Abstract
This article comprehensively investigates the dynamic performance of a helical gear pair under tooth breakage fault. Utilizing the potential energy method combined with the slicing technique, the time-varying meshing stiffness of the gear is analyzed, considering Hertz, bending, shearing, and axial compression effects. Based on the helical gear dynamic model, the Runge-Kutta method is adopted to study the impact of fault parameters on the system’s dynamic characteristics. The results demonstrate a significant reduction in the comprehensive effective meshing stiffness of gears with tooth breakage faults, resulting in periodic shock phenomena in dynamic transmission errors, vibration velocity, and acceleration. Additionally, side-frequency signals centered on the meshing frequency emerge in the frequency domain response, facilitating the effective identification of tooth breakage faults in gear transmission systems.
Introduction
Gear transmission systems play a crucial role in various mechanical applications, particularly in high-speed, high-temperature, and harsh loading conditions. Despite their reliability, gears are prone to failures, among which tooth breakage is a common and critical issue. Tooth breakage not only degrades the performance and lifespan of mechanical equipment but also poses safety hazards. Therefore, studying the dynamic behavior of helical gear pairs under tooth breakage faults is essential for ensuring the safety and reliability of gear transmission systems.
This paper aims to investigate the dynamic characteristics of helical gear pairs with tooth breakage faults. The potential energy method and slicing technique are utilized to analyze the time-varying meshing stiffness, and the Runge-Kutta method is employed to solve the dynamic equations. The impacts of fault parameters on system dynamics are comprehensively explored, providing insights into fault prediction and diagnosis of gear transmission systems.
Literature Review
Time-Varying Meshing Stiffness of Gears
Time-varying meshing stiffness is a vital parameter in gear dynamics analysis, reflecting the system’s vibration characteristics under different fault conditions. Researchers have conducted extensive studies on the time-varying meshing stiffness of gears using various methods.
- Finite Element Method (FEM): Lin et al. analyzed the effects of various crack parameters on the meshing stiffness of helical gear using FEM, concluding that the meshing stiffness decreases with crack size.
- Theoretical Modeling: Ma et al. established a theoretical model for a single gear pair, investigating crack formation and vibration characteristics. Zhu et al. proposed a slicing-based method to calculate the time-varying meshing stiffness of helical gear, considering slice coupling theory.
- Potential Energy Method: Wang et al. highlighted the need to consider axial bending, torsion, and substrate stiffness when calculating the time-varying meshing stiffness of helical gear.
Fault Identification in Gear Systems
Fault identification in gear systems has been extensively studied, focusing on vibration signals and dynamic responses. Yan et al. extracted fault features from vibration signals to determine the severity and number of broken teeth. Chen et al. analyzed the amplitude of meshing forces, which significantly increases with the extent of tooth breakage.
Despite these efforts, comprehensive analyses that integrate fault effects on system dynamics within a closed gearbox remain limited. This paper aims to bridge this gap by studying the dynamic performance of helical gear pairs under tooth breakage faults.
Modeling and Theoretical Analysis
Modeling of Helical Gear Pair with Tooth Breakage
A three-dimensional model of a helical gear pair with a broken tooth on the pinion is created using SolidWorks. The tooth breakage is simulated by removing a section of one tooth from the pinion.
Time-Varying Meshing Stiffness Calculation
The time-varying meshing stiffness is calculated using the potential energy method, considering Hertz contact energy, bending potential energy, shear potential energy, and axial compression energy.
Elastic Potential Energy Formulation
The total elastic potential energy U of the meshing gear teeth can be expressed as the sum of Hertz contact energy Uh, bending potential energy Ub, shear potential energy Us, and axial compression energy Ua:
U=Uh+Ub+Us+Ua
The individual energy components are calculated as follows:
- Hertz Contact Energy Uh:Uh=4(1−ν2)πEhLδ2where E is Young’s modulus, h is the contact width, L is the contact length, ν is Poisson’s ratio, and δ is the elastic deformation.
- Bending Potential Energy Ub:Ub=21∫xbxcEIxcM2(x)dxwhere M(x) is the bending moment at section x, Ixc is the moment of inertia of the cross-section at x, and xb and xc define the integration limits.
- Shear Potential Energy Us:Us=2G1∫Axcτ2(x)dAxcwhere G is the shear modulus, τ(x) is the shear stress at cross-section x, and Axc is the cross-sectional area.
- Axial Compression Energy Ua:Ua=2E1∫Axcσ2(x)dAxcwhere σ(x) is the normal stress at cross-section x.
Time-Varying Meshing Stiffness
The time-varying meshing stiffness k(t) is derived from the relationship between potential energy and stiffness:
k(t)=dδ2d2U
By substituting the expressions for the various potential energies, the comprehensive time-varying meshing stiffness k(t) can be calculated.
Dynamic Model and Equations of Motion
A one-degree-of-freedom dynamic model for the helical gear pair is established using the lumped mass method, considering time-varying meshing stiffness k(t), time-varying meshing damping cm, static transmission error e(t), and friction excitation. The dynamic equation of motion is given by:
Ipθ¨p+cm(θ˙p−θ˙g)+k(t)(θp−θg−e(t))=Tp−M1
Igθ¨g−cm(θ˙p−θ˙g)−k(t)(θp−θg−e(t))=−Tg+M2
where Ip and Ig are the inertia of the pinion and gear, respectively; θp and θg are the angular displacements; Tp and Tg are the applied torques; M1 and M2 are the time-varying friction torques; and e(t) is the static transmission error.
Simulation Results and Discussion
Time-Varying Meshing Stiffness under Tooth Breakage
The time-varying meshing stiffness under tooth breakage conditions is significantly lower than that under ideal conditions. The reduced meshing stiffness leads to altered dynamic behavior in the gear transmission system.
Dynamic Characteristics under Different Operating Conditions
Effect of Rotational Speed
The dynamic characteristics of the gear transmission system are analyzed under different rotational speeds, with a constant applied torque of 50,000 N·m. higher rotational speeds result in faster dynamic transmission error responses, with more intense vibration velocity and acceleration.
Effect of Applied Torque
The effect of applied torque on dynamic characteristics is investigated at a constant rotational speed of 3785 r/min. higher applied torques lead to more significant dynamic transmission errors, vibration velocities, and accelerations.
Frequency Domain Analysis
The frequency domain responses of dynamic transmission error, vibration velocity, and acceleration under tooth breakage conditions exhibit side-frequency signals centered on the meshing frequency. These side-frequency signals are indicative of tooth breakage faults.
Conclusion
This paper comprehensively investigates the dynamic performance of helical gear pairs under tooth breakage faults. Utilizing the potential energy method combined with the slicing technique, the time-varying meshing stiffness is analyzed, considering various elastic deformation effects. Based on the established dynamic model, the Runge-Kutta method is employed to solve the equations of motion and study the impact of fault parameters on system dynamics.
The results indicate that tooth breakage faults significantly reduce the comprehensive effective meshing stiffness of gears, leading to periodic shock phenomena in dynamic transmission errors, vibration velocities, and accelerations. Furthermore, side-frequency signals centered on the meshing frequency emerge in the frequency domain response, facilitating fault identification. This study provides valuable insights into the dynamic behavior of helical gear pairs under tooth breakage faults, contributing to improved fault prediction and diagnosis in gear transmission systems.