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Spalling alters a gear’s time-varying meshing stiffness (TVMS), thereby affecting its vibration characteristics. However, most studies focus on single-spalling gears and overlook the possibility of multi-spalling gears. Additionally, because most spalls are irregular, traditional analytical models neglect the torsional effects that are caused by asymmetric spalling. In this study, a shape-independent model for calculating the TVMS of multi-spalling gears, which considers torsional stiffness, was developed. A 16-degree-of-freedom dynamic model was established to analyze the dynamic response, incorporating the multi-spalling TVMS. The model was then validated through experiments. The results show that the proposed method accurately calculates the TVMS of a multi-spalling spur-gear system. Changes in the relative position of the spalling can significantly affect the TVMS. Multiple-tooth spalling influences the TVMS over several meshing cycles, while single-tooth multiple spalling affects the TVMS based on the specific spalling parameters. Different spalling patterns lead to substantial differences in the system’s dynamic behavior. Multiple spalling teeth generate several pulses, whereas a single tooth with multiple spalls only generates one significant pulse. This study provides a solid foundation for understanding the dynamic behavior of spalled gear systems, revealing their dynamic characteristics and failure mechanisms.

Introduction The electric drive system is usually composed of an electric motor and gear transmission system, and widely used for industrial applications. However, the dynamic behaviors of the coupled electromechanical system are still not well understood, especially when there are faults in the system. Purpose The faults such as gear spalling fault of the mechanical system or inter-turn short-circuit fault of electrical device will threaten the operation safety of the electric vehicle. Thus, the faulty dynamic analysis for the electric drive system is crucial for avoiding fatal catastrophes. Method In this paper, the permeance network motor model with inter-turn short-circuit fault and the dynamic planetary gear transmission model considering spalling faults are proposed, respectively. The electromechanical dynamic model integrating two models mentioned above is employed to acquire the fault characteristics. Then, the vibration characteristics of the electric drive system with and without gear fault or inter-turn fault under the effects of time-varying mesh stiffness, slot effect and magnetic saturation are distinguished at different operation conditions. Results The results show that the inter-turn short-circuit fault can trigger conspicuous harmonics within the low frequency domain of the mesh force components, which will predicatively obstruct the detection or diagnosis of the spalling fault. Moreover, the vibration characteristics and vibration mechanism concerning these faults are revealed by the frequency and statistical analysis, which facilitates the condition monitoring for the gear transmission system when the tooth spalling and inter-turn faults are coupled. Conclusion The research results bring theoretical reference for the dynamic analysis and vibration-based condition monitoring of the vehicles’ integrated electric drive system.

Tooth spall is a prevalent gear fault that reduces mesh stiffness and adversely affects transmission ability of gear systems. While plenty of research focuses on tooth spall faults in spur and helical gears, few analytical approaches were established to calculate the time-varying mesh stiffness in straight bevel gears, especially those affected by tooth spall. This deficiency can be attributed to the lack of an accurate tooth spall model. This paper proposes an approach for straight bevel gears mesh stiffness calculation with a curved-bottom spall. The spur gear spall model is modified to adapt to the tooth profile in straight bevel gears. The time-varying mesh stiffness calculation formulas are then revised in spall region. Using potential energy, Hertzian contact stiffness is calculated considering load distribution between gear teeth. The foundation stiffness calculation is updated considering practical gear shape. Tooth stiffness caused by axial forces is also considered. Finite element analysis is utilized for the theoretical method verification, which suggests a high consistency of results between two methods. The low error between the proposed approach and finite element approach implies that the proposed method is acceptable for practical use.

Gear dynamics analysis based on time-varying meshing stiffness (TMS) is an important means to understand the gear fault mechanism. Based on Jones bearing theory, a bearing statics model was established and introduced into a gear system. The lateral–torsion coupling vibration model of the gear shaft was built by using a Timoshenko beam element. The lumped parameter method was used to build the dynamic model of a gear pair. The dynamic model of a spur gear system was formed by integrating the component model mentioned above. The influence of rectangular and elliptical spalling on TMS was analyzed by the potential energy method (PEM). The fault feature of tooth spalling was studied by dynamic simulation and verified by experiments. It is found that the gear system will produce a periodic shock response owing to the periodic change of the number of meshing gear teeth. Due to the contact loss and the decrease of TMS, a stronger shock response will be generated when the spalling area is engaged. In the spectrum, some sidebands will appear in the resonance region. The results can provide a theoretical guide for the health monitoring and diagnosis of gear systems.

The TVMS of meshing gears is the important factor affecting the performance of gear systems. Obtaining accurate TVMS is a necessary condition for accurately modeling the dynamics of gear systems. This paper proposes a TVMS analysis method for the actual situation of non-coincidence among the root circle of the gear and base circle, the results of their calculations were compared and analyzed with the Ishikawa formula method and the base energy method, which were 10.73 % and 13.67 % for tooth number 20 and 8.13 % and 9.89 % for tooth number 60, respectively, and the error of the improved energy method was smaller. The method in this paper is used for establishing the calculation formula of TVMS of tooth root crack and flank spalling, and analyze the change rule of TVMS of gears when there are local defects and faults. The results show that the TVMS computation method of gears on the basis of the improved energy method presented in this paper is better than basic energy method, and TVMS obtained is closer to the reality. The proportion of stiffness loss for the maximum depth and inclination of through cracks is 23.6 % and 27 %, respectively, while that of shallow cracks is 8.37 % and 2.8 %, respectively. The proportion of stiffness loss in the interval of stiffness change under the longest spalling length decreases gradually with the increase in the number of teeth, from 7.45 % to 4.63 %, and the proportion of increase in the width of the interval of stiffness loss in the gear under the maximum spalling width rises gradually, from 50.96 % to 51.22 %, but they are all around 50 %.

