Abstract
Tooth pitting, one of the most common failures in gear transmission systems, significantly affects the time-varying meshing stiffness (TVMS) of spur gear pairs, thereby altering the dynamic characteristics of the entire system. In this study, each pit shape is approximated as a section of an ellipsoidal cylinder, and three damage levels—slight pitting, moderate pitting, and severe pitting—are defined based on the location and number of pits. The TVMS of undamaged gears and gears with varying degrees of pitting are calculated using the potential energy method, and the effects of pit location and size on TVMS are discussed. The fault dynamic characteristics of a single-stage spur gear transmission system are investigated, and the theoretical results are qualitatively validated using a comprehensive drive train dynamics simulator. The results show that the proposed pit model aligns better with actual pit morphology. As the pit location parameter increases, the pit area gradually shifts from the base circle towards the tip circle. The longer the major axis of the pit, the more pronounced the reduction in TVMS within the pit area. Additionally, for varying minor axis lengths, the reduction in TVMS caused by different levels of pitting damage remains consistent over the same range of driving gear angular displacement. The established model can predict the meshing stiffness and vibration behavior of a pitted gear system, and the corresponding vibration analysis provides a theoretical foundation for detecting and diagnosing tooth pitting faults.
1. Introduction
Spur gears, widely employed in various mechanical systems due to their precise transmission ratio, compact structure, high efficiency, reliability, and longevity, play a crucial role in ensuring smooth power transmission. However, their time-varying meshing stiffness (TVMS), influenced by alternating single and double tooth engagement and varying meshing positions, exhibits periodic fluctuations. Gear failures, particularly tooth pitting, can significantly alter the TVMS, subsequently impacting the dynamic behavior of the entire system . Consequently, studying the changes in TVMS provides valuable insights into the extent of gear damage.
Tooth pitting, a common form of gear wear, arises from localized stress concentrations that lead to fatigue spalling on the tooth surface. Pits exhibit random morphological features and complex geometries, which presents actual pitting observed during experiments or in operational gears. Accurate TVMS calculations in the presence of pitting are essential for understanding gear fault dynamics. Several studies have employed simplified geometric shapes, such as cuboids, cylinders, spheres, and ellipsoids, to model pits and study their impact on meshing stiffness .
To more accurately simulate pitting observed in practice, this study treats each pit as a section of an ellipsoidal cylinder. Based on the pits’ distribution and number, three damage levels are defined: slight pitting, moderate pitting, and severe pitting. By solving for the effective tooth width, cross-sectional area, and moment of inertia under pitting conditions and utilizing the potential energy method, the TVMS variations with the driving gear’s angular displacement under different pitting severities are investigated. Furthermore, the effects of pit location and size on TVMS are discussed. The stiffness models incorporating pitting faults are then integrated into a four-degree-of-freedom spur gear model that considers both torsional vibration around the gear axes and vertical translational vibration of the support system. The influence of tooth pitting on the system’s vibration response is analyzed in the frequency domain. The findings of this study aim to provide a theoretical basis for detecting and diagnosing tooth pitting faults.
2. Pitting Model and TVMS Calculation
2.1 Pitting Model
Tooth pitting reduces the meshing stiffness of gear teeth. To investigate its impact on TVMS, this study models each pit as a section of an ellipsoidal cylinder. Based on the pits’ distribution and number on the tooth surface, three damage levels are defined: slight pitting, moderate pitting, and severe pitting. As the number of pits increases or their centers shift from the pitch line to the tooth tip, the pit area expands accordingly.
2.2 TVMS Calculation
The potential energy method calculates the total energy stored during gear deformation, which comprises Hertz contact energy, bending deformation energy, axial compression deformation energy, shear deformation energy, and base flexibility deformation energy. Each spur gear tooth is treated as a tapered cantilever beam originating from the base circle, neglecting the mismatch between the base and root circles. Since the driving gear’s teeth engage more frequently than those of the driven gear in spur gear reducers, the driving gear is more susceptible to fatigue damage. Therefore, a single tooth on the driving gear is assumed to exhibit pitting, while all other teeth, including those on the driven gear, remain undamaged.
