1. Introduction
1.1 Background and Significance
Helical gear systems are widely used in various industries, such as aerospace, automotive, and marine, due to their advantages like smooth transmission, high load – carrying capacity, and quiet operation. However, vibration and noise in these systems remain significant issues. The time – varying mesh stiffness of gear pairs is a major internal excitation source, which can lead to vibrations that not only affect the performance and reliability of the equipment but also cause noise pollution. Reducing vibration and noise in helical gear systems is crucial for enhancing the overall quality of mechanical products and improving the working environment.
1.2 Research Objectives
The main objective of this research is to propose a low – fluctuation mesh stiffness design method for helical gear systems. By optimizing the design parameters of helical gears, we aim to minimize the mesh stiffness fluctuation, thereby reducing the vibration and noise levels of the system. This will provide a theoretical basis and practical guidance for the design and improvement of helical gear systems.
1.3 Research Status
Previous studies on reducing vibration and noise in gear systems mainly focused on methods like tooth surface modification and adding dampers. For example, some researchers optimized the structure of shafts to reduce vibration responses, while others proposed adding annular dampers or using squeeze – film dampers in gear systems. However, the impact of mesh stiffness fluctuation on the vibration characteristics of the system has received relatively little attention. This research fills this gap by specifically studying the relationship between mesh stiffness fluctuation and system vibration, and developing corresponding design methods.
2. Low Meshing Stiffness Fluctuation Gear Pair Parameter Optimization Design
2.1 Contact Line Fluctuation Calculation of Helical Gear Pairs
2.1.1 Meshing Characteristics of Helical Gears
Helical gears have distinct meshing characteristics compared to spur gears. The contact line length of helical gear pairs changes during the meshing process. As shown in Figure 1, in different meshing positions, the contact line lengths are different. When the helical gear enters and exits the meshing state, the contact line lengths vary, which is an important factor contributing to mesh stiffness fluctuations. [Insert Figure 1: Diagram of the helical gear meshing, similar to the one in the original paper but with clear labels in English]
2.1.2 Contact Line Length Calculation
The time – varying contact line length of helical gears can be expressed by specific formulas. For two types of helical gears (Type 1 with \(\varepsilon_{\alpha}>\varepsilon_{\beta}\) and Type 2 with \(\varepsilon_{\alpha}<\varepsilon_{\beta}\)), the contact line length W is related to parameters such as the movement distance u of the meshing point along the meshing line length direction, the face – width B, the helix angle \(\beta_{b}\), and the normal tooth pitch \(p_{b v}\) in the end – face. The formulas are as follows:
| Gear Type | Contact Line Length Formula |
|---|---|
| Type 1 | \(W_{1}=\begin{cases}u / \sin\beta_{b}&u < L_{\beta}\\B / \cos\beta_{b}&L_{\beta}\leq u\leq L_{\alpha}\\B / \cos\beta_{b}-(u – L_{\alpha}) / \sin\beta_{b}&L_{\alpha}<u\leq L_{\alpha}+L_{\beta}\end{cases}\) |
| Type 2 | \(W_{2}=\begin{cases}L_{\alpha} / \sin\beta_{b}&u < L_{\alpha}\\L_{\alpha} / \sin\beta_{b}&L_{\alpha}\leq u\leq L_{\beta}\\B / \cos\beta_{b}-(u – L_{\beta}) / \sin\beta_{b}&L_{\beta}<u\leq L_{\alpha}+L_{\beta}\end{cases}\) |
The 重合度 (contact ratio) \(\varepsilon_{k}\) of helical gear pairs is calculated by \(\varepsilon_{k}=\varepsilon_{\alpha}+\varepsilon_{\beta}\), where \(\varepsilon_{\alpha}=\frac{z_{p}(\tan\alpha_{\alpha p}-\tan\alpha_{t}) + z_{g}(\tan\alpha_{\alpha g}-\tan\alpha_{t})}{2\pi}\) and \(\varepsilon_{\beta}=\frac{B\sin\beta}{\pi m_{n}}\) (\(z_{p}\) and \(z_{g}\) are the numbers of teeth of the driving and driven wheels respectively, \(\alpha_{\alpha p}\) and \(\alpha_{\alpha g}\) are the pressure angles at the addendum circles of the driving and driven wheels, \(\alpha_{t}\) is the meshing angle, \(m_{n}\) is the normal module).
2.1.3 Conditions for Minimizing Contact Line Fluctuation
To minimize the mesh stiffness fluctuation, we need to minimize the contact line length fluctuation. Theoretically, when the total contact line length is in a dynamically balanced state, the contact line length fluctuation is minimized. For a complete meshing cycle, the number of contact lines in different regions (\(S_{1}\), \(S_{2}\), \(S_{3}\)) needs to satisfy certain conditions. The total contact line length l of Type 1 and Type 2 helical gear pairs can be expressed by complex formulas considering the number of contact lines in different regions.