Accurate assessment and modeling of the effects of tooth defects on the vibration response of gear systems is beneficial for the early detection and diagnosis of gear faults and failures. This paper presents a new dynamic model of a cylindrical gear pair with localized tooth spalling defects. Depending on the depth and extent of a spall, the resulting contribution of the spall to the gear dynamics can be classified in terms of mesh stiffness reduction and displacement excitation. The main improvement of the proposed model, relative to previous similar models which assume a linear line contact between mating tooth pairs, is that modification coefficients are introduced to account for the effects of nonlinear elliptical tooth surface contact pattern on the influencing mechanisms of the spalls on the gear dynamics. It was found that the modification coefficients are dependent on the transmitted load, the dimensions of the spall, and the amount of tooth crowning. Experimental vibration responses of several 1:1 ratio spur gear pair sets with different dimensions of spalling defects working under various load and speed conditions were measured to compare with the simulated results based on the proposed model and previous similar models in terms of the time-history acceleration, power spectrum and some statistical indicators. The results validate the superiority of the proposed model against previous similar models.

The dynamic characteristics and tooth spalling fault features are studied for the high-contact-ratio spur gear bearing system. The bending torsional dynamic model is proposed in this study for the gear bearing system with an ellipsoid spalling fault. This model also considers time-varying meshing stiffness, tooth friction, fractal gear backlash, and comprehensive transmission error. The meshing stiffness of the system is evaluated using the potential energy method. The bifurcation diagram, time-domain waveform, Poincaré map, phase map, frequency spectrum, and related three-dimensional map are used as tools to analyze the system’s dynamic response qualitatively. The results reveal that the system’s motion with ellipsoid tooth spalling defect exhibits rich dynamic behavior. The response of the proposed dynamic model is consistent with experimental results in the frequency domain. Therefore, the developed dynamic model can predict the system’s vibration behavior with localized spalling fault. Hence, it could also provide a theoretical foundation for future spall defect diagnosis of the gear transmission system.

For the analysis on the deformation of flexible ring gears in spur gear pairs, the complete flexible ring is discretized, and the boundary condition is added to the connecting points to develop a calculation method for the flexible deformation. The ovality index is used to describe the deformation degree of flexible ring gears, then the influences of ring-gear width and the spalling defects on the flexible deformation of ring gears are discussed. The result shows that the flexible deformation of ring gears is caused by the gear pair meshing force, and the deformed shape is close to an ellipse. In the single-tooth meshing interval of gear pairs, the main form of deformation is being stretched, and while in the double-tooth meshes, the main form is bending deformation. When the width of the ring gear rims is increased, the flexible deformation of the ring gears can be effectively suppressed, and the vibration amplitude of the gear pairs can be reduced. Additionally, when there is a localized spalling fault on gear pairs, the sudden changes in the deformation of flexible ring gears are generated by the shock of the meshing force. Finally, through the finite element analysis model and the experiment, the mathematical model of gear pairs with flexible rings is confirmed.

Helical gear teeth are subject to spalling, pitting and other failures under prolonged operation, which can contribute to a reduction in the time-varying meshing stiffness (TVMS) of the gear. The shape of the depression formed by the absence of the gear tooth surface is irregular in practice. Firstly, an irregular-shaped pitting model is constructed by the slicing method. On the premise of improving the transition curve, the TVMS calculation equations under the irregular pitting model are derived considering the effect of axial stiffness. Then, a randomly distributed tooth surface pitting evolution model was established by the random pitting generation function, and the effects of three different failure degrees from slight to severe pitting on the TVMS are evaluated. Eventually, the faulty helical gear pairs are constructed in Solidworks and simulated by the finite element method (FEM), verifying that the irregular pitting evolution model and calculation method proposed in this paper are effective.

Misalignment and spalling of bearing rings are typical errors and faults that significantly influence the vibration characteristics of a gear-rotor-bearing system. In order to investigate the coupling effects of misalignment and spalling, it is necessary to analyze the vibration characteristics of a gear-rotor-bearing system with misalignment and spalling. Firstly, considering the outer ring misalignment and spalling, the 5-DOF nonlinear bearing restoring force is calculated. Then, based on the loaded tooth contact analysis (LTCA) method, the meshing stiffness is calculated, and the dynamic model of the spur gear pair is established. Furthermore, the equivalent stiffness model of the spline is established based on the slicing method. Finally, by coupling the bearing restoring force model, the dynamic model of the spur gear pair, and the equivalent stiffness model of the spline with the finite element (FE) model of the rotor system, a dynamic model of a gear-rotor-bearing system is developed. The analysis shows that the contact force, contact angle, and system vibration are only affected when the spalling is located in load-bearing areas. The misalignment of the outer ring leads to the increase of the load-bearing area range of the bearing and the influence probability of the spalling on the system.