Abstract: Tooth pitting is a common fault in gear transmission that affects the time-varying meshing stiffness (TVMS) and dynamic characteristics of the gear pair. This article presents a study on the influence of tooth pitting on spur gears. The tooth pitting model is proposed, and the TVMS is calculated using the potential energy method. The effects of pitting position and size on TVMS are analyzed. Dynamic simulations and experimental validations are conducted. The results show that the proposed model can better predict the meshing stiffness and vibration behavior, providing a theoretical basis for the detection and diagnosis of tooth pitting faults.
1. Introduction
Gear transmission is widely used due to its advantages such as accurate transmission ratio, compact structure, high efficiency, reliable operation, and long service life. However, during operation, the meshing stiffness of gears changes periodically due to the alternating meshing of teeth and the change of meshing positions. When tooth faults occur, they mainly affect the dynamic characteristics of the system by influencing the TVMS of the gear pair. Therefore, studying the change of TVMS can reflect the degree of gear faults.
Tooth pitting is a common fault in gear transmission, which is caused by local stress concentration on the tooth surface and leads to fatigue peeling. The pitting morphology is complex and random. Previous studies have used various geometric shapes to simulate pitting for the research of gear meshing stiffness. However, these simplified models have limitations in accurately reflecting the real pitting state. In addition, most of the previous studies focused on single pitting or multiple pittings on a single tooth, and did not consider the overlap between pittings.
In this study, a more realistic tooth pitting model is proposed, and the influence of tooth pitting on the TVMS and vibration response of spur gears is analyzed, aiming to provide a theoretical basis for the detection and diagnosis of tooth pitting faults.
2. Tooth Pitting Model and Calculation of Time-Varying Meshing Stiffness
2.1 Tooth Pitting Model
The existence of tooth pitting reduces the meshing stiffness of gears. To study the influence of tooth pitting on TVMS, the potential energy method is used. In this method, the tooth is assumed to be a cantilever beam with variable cross-section starting from the base circle, and the non-coincidence between the base circle and the root circle is not considered. For a spur gear reducer, the driving gear with fewer teeth is more prone to fatigue damage as its teeth participate in meshing more frequently. It is assumed that there is a pitting fault on a certain tooth of the driving gear, while the other teeth of the driving gear and all teeth of the driven gear are intact.
The geometric characteristics of pitting. Each pitting is approximately a part of an ellipsoidal cylinder with a major axis length of , a minor axis length of , and a depth of . The entire pitting area is regarded as the union of several ellipsoidal cylinders. According to the size and distribution position of tooth pitting on the tooth surface, tooth pitting is defined as three fault levels: slight pitting, moderate pitting, and severe pitting, respectively, where represents the distance from the base circle to the pitch line.
2.2 Calculation of Time-Varying Meshing Stiffness
The potential energy method divides the total energy stored in the gear due to deformation during the transmission process into Hertz contact energy, bending deformation energy, axial compression deformation energy, shear deformation energy, and matrix flexible deformation energy. Each tooth of the spur involute cylindrical gear is regarded as a cantilever beam with variable cross-section. By calculating the Hertz contact energy, bending deformation energy, shear deformation energy, axial compression deformation energy, and matrix deformation energy generated by Hertz contact deformation, bending deformation, shear deformation, axial compression deformation, and gear matrix flexible deformation, the Hertz contact stiffness, bending stiffness, shear stiffness, axial compression stiffness, and matrix flexible deformation stiffness can be obtained, and finally the comprehensive TVMS of the gear can be calculated.
When tooth pitting exists, the effective tooth width and effective cross-sectional area change, which in turn affects the effective cross-sectional moment of inertia and ultimately leads to changes in Hertz contact stiffness, bending deformation stiffness, shear deformation stiffness, and axial compression deformation stiffness. The formulas for calculating the reduction in tooth width and the changes in effective cross-sectional area and moment of inertia are given in the paper.
The Hertz contact stiffness, bending deformation stiffness, shear deformation stiffness, and axial compression deformation stiffness of the gear with pitting faults are calculated respectively. The stiffness corresponding to the flexible deformation of the gear matrix is calculated using a specific formula regardless of whether the gear is healthy or not. Finally, the meshing stiffness of a pair of gears when pitting faults exist is calculated in different meshing situations.
3. Influence of Pitting on Time-Varying Meshing Stiffness
3.1 Basic Parameters and Pitting Information
The position and size parameters of pitting are set as μ = 1.9mm, α = 0.3mm, b = 0.2mm, and δ = 0.4mm. The distribution and contact information of severe pitting on the tooth surface and the change of the distance between the contact point and the base circle with the angular displacement of the driving gear. The meshing process of the gear with pitting can be divided into different stages according to the position of the contact point.