The conditions for minimizing the contact line length fluctuation are \(\varepsilon_{\alpha}=N_{+}\) or \(\varepsilon_{\beta}=N_{+}\) (\(N_{+}\) represents a positive integer). By designing the transverse contact ratio \(\varepsilon_{\alpha}\) or the longitudinal contact ratio \(\varepsilon_{\beta}\) as positive integers, the change in the contact line length during the meshing process can be minimized, thereby reducing the system’s stiffness fluctuation.
2.2 Meshing Stiffness Analytical Model of Helical Gear Pairs
2.2.1 Calculation Method Combining “Slice Method” and “Offset Method”
The “slice method” and “offset method” are commonly used to calculate the meshing stiffness of helical gear pairs. For a helical gear pair with a contact ratio in the range of 2 – 3, the meshing region can be divided into multiple zones, such as the three – tooth meshing zone, two – tooth meshing zone. The time intervals corresponding to different meshing regions can be calculated based on the rotational speed of the driving wheel \(\omega_{p}\) and other parameters. For example, for a helical gear pair, the time intervals \(t_{A}\), \(t_{B}\), \(t_{C}\), \(t_{D}\), \(t_{E}\), \(t_{F}\) and \(t_{0}\) can be calculated by the formulas:
| Time Interval | Formula |
|---|---|
| \(t_{A}\) | 0 |
| \(t_{B}\) | \(\frac{(\varepsilon_{A}-2)p_{b t}}{r_{b p}\omega_{p}}\) |
| \(t_{C}\) | \(\frac{p_{b t}}{r_{b p}\omega_{p}}\) |
| \(t_{D}\) | \(\frac{(\varepsilon_{h}-1)p_{b t}}{r_{b p}\omega_{p}}\) |
| \(t_{E}\) | \(\frac{2p_{b t}}{r_{b p}\omega_{p}}\) |
| \(t_{F}\) | \(\frac{\varepsilon_{b}p_{b t}}{r_{b p}\omega_{p}}\) |
| \(t_{0}\) | \(\frac{2\pi}{z_{p}\omega_{p}}\) |
2.2.2 Calculation of Key Parameters in the Meshing Stiffness Model
The meshing stiffness calculation involves many key parameters. The pressure angles at different positions, such as the maximum and minimum pressure angles \(\varphi_{b}\) and \(\varphi_{e}\) of the meshing region, and the pressure angles \(\varphi_{b}^{i}\) and \(\varphi_{e}^{i}\) at the start and end of the contact line, can be calculated by specific formulas related to the radii of the base circle \(r_{b j}\), the addendum circle \(r_{a j}\), and other parameters.
The 柔度 (flexibility) of the helical gear teeth is composed of axial compression flexibility, bending flexibility, and shear flexibility. The formulas for calculating these flexibilities are complex and involve integral operations on the tooth profile. The nonlinear Hertz contact stiffness on the actual contact line length slice and the meshing stiffness K and loaded static transfer error \(e_{LSTE}\) of the helical gear pair can also be calculated by corresponding formulas.
3. Model Verification
3.1 Verification Purposes
The main purposes of model verification are two – fold. First, to verify that minimizing the total contact line length fluctuation of helical gears can achieve a low – fluctuation meshing stiffness design. Second, to verify the proposed meshing stiffness analytical calculation model of helical gear pairs.
3.2 Optimization Schemes
Two optimization schemes are designed to study the influence of gear parameters on meshing characteristics.
| Optimization Scheme | Optimization Content |
|---|---|
| Scheme 1 | Keep the bearing capacity of the gear unchanged and optimize the helix angle \(\beta\) |
| Scheme 2 | Keep the center distance of the gear unchanged and optimize the face – width b |
The basic parameters of the helical gear pairs before and after optimization are shown in Table 1. [Insert Table 1: Basic parameters of the helical gear pairs, similar to the one in the original paper but with English labels]
3.3 Finite Element Model Establishment and Verification Results
A finite – element model of the helical gear pair is established. As shown in Figure 2, in the model, the red points represent the fixed constraint points of the two gear centers, and the yellow circles represent the coupling relationship between the gear hub and the constraint points. A load torque \(T_{eut}=200 N\cdot m\) is applied to the driven wheel, and an angular displacement UR is applied to the pinion to simulate the rotation of the gear pair. [Insert Figure 2: FE model of the helical gear pair, with clear labels in English]
The meshing stiffness fluctuation coefficient \(\varepsilon_{h}=\frac{K_{max}-K_{min}}{K_{min}}\times100\%\) is defined. By comparing the results calculated by the finite – element method (FE) and the analytical method (AM), the following conclusions can be drawn:
- After optimization, the stiffness fluctuation is significantly reduced. Before optimization, the stiffness fluctuation coefficient of the FE method is 18.97%. After optimizing the helix angle and face – width, the fluctuation coefficients are reduced to 2.02% and 2.96% respectively.
- The calculation results of the AM method are in good agreement with those of the FE method, with a maximum calculation error of 4.7%, which verifies the effectiveness of the model.
- When other parameters remain unchanged, the meshing stiffness after optimizing the helix angle is close to the average meshing stiffness before optimization, while the meshing stiffness after optimizing the face – width increases significantly.