The effective tooth width and cross-sectional area vary with pitting, altering the effective cross-sectional moment of inertia and, subsequently, the Hertz contact stiffness, bending stiffness, shear stiffness, and axial compression stiffness. For a detailed derivation of these stiffnesses under pitting conditions, refer to the potential energy method outlined in previous studies.
3. Effects of Pitting on TVMS
The fundamental parameters used in this study are presented in Table 1, and the pit location and size parameters are set to u=1.9 mm, a=0.3 mm, b=0.2 mm, and δ=0.4 mm. the pit distribution and contact information for severe pitting, the variation in the distance between the contact point and the base circle with the driving gear’s angular displacement.
The TVMS of undamaged and pitted gears with varying degrees of severity is plotted against the driving gear’s angular displacement. The variation in TVMS (\Delta k_t\)) with angular displacement. In intervals where no pitting exists (\(\theta_1 \in [0^\circ, 15.64^\circ] \cup [20.05^\circ, 31.17^\circ] and at \theta_1 = 17.99^\circ\)), the TVMS curves for all cases nearly overlap. Within \(\theta_1 \in (15.64^\circ, 17.99^\circ), the TVMS reductions for moderate and severe pitting are identical due to the same number and size of pits in this interval. However, slight pitting causes a smaller TVMS reduction due to having half the number of pits.
The influence of pit location on TVMS is analyzed by varying the distance from the base circle to the center of the first pit layer (parameter u\)). As \(u increases, the pit center shifts further from the base circle, gradually moving the pit area towards the tooth tip.
The impact of the pit’s major axis length (a\)) on TVMS. As \(a increases, the pit size along the tooth width expands, reducing the effective tooth width and consequently decreasing TVMS more significantly.
In contrast, varying the pit’s minor axis length (b\)) does not affect the effective tooth width but alters the pit’s extent along the tooth height. within the same angular displacement range, varying \(b does not significantly impact TVMS reduction.
4. Pitting Dynamics Simulation and Experimental Validation
To investigate the influence of varying pitting severities on gear system response, the derived stiffnesses of undamaged and pitted gears are incorporated into a multi-degree-of-freedom gear dynamics model. Numerical solutions are obtained using MATLAB, and the vertical amplitude-frequency responses of the driving gear are plotted for an input shaft rotation frequency of 30 Hz. The responses are dominated by the meshing frequency (f_m\)) and its higher harmonics (2\(f_m, 3fm, …, 10(f_m)). Pitting introduces sidebands around the meshing frequency and its harmonics, with the sideband amplitudes increasing with pitting severity.
To qualitatively validate the theoretical findings, experiments are conducted using a comprehensive drive train dynamics simulator. A single-stage parallel-axis spur gear reducer is designed, and driving gears with varying degrees of tooth pitting are manufactured using electrical discharge machining . Vibration signals are acquired using triaxial vibration accelerometers mounted on the driving gear’s bearing housing. The signals are then processed using a digital acquisition system developed by the Beijing Institute of Vibration and Acoustics.
The experimental setup, and the manufactured gears with varying pitting severities . The driving and driven gears have 18 and 47 teeth, respectively, with an input shaft rotation frequency of 30 Hz. Data are sampled at 51.2 kHz for 10 seconds after the test’s stable operation commences. The vertical acceleration signals’ frequency spectra within the range of [500, 2000] Hz are plotted. Similar to the simulation results, pitting introduces sidebands around the meshing frequency and its harmonics, with increasing sideband amplitudes corresponding to more severe pitting.
5. Conclusion
This study proposes a tooth pitting model that treats each pit as a section of an ellipsoidal cylinder, addressing limitations in previous studies that ignored pit overlap. As the driving gear rotates, pitting significantly affects TVMS. When other parameters remain constant, the pit location and the minor axis length of the ellipsoidal cylinder influence the pit’s extent along the tooth height but do not significantly alter TVMS reduction. In contrast, increasing the major axis length reduces the effective tooth width and consequently decreases TVMS more prominently. Both simulation and experimental results demonstrate that tooth pitting introduces sidebands around the meshing frequency and its harmonics in the frequency spectra of gear vibration responses. As pitting severity increases, the sideband amplitudes notably enlarge, providing a basis for detecting and diagnosing tooth pitting faults in spur gear systems.