Abstract
The time-varying meshing stiffness of spur gear is a critical parameter affecting the dynamic performance of gear transmission systems, playing a vital role in the construction of dynamic models and fault diagnosis. However, the traditional Ishikawa formula method exhibits certain deviations when calculating the time-varying meshing stiffness. To enhance calculation accuracy and efficiency, this study adopts an improved Ishikawa method for the analysis of time-varying meshing stiffness in spur gear.
1. Introduction
With the rapid development of modern industrial technology, gear transmission systems are increasingly used in fields such as aerospace, military industry, and new energy vehicles. The time-varying meshing stiffness of spur gear is a key parameter determining the dynamic performance of spur gear transmission systems. However, the traditional Ishikawa formula method fails to consider factors such as gear body deformation, resulting in deviations between calculation results and actual conditions.
2. Improved Ishikawa Deformation Formula Calculation Method
2.1 Improved Ishikawa Formula Considering Gear Body Deformation
The Ishikawa formula simplifies spur gear into a cantilever beam model composed of a trapezoid and a rectangle. However, the traditional Ishikawa formula does not include the deformation of spur gear body when subjected to load, which can lead to insufficient accuracy in calculating the time-varying meshing stiffness. In this study, spur gear body is simplified as a curvilinear trapezoid, and the deformation of spur gear body (δω) is introduced when calculating the deformation (δ) produced by the load along the meshing line direction.
The formula for the total deformation is:
δ = δBr + δBt + δs + δG + δω (1)
Where: δBr is the bending deformation of the rectangular part; δBt is the deformation of the trapezoidal part; δs is the deformation produced by shear force; δG is the deformation produced by the inclination of the base part; and δω is the deformation of spur gear body.
The expression for δω is derived as:
Where: Fn is the normal load on the tooth surface; ωx is the load angle at the meshing point; E is the elastic modulus; b is the tooth width; hDi is the height of the curvilinear trapezoid; rf is the radius of the tooth root circle; and hx is the distance from the load application point to the critical section.
Table 1: Summary of Gear Parameters
| Gear Parameter | Drive Gear | Driven Gear |
|---|---|---|
| Module (mm) | 4.0 | 4.0 |
| Number of Teeth | 25.0 | 33.0 |
| Tooth Width (mm) | 20.0 | 20.0 |
| Pressure Angle (°) | 25.0 | 25.0 |
| Elastic Modulus (GPa) | 210.0 | 210.0 |
| Poisson’s Ratio | 0.3 | 0.3 |
2.2 Determination of Unclear Parameters
The effective tooth root circle radius refers to the actual position of the tooth when it exits the tooth profile during meshing. The effective tooth root circle radii (rF1 and rF2) for the drive and driven gears are calculated as follows:
Where: rb1 and rb2 are the base circle radii of the drive and driven gears, respectively; αa1 and αa2 are the pressure angles at the tooth tips of the drive and driven gears, respectively; and B1B2 is the actual length of the line of action, which can be calculated using methods described in literature.
3. Mechanism of Time-Varying Meshing Stiffness in Spur Gear
The Ishikawa formula is an important tool for calculating the time-varying meshing stiffness of single-pair spur gear. However, in actual meshing processes, the meshing process can be divided into double-tooth and single-tooth meshing regions. The periodic change in the meshing region leads to periodic fluctuations in the time-varying meshing stiffness of the spur gear, resulting in periodic impacts and vibrations.
4. Case Study Analysis
To verify the effectiveness of the proposed method, spur gear parameters shown in Table 1 were used. Three calculation methods were compared: Method A using the unmodified Ishikawa formula, Method B using the improved Ishikawa formula (proposed in this study), and Method C referencing finite element method (FEM) results.
Table 2: Comparison of Calculation Errors in Time-Varying Meshing Stiffness
| Torque (N·m) | Calculation Method | Maximum Single-Tooth Meshing Stiffness (N·mm⁻¹) | Relative Error with Method C (%) | Average Time-Varying Meshing Stiffness (N·mm⁻¹) | Relative Error with Method A (%) |
|---|---|---|---|---|---|
| 30 | A | 2.61 | 1.95 | 3.82 | 1.87 |
| B | 2.59 | 1.17 | 3.79 | 1.06 | |
| C | 2.56 | 0 | 3.75 | 0 | |
| 200 | A | 2.72 | 2.64 | 4.23 | 2.42 |
| B | 2.69 | 1.51 | 4.18 | 1.21 | |
| C | 2.65 | 0 | 4.13 | 0 | |
| 400 | A | 2.79 | 3.33 | 4.47 | 2.76 |
| B | 2.75 | 1. |
5. Conclusion
Addressing the issues of the neglect of spur gear body deformation in the process of calculating time-varying meshing stiffness for spur gear pairs using the Ishikawa formula, this paper has derived the relevant parameters in detail. When calculating the time-varying meshing stiffness, spur gear body part is simplified as a curvilinear trapezoid, and its deformation is estimated. Finally, the calculation results are compared with those obtained from the unmodified Ishikawa formula and the finite element method (FEM). It is found that the improved Ishikawa method exhibits higher accuracy. Additionally, when a larger torque is applied to spur gear pair, the accuracy of the results remains guaranteed. Due to the significant reduction in computation time compared to the FEM, the improved Ishikawa method can be applied to the dynamic analysis of spur gears, thereby reducing the parametric excitation caused by stiffness.
The research highlights the importance of considering gear body deformation in meshing stiffness calculations and demonstrates the superiority of the improved Ishikawa method in terms of accuracy and computational efficiency. This methodology provides a valuable tool for spur gear dynamic analysis, enabling more accurate predictions of system behavior and potential shock issues. Furthermore, the findings contribute to the advancement of spur gear design and optimization, facilitating the development of more reliable and efficient gear transmission systems.
In summary, the improved Ishikawa method presented in this paper offers a refined approach to calculating time-varying meshing stiffness for spur gear, addressing the limitations of the traditional method. By incorporating gear body deformation and providing a clear derivation of relevant parameters, this method achieves higher accuracy and computational efficiency, paving the way for more accurate dynamic analysis of spur gear systems.