Abstract
In this study, the dynamic characteristics of helical gears with different modified designs were investigated. A comparative analysis was conducted between standard helical gears and modified helical gears. A dynamics model of the modified helical gear system was established based on Lagrange dynamics. An analytical method was used to solve the time-varying meshing stiffness (TVMS) of the modified helical gears, and the influence of different modification coefficients on the TVMS was analyzed. By combining the TVMS with the dynamics model, the influence of modified designs on the dynamic characteristics of the helical gear system was evaluated. The results show that positive drive modifications reduce the TVMS, thereby decreasing the meshing force, while negative drive modifications increase the TVMS, leading to an increase in the meshing force. This study provides valuable insights for designing and optimizing helical gear transmissions.
1. Introduction
Helical gears, as critical components in electromechanical equipment, are widely used in various industries such as aerospace, railway transportation, and marine equipment due to their excellent meshing performance, high contact ratio, and strong load-bearing capacity. To enhance the performance and durability of helical gear systems, modification designs, such as profile shifting, are often applied. Modification not only helps avoid undercutting but also improves the load-bearing capacity, adjusts the center distance, and reduces the overall dimensions of the gearbox.
The time-varying meshing stiffness (TVMS) of helical gears is a significant internal nonlinear excitation source that significantly affects the vibration, noise, and fatigue failure of gear systems. Therefore, accurately calculating the TVMS and studying its impact on the dynamic characteristics of helical gear systems is essential. This study aims to explore the effects of different modification designs on the TVMS and dynamic characteristics of helical gears.
2. Literature Review
Several methods exist for calculating the TVMS of gears, including experimental methods, finite element analysis (FEA), and analytical methods. Experimental methods are costly and require strict environmental control. FEA, while accurate, is computationally intensive and may not efficiently capture the TVMS variations across multiple structural parameters. In contrast, analytical methods offer a balance between accuracy and computational efficiency, making them a popular choice for TVMS calculations.
Previous studies have focused on the TVMS of standard and modified gears. For example, Liu et al. [1] proposed a stiffness correction algorithm based on the traditional potential energy method to improve the calculation accuracy of TVMS in helical gears. Zhu et al. [2] calculated the TVMS of helical gears using the slice coupling theory. Other researchers have analyzed the TVMS of modified spur gears [3, 4], helical gears [5, 6], and planetary gears [7, 8].
While significant progress has been made, studies on the TVMS and dynamic characteristics of helical gears with different modification designs are still limited. Therefore, this study aims to fill this gap by investigating the effects of modification coefficients on the TVMS and dynamic characteristics of helical gears.
3. Methodology
3.1 Time-Varying Meshing Stiffness Calculation
The TVMS of modified helical gears was calculated using the potential energy method combined with the slice method. The key geometric relationships between standard and modified gears .
The modified gear tooth profile can be described by the following equations:
hai=(ha∗+xni−Δyni)mn(1)
hfi=(ha∗+c∗−xni)mn(2)
ai=mt(z1+z2)cosα0t/2cosαt(3)
textinvαt=2tanα0tz1+z2xmc1+xmc2+invα0t(4)
where ha∗ is the addendum coefficient, c∗ is the clearance coefficient, xmc is the modification coefficient, Δyni is the addendum reduction coefficient, mn and mt are the normal and transverse module, respectively, z1 and z2 are the number of teeth of the pinion and gear, respectively, α0t is the transverse pressure angle, αt is the transverse contact angle, and inv is the involute function.
The TVMS of a helical gear pair can be calculated by considering the contributions from various stiffness components, including Hertz contact stiffness kh, fillet basis stiffness kf, bending stiffness kb, shear stiffness ks, and axial compression stiffness ka. The combined mesh stiffness k is obtained by the inverse of the sum of the inverse stiffnesses of these components [9]:
frac1k=kh1+kf1+kb1+ks1+ka1(5)
3.2 Dynamics Model Establishment
Based on Lagrange dynamics, a dynamics model of the modified helical gear system was established, considering three translational and one rotational degree of freedom for each gear. The generalized displacements and velocities are defined as:
mathbfq={xp,yp,zp,θp,xg,yg,zg,θg}
dotq={x˙p,y˙p,z˙p,θ˙p,x˙g,y˙g,z˙g,θ˙g}
The Lagrangian function L is defined as the difference between the kinetic energy Ek and potential energy Ep:
L=Ek−Ep=21q˙TMq˙−21qTKq(6)
where M is the mass matrix, and K is the stiffness matrix.
The Rayleigh dissipation function D is introduced to account for energy dissipation:
D=21q˙TCq˙(7)
where C is the damping matrix.
The equations of motion are derived from the Lagrange equations with dissipation:
fracddt(∂q˙∂L)−∂q∂L+∂q˙∂D=Q(8)
where Q is the generalized force vector, including meshing force Fm, friction force Ff, driving torque Tp, and load torque Tg.
4. Influence of Modification Designs on Dynamic Characteristics
4.1 Time-Domain Analysis
The time-domain responses of TVMS and meshing force were analyzed for different modification designs. The primary design parameters of the helical gears are listed in Table 1.
| Parameter | Value |
|---|---|
| Module (( m_n )) | 5.5 mm |
| Number of teeth | 17, 107 |
| Face width (( b )) | 70 mm |
| Helix angle | 17° |
| Pressure angle | 20° |
Table 1: Main design parameters of the helical gear system
The time-domain responses of TVMS and meshing force for standard and modified gears , respectively.
It can be observed that positive modifications (smaller TVMS) lead to reduced meshing force fluctuations, while negative modifications (larger TVMS) increase the meshing force fluctuations.
4.2 Frequency-Domain Analysis
The frequency-domain responses of TVMS and meshing force were analyzed to gain further insights. the frequency-domain responses of TVMS and meshing force, respectively.
In the frequency domain, the meshing frequency and its harmonics are clearly visible. With positive modifications, the meshing frequency amplitudes decrease, and the envelope period shortens, indicating a reduced dynamic load. Conversely, with negative modifications, the meshing frequency amplitudes increase, and the envelope period lengthens.
4.3 Statistical Analysis
Statistical indicators, including root mean square (RMS) and kurtosis value (KV), were used to quantify the dynamic behavior of the gear system. the variation of RMS and KV for different modification designs.
The results indicate that positive modifications tend to reduce the dynamic load, while negative modifications increase it. The statistical indicators vary systematically with the modification coefficients.
5. Conclusion
This study investigated the influence of modification designs on the dynamic characteristics of helical gear systems. The TVMS of modified helical gears was calculated using an analytical method based on the potential energy and slice methods. A dynamics model was established using Lagrange dynamics to analyze the system’s response under different modification coefficients.
The main findings are summarized as follows:
- TVMS and Meshing Force: Positive modifications reduce the TVMS, leading to a decrease in meshing force fluctuations, while negative modifications increase the TVMS and meshing force fluctuations.
- Frequency-Domain Analysis: In the frequency domain, positive modifications result in lower meshing frequency amplitudes and shorter envelope periods, indicating reduced dynamic loads. Negative modifications exhibit the opposite trends.
- Statistical Indicators: The statistical indicators (RMS and KV) vary systematically with the modification coefficients, providing a quantitative measure of the system’s dynamic behavior.
This study provides valuable insights into the effects of modification designs on the dynamic characteristics of helical gear systems, facilitating the design and optimization of gear transmissions.