Gear-rotor systems are one of the commonly used motion and power transmission mechanisms in industry. With the requirements of modern industry for gear transmission systems developing towards heavy load, high speed, and harsh working environments, gears are more prone to failures. Among the faults in gears and their transmission systems, the proportion of tooth root cracks is almost 40% of all faults. When a fault occurs, it may lead to system shutdown in mild cases, and cause significant economic losses and even casualties in severe cases. Therefore, tooth root crack faults in various gear systems must be accurately detected, controlled, and tracked.
The time-varying nature of meshing stiffness can reflect various factors that affect the meshing condition of gears. The Finite Element Method (FEM) is an effective tool for calculating the time-varying meshing stiffness, but it has a large computational cost and a complex mesh division and post-processing process. The energy method has high computational efficiency and is still the mainstream method for calculating the time-varying meshing stiffness.
In 2011, Chen et al. first proposed a crack model that expands simultaneously along the tooth width and tooth thickness. Verma et al. determined the crack propagation path of spur gears using the Extended Finite Element Method (XFEM) and studied the influence of the rim thickness on the crack propagation path of gears. Feng Nana et al. proposed a new analytical algorithm based on computer simulation to quantify the time-varying meshing stiffness of meshing gears under different fault conditions. Sun Guangyao et al. considered different initial parameters of cracks and explored the numerical curve distribution law of the stress intensity factor and J integral of helical gears with three-dimensional cracks using finite element software. Wang Jinwen et al. considered the non-coincidence of the base circle and the tooth root circle and used the improved energy method to calculate the influence of the sun gear crack fault on the meshing stiffness. Ning et al. proposed a dynamic model of tooth root cracks with non-uniform distribution based on the “slice method” and revealed the dynamic characteristics of the gear transmission system with non-uniformly distributed cracks. Wan Zhiguo et al. compared and analyzed the analytical calculation models and internal mechanisms of tooth root cracks and tooth surface spalling faults, and analyzed the differences in the vibration response characteristics of the two faults through dynamic simulation. Xie Fuqi et al. comprehensively considered the nonlinear excitations such as time-varying meshing stiffness and dynamic error, and analyzed the influence of the through-type tooth root crack along the tooth thickness on the dynamic characteristics of the system. Liu Qikun et al. considered the characteristics of the non-continuity and singularity of the actual crack tip, improved the XFEM and verified the accuracy and validity of the model. Lai Junjie et al. established a bilateral asymmetric tooth root crack model based on the full-tooth energy equation and derived the calculation formula for the meshing stiffness. Zhang et al. integrated the advantages of the finite element method, Hertz contact theory, and the slice method, and proposed a new calculation method for the time-varying meshing stiffness of spur and helical gear pairs. Yang et al. considered the crack opening state, re-evaluated the effective compression section and neutral layer of the cracked gear, and further studied the differences in the meshing stiffness and dynamic response statistical indicators between the proposed model and the traditional model. Zhao et al. derived the load distribution of the gear with a cracked tooth root using the coordinated deformation condition, obtained the influence of the tooth root crack on the tribological characteristics of the tooth surface based on the Elasto-Hydrodynamic Lubrication (EHL) model, and studied the frictional dynamics behavior of the gear under the action of the tooth root crack.
In general, most of the existing literature focuses on the crack propagation path and system response of the gear system at a specific crack level, and there is less research on the modeling and calculation principle analysis of fault gear teeth at different crack levels. Moreover, the tooth root transition curve of the cracked gear tooth is usually approximated by a circular arc, and the effective tooth thickness reduction limit line is a straight line parallel to the center line of the gear tooth, which still has a certain gap from the real situation. In addition, the starting point of the involute profile of the gear and the intersection of the involute and the base circle do not coincide, which is not mentioned in the above literature.
Based on the previous research, this paper starts from the geometry of gear cutting and the principle of gear meshing, improves the calculation formula of the time-varying meshing stiffness of large-tooth-number gears based on the energy method; introduces the parameter equation of the transition curve, considers a more realistic effective tooth thickness reduction limit line, and comprehensively establishes the fault gear tooth models at different crack levels; analyzes the influence of the gear tooth model correction, different crack degrees, and the specific form of the effective tooth thickness reduction limit line on the calculation of the time-varying meshing stiffness of the cracked gear, and gives relevant suggestions for the effective tooth thickness reduction limit line at different crack degrees. By comparing with the results of the finite element method, the effectiveness of the method in this paper is verified.
Conclusion
- For spur gears with more than 41 teeth, combined with the geometry of gear cutting and the principle of tooth profile envelope, the difference between the starting point of the involute profile and the intersection point of the involute profile and the base circle is clarified, the tooth root transition parameter curve and a more realistic effective tooth thickness reduction limit line are introduced, the calculation formula of the time-varying meshing stiffness of the traditional potential energy method gear tooth is improved, and a stricter and more comprehensive calculation model for the time-varying meshing stiffness of fault gear teeth with different crack degrees is established.
- Based on the improved cracked gear tooth model, the time-varying meshing stiffness at different crack levels is calculated. The calculation results show that when solving the time-varying meshing stiffness of large-tooth-number gears, not modifying the gear tooth model will lead to a smaller calculation result. The tooth root crack will lead to a reduction in meshing stiffness, and the reduction is greater in the second meshing cycle. As the crack degree increases, the reduction in meshing stiffness also increases.
- When the crack size is small, the specific form of the effective tooth thickness reduction limit line has a small influence on the calculation of the time-varying meshing stiffness. At this time, it is recommended to use a straight line as the effective tooth thickness reduction limit line; when the crack size is large, the straight line reduction limit line does not conform to the actual situation of the crack. Compared with the parabola, the diagonal line can meet the requirements of calculation accuracy to a certain extent, and has the advantages of simple modeling and convenient calculation, and has stronger practicability. At this time, it is recommended to use a diagonal line as the effective tooth thickness reduction limit line.