The loaded static transfer error \(e_{LSTE}\) is also analyzed. The results show that the proposed optimization design method can significantly reduce the fluctuation of \(e_{LSTE}\).
4. Results and Discussion
4.1 Dynamic Modeling of Helical Gear Systems
4.1.1 Establishment of the 8 – Degree – of – Freedom Dynamic Model
A concentrated – mass method is used to establish an 8 – degree – of – freedom helical gear system dynamic model, as shown in Figure 3. In the model, \(X_{i}\), \(Y_{i}\), Z represent the horizontal, vertical, and axial displacements respectively, and \(\theta\) represents the angular displacement around the rotation axis. [Insert Figure 3: Meshing transmission dynamics model of a helical gear pair, with clear labels in English]
The meshing effect of the gear pair is equivalent to an elastic element with time – varying meshing stiffness \(k_{m}(t)\) and time – varying damping \(c_{m}(t)\), and the influences of unloaded static transfer error \(e(t)\) and tooth profile clearance 2b are considered. The displacement vector q of the helical gear system can be expressed as \(q = [X_{p}, Y_{p}, Z_{p},\theta_{p}, X_{g}, Y_{g}, Z_{g},\theta_{g}]^{T}\).
The dynamic transfer error \(\delta_{m}\) along the meshing line direction, the dynamic meshing force \(F_{m}\), and the motion equations of the helical gear pair can be expressed by specific formulas based on Newton’s second law. The bearing – related parameters, such as bearing damping \(c_{i j}\) and stiffness \(k_{i j}\), are also determined, as shown in Table 2. [Insert Table 2: Parameters of the bearing, with English labels]
4.1.2 Analysis of Vibration Characteristics
- Dynamic Response: The dynamic characteristics of the gear system before and after optimization, including \(e_{DTE}\), vibration displacement, and vibration speed, are compared and analyzed. The root – mean – square value \(R_{RMS}\) of displacement and vibration speed are used to evaluate the vibration level of the system.
- For \(e_{DTE}\), optimizing the face – width has a significant impact, and it can reduce \(e_{DTE}\) by about [X]% under most rotational speeds. Optimizing the helix angle has little effect on \(e_{DTE}\).
- In terms of vibration displacement, the change trends of the two optimization schemes in the X and Y directions are basically the same. After optimizing the face – width, the \(R_{RMS}\) of vibration displacement at some rotational speeds slightly decreases. After optimizing the helix angle, the system vibration in the X and Y directions is reduced by about [X]%. In the Z direction, optimizing the face – width keeps the vibration basically unchanged, while optimizing the helix angle increases the Z – direction vibration by about [X]% because the increase in the helix angle makes the axial force of the helical gear system larger, increasing the axial displacement vibration amplitude.
- The vibration speed also changes significantly after optimization. The \(R_{RMS}\) of the vibration speed shows that the optimization reduces the vibration response of \(e_{DTE}\) and in the X, Y directions, and the system energy also decreases.
- Comparison of Two Optimization Schemes: For Gear Pair 1, when the helix angle is increased (positive optimization), the meshing stiffness fluctuation is significantly reduced, but the axial vibration is significantly aggravated. When the helix angle is decreased (negative optimization), this problem can be alleviated.
- The contact line fluctuation coefficient \(\varepsilon_{i}=\frac{l_{max}-l_{min}}{B}\) changes with the helix angle. When the helix angle is in the range of [7.6°, 21.84°], the contact line fluctuation coefficient first decreases and then increases, showing a V – shape.
- By comparing the time – varying meshing stiffness (TVMS) curves of different gear pairs, it can be seen that the stiffness fluctuation coefficient of Gear Pair 4 is 7.01%, and that of Gear Pair 2 is 0.82%. After negative optimization, the displacement and speed \(R_{RMS}\) of Gear Pair 2 and Gear Pair 4 are compared. The results show that the \(e_{DTE}\) and axial vibration are effectively suppressed, and the axial vibration is reduced by 26.7%. The derivative of \(e_{DTE}\) basically remains unchanged, and the axial vibration speed is reduced by 14.3%.
5. Conclusion
- A new method for reducing the meshing stiffness fluctuation of helical gear systems by parameter optimization design is proposed. By designing the transverse or longitudinal contact ratio of helical gear pairs to be an integer, the contact line fluctuation can be minimized, achieving the goal of low – fluctuation meshing stiffness design.
- The finite – element method is used to verify the optimization design method and the meshing stiffness analytical calculation model. Optimizing the face – width and helix angle can achieve low – fluctuation stiffness design, proving the accuracy and effectiveness of the calculation model. The optimized meshing stiffness fluctuation is much smaller than that before optimization.
- The system can significantly reduce the loaded static transfer error (LSTE) and dynamic response after parameter optimization design. Optimizing the face – width and helix angle can reduce the peak – to – peak value of LSTE. Positive optimization of the helix angle will reduce the radial dynamic response of the system but increase the axial dynamic response. Optimizing the face – width and negative optimization of the helix angle can reduce both the axial and radial dynamic responses of the system. The proposed parameter optimization design method can achieve the purpose of vibration reduction, providing valuable guidance for the design of gear systems.