3.2 Influence of Pitting on Meshing Stiffness
The TVMS of healthy gears and gears with different degrees of pitting varies with the angular displacement of the driving gear. The difference in meshing stiffness is also shown. In certain angular displacement ranges, the TVMS of gears without pitting faults is almost the same. The reduction in the TVMS of gears with moderate and severe pitting is the same in a specific interval because the number and size of pittings are the same in this interval. However, the reduction in the TVMS of gears with slight pitting is smaller than that of the other two cases in this interval because the number of slight pittings is half that of the other two fault levels.
3.3 Influence of Pitting Position on Meshing Stiffness
By changing the distance from the center of the first layer of pitting to the base circle, the influence of the pitting distribution position on the TVMS is analyzed. The center of the first layer of pitting moves away from the base circle, and the pitted area gradually moves from the base circle to the addendum circle.
3.4 Influence of Pitting Size on Meshing Stiffness
The influence of the length of the major axis of pitting on the TVMS. As increases, only the size of pitting in the tooth width direction is affected, and the size in the direction from the base circle to the addendum circle is not affected. Therefore, as increases, the reduction in tooth width increases, resulting in a greater reduction in the TVMS.
The influence of the length of the minor axis of pitting on the TVMS. The change in only affects the size of pitting in the direction from the base circle to the addendum circle, and the size in the tooth width direction does not change. Therefore, in the same range of angular displacement change, the reduction in the TVMS does not change with the change in .
4. Pitting Dynamics Simulation and Experimental Verification
4.1 Dynamics Simulation
To analyze the influence of different degrees of pitting on the gear system response, the calculated meshing stiffness of normal and gears with different degrees of pitting is substituted into the multi-degree-of-freedom gear dynamics model proposed by Hou et al. in 2021, and numerical solutions are obtained using MATLAB software. The vertical frequency response of the driving gear. It can be seen that each frequency response is mainly dominated by the meshing frequency , and there are also , , , and higher harmonics of the meshing frequency. When tooth pitting occurs, sidebands with intervals of the input shaft rotational frequency appear around the meshing frequency and its harmonics. As the severity of pitting increases, the amplitude of the sidebands increases more significantly.
4.2 Experimental Verification
To better interpret the dynamic response characteristics of the system from the healthy state to the occurrence of fatigue pitting, a single-stage parallel shaft gear reducer is designed based on the drivetrain dynamics simulator, and driving gears with different degrees of tooth pitting are machined. The test bench, and the three-axis vibration acceleration sensor is installed on the bearing end cover of the shaft where the driving gear is located. The vibration signal during gear operation is collected using the DASP intelligent data acquisition instrument developed by the Beijing Oriental Vibration and Noise Technology Research Institute. The schematic diagram of the test bench structure, and the gears with different degrees of pitting faults machined by electric spark technology.
The number of teeth of the driving gear and the driven gear is 18 and 47, respectively. The rotational frequency of the input shaft is 30 Hz, the sampling frequency is set to 51.2 kHz, and the sampling time is 10 s after the test starts and stabilizes. The collected vertical acceleration signal is subjected to Fourier transform, and the frequency domain signal in the range of [500, 2000] is intercepted. With the occurrence of pitting faults, sidebands with intervals of the input shaft rotational frequency appear around the meshing frequency and its harmonics in the amplitude-frequency response, and the amplitude of most sidebands increases as the fault severity increases. At the same time, the amplitude of the meshing frequency and its harmonics also gradually increases. It should be noted that due to the limitations of experimental conditions, there is a lack of a test bench completely consistent with the theoretical model, and the experimental results can qualitatively verify the correctness of the theoretical results.
5. Conclusion
In this study, a tooth pitting model is proposed, in which each pitting is regarded as a part of an ellipsoidal cylinder, and the entire pitting area is considered as the union of multiple ellipsoidal cylinders, overcoming the problem of ignoring the overlap between pittings in previous studies. The existence of pitting has a significant impact on the TVMS as the driving gear rotates. When other parameters are fixed, the position of pitting and the length of the minor semi-axis of the ellipsoidal cylinder affect the existence range of pitting in the direction from the base circle to the addendum circle, but do not reduce the reduction in meshing stiffness. When the length of the major semi-axis of the ellipsoidal cylinder increases, the effective contact tooth width decreases, and the reduction in stiffness increases, but the existence region of pitting in the direction from the base circle to the addendum circle does not change. By comparing the simulation and experimental results of the vertical direction of the spur gear system without faults and with tooth pitting faults, it is found that due to the existence of tooth pitting, sidebands with intervals of the input shaft rotational frequency appear around the meshing frequency and its harmonics, and the amplitude of the sidebands increases significantly as the fault severity increases. In addition, the amplitude of the meshing frequency and its harmonics also has an increasing trend.
The research results of this article can provide a theoretical basis for the detection and diagnosis of tooth pitting faults in spur gears, and help engineers better understand the influence of tooth pitting on the performance of gear systems, so as to take appropriate measures to improve the reliability and service life of gear transmissions. Future research can focus on further optimizing the tooth pitting model, considering more complex working conditions and fault combinations, and exploring more accurate fault diagnosis methods and preventive maintenance strategies.