1. Introduction
Helical gears play a crucial role in many mechanical transmission systems. However, the occurrence of cracks can significantly affect their performance. The meshing stiffness of helical gears is a key factor influencing the dynamic behavior of gear systems. In this article, we will explore the research on the meshing stiffness calculation model of helical gear systems under crack conditions.
1.1 Gearbox Transmission Systems and Failures
Gearbox transmission systems are widely used in various industries. They operate under complex and variable loads, which can lead to vibrations. These vibrations can cause damage to the internal structure of the gearbox and reduce its reliability. One of the common failure modes in gearboxes is gear crack failure.
1.2 Importance of Meshing Stiffness
The determination of time-varying meshing stiffness is fundamental for the study of gear system vibration responses and fault feature analysis. Among different gear failure types, gear cracks have the most significant impact on meshing stiffness.
1.3 Previous Research Review
Many researchers have conducted studies on gear transmission system crack modeling and vibration fault characteristics. For example, YANG et al. considered clearance nonlinearity to study the dynamic characteristics of spur gear pairs with tooth cracks. Kong Yiyi et al. established a double-tooth stiffness model for gear transmission systems considering gear wear and crack faults. However, most of these studies have limitations. They often do not consider the impact of gear cracks on the gear fillet foundation stiffness or the changes in axial meshing stiffness during the meshing process of helical gear pairs.
2. Improved Meshing Stiffness Calculation Model for Cracked Helical Gear Pairs
2.1 Helical Gear Space Crack Propagation Path Modeling
The crack propagation path in helical gears can be modeled as a space curve. In this study, it is assumed to be a parabola. The crack can be divided into two main types: tooth tip propagation crack and end face propagation crack. The equations for crack depth and crack length along different directions are presented. For example, the non-penetrating tooth tip crack depth can be expressed as , and the penetrating tooth tip crack depth as .
2.2 Helical Gears End Face Crack Propagation Path Modeling
For end face cracks, the effective tooth thickness in different directions is calculated based on the crack starting point and angle. The equations for effective tooth thickness are given as when , and different expressions for when .
2.3 Meshing Stiffness Calculation Model using the “Slice” Method
The meshing stiffness of helical gear pairs can be represented by the comprehensive meshing stiffness of small tooth width “slice” gear pairs. The total meshing stiffness of helical gear pairs is composed of transverse meshing stiffness and axial meshing stiffness. The calculation of these stiffness components involves considering various factors such as gear tooth stiffness and gear foundation stiffness.
| Meshing Stiffness Component | Calculation Method |
|---|---|
| Transverse meshing stiffness | Calculated by integrating “cut teeth” spur gear pairs with a certain dislocation angle. Involves considering transverse gear tooth stiffness and transverse gear foundation stiffness. |
| Axial meshing stiffness | Calculated based on the axial force component during the gear meshing process, considering axial gear tooth stiffness and axial gear foundation stiffness. |
3. Consideration of Different Stiffness Components
3.1 Transverse Gear Tooth Stiffness Calculation
The transverse gear tooth stiffness is calculated using the “cut teeth” method. The deflection of the gear tooth is calculated considering bending, shear, and axial compression deformations. The forces and moments acting on the gear tooth are analyzed, and the equations for calculating the stiffness components are derived. For example, the equations for calculating the bending deformation , shear deformation , and axial compression deformation of the gear tooth are presented.
3.2 Transverse Gear Foundation Stiffness Calculation
The transverse gear foundation stiffness is calculated considering the interaction between the gear and the foundation. The position and orientation of the force acting on the gear are analyzed, and the equations for calculating the foundation stiffness are derived. The modified gear foundation stiffness due to gear tooth root cracks is calculated, and the relevant parameters and equations are given.
3.3 Axial Gear Tooth Stiffness Calculation
The axial gear tooth stiffness is calculated based on the axial force component during the gear meshing process. The moments acting on the gear tooth due to the axial force are analyzed, and the equations for calculating the axial gear tooth bending stiffness and axial gear tooth torsional stiffness are derived.
3.4 Axial Gear Foundation Stiffness Calculation
The axial gear foundation stiffness is calculated considering the deformation of the gear foundation under the axial force. The section inertia moment of the gear with cracks is calculated, and the equation for calculating the axial gear foundation stiffness is derived.
4. Results and Discussion
4.1 Verification of the Model
A typical double helical gear pair is used to verify the improved meshing stiffness calculation model. The basic parameters of the gear pair are given in a table.
| Parameter | Driving Wheel | Driven Wheel |
|---|---|---|
| Number of teeth | 42 | 43 |
| Tooth width (mm) | 30 | 30 |
| Module (mm) | 3.5 | 3.5 |
| Pressure angle (°) | 22.5 | 22.5 |
| Helical angle (°) | 17 | 17 |
| Elastic modulus E (Pa) | ||
| Poisson’s ratio | 0.3 | 0.3 |
| Input torque (N·m) | 119 | – |
| Input rotational speed (r/min) | 1500 | – |
4.2 Meshing Stiffness Calculation for Tooth Tip Propagation Crack
The meshing stiffness of helical gear pairs with tooth tip propagation cracks is calculated. Different crack cases are considered, including penetrating cracks at different locations, non-penetrating cracks of different crack lengths, and penetrating cracks with different crack depths. The results show that as the crack length and depth increase, the meshing stiffness decreases. The reduction in meshing stiffness is related to the effective crack area. The finite element simulation results of the meshing stiffness are in good agreement with the calculated results, validating the accuracy of the model.
4.3 Meshing Stiffness Calculation for End Face Propagation Crack
The meshing stiffness of helical gear pairs with end face propagation cracks is calculated. Similar to the tooth tip propagation crack, different crack cases are considered. The results show that the end face propagation crack has a more significant impact on the meshing stiffness than the tooth tip propagation crack when the crack length and depth are the same. This is because the end face propagation crack causes a larger effective crack area. The finite element simulation results also verify the accuracy of the calculated results.
5. Conclusions
In this article, an improved meshing stiffness calculation model for cracked helical gear pairs is proposed. The model comprehensively considers the transverse and axial gear tooth stiffness and gear foundation stiffness. The meshing stiffness of gear systems under tooth tip propagation crack and end face propagation crack conditions is studied. The following conclusions are drawn:
- The reduction in meshing stiffness is mainly related to the effective crack area. The larger the crack area, the greater the decrease in meshing stiffness.
- When the crack length and depth are the same, the end face propagation crack has a more significant impact on the meshing stiffness than the tooth tip propagation crack.
- The finite element method simulation results of the meshing stiffness are in good agreement with the calculated results, validating the accuracy and effectiveness of the proposed method.
This research provides a more accurate method for calculating the meshing stiffness of cracked helical gear pairs, which is beneficial for the design and maintenance of gear transmission systems. Future research can focus on further improving the model and applying it to more complex gear systems.