This paper focuses on the meshing behavior of double helical gears, considering lead crowning modification and stagger angle. A new axial displacement iteration method is proposed, and a load-carrying contact analysis (LTCA) model for double helical gears is established. The model is based on the finite element method and the finite long line contact theory, incorporating the differences in helix angles at different meshing points into the load-deformation coordination equation. The Newton iteration method is used to calculate the axial displacement of the driving gear. The LTCA model is employed to analyze the influence of lead crowning modification and tooth surface stagger angle on meshing excitations and load distribution. The results indicate that the Newton iteration method is applicable to LTCA models of double helical gears with different types of errors, and it significantly reduces calculation time compared to traditional iteration methods under small errors. With the increase in lead crowning modification, the mean time-varying mesh stiffness decreases, while the mean meshing error and the maximum normal meshing force at the mid-tooth surface increase. As the tooth surface stagger angle increases, the mean time-varying mesh stiffness increases, and the mean meshing error decreases. The findings of this study provide new insights for improving the meshing stability of double helical gears.
1. Introduction
Double helical gears, also known as herringbone gears, are widely used in high-speed and heavy-load transmissions due to their good axial load balancing capabilities and stability. However, the complex meshing behavior of double helical gears, influenced by factors such as lead crowning modification and tooth surface stagger angle, poses challenges for their precise design and analysis.
In recent years, with the development of computer technology and finite element methods, load-carrying contact analysis (LTCA) models have been increasingly applied to study the meshing characteristics of gears. Researchers have explored the impact of various factors, such as torque, tooth surface geometry, and modification, on the meshing behavior of gears. However, there is still a lack of research on the influence of tooth surface stagger angle on the meshing excitations and load distribution of double helical gears.
In this paper, a new axial displacement iteration method is proposed for double helical gears, and an LTCA model is established to analyze the influence of lead crowning modification and tooth surface stagger angle. The results provide theoretical support and guidance for optimizing the design and improving the performance of double helical gears.
2. LTCA Model for Double Helical Gears
2.1 Overview of the LTCA Model
The LTCA model for double helical gears considered in this paper is based on the finite element method and the finite long line contact theory. It incorporates the differences in helix angles at different meshing points into the load-deformation coordination equation and uses the Newton iteration method to calculate the axial displacement of the driving gear. The model allows for the analysis of the influence of lead crowning modification and tooth surface stagger angle on meshing excitations and load distribution.
2.2 Calculation of Axial Force for Double Helical Gears
The meshing force analysis of helical gears is illustrated in Figure 3. The calculation formulas for circumferential force (pt), normal force (pn), and axial force (pa) are as follows:
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Figure 3: Illustration of meshing force on helical gear
In the formula, T1 is the torque of the gear; R is the pitch circle radius; α is the helix angle at the pitch circle; βn is the normal pressure angle at the pitch circle; and γ is the angle axial.
It should be noted that the helix angle mentioned here usually refers to the helix angle at the pitch circle of the helical gear, but in reality, the helix angles at different meshing points on the tooth surface are not completely the same.
2.3 Calculation of Axial Displacement for Double Helical Gears
The calculation of axial displacement is crucial for ensuring the accuracy of the LTCA model. In this paper, a new axial displacement iteration method is proposed, which is suitable for double helical gears with large tooth surface stagger angles. The method uses the Newton iteration method to balance the axial loads of the left and right helical gear pairs, thereby accurately calculating the normal meshing excitations and axial displacements of the double helical gear with large tooth surface stagger angles.
3. Influence of Lead Crowning Modification on Meshing Behavior
3.1 Overview of Lead Crowning Modification
Lead crowning modification is a common method to improve the load distribution on the tooth surface and enhance the carrying capacity of gear transmissions. It is also one of the important factors affecting the meshing behavior of gears. To avoid stress concentration at the tooth edge under high-speed and heavy-load conditions, as well as uneven load distribution along the tooth width caused by gear and support shaft deformation, high-speed train gears usually require tooth profile modification. Common methods include lead crowning modification, spiral modification, surface modification, etc. Although the actual error does not distribute according to an arc curve, considering the impact of the modification method on load distribution and processing technology, lead crowning modification is often adopted in practical applications.
3.2 Influence of Lead Crowning Modification on Meshing Excitations
Taking a high-speed train double helical gear pair with a rated torque of 500 N·m as an example, the influence of lead crowning modification on the meshing excitations of the double helical gear pair was studied. Figure 9 shows the change curve of axial displacement under different maximum lead crowning modifications (Cmax). It can be seen from the figure that the axial displacement increases slightly with the increase in modification amount, but the influence on axial movement is relatively small and can be ignored.
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Figure 9: Axial displacement under different maximum lead crowning modification amounts
Figure 10 presents the variation laws of time-varying mesh stiffness (km) and composite meshing error (em) of the double helical gear with different maximum lead crowning modifications over a dimensionless time. It can be observed that as the modification amount increases, km tends to decrease, while em exhibits an increasing trend in the double-tooth meshing zone and first increases and then decreases in the triple-tooth meshing zone.
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Figure 10: Meshing excitations under different maximum lead crowning modification amounts
Table 4 lists the mean and amplitude values of meshing excitations under different maximum lead crowning modifications.
| Cmax / μm | km /(kN·mm⁻¹) | em / μm | Mean | Amplitude |
|---|---|---|---|---|
| 2 | 1622.15 | 1.61 | 282.67 | 1.94 |
| 4 | 1403.73 | 2.09 | 203.43 | 2.15 |
| 6 | 1263.28 | 2.18 | 303.84 | 2.27 |
| 8 | 1170.47 | 2.20 | 431.09 | 2.34 |
| 10 | 1104.83 | 2.22 | 422.13 | 2.45 |
| 15 | 1004.10 | 2.25 | 426.93 | 2.55 |
Table 4: Mean value and amplitude of meshing excitations under different maximum lead crowning modification amounts
3.3 Influence of Lead Crowning Modification on Load Distribution
The load distribution of the double helical gear pair under different maximum lead crowning modifications is shown in Figure 11. It can be seen from the figure that the meshing process of the two helical gear pairs is from point B1 entering the meshing until point B3 exiting the meshing. When Cmax = 2 μm, the meshing-in and meshing-out positions are close to the tooth root and tooth tip, and the normal meshing force at the tooth edge is higher than that at the tooth center, resulting in a relatively dispersed load distribution on the tooth surface.
As Cmax increases, the tooth edge no longer participates in the meshing. Although there is still meshing impact, the load gradually concentrates towards the middle of the tooth surface. As Cmax increases from 2 μm to 15 μm, the maximum normal meshing force at the mid-tooth surface increases from approximately 100 N to 370 N.
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