Abstract
Cumulative pitch errors are common manufacturing defects in gear transmission systems. This study aims to investigate the dynamic wear behavior of cylindrical gears with cumulative pitch errors. A load distribution model is developed based on loaded tooth contact analysis, and the mesh stiffness and non-loaded transmission error are obtained. These parameters are then incorporated into a gear rotor dynamic model to analyze the system’s vibration responses. Furthermore, a dynamic wear prediction model is established using Archard’s wear theory and the dynamic load distribution. The proposed model is verified by comparing the simulated results with experimental data from the literature. The analysis reveals that cumulative pitch errors lead to non-uniform load and wear distributions among gear teeth, introducing shaft frequency, hunting tooth frequency, and assembly phase frequency components in the vibration responses. Mild wear mitigates non-uniform wear, whereas severe wear exacerbates contact conditions, leading to increased vibration. Adopting a hunting tooth design can reduce the wear non-uniformity coefficient by approximately 30%. This study provides valuable insights into gear wear mechanisms and parameter design.
Introduction
Cylindrical gears are essential components in various industrial applications, transmitting power and motion through their meshing teeth. However, manufacturing errors, including cumulative pitch errors, can significantly impact the gear’s dynamic behavior and ultimately lead to wear and failure. Understanding the relationship between cumulative pitch errors, dynamic behavior, and wear is crucial for the design and maintenance of cylindrical gear systems.
Background and Motivation
Cumulative pitch errors are deviations from the nominal tooth spacing, resulting in non-uniform load distribution during meshing. This non-uniformity can cause additional dynamic excitations and increase wear rates on specific teeth. Traditional wear models often assume uniform wear across all teeth, which is not accurate for gears with cumulative pitch errors.
Several studies have investigated the influence of manufacturing errors and wear on cylindrical gears. Wang et al. analyzed the effect of pitch deviations on the vibration characteristics of spur gear rotor systems. Yuan et al. established a dynamic model for helical gears considering cumulative pitch errors. Ding et al. coupled a gear dynamic model with Archard’s wear theory to investigate the interactions between dynamics and wear. However, few studies have systematically analyzed the dynamic wear behavior of cylindrical gears with cumulative pitch errors.
Objectives
The primary objectives of this study are:
- Develop a load distribution model for cylindrical gears with cumulative pitch errors based on loaded tooth contact analysis.
- Incorporate the load distribution model into a gear rotor dynamic model to analyze the system’s vibration responses.
- Establish a dynamic wear prediction model using Archard’s wear theory and the dynamic load distribution.
- Analyze the effects of cumulative pitch errors on meshing characteristics, response characteristics, and wear distribution.
- Investigate the influence of different gear tooth number combinations on wear uniformity.
Methodology
The proposed methodology consists of three main parts: load distribution modeling, dynamic response analysis, and wear prediction.
Load Distribution Modeling
Loaded tooth contact analysis (LTCA) is employed to model the load distribution in cylindrical gears with cumulative pitch errors. LTCA separates the overall deformation from the contact deformation, improving computational efficiency.
Cumulative Pitch Errors
Cumulative pitch errors are defined as deviations from the nominal tooth spacing. These errors are incorporated into the LTCA model by transforming them into tooth surface normal displacements.
Finite Element Model
The finite element (FE) model used in LTCA is constructed in MATLAB to simulate the gear contact. The local contact flexibility matrix and overall flexibility matrix are calculated using analytical and FE methods, respectively.
The LTCA iteration equation is solved to obtain the normal contact forces and static transmission error, which are then used to calculate the mesh stiffness.
Dynamic Response Analysis
The load distribution model is integrated into a gear rotor dynamic model to analyze the system’s vibration responses. The dynamic model considers bending, torsion, and axial degrees of freedom.
Gear Rotor Dynamic Model
The gear rotor dynamic model is established using Timoshenko beam elements to simulate the shafts and linear springs to model the bearing supports. The gear meshing is simulated using a lumped parameter model, with the mesh stiffness and damping calculated from the load distribution model.
The dynamic response is obtained by solving the overall system’s equations of motion, considering various excitation frequencies, including the shaft rotation frequency, mesh frequency, and hunting tooth frequency.
Wear Prediction Model
A wear prediction model is developed based on Archard’s wear theory and the dynamic load distribution. The model calculates the wear depth at each meshing point using the dynamic contact forces and sliding velocities.
Archard’s Wear Theory
Archard’s wear theory relates the wear volume to the normal load, sliding distance, and material properties:
DeltaV=K⋅HP⋅L
where ΔV is the wear volume, K is the wear coefficient, P is the normal load, L is the sliding distance, and H is the material hardness.
The wear depth Δh can be approximated as:
Deltah=AΔV
where A is the contact area.
Dynamic Wear Simulation
The dynamic wear simulation iterates through multiple wear cycles, updating the tooth profiles as wear progresses. The load distribution and dynamic responses are recalculated after each significant wear update.
Results and Discussion
The proposed model is validated using experimental data from the literature and analyzed to investigate the effects of cumulative pitch errors on cylindrical gears.
Model Validation
The simulated dynamic transmission error spectrum is compared with experimental data from Guo and Fang. The results show good agreement, validating the model’s effectiveness in predicting the frequency components introduced by cumulative pitch errors.
Meshing Characteristics
The time-varying mesh stiffness is analyzed for gears with and without cumulative pitch errors. The cumulative pitch errors introduce fluctuations in the mesh stiffness during the hunting tooth cycle, indicating non-uniform load distribution.
Vibration Characteristics
The dynamic transmission error spectrum for a worn gear with cumulative pitch errors exhibits additional frequency components, including the shaft frequency, hunting tooth frequency, and assembly phase frequency. These frequencies reflect the non-uniform load distribution and dynamic interactions.
The dynamic load factor, defined as the ratio of the maximum dynamic mesh force to the static mesh force, increases with wear. However, mild wear initially mitigates load fluctuations, acting as a passive modification.
Wear Distribution
The wear depth distribution across gear teeth shows significant non-uniformity due to cumulative pitch errors. The wear is more pronounced at the tooth roots and tips, where the contact stresses are highest.
The wear non-uniformity coefficient decreases initially but increases significantly as wear progresses. This trend highlights the complex interplay between dynamics and wear.
Influence of Gear Tooth Number Combinations
The wear non-uniformity coefficient varies with different gear tooth number combinations (Table 3). The hunting tooth coefficient, defined as the ratio of the combination state coefficient to the maximum gear tooth number, significantly affects wear uniformity.
Conclusion
This study proposes a comprehensive model for analyzing the dynamic wear behavior of cylindrical gears with cumulative pitch errors. The model integrates loaded tooth contact analysis, gear rotor dynamics, and Archard’s wear theory.
The analysis reveals that cumulative pitch errors introduce non-uniform load and wear distributions, affecting the system’s vibration responses. Mild wear initially mitigates load fluctuations but exacerbates them at later stages. The hunting tooth coefficient provides a useful metric for assessing wear uniformity, with lower coefficients leading to more uniform wear.
The proposed model can guide the design and maintenance of cylindrical gear systems, enabling engineers to optimize gear parameters to reduce wear and improve system performance.