaction. It has been experimentally proved that the improved model outperforms traditional methods in terms of deformation accuracy (linear response ±3%), recovery (ceramic improved by 4.6 times), loss prediction (error <6%) and real-time performance (FPS improved by 14%–26%), providing an efficient and reliable tool for dynamic simulation of ceramic bearings.[Conclusions] An improved spring harmonic oscillator model was proposed for efficiently simulating the deformation and loss characteristics of ceramic rolling bearings. Tetrahedral mesh discrete ceramic bearings are adopted and a volume spring mechanism (Formula 3) is introduced. The stability of the model is restored by enhancing the volume force directed towards the center of gravity of the tetrahedron. In terms of loss modeling, a segmented wear depth model (Formula 8) was proposed, integrating pressure, speed and time to precisely describe the loss process during the running-in period (exponential growth), stable period (linear Archard model) and accelerated period (fatigue-wear coupling). For real-time solution, based on Newtonian mechanics (Formula 4–7) and explicit Euler integral, efficient update of particle motion is achieved. The tensile test results of alumina/zirconia/silicon carbide showed that, in the small deformation stage, the force-displacement relationship is strictly linear (error <3%), while the fracture after exceeding the threshold conforms to the brittle characteristics of ceramics. The body spring increases the recovery rate of ceramic materials by 4.6 times (from 9.7% to 45%), while rubber achieves nearly complete recovery. The wear simulation of silicon nitride bearings is consistent with the real experiment (error ≤5.8%), while the failure threshold of 0.02 mm is successfully predicted. The collision detection rate of the hierarchical bounding box reaches 281 FPS, which is 26% higher than that of the finite element model. This model takes into account both accuracy and real-time performance, providing an effective tool for the design optimization and life prediction of ceramic bearings.

Key words: ceramic bearing; spring resonator; virtual simulation; loss characteristic; elastic deformation