Abstract:
This paper focuses on spur gears with more than 41 teeth. Based on cutting geometry and gear meshing principles, the distinction between the starting point of the involute profile and the point of intersection of the involute with the base circle is clarified. By introducing the parametric equation of the transition curve, the time-varying meshing stiffness calculation formula based on the energy method is improved. Faulty gear tooth models with different crack levels are comprehensively established. On this basis, an oblique line is used as the effective tooth thickness reduction limit line, making the crack tooth model stricter and the considerations more comprehensive.
Keywords: spur gear; tooth root crack; different crack degrees; effective tooth thickness reduction limit line; time-varying meshing stiffness
1. Introduction
Gear-rotor systems are commonly used motion and power transmission mechanisms in industry. With the development of modern industrial requirements for gear transmission systems towards heavy loads, high speeds, and harsh working environments, gears are more prone to failures. Among gear and transmission system failures, tooth root cracks account for almost 40% of all failures. Therefore, tooth root crack faults in various gear systems must be accurately detected, controlled, and tracked. The time-varying nature of meshing stiffness can reflect various factors affecting gear meshing conditions. The Finite Element Method (FEM) is an effective tool for calculating time-varying meshing stiffness, but it has a large calculation volume and complex mesh generation and post-processing procedures. The energy method, with its high computational efficiency, is still the mainstream method for calculating time-varying meshing stiffness.
Table 1: Summary of Previous Research on Gear Crack Modeling
| Author | Year | Method | Focus |
|---|---|---|---|
| Chen et al. | 2011 | Crack model along tooth width and thickness | Crack propagation path |
| Verma et al. | N/A | Extended Finite Element Method (XFEM) | Influence of hub thickness on crack propagation |
| Feng Nana et al. | N/A | Computer simulation-based analytical algorithm | Quantification of time-varying meshing stiffness under different fault conditions |
| Sun Guangyao et al. | N/A | Finite element software | Stress intensity factor and J-integral distribution of 3D cracked helical gears |
| Wang Jinwen et al. | N/A | Improved energy method | Influence of cracked sun gear on meshing stiffness |
| Ning et al. | N/A | “Slice method”-based dynamics model | Dynamics characteristics of gear transmission systems with unevenly distributed cracks |
| Wan Zhiguo et al. | N/A | Analytical calculation models and dynamic simulations | Comparison of vibration response characteristics between tooth root cracks and tooth surface spalling |
| Xie Fuqi et al. | N/A | Comprehensive consideration of nonlinear excitations | Influence of through-thickness tooth root cracks on system dynamic characteristics |
| Liu Qikun et al. | N/A | Improved XFEM | Accuracy and validity of crack tip discontinuity and singularity models |
| Lai Junjie et al. | N/A | Full-tooth energy equation | Bilateral asymmetric tooth root crack model and meshing stiffness calculation |
| Zhang et al. | N/A | Combination of FEM, Hertz contact theory, and slice method | New calculation method for time-varying meshing stiffness of spur and helical gear pairs |
| Yang et al. | N/A | Crack opening state consideration | Re-assessment of effective compressed section and neutral layer of cracked gears |
| Zhao et al. | N/A | Elastic fluid dynamic lubrication (EHL) model | Influence of tooth root cracks on tribological characteristics and friction dynamics |
2. Improved Calculation of Time-varying Meshing Stiffness for Spur Gears with Large Tooth Numbers
2.1 Basic Principles
A Cartesian coordinate system is established. The gear teeth processed by a rack-type cutter are studied, with a transition curve that is the equidistant line of the involute of a prolate involute. The parametric equation is given by:
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Where, γ ∈ (α0, π/2) is the variable parameter; r is the pitch circle radius; ϕ = (a/tanγ + b)/r; a = ha_m + cm – ρ is the distance between the center of the cutter fillet and the tooth midline; b = πm/4 + ha_m tanα0 + ρcosα0 is the distance from the center of the cutter tooth groove to the center of the cutter fillet; m is the module; α0 is the pressure angle; ha and c* are the addendum coefficient and top clearance coefficient, respectively; ρ = c*m/(1 – sinα0) is the cutter fillet radius.
2.2 Improved Algorithm for Time-varying Meshing Stiffness
The total potential energy of the meshing gear pair consists of five components, corresponding to Hertz contact stiffness k_h, bending stiffness k_b, shear stiffness k_s, axial compression stiffness k_a, and tooth base stiffness k_f. The calculation formulas for k_h and k_f are:
Where the parameters’ meanings are detailed in literature.
The comprehensive time-varying meshing stiffness of the gear is given by:
Where subscripts 1 and 2 represent the driving gear and driven gear, respectively, during the meshing process.
3. Calculation of Meshing Stiffness for Spur Gears with Large Tooth Number Cracks
3.1 Meshing Stiffness Calculation Model for Different Crack Degrees
According to literature, tooth root cracks do not affect the integrity of the tooth profile curve and effective tooth width, but only affect bending stiffness k_b and shear stiffness k_s. It is assumed that the crack occurs at point H, penetrating the entire tooth and making an angle υ with the tooth midline.
3.2 Further Correction of the Effective Tooth Thickness Reduction Limit Line
The presence of tooth root cracks will result in a reduction in effective tooth thickness. The effective tooth thickness reduction limit line is defined as the “Dead line,” and the area of tooth thickness failure is called the “Dead zone,”.
Considering the difficulty in modeling and complexity of calculations with a parabolic limit line, this paper proposes using an oblique line connecting the crack tip point A and the crack-side tooth point K as the effective tooth thickness reduction limit line based on the stress gradient distribution under load.
4. Calculation Results and Analysis
4.1 Comparison of Meshing Stiffness Calculation Methods
Table 2: Meshing Stiffness Solution Methods
| Method | Real Tooth Root Transition Curve | Gear Tooth Model Correction | Effective Tooth Thickness Reduction Limit Line |
|---|---|---|---|
| Method 1 | Yes | No | Straight line |
| Method 2 | Yes | Yes | Straight line |
Table 3: Comparison of Time-varying Meshing Stiffness Calculation Results
| Crack Level | Method 1 | Stiffness Reduction (%) | Method 2 | Stiffness Reduction (%) | FEM | Stiffness Reduction (%) |
|---|---|---|---|---|---|---|
| 0% | 3.121 | – | 3.252 | – | 3.322 | – |
| 20% | 3.064 | -1.93 | 3.214 | -1.23 | 3.197 | -3.88 |
| 40% | 2.946 | -5.77 | 3.037 | -6.76 | 3.0 |